General Relativistic Simulations of Binary Neutron Star Mergers … Summary Binary neutron star (‘BNS’) mergers are violent cosmic events whose rst detec-tion (on 17th August 2017,

Universita degli Studi di Milano - Bicocca

Dipartimento di Fisica G. Occhialini˝Corso di Laurea Magistrale in Fisica

General Relativistic Simulationsof Binary Neutron Star Mergers

with the Spritz Code

Advisor: Candidate:Prof. Bruno Giacomazzo Lorenzo Ennoggi

Student ID : 776335

Co-advisor:Prof.ssa Monica Colpi

A.A. 2019⁄2020

Summary

Binary neutron star (‘BNS’) mergers are violent cosmic events whose first detec-tion (on 17th August 2017, event GW170817 [1]) marked a turning point in theera of multi-messenger astrophysics, since these phenomena are strong sources ofgravitational and electromagnetic radiation, as well as neutrinos and cosmic rays.Given that the extreme physical conditions characterising BNS mergers are not(currently) reproducible on Earth, one of the few chances of getting some insightinto their complicated dynamics comes from numerical simulations in full generalrelativity.

Spritz [19, 20] is the name of a new computer code whose purpose is to solve,by means of numerical techniques, the full set of partial differential equationsarising in general relativistic magnetohydrodynamics (GRMHD), the frameworkdescribing the physics of electrically conducting fluids moving at relativistic speedin presence of a strong gravitational field. This code can be used effectively tostudy a number of interesting astrophysical scenarios, including (but not limitedto) BNS mergers.

Spritz solves the hydrodynamics equations in flux-conservative form by meansof a high-resolution-shock-capturing (HRSC) scheme, typically employing PPMreconstruction coupled to a HLLE approximate Riemann solver. Maxwell’s equa-tions are tackled in the ideal MHD approximation by evolving the electromagnetic4-potential, thus ensuring the divergence-free character of the magnetic field (upto machine precision) when retrieving the latter as the curl of the vector potential.Furthermore, because all electromagnetic variables in Spritz are stored on a fully-staggered mesh, evolving the 4-potential as outlined above on uniform numericalgrids is in fact equivalent to the standard constrained-transport scheme [8, 29].

The rest mass density in the inner core of neutron stars can be as high as∼ 1014 g

cm3 , or even higher than that. The properties of matter in such a regimeare still largely unknown, so several microphysical models describing such scenar-ios can be found on the market. These models usually arise as the result of so-phisticated calculations involving effective internuclear interactions based on QCDand statistical-physics considerations and typically translate into tabulated (thus,non-analytical) equations of state (‘EOSs’) relating fundamental thermodynamicquantities such as pressure, rest mass density, specific internal energy, tempera-ture, and electron fraction to each other. Spritz supports tabulated EOSs —as well as more idealised, analytical ones — thus being able to simulate realisticBNS systems properly. Conversely, numerical simulations of realistic BNS systems

1

(coupled to experimental observations) can provide critical information about thephysical properties of neutron stars and, as such, are a key tool which can helpconstrain the EOS of nuclear matter at extremely high densities.

The Spritz code is designed to be included as a module (or ‘thorn’) into theEinstein Toolkit, a ‘community-driven software platform of core computationaltools to advance and support research in relativistic astrophysics and gravitationalphysics’ [38, 46, 72]. To this end, the behaviour of Spritz needs be assessescarefully in all of its parts, with particular attention to the components developedmost recently. The project presented here is intended to test two of these newpieces of software included in Spritz:

1. the implementation of the generalised Lorenz gauge condition for the evolu-tion of the electromagnetic 4-potential;

2. the ability of the code to handle realistic, tabulated EOSs for nuclear matterat extremely high densities.

The work done within the scope of this thesis led to significant improvement incode performance on both fronts. In particular, a number of flaws were fixedwhich prevented the correct evolution of magnetised BNS systems with generalisedLorenz gauge; Spritz is now able to handle that configuration, at least if someamount of Kreiss-Oliger dissipation is applied to the electromagnetic potentials. Inaddition, an issue related to the conservatives–to-primitives routine dealing withtabulated EOSs was solved and some BNS simulations with microphysical SLy4EOS could then run successfully.

A number of new features, which are not discussed in detail in this thesis,have recently been added to Spritz. These include neutrino physics via a leakagescheme based on the ZelmaniLeak code (see [20] and references therein), high-orderreconstruction techniques, and a new conservatives-to-primitives scheme which willsoon be the default one when tabulated EOSs are involved [40].

2

https://einsteintoolkit.org/

Sommario

Le fusioni di sistemi binari di stelle di neutroni (BNS˝) sono eventi cosmici violentila cui prima osservazione (17 agosto 2017, evento GW170817 [1]) ha segnato ilpassaggio definitivo verso l’era dell’astrofisica multi-messaggero, dal momento chequesti fenomeni sono fonti copiose di onde gravitazionali ed elettromagnetiche,oltre che di neutrini e di raggi cosmici. Siccome le condizioni fisiche estreme checaratterizzano i sistemi BNS non sono (al momento) riproducibili sulla terra, unadelle poche possibilita di conoscere qualcosa riguardo alla loro dinamica complicatae costituita da simulazioni numeriche che tengano conto della relativita generalein modo completo.

Spritz [19, 20] e il nome di un nuovo codice il cui scopo e risolvere, tramitetecniche numeriche, il sistema completo delle equazioni a derivate parziali dellamagnetoidrodinamica in relativita generale (GRMHD˝), ovvero la teoria che de-scrive la fisica dei fluidi magnetizzati in moto a velocita relativistiche in presenza diforti campi gravitazionali. Questo codice si presta bene a simulare diversi scenariastrofisici interessanti, tra cui la fusione di sistemi BNS.

Spritz risolve le equazioni dell’idrodinamica in forma conservativa tramiteschemi high-resolution-shock-capturing (HRSC˝), utilizzando solitamente una tec-nica di ricostruzione PPM accoppiata a un risolutore di Riemann approssimato ditipo HLLE. Le equazioni di Maxwell sono trattate nell’approssimazione di MHDideale evolvendo nel tempo il quadripotenziale elettromagnetico, di modo che ladivergenza del campo magnetico, calcolato come rotore del potenziale vettore, siauguale a zero (o quasi, tenuto conto della precisione finita di macchina). In piu,siccome tutte le variabili elettromagnetiche in Spritz si trovano in posizioni sfal-sate tra loro sulla griglia numerica, evolvere il quadripotenziale come descrittoin precedenza su griglia uniforme e in realta equivalente allo schema standard ditrasporto vincolato [8, 29].

La densita di massa a riposo all’interno delle stelle di neutroni puo raggiungerevalori di ∼ 1014 g

cm3 o superiori. Le proprieta della materia in queste condizionisono ad oggi perlopiu sconosciute, percio e stato sviluppato un gran numero dimodelli microfisici atti a descrivere tali scenari. Generalmente, questi modelli sonoil risultato di calcoli sofisticati costruiti a partire da interazioni effettive tra i nucleibasate su QCD e su considerazioni di fisica statistica e si concretizzano di solito inequazioni di stato (EOS˝) tabulate (cioe non analitiche) che mettono in relazionetra loro quantita termodinamiche fondamentali quali pressione, densita di massaa riposo, energia interna specifica, temperatura e frazione elettronica. Spritz

supporta, oltre a EOS idealizzate e analitiche, anche EOS tabulate, ed e cosı in

3

grado di simulare correttamente sistemi BNS realistici. Viceversa, le simulazioninumeriche di sistemi BNS realistici (assieme alle osservazioni sperimentali) possonofornire informazioni cruciali sulla fisica delle stelle di neutroni e, come tali, sonostrumenti chiave per porre dei vincoli sulla EOS della materia nucleare ad altissimadensita.

Il codice Spritz e progettato per essere incluso come modulo (thorn˝) inEinstein Toolkit, un’infrastruttura software che e gestita dalla comunita e ilcui obiettivo e supportare e far progredire la ricerca nei campi dell’astrofisicarelativistica e della fisica computazionale [38, 46, 72]. Per questo, e necessarioaccertarsi del buon funzionamento di Spritz in tutte le sue parti, con particolareattenzione a quelle sviluppate piu di recente. Il progetto presentato qui ha lo scopodi collaudare due di questi nuovi componenti:

1. l’implementazione della condizione di gauge di Lorenz generalizzata per l’e-voluzione del quadripotenziale elettromagnetico;

2. l’abilita del codice di lavorare con EOS tabulate realistiche che descrivono lamateria nucleare ad altissima densita.

Il lavoro svolto nell’ambito di questa tesi ha generato un miglioramento consi-derevole delle prestazioni del codice sotto entrambi gli aspetti. In particolare, sonostati corretti alcuni difetti che impedivano la corretta evoluzione di sistemi BNSmagnetizzati con gauge di Lorenz generalizzato; Spritz e ora in grado di gestirequeste configurazioni correttamente, a patto che vanga applicata della dissipazionealla Kreiss-Oliger ai potenziali elettromagnetici. In piu, e stato risolto un problemanella routine che recupera le variabili primitive a partire da quelle conservate nelcaso di EOS tabulata e, grazie a questo, si e riusciti a portare a termine consuccesso alcune simulazioni con EOS microfisica di tipo SLy4.

Di recente, diverse nuove funzionalita, che non sono discusse nel dettaglio inquesta tesi, sono state aggiunte a Spritz. Innanzitutto e stato inserito uno schemadi leakage per la fisica dei neutrini basato sul codice ZelmaniLeak (si vedano[20] e i riferimenti ivi contenuti); inoltre, sono state introdotte delle tecniche diricostruzione ad alto ordine ed e stato sviluppato un nuovo schema di recuperodelle variabili primitive, che sara presto il predefinito quando si lavori con EOStabulate.

4


Contents

1 Introduction and purpose 71.1 Why numerical relativity? . . . . . . . . . . . . . . . . . . . . . . . 71.2 Gravitational waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3 Gamma-ray bursts . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.4 Outline of this work . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Geometric preliminaries 122.1 Review of basic differential geometry . . . . . . . . . . . . . . . . . 12

2.1.1 Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . 122.1.2 Vector fields on a smooth manifold . . . . . . . . . . . . . . 132.1.3 1-form fields on a smooth manifold . . . . . . . . . . . . . . 162.1.4 Tensor fields on a smooth manifold . . . . . . . . . . . . . . 172.1.5 The metric tensor . . . . . . . . . . . . . . . . . . . . . . . . 212.1.6 The Levi-Civita pseudotensor . . . . . . . . . . . . . . . . . 232.1.7 Connections and covariant differentiation . . . . . . . . . . . 242.1.8 Lie differentiation . . . . . . . . . . . . . . . . . . . . . . . . 292.1.9 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.2 Geometry of hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . 322.2.1 Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . 322.2.2 Spacelike hypersurfaces in spacetime . . . . . . . . . . . . . 352.2.3 The Gauss-Codazzi relations . . . . . . . . . . . . . . . . . . 40

2.3 Foliations of spacetime . . . . . . . . . . . . . . . . . . . . . . . . . 412.3.1 The lapse function . . . . . . . . . . . . . . . . . . . . . . . 412.3.2 Eulerian observers . . . . . . . . . . . . . . . . . . . . . . . 422.3.3 The shift vector and the ADM metric . . . . . . . . . . . . . 432.3.4 Some useful relations . . . . . . . . . . . . . . . . . . . . . . 45

3 3+1 formulation of general relativity 483.1 3+1 decomposition of Einstein’s equations . . . . . . . . . . . . . . 48

3.1.1 ADM formalism . . . . . . . . . . . . . . . . . . . . . . . . . 483.1.2 BSSN formalism . . . . . . . . . . . . . . . . . . . . . . . . 503.1.3 Slicing conditions . . . . . . . . . . . . . . . . . . . . . . . . 53

3.2 3+1 decomposition of the hydrodynamics equations . . . . . . . . . 593.2.1 Conservation of energy and momentum . . . . . . . . . . . . 593.2.2 Ideal fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.2.3 Equations of state (EOS) . . . . . . . . . . . . . . . . . . . . 643.2.4 Relations among quantities as measured by the Eulerian and

comoving observers . . . . . . . . . . . . . . . . . . . . . . . 66

5

CONTENTS

3.2.5 Total rest mass (or baryon number) conservation and elec-tron fraction advection . . . . . . . . . . . . . . . . . . . . . 70

3.3 3+1 decomposition of Maxwell’s equations . . . . . . . . . . . . . . 733.3.1 Some basic facts about the electromagnetic field . . . . . . . 733.3.2 3+1 treatment of the electromagnetic field . . . . . . . . . . 753.3.3 Ideal MHD limit . . . . . . . . . . . . . . . . . . . . . . . . 78

3.4 Flux-conservative formulation and vector potential evolution . . . . 803.4.1 3+1 ideal-fluid magnetohydrodynamics in the ideal MHD

limit in flux-conservative form . . . . . . . . . . . . . . . . . 803.4.2 3+1 Maxwell equations in flux-conservative form . . . . . . . 833.4.3 Evolution of the electromagnetic 4-potential . . . . . . . . . 87

4 Numerical techniques and the Spritz code 904.1 A brief introduction to hyperbolic PDEs . . . . . . . . . . . . . . . 90

4.1.1 Notions of hyperbolicity . . . . . . . . . . . . . . . . . . . . 904.1.2 A grasp at non-linear dynamics: shocks and rarefaction waves 93

4.2 Numerical approaches to hyperbolic PDEs . . . . . . . . . . . . . . 974.2.1 Finite-difference schemes . . . . . . . . . . . . . . . . . . . . 974.2.2 Finite-volume techniques for conservation laws . . . . . . . . 1024.2.3 Approximate Riemann solvers . . . . . . . . . . . . . . . . . 1034.2.4 Reconstruction techniques . . . . . . . . . . . . . . . . . . . 1064.2.5 Importance of flux-conservative formulation and strong hy-

perbolicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1094.3 The Spritz code . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

4.3.1 The Einstein Toolkit environment framework . . . . . . . 1114.3.2 How Spritz works . . . . . . . . . . . . . . . . . . . . . . . 1124.3.3 Evolution of electromagnetic variables . . . . . . . . . . . . 1134.3.4 Conservatives-to-primitives (‘C2P’) solvers . . . . . . . . . . 117

5 BNS simulations with Spritz 1225.1 Choice of electromagnetic gauge . . . . . . . . . . . . . . . . . . . . 122

5.1.1 Special-relativistic shocktube test with algebraic gauge . . . 1225.1.2 Special-relativistic magnetic rotor test with algebraic gauge . 1255.1.3 Magnetised, equal-mass BNS with polytropic EOS . . . . . . 1275.1.4 Magnetised TOV stars with generalised Lorenz gauge . . . . 139

5.2 Realistic, tabulated EOS handling . . . . . . . . . . . . . . . . . . . 1425.2.1 Non-magnetised, equal-mass BNS with SLy4 EOS . . . . . . 1425.2.2 Non-magnetised, unequal-mass BNS with SLy4 EOS . . . . 145

6 Conclusions and future work 152

6

Chapter 1Introduction and purpose

1.1 Why numerical relativity?

General relativity is our current theory of gravity and is the foundation of ourunderstanding of cosmic phenomena. Since its publication by Albert Einstein in1915, several tests have been proposed and performed in order to establish ob-servational evidence for it. Although generally successful, these tests were mainlymeant to explore the weak-field regime of the theory, leaving the strong-field, high-velocity regime mostly untested. Furthermore, the list of known exact solutionsto Einstein’s equations is very short and many of these rely on strong symmetryassumptions: thus, such solutions are only capable of describing highly idealisedgravitational systems. As a result, many of the more interesting scenarios whichare thought to occur in Nature cannot be dealt with analytically: examples include,but are not limited to, the coalescence of compact binary systems, the gravitationalcollapse to a black hole, the generation and propagation of gravitational waves,and supernovae explosions.

Given these facts, it is quite natural to try to solve the Einstein’s equationsnumerically. However, this turns out to be a formidable task: they are a set of non-linear, coupled partial differential equations in space and time, with the additionalcomplication of the spacetime metric being unknown until the equations have beensolved. This means, for example, that singularities may arise as the evolutiongoes on: in this case, some of the metric-related terms in the equations becomeinfinite, causing the numerical integration to fail. Sometimes, a clever choice ofcoordinates can remove the singularity; however, in case the latter is a physicalone, this is not possible and other strategies must be adopted in order to achievestable, long term simulation. In addition to this, numerical solutions to Einstein’sequations in realistic conditions typically require a huge amount of computationalresources, both in terms of power and memory, so that supercomputers becomeessential. Therefore, considerable skills are needed in order to develop algorithmswhich are both effective and efficient in performing numerical relativity simu-lations.

7

Chapter 1 Introduction and purpose

1.2 Gravitational waves

On 14th September 2015, a gravitational wave (‘GW’ ) signal was detected for thefirst time by the two LIGO interferometers in the USA; the Virgo collaboration,whose interferometer is located near Pisa, Italy, contributed substantially to theanalysis of the related observational data. The event, named GW150914, wasattributed to the inspiral and merger of two black holes with masses approximatelyequal to 29 and 36 times the mass of the Sun [2]. From that moment on, many othersources of gravitational radiation were detected by the LIGO-Virgo collaboration.Most notably, on 17th August 2017, the three interferometers measured the firstGW signal to be ascribed to the coalescence of two neutron stars; this event wasnamed GW170817 [1]. Just about 1.7 s after merger, the Fermi satellite observeda short gamma-ray burst, so this was also the first GW event for which anelectromagnetic counterpart could be measured. This detection can be marked asa milestone in the modern era of multimessenger astrophysics : many signals,typically of very different nature (e.g., gravitational and electromagnetic radiation,neutrinos, cosmic rays), are observed and interpreted collectively in order to getthe best possible picture of the cosmic phenomenon under consideration.

Identifying the source of any observed gravitational signal requires a compar-ison between the measured signal itself and a waveform whose origin is known.However, the extreme physical conditions which are thought to produce the grav-itational radiation detected by LIGO and Virgo are not (at least, at present) re-producible on Earth. Therefore, computer simulations of compact binary mergersare gaining increasing popularity and interest as regards the generation of grav-itational wave templates starting from different configurations of magnetic field,neutron star equation of state, and spin and mass of the spiraling objects. Onthe other hand, observation of gravitational wave signals can help constrain theequation of state for baryonic matter at ultra-high density, which is believed toconstitute the core of neutron stars and the properties of which are still mostlyunknown.

1.3 Gamma-ray bursts

During the merger of two neutron stars, a shear layer is produced at the contactsurface between the two objects, and here a Kelvin-Helmoltz instability istriggered [42]. Assume the two stars initially possess a purely poloidal magneticfield: then, the turbulent nature of the flow during coalescence, consisting in small-to large-scale eddies and vortices, causes part of the kinetic energy of the fluid tobe transformed into magnetic energy, thus amplifying the magnetic field by severalorders of magnitude until the kinetic and magnetic energy contributions are equal

8


[73]. After that, if the system is able to generate a differentially-rotating accretiontorus around the merger remnant, the poloidal component of the magnetic field candrive a magneto-rotational instability which further amplifies the magneticfield itself and causes it to organise into a dual structure: the field is mainlytoroidal into the disk and mainly poloidal in the central region [7, 41, 58]. Now,suppose the merger remnant collapses to a (spinning) black hole and the poloidalmagnetic field lines penetrate the ergosphere of the latter. Then, an electric fieldcan be induced gravitationally into the ergosphere itself and, if charged particlesare injected there somehow, poloidal electric currents can be established whichcan eventually launch a relativistic jet in both directions orthogonal to the orbitalplane of the black hole; this is believed to be the essence of the Blandford-Znajekmechanism [12, 43, 58, 60].

Blandford and Znajek proposed that e–- e+ pairs can be generated in cascadedue to the strong electric field in the ergosphere of the rotating black hole [12].On the other hand, Ruiz and collaborators have shown, by means of numericalsimulations, that charged particles can be driven into the ergosphere from thesurrounding torus [60]. A third possibility is that the torus is hot enough totrigger reactions involving neutrinos [58, 59], such as:

1. Electron and positron capture on nucleons, which can produce electronicneutrinos and antineutrinos:

e− + p→ n + νe (1.3.1)

e+ + n→ p + νe (1.3.2)

2. Electron-positron pair annihilation, which can produce neutrinos and an-tineutrinos of all flavours:

e− + e+ → νi + νi (1.3.3)

3. Plasmon decay :γ → νi + νi (1.3.4)

Being electrically neutral, neutrinos and antineutrinos resulting from the abovereactions can escape the accreting torus and may annihilate each other into anhighly energetic e–- e+ pair, thus powering a relativistic jet. It may well be possiblethat jets are powered by a combination of the Blandford-Znajek and neutrinoemission mechanisms [58].

After matter has been expunged from the binary system, shocks are likely tooccur into the jet, thus causing the emission of gamma rays. After that, the jetinteracts with the interstellar medium, again resulting in shocks which give rise to

9


the afterglow, consisting in x-rays, visible light, and radio emission. This is (inextreme synthesis) the fireball model of gamma-ray burst events [55].

In view of the above discussion and of the electromagnetic counterpart of thedetection event GR170817, binary neutron star merger simulations can help shedlight onto the physical origin of short gamma-ray bursts, dealing with the highlyinvolved spacetime, matter, and electromagnetic dynamics consistently. More-over, results from these simulations can be used to follow the evolution of thepost-merger phase by means of other numerical codes in order to investigate theevolution of the relativistic jets produced by the coalescence and the possible for-mation of gamma-ray and afterglow emission.

1.4 Outline of this work

In this work, numerical simulations of binary neutron stars (‘BNS’ ) systems areperformed from the inspiral stage to the merger and post-merger phases. Theprimary goals are:

1. to investigate the effects of the electromagnetic gauge choice for the evolutionof magnetised BNS configurations. While irrelevant on the physics from ananalytical perspective, this turns out to play a crucial role when coming tonumerical simulations, and poses a number of non-trivial problems;

2. to explore the dynamics of BNS systems when a realistic equation of statefor the matter which neutron stars are made of is employed.

Simulations are performed using Spritz, a new, fully general relativistic magne-tohydrodynamics (‘GRMHD’ ) code developed by Prof. Bruno Giacomazzo andcollaborators [19, 21] and based upon the Einstein Toolkit [38, 46, 72]. Thiswork is also an opportunity to test Spritz against possible bugs before it can befinally included into the Einstein Toolkit as a module (‘thorn’), and the hopeis that the improvements in code performance achieved within the scope of thisthesis may help accelerate this process.

The work is organised as follows:

1. in part 2, the basic concepts in differential geometry needed for the sub-sequent developments are reviewed. Then, the geometry of spacelike hy-persurfaces embedded in spacetime is studied and the concept of spacetimefoliation is presented;

2. in part 3, the 3+1 formulation of general relativity is introduced and theADM and BSSN formalisms are exposed; contextually, different possible

10


gauge conditions on the metric (‘slicing conditions’) are discussed. Then,the GRMHD equations in the ideal MHD limit are cast in 3+1 form as well,in a flux-conservative fashion. In passing, the basic properties of ideal fluidsand their equations of state (‘EOS’) are outlined;

3. in part 4, the main numerical techniques used to solve the GRMHD equationsare analysed and the main working principles of Spritz are explained;

4. finally, part 5 presents the results of the numerical simulations done withSpritz.

Notations and conventions

The Einstein summation convention is understood throughout this work,unless specifically stated. Moreover, geometric units are adopted, so that thespeed of light c and the universal gravitational constant G are set to unity,

c = G = 1 .

11

Chapter 2Geometric preliminaries

The basic ideas in differential geometry of manifolds and submanifolds found in thischapter are presented following somewhat closely chapters 1 — 4 of [37], althoughclassical references such as [71], [50], [15], and [17], as well as the lecture notes[68], have been referred.

2.1 Review of basic differential geometry

2.1.1 Fundamentals

A homeomorphism f :X → Y between topological spaces X and Y is a one-to-one, continuous map whose inverse is continuous. Given D∈N\0, a topologicalD-manifold is a Hausdorff, second-countable topological space M which is lo-cally homeomorphic to RD, that is, every point in RD has an open neighbourhoodwhich is homeomorphic to some open set in RD. A chart or coordinate systemon M is a pair (U ,Φ) where U ∈M is open and Φ is a homeomorphism

Φ : U → RD

U 3 p 7→(x0, x1, . . . xD−1

)∈ RD .

(2.1.1)

Given K ∈ N \ 0, an atlas onM is a finite set (Uk,Φk)k∈1, ... ,K of charts on

M such that⋃

k∈1, ... ,KUk =M. The atlas is called smooth if, for all Ui∩Uj 6= ,

the transition maps

Φi Φ−1j : Φj (Ui ∩ Uj) ⊂ RD → Φi (Ui ∩ Uj) ⊂ RD (2.1.2)

are smooth in the sense of RD; for any i, j∈1, . . . , K, the map Φi Φ−1j can be

regarded as a change of coordinates on M. A chart (U ,Φ) on M is compat-ible with a smooth atlas on M if the inclusion of (U ,Φ) into that atlas results ina smooth atlas; therefore, any smooth atlas onM defines a maximally smoothatlas on M as the collection of all charts on M which are compatible with thegiven atlas. Finally, a smooth manifold is a manifoldM endowed with a max-imally smooth atlas.

12

Chapter 2 Geometric preliminaries

2.1.2 Vector fields on a smooth manifold

A curve on a smooth manifold M is a map

s : [0, 1]⊂R→Mλ 7→ s(t) .

(2.1.3)

Given a chart (U ,Φ) on M and p∈M, let

S(0)p := s : [0, 1]⊂R→M| s(0) = p (2.1.4)

and introduce an equivalence relation ∼ on S(0)p such that

s1 ∼ s2 ifd

dλ(Φ s1)(0) =

d

dλ(Φ s2)(0) ∀s1 , s2 ∈ S(0)

p . (2.1.5)

Notice that this definition is in fact independent of the particular choice of chart.Indeed, if (V ,Ψ) is any other chart onM, then ΦΨ−1 is smooth and, on (U ∩ V)(2.1.5) can be rewritten as

d

dλ

((Φ Ψ−1

) (Ψ s1)

)(0) =

d

dλ

((Φ Ψ−1

) (Ψ s2)

)(0) . (2.1.6)

The tangent space at p∈M is the quotient

TpM := S(0)p ∼ ≡ span

[s]∼ | s∈S

(0)p

(2.1.7)

and elements of TpM are called tangent vectors at p∈M. TpM can be easilyshown to be a vector space in the usual sense. A vector field on M is a mapwhich assigns an element Xp ∈ TpM to each point p ∈ M and the space of allvector fields on M is denoted TM.

Actually, another approach to vectors on a smooth manifold is possible. Afunction f :M → R is called smooth if there exists a chart (U ,Φ) on M suchthat f Φ−1 ∈ C∞

(Φ(U)⊂RD → R

)is smooth; if the smooth atlas on M is

maximal, then f is smooth with respect to any chart in it and one writes f ∈C∞ (M→ R). Given a family φε :M→Mε∈[0,1] of diffeomorphisms (i.e. ,smooth homeomorphisms) on M, a derivation is defined as the operator

Dφε :f ∈C∞ (M→ R) → f ∈C∞ (M→ R)

Dφε [f ](p) := limε→0

f(φε(p))− f(p)

ε.

(2.1.8)

13


Let (U ,Φ) be a chart on M and take p, q ∈U in such a way that q = φε(p) andΦ(q) = Φ(p) + εXp, where Xp =

(X0p , . . . , X

D−1p

)∈ RD and ‖Xp‖ = 1. In this

case, calling X := Dφε and Φ(p) =(x0, . . . , xD−1

)∈RD, (2.1.8) reads

X[f ](p) = Dφε [f ](p) := limε→0

f(φε(P ))− f(p)

ε= lim

ε→0

f(p′)− f(p)

ε=

= limε→0

(f Φ−1) (Φ(p′))− (f Φ−1) (Φ(p))

ε=

= limε→0

(f Φ−1) (Φ(p) + εXp)− (f Φ−1) (Φ(p))

ε=

=∂

∂Xp

(f Φ−1

)(Φ(p)) = Xµ

p

∂

∂xµ(f Φ−1

)(Φ(p)) =: Xµ

p ∂µf(p) .

(2.1.9)

The operator X is called a vector field on M. The expression

X[f ](p) =: Xµp ∂µf(p) (2.1.10)

gives a very ‘concrete’ idea of vector fields as directional derivatives (in the usual,Euclidean sense) on any chart. Indeed, following the same steps as those leading

to (2.1.9), but taking e(µ) :=

0, . . . , 1↑

µ-th direction

, . . . , 0

, in place of the generic

Xp , one can define, for all µ∈0, . . . , D − 1, the partial derivative operator

∂µ (which is a vector field on M) as

∂µ[f ](p) :=∂

∂xµ(f Φ−1

)(Φ(p)) ≡ ∂µf(p) ; (2.1.11)

thus X, just like a ‘directional derivative’, is a linear combination of the ∂µ’ s,

X = Xµp ∂µ ∀p∈M . (2.1.12)

Now, repeat the same passages that led to (2.1.9), but take another chart (V ,Ψ)on M such that p, q∈V , Ψ(p) =

(x′ 0, . . . , x′D−1

)∈RD,

Ψ(q) = Ψ(p) + tX′p, where X′p =(X ′ 0p , . . . , X

′D−1p

)∈ RD and

∥∥X′p∥∥ = 1; thisresults in X[f ](p) = Xµ∂µ(f Ψ−1) (Ψ(p)). Thus, on U ∩ V , consistency with(2.1.9) requires

X ′µp∂

∂x′µ(f Ψ−1

)(Ψ(p))

!= Xν

p

∂

∂xν(f Φ−1

)(Φ(p)) =

=Xνp

∂x′µ

∂xν∂

∂x′ µ(f Φ−1

)(Φ Ψ−1

)(Ψ(p)) ,

(2.1.13)

14


or, in other words,

X ′µp =∂x′µ

∂xνXνp . (2.1.14)

(2.1.14) is the transformation law for the components of a vector field under achange of coordinates x 7→ x′.

The two notions of vector field outlined above are intimately linked to oneanother. Indeed, use the same derivation as in (2.1.9), but take s∈S(0)

p in such away that p = s(0) and p′ = s(t) for some t∈ [0, 1]. Then,

X[f ](p) = limt→0

f(φt(P ))− f(p)

t= lim

t→0

f(p′)− f(p)

t= lim

t→0

f(s(t))− f(s(0))

t=

= limt→0

(f s)(t)− (f s)(0)

t=

d

dt(f s)(0) .

(2.1.15)

On the other hand, because M is a smooth manifold, then the implication

d

dt(Φ s1)(0) =

d

dt(Φ s2)(0)

⇓d

dt(f s1)(0) =

d

dt(f s2)(0)

(2.1.16)

holds for any s1 , s2∈S(0)p and for any smooth function f :M→ R, so that (2.1.5)

can be written as

s1 ∼ s2 ifd

dt(f s1)(0) =

d

dt(f s2)(0) ∀s1 , s2 ∈ S(0)

p . (2.1.17)

Together, (2.1.15) and (2.1.17) show that each tangent vector at p∈M uniquelydefines a derivation at p; thus, one can think of tangent vectors at p either as equiv-alence classes of curves having the same ‘time derivative’ at p, or as derivations atp. Thus, for any p∈M, one can then define the tangent vectors

Xp∈TpM s.t. Xp[f ]!

= X[f ](p) (2.1.18)

∂(p)µ ∈TpM s.t. ∂(p)

µ [f ]!

= ∂µ[f ](p) (2.1.19)

for all f :M → R smooth, and the above identification, together with (2.1.12),

allows one to think of the set of partial derivative operators∂

(p)µ

D−1

µ=0as a basis

for TpM; therefore, dim(TpM) = D and one can expand any Xp∈TpM as Xp ≡Xµp ∂

(p)µ . One refers to

∂

(p)µ

D−1

µ=0as to the natural or coordinate basis of TpM

and to∂µ

D−1

µ=0as a frame field on M.

15


Given a vector field X onM and a smooth map f :M→ R, denote X[f ] :M→R the map p 7→ X[f ](p) for all p∈M. Given another vector field Y on M, thecommutator or Lie bracket of X with Y is the vector field [X, Y ] ≡ [X, Y ]µ ∂µsuch that

[X, Y ][f ](p) := (X[Y [f ]]− Y [X[f ]])(p) = (Xν∂νYµ∂µf − Y ν∂νX

µ) ∂µf

[X, Y ]µ = Xν∂νYµ − Y ν∂νX

µ .(2.1.20)

2.1.3 1-form fields on a smooth manifold

The cotangent space at p∈M is the space dual to TpM, that is,

T ∗pM := (TpM)∗ = ωp :TpM→ R |ωp is linear . (2.1.21)

T ∗pM is a vector space with the obvious addition and scalar multiplication of linearmaps. Elements of T ∗pM are called covectors or 1-forms and a covector fieldof 1-form field overM is a map which assigns a 1-form ωp∈T ∗pM to each pointp∈M. The space of all 1-form fields on M is denoted T ∗M.

The coordinate basis of TpM induces the canonical dual basisλµpD−1

µ=0on

T ∗pM,

λµp

(∂(p)ν

)!

= δµν ⇒ ωp ≡ (ωp)µ λµp = ωp

(∂(p)µ

)λµp ∀ωp∈T ∗pM . (2.1.22)

Some insight can be gained about the λµp ’s by defining a special covector, thedifferential dfp of a smooth map f :M→ R at a point p∈M,

dfp(Xp) := Xp[f ](p) ∀Xp∈TpM . (2.1.23)

Given a chart (U ,Φ) on M such that p∈U , one has

dfp ≡ (dfp)µ λµp = dfp

(∂(p)µ

)λµp = ∂(p)

µ [f ]λµp =

=∂

∂xµ(f Φ−1

)(Φ(p))λµp .

(2.1.24)

where (2.1.11) has been used. Define the coordinate functions xµp :M→ R in

such a way that xµp(p) := xµ, where Φ(p) ≡(x0, . . . , xD−1

)⊂ RD. Application of

(2.1.24) with xµp in place of f gives

dxµp =∂

∂xν(xµp Φ−1

)(Φ(p))λνp =

=∂

∂xν(xµp Φ−1

)(x0, . . . , xD−1

)λνp = δµνλ

νp = λµp ,

(2.1.25)

16


that is, the canonical basis of T ∗pM induced by the natural basis of TpM is nothing

but the setdxµpD−1

µ=0. Letting dxµ be the 1-form field on M whose value at each

p∈M is dxµp , the set dxµD−1µ=0 is referred to as a dual frame field on M.

Notice that invariance of a vector field operator under change of coordinatescan be regarded as inducing a transformation either in the components of thevector field itself or in the elements of the frame field, defined by (2.1.11); in thelatter case, (2.1.13) gives (interchanging x↔ x′ and µ↔ ν)

∂µ′ =

∂xν

∂x′µ∂ν′ . (2.1.26)

Let ω ≡ ωµdxµ∈T ∗M, where ωµ ≡ ω

(∂µ

):M→ R is defined in such a way that

(ωp)µ = ωp

(∂

(p)µ

)at each point p ∈M. Requiring ω ≡ ωµdx

µ != ω′µdx

′µ ∈ T ∗Mgives

ω′µ ≡ ω(∂µ′)

= ω

(∂xν

∂x′µ∂ν′)

=∂xν

∂x′µω(∂ν′)

=∂xν

∂x′µων , (2.1.27)

so that

ω′µ =∂xν

∂x′µων (2.1.28)

is the transformation law of the components of the 1-form ω under a change ofcoordinates x 7→ x′(x).

2.1.4 Tensor fields on a smooth manifold

Given p∈M, define the space

(Tp)klM :=

τp :T ∗pM× · · · × T ∗pM︸︷︷︸k times

×TpM× · · · × TpM︸︷︷︸l times

→ R | τp is multilinear

.

(2.1.29)Elements of (Tp)

klM are called

(kl

)-tensors at p and a tensor field over M is

a map which assigns a tensor τp∈ (Tp)klM to each point p∈M. The space of all

tensor fields on M is denoted T klM.

For example,(

01

)-tensors at p are 1-forms and

(10

)-tensors at p are maps from

T ∗pM to R, i.e., elements of T ∗∗p M; however, one can identify elements of T ∗∗p Mwith elements of TpM (tangent vectors at p) by means of the isomorphism

θXp :T ∗pM→ R

θXp(ωp)!

= ωp(Xp) ∀Xp∈TpM ,(2.1.30)

17


and this identification will be understood in what follows.

Given the coordinate basis∂

(p)µ

D−1

µ=0⊂ TpM and the canonical dual basis

dxµpD−1

µ=0⊂T ∗pM, the set

∂(p)µ1⊗ · · · ⊗ ∂(p)

µk⊗ dxν1p ⊗ · · · ⊗ dxµlp

D−1

µ1, ... , µk, ν1, ... , νl=0⊂ (Tp)

klM (2.1.31)

can be easily shown to be a basis for (Tp)klM; any

(kl

)-tensor field τ on M can

then be expanded onto the set∂µ1 ⊗ · · · ⊗ ∂µk ⊗ dxν1 ⊗ · · · ⊗ dxµl

D−1

µ1, ... , µk, ν1, ... , νl=0, (2.1.32)

called the tensorial frame field on M, as

τ ≡ τµ1... µkν1... νl ∂µ1 ⊗ · · · ⊗ ∂µk ⊗ dxν1 ⊗ · · · ⊗ dxνl , (2.1.33)

where

τµ1... µkν1... νl ≡ τ(dxµ1 , . . . , dxµk , ∂ν1 , . . . , ∂νl

):M→ R (2.1.34)

is defined in such a way that

(τp)µ1... µk

ν1... νl= τp

(dxµ1p , . . . , dx

µkp , ∂

(p)ν1, . . . , ∂(p)

νl

)(2.1.35)

for all p∈M.

Given a(kl

)-tensor field τ and a

(rs

)-tensor field π on M, the outer product

or tensor product of τ and π is the(k+rl+s

)-tensor field τ ⊗ π defined by

(τ ⊗ π)(ω(1), . . . , ω(k+r);X(1), . . . X(l+s)

):=

:= τ(ω(1), . . . , ω(k);X(1), . . . X(l)

)π(ω(k+1), . . . , ω(k+r);X(l+1), . . . X(l+s)

)(2.1.36)

for any X(1) . . . , X(l+s) ∈ TM and ω(1), . . . , ω(k+r) ∈ T ∗M. The components ofτ ⊗ π on the tensorial frame field (2.1.32) are

(τ ⊗ π)µ1... µk µk+1... µk+rν1... νl νl+1... νl+s

=

= τµ1... µkν1... νlπµk+1... µk+r

νl+1... νl+s .(2.1.37)

Tensor fields which can be written as tensor products of vector fields and 1-formfields are called simple.

18


The transformation law of a(kl

)-tensor field τ onM under a change of coordi-

nates x→ x′(x) are inferred from (2.1.14) and (2.1.28),

τ ′µ1 ... µk

ν1 ... νl =∂x′µ1

∂xρ1· · · ∂x

′µk

∂xρk∂xσ1

∂x′ν1· · · ∂x

σl

∂x′νlτ ′ρ1 ... ρk

σ1 ... σl . (2.1.38)

The contraction of a tensor field τ ∈ T klM with respect to the index ρ ∈1, . . . ,min(k, l) is a map

C :T klM→ T k−1l−1M

C(τ) := τ(ω(1), . . . , dxρ, . . . , ω(k−1);X(1), . . . , ∂ρ, . . . , X

(l−1)) (2.1.39)

for any ω(1), . . . , ω(k−1)∈T ∗M and for any X(1) . . . , X(l−1)∈TM; on the tensorialframe field (2.1.32), one has

(C (τ))µ1... µk−1

ν1... νl−1= τµ1... ρ... µkν1... ρ... νl . (2.1.40)

A(

0l

)-tensor field σ on M is called symmetric in the indices i and j if

σ(X(1), . . . , X(i), . . . , X(j), . . . , X(l)

)=

= σ(X(1), . . . , X(j), . . . , X(i), . . . , X(l)

) (2.1.41)

for any X(1), . . . , X(k) ∈ TM. Likewise, a(

0l

)-tensor field α on M is called

antisymmetric in the indices i and j if

α(X(1), . . . , X(i), . . . , X(j), . . . , X(l)

)=

= − α(X(1), . . . , X(j), . . . , X(i), . . . , X(l)

) (2.1.42)

for any X(1), . . . , X(k) ∈ TM. The components of (2.1.41) and (2.1.42) on thetensorial frame field (2.1.32) read, respectively,

σν1... νi... νj ... νl = σν1... νj ... νi... νl (2.1.43)

αν1... νi... νj ... νl = −αν1... νj ... νi... νl . (2.1.44)

The symmetrisation of a tensor field τ ∈ T 0kM is the totally symmetric(

0k

)-tensor field

sym(τ) :=1

k!

∑π∈Sk

πτ , (2.1.45)

where Sk denotes the group of permutations of k elements and

πτ(X(1), . . . , X(k)

):= τ

(X(π(1)), . . . , X(π(k))

). (2.1.46)

19


The symmetric product of the totally symmetric(

0k

)-tensor field σ with the

totally symmetric(

0l

)-tensor field ζ is the totally symmetric

(0kl

)-tensor field

σζ := sym(σ ⊗ ζ) . (2.1.47)

A k-form field (or differential or exterior k-form field) on M is a totallyantisymmetric

(0k

)-tensor field onM. The antisymmetrisation of a tensor field

τ ∈ T 0kM is the k-form field

alt(τ) :=1

k!

∑π∈Sk

sgn(π) πτ , (2.1.48)

where sgn(π) is the sign of the permutation π∈Sk . The wedge or antisymmet-ric product of a k-form field α with an l-form field β is defined as

α ∧ β := alt(α⊗ β) , (2.1.49)

which implies α ∧ α = 0.

As important applications of (2.1.49) and (2.1.47), notice that, thanks to(2.1.45) and (2.1.48), the dual frame field dxµD−1

µ=0 ⊂T ∗M satisfies

dxµdxν =1

2(dxµ ⊗ dxν + dxν ⊗ dxµ) (2.1.50)

dxµ ∧ dxν =1

2(dxµ ⊗ dxν − dxν ⊗ dxµ) (2.1.51)

and, more generally,

dxµ1 · · · dxµk =1

k!

∑π∈Sk

dxπ(µ1) ⊗ · · · ⊗ dxπ(µk) (2.1.52)

dxµ1 ∧ · · · ∧ dxµk =1

k!

∑π∈Sk

sgn(π) dxπ(µ1) ⊗ · · · ⊗ dxπ(µk) . (2.1.53)

Every totally symmetric(

0k

)-tensor field σ and every k-form α on M can be ex-

panded, respectively, as

σ ≡ σµ1... µkdxµ1 ⊗ · · · ⊗ dxµk = σµ1... µkdx

µ1 · · · dxµk (2.1.54)

α ≡ αµ1... µkdxµ1 ⊗ · · · ⊗ dxµk = αµ1... µkdx

µ1 ∧ · · · ∧ dxµk ; (2.1.55)

in particular, given the(

0k

)-tensor field τ on M, one has

sym(τ) ≡ τ(µ1... µk) dxµ1 · · · dxµk (2.1.56)

alt(τ) ≡ τ[µ1... µk] dxµ1 ∧ · · · ∧ dxµk , (2.1.57)

20


where

τ(µ1... µk) := (sym(τ))µ1... µk =1

k!

∑π∈Sk

τπ(µ1), ... , π(µk) (2.1.58)

τ[µ1... µk] := (alt(τ))µ1... µk =1

k!

∑π∈Sk

sgn(π) τπ(µ1), ... , π(µk) . (2.1.59)

2.1.5 The metric tensor

A metric g on a smooth manifoldM is a symmetric, non-degenerate(

02

)-tensorial

field on M. Expanding

g ≡ gµν dxµ ⊗ dxν = gµν dx

µdxν , (2.1.60)

the symmetry and non-degeneracy requirements read, respectively,

1. g(X, Y )!

= g(Y,X) ∀X, Y ∈TM , i.e. gµν = gνµ (2.1.61)

2. For all Y ∈TM , g(X, Y ) = 0 ∀x∈TM ⇒ Y = 0 . (2.1.62)

Notice that (2.1.61) and (2.1.62) make g into an inner product,

gp ≡ 〈· , ·〉p :TpM× TpM→ R ∀p∈M . (2.1.63)

For any p∈M, the norm of a vector Xp∈TpM is defined as

‖Xp‖2 := 〈Xp, Xp〉p = (gp)µν XµpX

νp , (2.1.64)

Given any p∈M, by means of the Gram-Schmidt procedure, one can always

find a basisep(µ)

D−1

µ=0⊂TpM which is orthonormal with respect to the metric

g, meaning that ⟨ep(µ), ep(ν)

⟩p

= ±δµν ∀µ∈0, . . . , D − 1 . (2.1.65)

The set of vector fieldse(µ)

D−1

µ=0⊂ TM such that, for all µ ∈ 0, . . . , D − 1,

the value of e(µ) at any p∈M is ep(µ) is commonly referred to as an orthonor-mal tetrad on M. Though there is an infinite number of orthonormal tetradson M, the signature of g, that is, the number of vector fields e(µ) such that⟨ep(µ), ep(ν)

⟩p

= −1, 0, 1 for any p ∈M, can be shown to be independent of the

orthonormal tetrad chosen. A smooth manifold M endowed with a metric whose

21


signature is (+, +, . . . , +) is called a Riemannian manifold, while the metricon a Lorentzian manifold has signature (-, +, . . . , +).

In this work, only smooth Lorentzian manifolds will be considered. A Lorentzianmetric always admits a decomposition as

g ≡ −(θ0)2

+(θ1)2

+ · · ·+(θD−1

)2, θµ ≡ cµν dx

µ , cµν∈R , (2.1.66)

where (θµ)2 ≡ θµθµ ≡ sym(θµ ⊗ θµ) = θµ ⊗ θµ. Because of (2.1.66), at any pointp∈M, any vector Xp∈TpM such that ‖Xp‖2 ≤ 0 defines a double cone C(Xp) inTpM, the causal cone of Xp; the boundary of Cp is determined by the condition‖Xp‖2 = 0 and is called the light cone of Xp . The causal cone C(Xp) splits intotwo convex cones C+(Xp) and C-(Xp), defined by the relations

C+(Xp) : X0p > 0 Future of Xp (2.1.67)

C-(Xp) : X0p < 0 Past of Xp . (2.1.68)

The Lorentzian manifold M is said to be time-orientable if the future-pastsplitting in (2.1.67) can be carried out in a continuous way over the entire M;a Lorentzian, time-orientable, smooth manifold is called a spacetime. Given aspacetime M and any point p∈M, a vector Xp ∈ TpM is called timelike, nullor spacelike if ‖Xp‖2 < 0, ‖Xp‖2 = 0 or ‖Xp‖2 > 0, respectively; hence, a curvein spacetime is timelike, null or spacelike if its tangent vectors are everywheretimelike, null or spacelike, respectively. Any vector or curve which is not spacelike(i.e. , it is timelike or null) is called causal.

One can use the metric to ‘lower the indices’ of vector fields onM by defininga map

g :TM→ T ∗M

g(X)!

= g(X, ·) = gµν (dxµ ⊗ dxν)(Xρ ∂ρ , ·

)=

= gµνXρ dxµ

(∂ρ

)dxνp = gµνX

ρ δµρdxν = (gµνX

µ) dxν =: Xν dxν

(2.1.69)

for all X ∈ TM and for all p ∈M. Because g is non-degenerate, the map g isinvertible and the inverse g−1 :T ∗M→ TM satisfies g−1 g = 1TM. This meansthat, for all X∈TM,

X = g−1(g(X)) = g−1(Xµdxµ) = Xµ g

−1(dxµ) . (2.1.70)

Now g−1(dxµ) ∈ TM, so g−1(dxµ) ≡ ξµν ∂ν for some coefficients ξµν . Therefore,using Xµ := gµνX

ν and X ≡ Xµ∂µ , (2.1.70) reduces to

Xν ∂ν = X = gµνXνξµρ∂ρ , (2.1.71)

22


which implies gµνξµρ !

=δρν . One can think of ξµν as the components of a(

20

)-tensor

field on M; it is standard practice to call such field the inverse or dual of themetric g, denote such field as g−1 and rename (g−1)

µν ≡ gµν . Then,

gµνgνρ!

= δµρ (2.1.72)

is the defining relation of g−1. The inverse metric can be used to ‘raise the indices’of any 1-form field ω∈T ∗M:

g−1(ω) ≡ g−1(ωµdxµ) = ωµg

−1(dxµ) =

= ωµgµν ∂ν := ων ∂ν .

(2.1.73)

The maps g and g−1 are called musical isomorphisms, because they are indeedisomorphisms and they are used to raise and lower indices, just like the sharp andflat symbols are used to raise and lower the pitch of musical notes by one semitone.

2.1.6 The Levi-Civita pseudotensor

The Levi-Civita symbol in a spacetime M is the quantity

εµ0... µD−1= εµ0... µD−1 =

+1 even permutations of (µ0 . . . µD−1)

−1 odd permutations of (µ0 . . . µD−1)

0 otherwise

, (2.1.74)

with the convention ε0 ... D−1 = ε 0 ... D−1 = +1. For every D×D square matrix M ,it holds that

εµ0... µD−1det(M) = εα0... αD−1

Mα0µ0 . . .M

αD−1µD−1

; (2.1.75)

Given a point p ∈ M, take charts (U ,Φ) and (V ,Ψ) such that p ∈ U ∩V , with

Φ(p) =(x0, . . . , xD−1

)and Ψ(p) =

(x′0, . . . , x′D−1

). Let Jµα := ∂x′µ

∂xαbe the

Jacobian of the map x 7→ x′ and set Mαiµi

!= (J−1)

αiµi≡ ∂xαi

∂x′µiin (2.1.75) to write

εµ0... µD−1= det(J)

∂xα0

∂x′µ0· · · ∂x

αD−1

∂x′µD−1εα0... αD−1

. (2.1.76)

Equation (2.1.76) shows that εµ0... µD−1is a tensor density of weight w = 1

(it transforms ‘almost’ as a tensor, the difference being in the prefactor det(J)w).On the other hand, because the metric g is a

(02

)-tensor field, its determinant

transforms asdet(g′) = det(J)−2 det(g) , (2.1.77)

23


so that the quantity

εµ0... µD−1:=√− det(g) εµ0... µD−1

(2.1.78)

is the component of a pseudotensor, as its transformation law is

ε′µ0... µD−1= sgn(det(J))

∂xα0

∂x′µ0· · · ∂x

αD−1

∂x′µD−1εα0... αD−1

. (2.1.79)

ε is called the Levi-Civita pseudotensor and, because it is a (pseudo)tensor,it makes sense to raise its indices using the inverse metric g−1,

εµ0... µD−1 ≡ gµ0α0 · · · gµD−1αD−1εα0... αD−1=

=√− det(g) gµ0α0 · · · gµD−1αD−1 εα0... αD−1

=

=√− det(g) εµ0... µD−1 det

(g−1)

= −εµ0... µD−1√− det(g)

.

(2.1.80)

From now on, the spacetime M will be assumed to be orientable, that is, thereexists a D-form field on M which is continuous and nowhere vanishing. Then,one can prove that there only exist two D-form fields ε such that, given anyorthonormal tetrad eµD−1

µ=0 ⊂TM,

ε(e0, . . . , eD−1

)≡ εµ0... µD−1

= ±1 ; (2.1.81)

choosing +1 in (2.1.81) defines a right-handed basis g, while choosing −1 definesa left-handed basis. Comparing with (2.1.78) and (2.1.74), it’s clear that theD-form field under consideration is the Levi-Civita pseudotensor, where a right-handed choice of basis has been made.

2.1.7 Connections and covariant differentiation

Defining a derivative operation acting on vector fields on smooth manifold Mrequires to compare vectors belonging to different tangent spaces; however, directcomparison of them does not make any sense, and a rule must be provided tellinghow to map a tangent vector at p∈M to a tangent vector at p′∈M, with p′ 6= p.One such a rule (not the only one: see the next section) involves the introductionof an affine connection ∇ on M, defined as a bilinear map

∇ :TM× TM→ TM , (X, Y ) 7→ ∇XY (2.1.82)

satisfying

1. ∇fXY = f∇XY Linearity on scalar functions (2.1.83)

2. ∇X (fY ) = df(X)Y + f∇XY Leibniz rule (2.1.84)

24


for all X, Y ∈TM and for all smooth maps f :M→ R ; in (2.1.84), df stands forthe 1-form field onM which is equal to dfp (as defined in (2.1.23)) for each p∈M.The vector field ∇XY is called the covariant derivative of X along Y . Noticethat, for any p∈M, (2.1.83) ensures that the value of (∇XY )p∈TpM, defined by

(∇XY )p[f ]!

= (∇XY ) [f ](p) on every smooth function f :M → R, only dependson Xp , being independent of the behaviour of X around p: because of this, onecan denote (∇XY )p ≡ ∇XpYp .

The expansion∇∂µ

∂ν ≡ Γρµν ∂ρ , (2.1.85)

where∂µ

D−1

µ=0is the frame field defined in (2.1.11), defines the connection

coefficients Γρµν :M → R. Notice that these are not the components of any(12

)-tensor field on M, as one may wrongly infer based on the index structure in

Γρµν . Indeed, using that ∂µ = ∂x′µ

∂xν∂ν (see (2.1.26)) and Γρµν ∂ρ = ∇∂µ

∂ν!

= ∇∂′µ∂′ν =

Γ′ρµν ∂′ρ under a change of coordinates x 7→ x′(x), one can easily verify that

Γ′ρµν =

∂xα

∂x′µ∂xβ

∂x′ν∂x′ρ

∂xγΓγαβ +

∂xα

∂x′µ∂x′ρ

∂xγ∂2xγ

∂x′ν∂x′α(2.1.86)

while the first term in the RHS of (2.1.86) is ‘tensorial’, the second one is not.

Properties (2.1.83) and (2.1.84), together with the expansion

∇XY ≡ ∇Xµ∂µ

(Y ν ∂ν

), give

∇XY ≡ (∇XY )µ ∂µ = Xµ(∇∂µ

Y)ν∂ν(

∇∂µY)ν

= ∂µYν + Γνµρ Y

ρ =: ∇µYν

(2.1.87)

for all X, Y ∈ TM. In (2.1.87), ∂µYν is shorthand for the map M 3 p 7→

∂∂xµ

(Y νp Φ−1

)(Φ(p)) , where (U ,Φ) is chart on M such that p ∈ U and Φ(p) =(

x0, . . . , xD−1). One can think of ∇µY

ν as the components of a(

11

)-tensor field

∇Y ≡ (∇Y )νµ ∂ν ⊗ dxµ on M such that

∇Y (ω,X)!

= ∇XY (ω)

(∇Y )νµ ∂ν ⊗ dxµ =: ∇µY

ν ∂ν ⊗ dxµ .(2.1.88)

for all X, Y ∈TM and for all ω∈TM.

Given vector fields X, Y ∈ TM and a curve γ : [0, 1] → M such that X istangent to γ, Y is said to be parallelly transported along γ if

∇XY = 0 . (2.1.89)

25


Choosing a connection means selecting a way of ‘parallelly transporting’ vectorsalong M. The curve γ is called a geodesic in M if

∇XX = 0 . (2.1.90)

Let λ∈ [0, 1] be the parameter running along γ and give a chart (U ,Φ) inM suchthat γ(λ)∈U for all λ∈ [0, 1]. Calling Φ(γ(λ)) =: x(λ) ≡

(x0(λ) , . . . , xD−1(λ)

)∈

RD, (2.1.90) can be recast as the geodesic equation,

d2xµ

dλ2+ Γµνρ

dxν

dλ

dxρ

dλ= 0 . (2.1.91)

The concept of affine connection can be extended to 1-form fields. Define amap, again called ∇,

∇ :TM× T ∗M→ T ∗M , (X,ω) 7→ ∇Xω , (2.1.92)

such that∇X (ω(Y )) =: (∇Xω)Y + ω(∇XY ) (2.1.93)

for all X, Y ∈ TM and for all ω ∈ T ∗M. Again, the image ∇Xω is called thecovariant derivative of ω along X. Using ω(Y ) = ωµY

µ gives

∇Xω = (∇Xω)ν dxν = Xµ

(∇∂µ

ω)νdxν(

∇∂µω)ν

= ∂µων − Γρµν ωρ =: ∇µων(2.1.94)

for all X ∈ M and for all ω ∈ T ∗M. In (2.1.94), ∂µων is shorthand for themap M3 p 7→ ∂

∂xµ

((ωp)ν Φ−1

)(Φ(p)) , where Φ is the chart on M giving the

coordinates x. One can think of ∇µων as the components of a(

02

)-tensor field

∇ω ≡ (∇ω)µν dxµ ⊗ dxν on M such that

∇ω (X, Y )!

= (∇Xω)(Y )

(∇ω)µν dxµ ⊗ dxν =: ∇µων dx

µ ⊗ dxν .(2.1.95)

At this point, generalising the affine connection for arbitrary(kl

)-tensors is

trivial. Define∇ :TM× T klM→ R , (X, τ) 7→ ∇Xτ (2.1.96)

in such a way that, for all τ, τ ∈T klM and for all σ∈T rsM,

1. ∇X(τ ⊗ σ)!

= (∇Xτ)⊗ σ + τ ⊗ (∇Xσ) (2.1.97)

2. ∇X (τ + τ)!

= ∇Xτ +∇X τ . (2.1.98)

26


Conditions (2.1.97) and (2.1.98) imply that

∇Xτ ≡ (∇Xτ)µ1 ... µkν1 ... νl ∂µ1 ⊗ · · · ⊗ ∂µk ⊗ dxν1 ⊗ · · · ⊗ dxνl =

= Xρ(∇∂ρ

τ)µ1 ... µk

ν1 ... νl∂µ1 ⊗ · · · ⊗ ∂µk ⊗ dxν1 ⊗ · · · ⊗ dxνl

∇ρτµ1 ... µk

ν1 ... νl :=(∇∂ρ

τ)µ1 ... µk

ν1 ... νl= ∂ρτ

µ1 ... µkν1 ... νl

+ Γµ1ρσ τσµ2 ... µk

ν1 ... νl + . . . + Γµkρσ τµ1 ... µk−1 µkσ

ν1 ... νl+

− Γσρν1 τµ1 ... µk

σν2 ... νl + . . . − Γσρνl τµ1 ... µk

ν1 ... νl−1σ .

(2.1.99)

In (2.1.99), ∂ρτµ1 ... µk

ν1 ... νl is shorthand for the map

M3p 7→ ∂∂xρ

((τp)

µ1 ... µkν1 ... νl

Φ−1)

(Φ(p)) , where Φ is the chart onM giving the

coordinates x.

Alternatively, the covariant derivative of the(kl

)-tensor field τp on M can be

defined in such a way as to meet the constraints (2.1.97) and (2.1.98),

∇X

(τ(ω(1), . . . , ω(k), Y (1), . . . , Y (l)

))=:

=: ∇Xτ(ω(1), . . . , ω(k), Y (1), . . . , Y (l)

)+ (. . . ) +

+ τ(∇Xω

(1), ω(2), . . . , ω(k), Y (1), . . . , Y (l))

+ (. . . ) +

+ τ(ω(1), . . . , ω(k−1),∇Xω

(k), Y (1), . . . , Y (l))

+ (. . . ) +

+ τ(ω(1), . . . , ω(k),∇XY

(1), Y (2), . . . , Y (l))

+ (. . . ) +

+ τ(ω(1), . . . , ω(k), Y (1), . . . , Y (l−1),∇XY

(k))

;

(2.1.100)

then, (2.1.99) follows from (2.1.100) straightforwardly.

∇ρτµ1 ... µk

ν1 ... νl in (2.1.99) can be thought of as the components of a(kl+1

)-

tensor field ∇τ ≡ ∇τµ1 ... µkν1 ... νl ∂µ1 ⊗ · · · ⊗ ∂µk ⊗ dxν1 ⊗ · · · ⊗ dxνl ⊗ dxνρ on Msuch that

∇τ(ω(1), . . . , ω(k), Y (1), . . . , Y (k);X

) !=

!= ∇Xτ

(ω(1), . . . , ω(k), Y (1), . . . , Y (k)

).

(2.1.101)

Up to this point, it is not obvious at all that an affine connection exists. How-ever, if ∇ and ∇ are two affine connections on M, then ∇− ∇ is easily shown tobe a tensor (use (2.1.86)); therefore, given an affine connection ∇ on M, one canbuild infinitely many other connections simply by addition of a

(12

)-tensor field to

∇. One can then try and build one connection explicitly, imposing two sensiblerequirements.

27


1. The antisymmetrised derivative of any k-form field α on M, whose compo-nents are defined by ∂[µαν1... νk], is easily shown to be a

(0

k+1

)-tensor field.

Therefore, requiring that

∇[µαν1... νk]!

= ∂[µαν1... νk] (2.1.102)

seems reasonable. Taking k = 1 and using (2.1.94), this translates into

Tρµν := 2Γρ[µν]

!= 0 , (2.1.103)

where T is the(

12

)-tensor field on M called torsion, whose action on any

ω∈T ∗M and X, Y ∈TM is

T(ω,X, Y ) = (∇XY −∇YX)(ω) = XµY νωρ · 2 Γρ[µν] . (2.1.104)

2. Parallel transport should preserve the norm of vectors and angles betweenthem; in other words, Γ must be compatible with the metric,

∇g != 0 . (2.1.105)

Requirements 1 and 2 single out the Levi-Civita connection, whose coefficientsare called Christoffel symbols and are given by

Γρµν =1

2gρσ(∂µgσν + ∂νgµσ − ∂σgµν) . (2.1.106)

The last relation implies

Γνµν =1

2gνρ∂µgνρ =

1

2Tr(g−1∂µg

)=

1

2∂µ(Tr(log(g))) =

1

2∂µ(log(− det(g))) =

= ∂µ

(log(√− det(g)

))=

1√− det(g)

∂µ

(√− det(g)

), (2.1.107)

where the following standard identity

log(|det(M)|) = Tr(log(M)) , (2.1.108)

valid for any non-degenerate matrix M , has been applied to g. Therefore, thefollowing very useful relation holds for any vector field X∈TM,

∇µXµ = ∂µX

µ + ΓνµνXν =

1√− det(g)

∂µ

(√− det(g)Xµ

). (2.1.109)

28


2.1.8 Lie differentiation

Another rule can be provided to define a derivative operation acting on tensor fieldson a smooth manifoldM. While covariant differentiation requires the introductionof an affine connection on M, the following rule does not require any additionalstructure.

Let X and Y be two vector fields on M and let φε :M→Mε∈[0,1] andψλ :M→Mλ∈[0,1] be the families of diffeomorphisms on M induced by X andY , respectively (see (2.1.9)). Let ε, λ 1, let (U ,Φ) be a chart on M and takefour points p, q, p′, q′∈M. Let

q = φε(p) Φ(q) = Φ(p) + εXp

p′ = ψλ(p) Φ(p′) = Φ(p) + λYp (2.1.110)

q′ = φε(p′) = φε(ψλ(p)) Φ(q′) = Φ(p′) + εXp′ = Φ(p) + λYp + εXp′ ,

where Xp =(X0p , . . . , X

D−1p

), Yp =

(Y 0p , . . . , Y

D−1p

)and

Xp′ =(X0p′ , . . . , X

D−1p′

)are the components of the vectors Xp , Yp and Xp′ on the

coordinate bases of TpM and Tp′M. Then, the distance between Φ(q) and Φ(q′)in RD is measured by the vector λYp + εXp′ − εXp.

On the other hand, take a fifth point q′′∈U such that

q′′ := ψλ(q) = ψλ(φε(p))

Φ(q′′) = Φ(q) + λYq = Φ(p) + εXp + λYq ,(2.1.111)

where Yq =(Y 0q , . . . , Y

D−1q

)are the components of the vector Yq on the coordinate

basis of TqM. The distance between Φ(q′′) and Φ(q) in RD is then measured bythe vector λYq .

The difference in RD between the vectors λYq and λYp + εXp′ − εXp (i.e. ,the difference between the vector connecting points Φ(q) and Φ(q′) and the vec-tor connecting points Φ(q) and Φ(q′′)) describes how the vector field Y is be-ing ‘transported’ along the vector field X. In order to make things simpler andwithout losing generality, one can always set X ≡ ∂0 for some coordinate choiceΦ(p) =

(x0, x1, . . . , xD−1

. In this case,

Xp = Xp′ = e(0) := (1, 0, . . . , 0) (2.1.112)

and the distance between Φ(q) and Φ(q′) in RD is simply λYp. The above choiceof coordinates implies that

Φ(q) =(x0 + ε, x1, . . . , xD−1

Φ(p′) =

(x0 + λY 0

p , x1 + λY 1

p , . . . , xD−1 + λY D−1

p

Φ(q′) =

(x0 + λY 0

p + ε, x1 + λY 1p , . . . , x

D−1 + λY D−1p

.

(2.1.113)

29


The interesting quantity is then the vector λYq − λYp in RD. More precisely,

one defines the Lie derivative of Y along X ≡ ∂0 as the vector field L∂0Y ≡(LXY )µ ∂µ such that, at point p,(

L∂0Y)µp

:=1

λlimε→0

∂0

[λY µ

p

](q)− ∂0

[λY µ

p

](p)

ε≡

≡ limε→0

(Y µp Φ−1

)(Φ(q))−

(Y µp Φ−1

)(Φ(p))

ε=

= limε→0

(Y µp Φ−1

)(Φ(p) + ε e(0)

)−(Y µp Φ−1

)(Φ(p))

ε≡

≡ ∂

∂x0

(Y µp φ−1

)(φ(p)) ≡ ∂0Y

µp (p) .

(2.1.114)

Notice that setting X ≡ ∂0 in (2.1.20) gives[∂0 , Y

]µ= ∂0Y

µp , (2.1.115)

so that combinining (2.1.114) with (2.1.115) and substituting ∂0 for the genericvector field X yields

LXY = [X, Y ] . (2.1.116)

The Lie derivative along a vector field X can be defined for any(kl

)-tensor field τ

by requiring that, for any smooth map f :M→ R,

1. LXf!

= df(X) Lie derivative of scalar functions (2.1.117)

2. LfXτ!

= fLXτ Linearity on scalar functions (2.1.118)

3. LX+Y τ!

= LXτ + LY τ Linearity over flow (2.1.119)

4. LX(τ ⊗ σ)!

= (LXτ)σ + τLXσ , Leibniz rule (2.1.120)

where σ is any other(kl

)-tensor field on M. On the

(kl

)-tensorial frame field on

M, this results in

(LXτ)µ1... µkν1... νl = Xρ∂ρτµ1... µk

ν1... νk+

− τσµ2... µkν1... νk∂σXµ1 + · · · − τµ1... µk−1σ

ν1... νk∂σXµk+

+ τµ1... µkσν2... νl∂ν1Xσ + · · ·+ τµ1... µkν1... νl−1σ∂νlX

σ ;

(2.1.121)

the torsion-free character of the Levi-Civita connection allows to trade all partialderivatives in (2.1.121) with covariant ones, making the tensorial nature of LXτclear,

(LXτ)µ1... µkν1... νl = Xρ∇ρτµ1... µk

ν1... νk+

− τσµ2... µkν1... νk∇σXµ1 + · · · − τµ1... µk−1σ

ν1... νk∇σXµk+

+ τµ1... µkσν2... νl∇ν1Xσ + · · ·+ τµ1... µkν1... νl−1σ∇νlX

σ .

(2.1.122)

30


In particular, the following very important identity holds for any X∈TM:

(LXg)µν = 2∇(µXν) . (2.1.123)

2.1.9 Curvature

Let M be a smooth manifold, p, q, r, s, t∈M and X ∈TM. Let X be parallellytransported from p to q along the vector ∂

(p)µ , then from q to r along ∂

(q)ν ; conversely,

let X be parallelly transported from p to s along ∂(p)ν , then from p to t along ∂

(p)µ .

In general, r 6= t and this difference can provide a means of characterising thecurvature of M around p without making reference to any ambient space. Theright tool to use is the Riemann tensor, the

(13

)-tensor field R on M whose

action on any ω∈T ∗M and X, Y, Z∈TM is

R(ω,X, Y, Z) := ω([∇X ,∇Y ]Z) ≡ ω(∇XZ −∇YZ) . (2.1.124)

In terms of components over the tensor frame field∂ρ, dx

σ, dxµ, dxν

, (2.1.124)

reads

RρσµνZ

σ := [∇µ,∇ν ]Zρ =

(2∂[µΓρν]σ + 2Γρ[µ |τ Γτν]σ

)Zσ . (2.1.125)

Letting Rρσµν ≡ gρτRρσµν , the Riemann tensor satisfies a number of identities:

1. Rρσµν = −Rρ

σνµ ; Rρσµν = −Rσρµν ; Rµνρσ = Rρσµν (2.1.126)

2. Rρ[σµν] = 0 First Bianchi Identity (2.1.127)

3. ∇[µRνρ]στ = 0 Second Bianchi Identity . (2.1.128)

The Ricci tensor is the symmetric(

02

)-tensor field on M, again denoted by R,

obtained by contraction of the Riemann tensor over the first and the third indices,

R(X, Y ) :=R(dxµ, X, ∂µ, Y

)R ≡Rµν dx

µ ⊗ dxν ;(2.1.129)

using (2.1.125), one gets

Rµν = Rρµρν = ∂[ρΓρν]µ + 2Γρ[ρ |σΓσν]µ . (2.1.130)

The scalar curvature R is the contraction of the Ricci tensor with the inversemetric of (2.1.72) over all indices, namely

R :=((g−1)⊗R

) (dxµ, dxν , ∂µ, ∂ν

)= gµνRµν . (2.1.131)

31


2.2 Geometry of hypersurfaces

From now on, a tensor field and its components over the tensorial frame field(2.1.32) may be used interchangeably to denote the same object.

2.2.1 Fundamentals

Consider a Lorentzian manifold M of dimension D = 4, i.e. , dim(TpM) = 4 ∀p∈M. Take a 3-dimensional manifold Σ and define an embedding of Σ into Mas an homeomorphism Φ : Σ → M; the image Σ := Φ

(Σ)⊂ M is called the

hypersurface in M generated by Σ.For all p∈ Σ, the pushforward map

Φ(p)∗ :TpΣ→ TΦ(p)M(Φ(p)∗ Xp

)[f ] := Xp[f Φ]

(2.2.1)

takes the tangent vector Xp∈TpΣ to the tangent vector Φ(p)∗ Xp∈TΦ(p)M in such

a way that the action of Xp and Φ(p)∗ Xp∈TΦ(p)M on any smooth map f :M→ R

is the same. Conversely, the pullback map

Φ∗(p) :(TΦ(p)

)0

kM→ (Tp)

0k Σ(

Φ∗(p)ωΦ(p)

) (X(1)p , . . . , X(k)

p

):= ωΦ(p)

(Φ(p)∗ X

(1)p , . . . ,Φ(p)

∗ X(k)p

) (2.2.2)

takes the(

0k

)-tensor ωΦ(p) at Φ(p) ∈ M to the

(0k

)-tensor Φ∗(p)ωΦ(p) at p ∈ Σ in

such a way that the action of ωΦ(p) and Φ∗(p)ωΦ(p) on any set of tangent vectorsX

(1)p , . . . , X

(k)p

⊂TpΣ is the same.

The pushforward and pullback operations serve to identify vectors and(

0k

)-

tensors on Σ andM. Therefore, from now on, Σ and Σ will be identified; accord-ingly, the point p∈ Σ will be the same as Φ(p)∈M and no distinction will be made

neither between Xp∈TpΣ and its pushforward Φ(p)∗ Xp∈TΦ(p)M, nor between ωΦ(p)

and its pullback Φ∗(p)ωΦ(p) .

A crucial example of pullback is provided by the first fundamental form ofΣ (or the 3-metric on M), that is, the metric γ induced on Σ by the spacetimemetric g as

γp := Φ∗(p)gp ∀p∈ Σ . (2.2.3)

Using (2.2.2) and identifying Σ with Σ, (2.2.3) gives

γij = gij ∀i, j∈1, 2, 3 . (2.2.4)

The hypersurface Σ is called:

32


• spacelike if γ is Riemannian (i.e. , γ has signature (+, +, +));

• timelike if γ is Lorentzian (i.e. , γ has signature (-, +, +));

• null if γ is degenerate (i.e. , γ has signature (0, +, +)).

One can think of Σ as the level set of some smooth map t :M→ R,

Σ ≡ p∈M| t(p) = t0∈R⊂M ; (2.2.5)

then, given any vector field X∈TΣ, the differential 1-form field dt satisfies dt(X) =

0 (use (2.1.23) and (2.1.10)). Therefore, the vector field−→dt , whose components

are−→dtµ ≡ gµνdtν , is orthogonal to X:⟨−→

dt,X⟩

= (dt)µXµ = dt(X) = 0 . (2.2.6)

The following properties are satisfied:

• Σ is spacelike ⇔−→dt is timelike:

∥∥∥−→dt∥∥∥2

< 0;

• Σ is timelike ⇔−→dt is spacelike:

∥∥∥−→dt∥∥∥2

> 0;

• Σ is null ⇔−→dt is null:

∥∥∥−→dt∥∥∥2

= 0.

If Σ is null, then−→dt is orthogonal to itself and (2.2.6) shows that

−→dt belongs to

TpΣ: in other words,−→dt is both orthogonal and tangent to Σ. If not null,

−→dt can

be normalised to the unit vector n orthogonal to Σ,

n :=

−→dt√∣∣∣∣ ∥∥∥−→dt∥∥∥2

∣∣∣∣, (2.2.7)

so that ‖n‖ = −1 if Σ is spacelike and ‖n‖ = +1 if Σ is timelike. Notice that

0 =1

2∇(‖n‖2) = 2nµ∇νnµ ⇒ nµ∇νnµ = 0 , (2.2.8)

wheren ≡ nµdx

µ ≡ gµνnνdxµ (2.2.9)

is the 1-form field dual to n.

33


If Σ is not null, then γ is non-degenerate and there is a unique Levi-Civitaconnection D on Σ. Denoting 3Γkij the Christoffel symbols for D, one has

Dγ = 0 (2.2.10)

3Γkij =1

2γkl(∂iγlj + ∂jγil + ∂lγij) , (2.2.11)

with the inverse 3-metric defined by γijγjk!

= δik . Likewise, one defines the Rie-mann and Ricci tensors and the scalar curvature of Σ associated to D as

3RklijX

k := [Di , Dj]Xk =

(2∂[i

3Γkj]l + 23Γ

k[i |m

3Γmj]l

)X l (2.2.12)

3Rij := 3Rkikj = 2∂[k

3Γkj]i + 2 · 3Γ

k[k |l

3Γlj]i (2.2.13)

3R := γij 3Rij . (2.2.14)

For all p∈Σ, define the Weingarten map or shape operator χp as

χp :TpΣ→ TpΣ

χp(Xp) := ∇Xpnp ,(2.2.15)

where ∇ is the spacetime’s Levi-Civita connection and np is the unit vector or-thogonal to Σ. The map χp tells how np ‘changes its orientation’ as it moves uponΣ, so it contains information about how Σ ‘lies’ into M. Notice that

〈n, χ(X)〉 ≡ 〈n,∇Xn〉 =1

2∇X

(‖n‖2) = 0 , (2.2.16)

which shows that indeed χp(Xp)∈TpΣ ; recall that no distinction is made betweentangent vectors in TpΣ and TpM, so here 〈· , ·〉 can either be regarded as the innerproduct induced by the spacetime metric g or by the induced metric γ. Moreover,notice that, if Σ is not null, then χ(X) ≡ ∇Xn does not depend on the particularn chosen.

It is not difficult to show that χ is self-adjoint with respect to the inner productinduced by the metric, i.e.

γ(X,χ(Y )) = γ(χ(X) , Y ) . (2.2.17)

This means that the three eigenvalues of χ are real; they are called the princi-pal curvatures of Σ and the corresponding eigenvectors are referred to as theprincipal directions of Σ. Because χ is self-adjont, the

(02

)-tensor

Kp :TpΣ× TpΣ→ RKp(Xp, Yp) := −Xpχp(Yp) ∀Xp, Yp∈TpΣ

(2.2.18)

34


is symmetric,

K(X, Y ) = −Xχ(Y ) = −Y χ(X) = K(Y,X) ∀X, Y ∈T Σ

⇒ K ≡ Kijdxi ⊗ dxj, Kij = Kji .

(2.2.19)

The(

02

)-tensor field K on Σ is called the extrinsic curvature or second fun-

damental form of Σ and one usually denotes

K :=(γ−1 ⊗K

)(dxi, dxj, ∂i, ∂j

)= γijKij . (2.2.20)

2.2.2 Spacelike hypersurfaces in spacetime

From now on, only spacelike hypersurfaces will be considered. For any p∈Σ⊂M,one can decompose

TpM≡ TpΣ⊕ span(n) (2.2.21)

and define the orthogonal projector Pp at p∈Σ as

Pp :TpM→ TpΣ

Pp(Xp) := Xp + 〈np , Xp〉np ∀Xp∈TpM .(2.2.22)

Were Σ null, then span(n)⊂ TpΣ and (2.2.21) would not make sense. Expanding

X ≡ Xµ∂µ , n ≡ nµ∂µ and 〈n,X〉 ≡ gµνnµXν ≡ nνx

ν , one gets

P (X) ≡ P µν X

ν ∂µ

P µν = δµν + nµnν ,

(2.2.23)

so that any vector field X onM can be uniquely decomposed along the directionsparallel and orthogonal to Σ by means of (2.2.22) as

X = P (X)− n(X)n

Xµ = P µνX

ν − nνXνnµ .(2.2.24)

The operator P is indeed a projector onto Σ, since it vanishes when applied to nand it is idempotent,

1.P (n) = n+ 〈n, n〉n = n− n = 0

P µνn

ν = 0(2.2.25)

2.P 2(X) ≡ P (P (X)) = P (X) ∀X∈TMP µ

νPνρ = P µ

ρ .(2.2.26)

35


One can use P to build a ‘reverse pullback’ operation P ∗ as

P ∗p :(T ∗p)0

kΣ→

(T ∗p)0

kM

P ∗p (ωp)(X(1)p , . . . , X(k)

p

)= ωp

(Pp(X(1)p

), . . . , Pp

(X(k)p

)) (2.2.27)

for all ωp ∈ T ∗p(T ∗p)0

kΣ and X

(1)p , . . . , X

(k)p ∈ TpM. The 3-metric γ can then be

‘extended’ to a symmetric(

02

)-tensor field P ∗γ =: P on M such that

P (X, Y ) := (P ∗γ)(X, Y ) = γ(P (X) , P (Y )) ∀X, Y ∈TM . (2.2.28)

The ‘extended 3-metric’ has the property that

P = g + n⊗ nPµν = gµν + nµnν .

(2.2.29)

Indeed, if X, Y ∈ T Σ, then (2.2.6) and (2.2.7) show that γ(X, Y ) = g(X, Y ); onthe other hand, if X ≡ λn for some λ 6= 0 (i.e. , X ∈ span(n)) and Y ∈ TM (orviceversa), then γ(X, Y ) = 0; because of (2.2.22), (2.2.29) follows. The extrinsiccurvature (2.2.18) can be extended in a similar way to the

(02

)-tensor field P ∗K =:

K on M such that

K(X, Y ) := (P ∗K)(X, Y ) = K(P (X) , P (Y )) ∀X, Y ∈TM . (2.2.30)

Finally, the orthogonal projector (2.2.22) defines the projection Pτ of any(kl

)-

tensor field τ on M as

Pτ ≡ (Pτ)µ1... µkν1... νl ∂µ1 . . . ∂µkdxν1 . . . dxνl

(Pτ)µ1... µkν1... νl := P µ1ρ1 . . . P

µkρ1P

σ1ν1 . . . P

σlνlτ

ρ1... ρkσ1... σk .

(2.2.31)

Any vector field X∈TM may be decomposed along the directions parallel andorthogonal to Σ: indeed,

Xµ = δµνXν = (P µ

ν − nµnν)Xν = (−nνXν)nµ + P µνX

ν . (2.2.32)

It will turn out to be useful to have the same decomposition for a(

02

)-tensor field

τ : writing τµν = gµρgρστρσ and applying (2.2.29), one gets

τµν = PµρPνστρσ − Pµρnνnστ ρσ − Pνρnµnστσρ + nρnστ

ρσnµnν . (2.2.33)

In case τ is symmetric, then all the terms in the decomposition (2.2.33) are fine.In case a

(02

)-tensor field α is antisymmetric, instead, the term nρnστ

ρσnµnν is notadmissible. Moreover, the term PµρPνσα

ρσ must be antisymmetric under µ ↔ ν,so it can be expressed in terms of the Levi-Civita pseudotensor (2.1.78) as

P µρP

νσα

ρσ ≡ εµνρσnρBσ (2.2.34)

36


for some suitable choice of the 1-form B ≡ Bµdxµ. In order to find B, contract

(2.2.34) with εµναβ and use (2.2.22) and the standard identity (recall (2.1.78) and(2.1.80))

εµναβ εµνρσ = −

√− det(g) εµναβ

εµνρσ√− det(g)

=

= −εµναβ εµναβ = −2(δραδ

σβ − δ

ρβδ

σα

) (2.2.35)

to writeεµναβα

µν + 2εµναβαµρnνnρ = −2 (nαBβ − nβBα) , (2.2.36)

then contract (2.2.36) with nα or nβ to get

Bµ = −1

2εµνρσn

ναρσ − nµnνBν . (2.2.37)

However, the second term in the RHS of (2.2.37) is useless, because in (2.2.34)there would be the term

εµνρσnρnσnτBτ = 0 ; (2.2.38)

indeed, the quantity P µρP

νσα

ρσ in (2.2.34) defines the full projection of α ontothe hypersurface Σ, so B must ‘lie tangent to Σ’,

B∈T ∗Σ ⇒ B(n) ≡ Bµnν = 0 . (2.2.39)

The quantity εµνρσnρ appearing in (2.2.34) must ‘lie tangent to Σ’, as well. This isindeed the case: if e1, e2, e3 is any orthonormal triad of vector fields on Σ (i.e. ,the set

e1p, e

2p, e

3p

is an orthonormal basis for TpΣ for all p∈Σ), then n, e1, e2, e3

is an orthonormal tetrad onM, since n is orthogonal to Σ. Therefore, the 3-form

3ε := ε(n, ·, ·, ·)3εµνρ = nαεαµνρ

(2.2.40)

is the Levi-Civita pseudotensor of Σ, because

3ε(e1, e2, e3

)= ε(n, e1, e2, e3

)= ±1 (2.2.41)

(compare with (2.1.81)). Notice, in passing, that, following the same line ofthoought which led to (2.1.78) and (2.1.80), one has

3εijk :=√

det(γ) 3εijk ⇒ 3εijk

=3ε

ijk√det(γ)

, (2.2.42)

37


where 3εijk = 3εijk

is the three dimensional Levi-Civita symbol given by (2.1.74).To summarise, the decomposition of the antisymmetric

(02

)-tensor field α along the

directions orthogonal and parallel to Σ is (recall that εµνρσ = −εµνρσ)

αµν =1

2εµνρσn

ρBσ − Pρ[µnν]nσαρσ

Bµ =1

23εµνρ α

νρ ⇔ Bµ = −1

23εµνρ

ανρ .(2.2.43)

In principle, the unit vector n orthogonal to the hypersurface Σ is ony defined

on Σ. However, because n ∝−→dt, it can be extended to an open neighbourhood of

any point p∈Σ and thus to all M. Therefore, the vector field

a := ∇nn

aµ = nν∇νnµ (2.2.44)

is well-defined over all M. If Σ is a spacelike hypersurface, then n is a timelikevector and can be regarded as the velocity vector of some observer, a being the4-acceleration associated to the latter. Notice that a is orthogonal to n (and hencetangent to Σ):

〈a, n〉 = 〈a,∇nn〉 =1

2∇(‖n‖2) = 0 . (2.2.45)

Application of (2.2.18), (2.2.22) and bilinearity of the Levi-Civita connection easilyprovides a useful relation among n (see (2.2.9)), the 1-form field

a ≡ aµdxµ ≡ gµνa

νdxµ (2.2.46)

dual to a and the extrinsic curvature K. It reads

K = −∇n− a⊗ nKµν = −∇µnν − nµaν

(2.2.47)

or, which is the same,

K = −P ∗∇nKµν = −P ρ

µPσν∇ρnσ ,

(2.2.48)

from which it follows (using (2.2.26)) that

K(n, ·) = 0 ⇒ Kµνnν = 0 . (2.2.49)

Let τ be a(kl

)-tensor field on Σ and let Dτ be its covariant derivative with

respect to the Levi-Civita connection (2.2.11) of Σ; then, P ∗Dτ is the extension

38


of Dτ to a tensor field over all M, as prescribed by (2.2.27). Moreover, let P ∗τbe the extension of τ to all M and let P (P ∗τ) be the projection of P ∗τ onto Σ,as in (2.2.31). Then, one can show that

P ∗Dτ = P (∇(P ∗τ)) (2.2.50)

or, avoiding making tensor extensions explicit and using (2.2.31),

Dτ = P (∇τ)

Dατµ1... µk

ν1...,νl = P µ1ρ1 . . . P

µkρkP

σ1ν1 . . . P

σlνlP

βα∇βτ

ρ1... ρkσ1... σl

.(2.2.51)

Indeed, let TP∇ be the torsion associated to the projection P∇ of the spacetimeconnection ∇, as in (2.2.50); it is easy to show that P∇ satisfies all the definingproperties of a connection, namely bilinearity, linearity on scalars (2.1.83), and theLeibniz rule (2.1.84). Furthermore, for any smooth map f :M→ R,

(TP∇)ρµν∂ρf = TP∇

(dxρ, ∂µ, ∂ν

)∂ρf =

[(P∇)∂µ ∂ν − (P∇)∂ν ∂µ

](dxρ) ∂ρf =

= P µαP

νβΓρµν∂ρf − P µ

αPνβΓρµν∂ρf = 0 , (2.2.52)

which shows that P∇ is torsion-free. Finally, (2.2.25) yields compatibility of P∇with the 3-metric γ,

(P (∇γ))µνρ = PαµP

βνP

γρ∇γγαβ = Pα

µPβνP

γρ∇γ(gαβ + nαnβ) =

= PαµP

βνP

γρ (nα∇γnβ − nβ∇γnα) = 0 .

(2.2.53)

Together, (2.2.52) and (2.2.53) show that P∇ is a Levi-Civita connection on Σ;because the Levi-Civita connection is unique and D is a Levi-Civita connection onΣ as well, (2.2.50) (or (2.2.51)) follows. Moreover, because of the compatibility ofP∇ with γ, the trace K of K satisfies

K ≡ P µνKµν = gµνKµν = ∇µnµ , (2.2.54)

where P µν are the components of the(

20

)-tensor field on M acting as the inverse

of the extended 3-metric,

P µνPνρ!

= δµρ ⇒ P µν = gµν + nµnν ; (2.2.55)

contraction with P µν or with gµν yields the same result, thanks to (2.2.29) and(2.2.49).

For any two vector fields X, Y ∈ T Σ, application of (2.2.22) and (2.2.18) to-gether with the relations P (X) = P µ

νXν = 0 and 〈n, Y 〉 = nµY

µ = 0, yields

DXY = ∇XY +K(X, Y )n

XνDνYµ = Xν∇νY

µ + nµXνY ρKνρ ,(2.2.56)

39


which gives a new perspective on the extrinsic curvature K: it is a measure of thedifference between taking covariant derivatives of vector fields on Σ along vectorfields on Σ with the D (i.e. , Σ’s) or the ∇ (i.e. , the spacetime’s) connections.More precisely, let X run along a geodesic γ of Σ, so that DXX = 0 and (2.2.56)give ∇XX = −K(X,X)n; on the other hand, if γ is a geodesic ofM (spacetime),then ∇XX = λX for some λ∈R. Because X cannot be parallel to n, K measuresthe failure of geodesics of Σ to be geodesics of M. When K = 0, the two notionsof geodesics coincide and Σ is called a totally geodesic hypersurface.

2.2.3 The Gauss-Codazzi relations

The last points which remain to be addressed are the relations between the intrinsiccurvatures of M and Σ. Start from (2.2.12), but extend all Σ-related quantitiesto 4-dimensional ones using (2.2.27) to write

3RρσµνX

ρ := [Dµ , Dν ]Xρ = 2

(∂[µ

3Γρν]σ + 3Γ

ρ[µ |τ

3Γτν]σ

)Xσ (2.2.57)

for any vector field X∈T Σ. Applying (2.2.56) two times gives

DµDνXρ = Pα

µPβνP

ργ∇α

(P δ

βPγλ∇δX

λ)

(2.2.58)

and, using (2.2.22), (2.2.25), (2.2.26), (2.2.48) and the symmetry of K yields

[Dµ , Dν ]Xρ = 2Kα[µKν]

ρXα + PαµP

βνP

ργR

γδαβX

δ . (2.2.59)

Combining (2.2.59) with (2.2.57) and replacing X ∈ T Σ with the more genericX ∈ TM (thanks to (2.2.49) and the presence of the orthogonl projector P in(2.2.59)) finally gives the Gauss relation,

PαµP

βνP

ργP

δσR

γδαβ = 3Rρ

σµν + 2Kρ[µKν]σ . (2.2.60)

Contraction of (2.2.60) over the indices µ and ρ, together with (2.2.26), yields thecontracted Gauss relation,

PαµP

βνRαβ + Pµαn

βP γνn

δRγδαβ =3Rµν +KKµν −KµαK

αν , (2.2.61)

and contraction of (2.2.61) with P µν gives the scalar Gauss relation,

R+ 2Rµνnµnν = R+K2 −KijK

ij . (2.2.62)

Now take the defining relation of the Riemann tensor on M (2.1.125) applied toΣ and project it onto the hypersurface as

PαµP

βνP

ργR

γδαβn

δ = PαµP

βνP

ργ[∇α,∇β]nγ . (2.2.63)

40


Using (2.2.47), (2.2.51), (2.2.26), the identities

P µνa

ν = aµ (2.2.64)

PαµP

βν∇αnβ = Kµν , (2.2.65)

and the symmetry of K gives the Codazzi relation,

PαµP

βνP

ργR

γδαβn

δ = 2D[µKρν] ; (2.2.66)

contracting (2.2.66) over the indices µ and ρ and using (2.2.22) yields the con-tracted Codazzi relation,

Pαµn

βRαβ = DµK −DαKαµ . (2.2.67)

2.3 Foliations of spacetime

2.3.1 The lapse function

Take again a spacetime M of dimension D = 4. A Cauchy surface Σ inspacetime is a spacelike hypersurface in M such that all causal curves in Mwithout endpoints (i.e. , curves γ : (a, b) →M with (a, b)∈R is open) intersect Σonce and only once. A spacetime admitting a Cauchy surface is called globallyhyperbolic, because the scalar wave equation in such a spacetime can be shownto yield a well-posed problem (i.e. , it has a unique solution and this dependssmoothly on initial data). Clearly, globally hyperbolic spacetimes do not admitclosed causal curves. Finally, a global hyperbolic spacetime must have topologyR× Σ, where Σ is the Cauchy surface defined above.

Consider a globally hyperbolic spacetime M, a smooth map t :M→ R and afamily of Cauchy surfaces Σtt∈R in M such that⋃

t∈R

Σt =M

Σt1∩Σt2 = ∀t1 , t2∈ R , t1 6= t2

Σt =p∈M| t(p) = t

∀t∈ R ,

(2.3.1)

The above defines a foliation or slicing ofM and each hypersurface Σt is calleda slice in the foliation.

In the present context, the unit vector field n M orthogonal to the slices,defined in (2.2.7), is timelike and is written

n =

−→dt√−∥∥∥−→dt∥∥∥2

=: −α−→dt , (2.3.2)

41


where α := − 1√−‖−→dt‖2

is the lapse function. If n is the 1-form field dual to n of

(2.2.9), then (2.3.2) says thatn = −αdt . (2.3.3)

The normal evolution vector m := αn satisfies⟨−→dt,m

⟩= (dt)µm

µ = dt(m) = 1 , (2.3.4)

which means that a slice Σt is mapped to the slice Σt+δt simply by sending p 7→p+ φ

(m)δt (p) for every point p∈Σt , where φ

(m)δt is the derivation map associated to

the vector field m with displacement δt in any local chart (see (2.1.9)) Indeed,

t(p+ φ

(m)δt (p)

)= t(p) + δt · dt(m) = t(p) + δt . (2.3.5)

Following the line of thought given in section 2.1.8, one can easily check that, givenany hypersurface Σt in the foliation, LmX ∈ Σt for any X ∈ Σt : Σt is said to beLie dragged along m.

2.3.2 Eulerian observers

Because n is a timelike unit vector, it can be regarded as the velocity vector tan-gent to the worldline of some observer in spacetime, the Eulerian or fiducialobserver. Let p∈Σt and p′ ≡ p + φ

(m)δt (p)∈Σt+δt, as above, lie on the worldline

of the Eulerian observer; then, the proper time interval δτ between the eventsp and p′ as measured by the Eulerian observer is

δτ :=√−〈δt ·m, δt ·m〉 = αδt . (2.3.6)

Therefore, the lapse function α relates the coordinate time t to the proper timeτ as measured by the Eulerian observer.

The 4-acceleration of the Eulerian observer is the vector field a := ∇nndefined in (2.2.44). Using (2.2.51), (2.2.22) and the torsion-free character of thespacetime connection ∇, the 1-form field a ≡ aµdx

µ ≡ gµνdxµdxν dual to a is

easily shown to satisfy

a = D(log(α)) =1

αDα

aµ = Dµ(log(α)) =1

αDµα .

(2.3.7)

Substituting (2.3.7) into (2.2.47), one gets

∇n = −K − (D(log(α)))⊗ n∇µnν = −Kµν −Dµ(log(α))nµ ,

(2.3.8)

42


so

∇m = −α−→K −

(−→Dα)⊗ n+ n⊗∇α

∇µmν = −αKµ

ν − nµDνα + nν∇µα(2.3.9)

and, contracting (2.3.8) by the inverse metric and using the compatibility of theLevi-Civita spacetime connection with the metric,

∇n = −K∇µn

µ = −K .(2.3.10)

2.3.3 The shift vector and the ADM metric

It is convenient to introduce coordinates onM which are ‘adapted to the foliation’.On each hypersurface Σt, introduce a maximal smooth atlas giving the spatialcoordinates (x1, x2, x3). If these coordinates vary smoothly with t (i.e. , alongthe foliation), then (t, x1, x2, x3) is a legitimate set of coordinates on M, whoseassociated partial derivative operators are ∂t , ∂1 , ∂2 , ∂3 , respectively. The vectors∂i

i∈1,2,3

are tangent to Σt for all t∈R, while the vector ∂t is called the time

vector and is tangent to the curves of constant spatial coordinates; moreover,

because dt(∂t

)≡ dtµ

(∂t

)µ= 1, ∂t Lie-drags the hypersurfaces in the foliation,

as the vector m := αn does (compare with (2.3.5)). However, in general, ∂t isdifferent from m, their difference being called the shift vector β,

∂t ≡ m+ β = αn+ β . (2.3.11)

For all t∈R, β is tangent to Σt, since

dt(β) = dt(∂t

)− dt(m) = 0 ; (2.3.12)

this can be written asn(β) ≡ nµβ

µ = 0 . (2.3.13)

Because β is tangent to Σt for all t∈R, (2.3.11) is the 3+1 decomposition of thetime vector ∂t along the directions orthogonal and parallel to each slice in thefoliation; hence

β = P(∂t

)βµ = P µ

ν

(∂t

)ν,

(2.3.14)

43


where ∂t ≡(∂t

)µ∂µ with

(∂t

)µ= (1, 0, 0, 0). Notice that the time vector ∂t is

not necessarily timelike, as it is not associated with the 4-velocity of any observer(as m is, instead). In particular, since ‖n‖2 = −1,∥∥∥∂t∥∥∥2

= −α2 + ‖β‖2 (2.3.15)

implies that

∂t is timelike ⇔ ‖β‖2 < α2

∂t is null ⇔ ‖β‖2 = α2

∂t is spacelike ⇔ ‖β‖2 > α2 .

(2.3.16)

Writing β ≡ βi∂i, β ≡ βidxi (1-form dual to β), (2.3.11) and (2.3.3) easily yield

nµ =

(1

α,−β

1

α,−β

2

α,−β

3

α

)(2.3.17)

nµ = (−α, 0, 0, 0) . (2.3.18)

The spacetime metric g ≡ gµνdxµdxν can be written in terms of the 3-metric

γ ≡ γijdxidxj (in both cases, the symmetric product notation has bee used: see

(2.1.52)) in terms of the coordinates ‘adapted to the foliation’. Indeed, thanks to(2.3.15), (2.3.11) and the fact that β is tangent to each slice in the foliation,

g00 = g(∂t , ∂t

)=∥∥∥∂t∥∥∥2

= −α2 + ‖β‖2 (2.3.19)

g0i = g(∂t , ∂i

)=⟨m+ β, ∂i

⟩=⟨β, ∂i

⟩= βi (2.3.20)

gij = γij ; (2.3.21)

the line element on M then reads

gµνdxµdxν = −α2dt2 + γij

(dxi + βidt

) (dxj + βjdt

). (2.3.22)

In matrix form,

gµν =

(g00 g0j

gi0 gij

)=

(−α2 + ‖β‖2 βj

βi γij

), (2.3.23)

from which the inverse (i.e. , dual) metric is deduced,

gµν =

(g00 g0j

gi0 gij

)=

(− 1α2

βj

α2

βj

α2 γij − βiβj

α2

). (2.3.24)

44


Finally, Cramer’s rule for inverse matrices implies that

− 1

α2= g00 !

=det(γ)

det(g)⇒

√− det(g) = α

√− det(γ) . (2.3.25)

Together, (2.3.22) , (2.3.23), (2.3.24) and (2.3.25) define the Arnowitt-Deser-Misner (ADM) decomposition of the spacetime metric with respect to a givenspacetime foliation.

Let τ be any(kl

)-tensor field onM. Then, thanks to (2.3.11) and the definition

of Lie derivative (2.1.122),

(Lmτ)µ1... µkν1... νl = ∂tτµ1... µk

ν1... νl − (Lβτ)µ1... µkν1... νl . (2.3.26)

2.3.4 Some useful relations

Equations (2.1.122) and (2.3.9) allow to express the Lie derivative of the ‘extended3-metric’ P in (2.2.29) as

(LmP )µν = −2αKµν . (2.3.27)

Because m := αn and LmP = αLnP , (2.3.27) can be recast as

Kµν = −1

2(LnP )µν ; (2.3.28)

on a hypersurface Σ, where Pij = γij, (2.3.28) reduces to

Kij = −1

2(Lnγ)ij . (2.3.29)

Besides the definition (2.2.18) and the relation (2.2.48), equation (2.3.29) gives athird point of view on the extrinsic curvature: it measures how the 3-metric isdragged along the foliation. Notice that this property is only valid in presence ofa foliation, and it does not make any sense when considering a single hypersurfacein spacetime.

Using again (2.1.122) and the definition of orthogonal projector P on hyper-surfaces (2.2.22), it is easily found that

LmP = 0 . (2.3.30)

Let Σt be a hypersurface in the foliation and let τ ∈ T kl Σ ; then, the extensionP ∗τ ≡ τ to a tensor field over all M by means of (2.2.27) coincides with τ whenprojected onto Σ,

Pτ = τ

P µ1α1 . . . P

µkαkP

β1ν1 . . . P

βlνlτ

α1... αkβ1... βl = τµ1... µkν1... νl .

(2.3.31)

45


Therefore, using (2.3.30) and (2.3.31) yields

P (Lmτ) = LτP µ1

α1 . . . PµkαkP

β1ν1 . . . P

βlνl(Lmτ)α1... αk

β1... βl= (Lmτ)µ1... µkν1... νl .

(2.3.32)

Recall that the Gauss relation (2.2.60) involves the full projection of the spacetimeRiemann tensor R onto a spacelike hypersurface Σ and that the Codazzi relation(2.2.66) involves three projections of R onto Σ and one onto the unit normal vectorn. Now that the concept of spacetime foliation has been defined, this picture canbe completed by projecting R twice onto a slice Σt and twice onto the unit vector northogonal to the latter; this will involve the derivative of the extrinsic curvature ofΣt with respect to n, so that the whole construction only makes sense in presenceof a foliation, and not for one single hypersurface in spacetime. Furthermore,projecting R three times along n gives zero, due to the antisymmetry of R in itslast two indices.

Start from (2.1.125) and write

PαµPγρn

βRγδαβn

δ = PαµPγρn

β [∇α,∇β]nρ , (2.3.33)

then use ‖n‖2 = −1 together with (2.2.25), (2.2.8), (2.2.49), (2.3.7) and (2.2.47)to get

PαµPγρn

βRγδαβn

δ = −KµαKαρ +

1

αDρDµα + Pα

µPγρn

δ∇δKµρ . (2.3.34)

Using the definition of Lie derivative (2.1.121) and relation (2.3.9), one has

(LmK)µρ = αnα∇αKµρ − 2αKµαKαρ − 2n[µKρ]αD

α . (2.3.35)

Because K ∈ T 02 Σ and thanks to (2.3.32), the Lie derivative LmK ‘lies on Σ’ as

well, and

P (LmK) = LmKPα

µPβν (LmK)αβ = (LmK)µν ;

(2.3.36)

therefore, full projection of (2.3.34) onto Σt yields

(LmK)µρ = αPαµP

γρn

δ∇δKµρ − 2αKµαKαρ (2.3.37)

and substitution of (2.3.35) in place of the last term in the RHS of (2.3.34) finallygives the Ricci equation,

PαµPγρn

βRγδαβn

δ =1

α(Lm)µρ +

1

αDµDρα +KµαK

αρ . (2.3.38)

46


Contracting (2.3.38) with P µρ and using (2.3.27) to write(Lmγ−1

)ij= 2αKij , (2.3.39)

the Ricci equation (2.3.38) becomes

R+Rµνnνnµ =3R+K2 − 1

αLmK −−

1

αDiD

iα . (2.3.40)

47

Chapter 33+1 formulation of generalrelativity

Einstein’s equations must be recast in 3+1 form, i.e. , as an initial value problem,in order to be tackled numerically. Section 3.1 deals with the geometric part ofthese equations, while sections 3.2 and 3.3 deal with matter and electromagneticfields, respectively. Finally, in section 3.4, Einstein’s equations are re-formulatedas a system of non-linear conservation laws.

For this chapter, the main reference is again [37], although part of the discussionfollows [10], [50], [19], and [27].

3.1 3+1 decomposition of Einstein’s equations

3.1.1 ADM formalism

Einstein’s equations describe how the dynamics of matter and fields is deter-mined by the geometry of the spacetime M they live in and, conversely, how Mitself is shaped by the presence of matter and fields. In components, Einstein’sequations read

Rµν −R2gµν = 8πTµν , (3.1.1)

where the stress-energy tensor T is the symmetric tensor encoding all theinformation about matter and fields in M. If the spacetime M is assumed to beglobally hyperbolic, then it can be foliated by means of a family Σtt∈R of spacelikehypersurfaces, with respect to which T can be decomposed using (2.2.33),

T = E n⊗ n+ n⊗ p+ p⊗ n+ S

Tµν = E nµnν + 2p(µnν) + Sµν .(3.1.2)

In (3.1.2), the following quantities have been defined as measured by the Eulerianobserver (see section 2.3.2):

E := T (n, n) = Tµνnµnν Total energy density (Eul. obs.) (3.1.3)

pµ := −PµνnρTνρ Momentum density (Eul. obs.) (3.1.4)

Sµν := PµρPν

σTρσ Stress tensor (Eul. obs.) . (3.1.5)

48

Chapter 3 3+1 formulation of general relativity

Contracting expression (3.1.2) with respect to the spacetime metric g yields

T = S − ET := gµνTµν S := gµνSµν .

(3.1.6)

Furthermore, contracting Einstein’s equations (3.1.1) with respect to gµν gives thealternative form

Rµν = 8π

(Tµν −

T2gµν

). (3.1.7)

Einstein’s equations can be projected onto any slice Σt in the foliation and/oronto the vector n using the results of sections 2.2.3 and (2.3.4). For example,using (2.3.40) and (2.3.26) yields the full projection of (3.1.7) onto Σt, whichconstitutes the time evolution (i.e. , evolution along the parameter spanning thespacetime foliation) of the extrinsic curvature K,

(LmK)ij = ∂tKij − (LβK)ij!

= −DiDjα+

+ α

3Rij +KKij −KikKkj + 4π[(S − E) γij − 2Sij]

.

(3.1.8)

On the other hand, one can apply (3.1.7) to the couple (n, n),

R(n, n)− R2g(n, n)︸︷︷︸

=‖n‖2=−1

= 8πT (n, n)

Rµνnνnν +

R2

= 8πTµνnµnν ,

(3.1.9)

and use the scalar Gauss relation (2.2.62) to obtain the Hamiltonian constraint

3R +K2 −KijKij = 16πE . (3.1.10)

Finally, the contracted Codazzi relation (2.2.67), together with (3.1.4) produce the‘mixed’ projection of (3.1.1) onto Σt and onto n, also known as the momentumconstraint,

DjKji −DiK = 8πpi . (3.1.11)

The last missing piece is the time evolution for the 3-metric, which can be recoveredby application of (2.1.123) to the 3-metric γ to write

(Lβγ)ij = 2D(iβ j) (3.1.12)

and using (2.3.29) and (2.3.26) to yield

∂tγij = −2αKij + 2D(iβ j) . (3.1.13)

49


The time evolution relations (3.1.13) and (3.1.8) subject to the constraints (3.1.10)and (3.1.11) form the Arnowitt-Deser-Misner (ADM) system equations [3,6], which is summarised as

∂tγij = −2αKij + 2D(iβ j) (3.1.14)

(LmK)ij = ∂tKij − (LβK)ij!

= −DiDjα+

+ α

3Rij +KKij −KikKkj + 4π[(S − E) γij − 2Sij]

(3.1.15)

subjected to the constraints

3R +K2 −KijKij = 16πE (3.1.16)

DjKji −DiK = 8πpi (3.1.17)

The Hamiltonian and momentum constraints may also be found as constraintequations for the Hamiltonian built from the classical Einstein-Hilbert action ofgeneral relativity (hence the names); however, further details about this are notnecessary here.

3.1.2 BSSN formalism

Unfortunately, the ADM system is not suitable for numerical simulations, as it isonly weakly hyperbolic (see section 4.2.5). Therefore, some suitable transformationof variables to make the system strongly hyperbolic is in order; in particular, stronghyperbolicity is obtained by means of a set of auxiliary variables whose purposeis to absorb terms involving mixed second derivatives of the metric, as will beexplained below.

Begin by performing a conformal transformation of the 3-metric γ,

γij =: e4φ γij ⇔ γij = e−4φγij

γij = e−4φ γij ⇔ γij = e4φγij ,(3.1.18)

with

e4φ := (det(γ))13 ⇔ φ =

1

12log(det(γ)) =

1

12Tr(log(γ)) ; (3.1.19)

in (3.1.19), the standard identity

det(M) = exp(Tr(log(M))) , (3.1.20)

valid for any square matrix M , has been used. In this way,

det(γ) = det(

(det(γ))−13 γ)

= (det(γ))−1 det(γ) = 1 . (3.1.21)

50


Define the trace-free extrinsic curvature A and its conformal partner A as

A := K − K3γ ≡ e4φA ⇒ γijAij = γijAij = 0 , (3.1.22)

which satisfyAijAij = AijAij . (3.1.23)

Contract (3.1.13) with γij to write

∂

∂t(log(det(γ))) = −αK +Diβ

i (3.1.24)

and use (3.1.18) to get the time evolution equation for φ,

∂tφ =1

6

(∂iβ

i − αK)

+ βi∂iφ . (3.1.25)

Notice that γij∂tγij = −γij∂tγij and that combination of this result with (3.1.13)

gives∂tγ

ij = 2αKij − 2D(iβ j) ; (3.1.26)

therefore, contracting (3.1.8) with γij yields the time evolution for the trace of theextrinsic curvature,

∂tK = −DiDiα + α

AijAij +K2

3︸︷︷︸=KijKij

+4π(S + E)

+ βi∂iK . (3.1.27)

After some algebra, (3.1.25) and (3.1.27) allow to write the time evoltion equations

for the conformal metric γij and the conformal trace-free extrinsic curvature Aij,

∂tγij = −2αAij + βk∂kγij + 2γk(i∂ j)βk − 2

3∂kβ

kγij (3.1.28)

∂tAij = e−4φ(− (DiDjα)TF + α

(3RTF

ij − 8πSTFij

))+ α

(KAij − 2AikA

kj

)+

+ βk∂kAij + 2Ak(i∂ j)βk − 2

3Aij∂kβ

k ; (3.1.29)

here, the superscript TF stands for trace-free, i.e. , for example,

3RTFij := 3Rij −

R3γij . (3.1.30)

51


In (3.1.29), the second derivatives of the metric appear into 3RTFij . Mixed second

derivatives should be absorbed into some auxiliary term, since they are more diffi-cult to deal with numerically than first derivatives and non-mixed second deriva-tives. To this end, introduce the Levi-Civita connection D associated to the con-formal 3-metric, whose Christoffel symbols are

3Γkij :=1

2γkl (∂iγlj + ∂j γil − ∂lγij) ≡ 3Γkij − 2

(2δk(i∂ j)φ− γkl∂lγij

); (3.1.31)

the corresponding Ricci tensor and scalar curvature read

3Rij := 2∂[k3Γ

k

j]i + 2 · 3Γk

[k |l3Γ

l

j]i (3.1.32)

3R := γij 3Rij (3.1.33)

and the relation between 3Rij and 3Rij is

3Rij = 3Rij + 3R(φ)ij

3R(φ)ij := − 2

(DiDjφ+ γij γ

klDkDlφ)

+ 4(DiφDjφ+ γij γ

klDkφDlφ).

(3.1.34)

Define the conformal connection functions as

Γi := γjk 3Γijk ; (3.1.35)

using (3.1.21) in the identity

γjk∂lγjk = Tr(γ−1∂lγ

)= ∂l Tr(log(γ)) = ∂l log(det(γ)) = 0 , (3.1.36)

results inΓi = −∂j γij . (3.1.37)

The conformal Ricci tensor can then be expressed in terms of the Γi’s as

3Rij = −1

2γkl∂k∂lγij + γk(i∂ j)Γ

k + γklΓk 3Γlij + 2 · 3Γkl(i

3Γlj)k + 3Γkil3Γljk (3.1.38)

so that now the only second-derivative operator in 3Rij is the Laplacian γkl∂k∂land no mixed second derivatives appear. The price to pay for that is having tosolve the time evolution equation for the auxiliary variable Γi, which reads

∂tΓi + ∂j

(2αAij − 2γk(i∂kβ

i) +2

3γij∂kβ

k + βk∂kγij

)= 0 , (3.1.39)

52


or, eliminating the term ∂jAij by means of the momentum constraint (3.1.11),

∂tΓi = − 2Aij∂jα + 2α

(ΓijkA

jk − 2

3γij∂jK − 8πγijSj + 6Aij∂jφ

)+

+ βj∂jΓi − Γj∂jβ

i +2

3Γi∂jβ

j +1

3γij∂j∂kβ

k + γjk∂j∂kβi .

(3.1.40)

To summarise, (3.1.28), (3.1.29), (3.1.25), (3.1.27) and (3.1.40) allow to recast theADM system (3.1.14) — 3.1.15 as

∂tγij = −2αAij + βk∂kγij + 2γk(i∂ j)βk − 2

3∂kβ

kγij (3.1.41)

∂tAij = e−4φ(− (DiDjα)TF + α

(3RTF

ij − 8πSTFij

))+ α

(KAij − 2AikA

kj

)+

+ βk∂kAij + 2Ak(i∂ j)βk − 2

3Aij∂kβ

k (3.1.42)

∂tφ =1

6

(∂iβ

i − αK)

+ βi∂iφ (3.1.43)

∂tK = −DiDiα + α

AijAij +K2

3︸︷︷︸=KijKij

+4π(S + E)

+ βi∂iK (3.1.44)

∂tΓi = −2Aij∂jα + 2α

(ΓijkA

jk − 2


)+


i +2

3Γi∂jβ

j +1

3γij∂j∂kβ

k + γjk∂j∂kβi ;

(3.1.45)

(3.1.41) — (3.1.45) is called the Baumgarte-Shapiro-Shibata-Nakamura(BSSN) system of equations [9, 52, 63] and it is subjected to the constraints(see (3.1.16) — (3.1.17) and (3.1.35))

3R +K2 −KijKij = 16πE (3.1.46)

DjKji −DiK = 8πpi (3.1.47)

Γi = γjk 3Γijk . (3.1.48)

3.1.3 Slicing conditions

In extreme synthesis, general relativity describes Nature as a manifold M withtensor fields T (i) defined on it. If N is a second manifold and Φ :M → N is adiffeomorphism, then the structures

(M, T (i)

)and

(N ,Φ∗T (i)

), where Φ∗ is the

pushforward map defined in (2.2.1), must represent the same physical scenario:in this sense, general relativity is invariant under diffeomorphisms, which act as

53


gauge transformations. To see this ‘passively’, let p∈M and let (U , φ) a coordinatechart on M such that p ∈ U ; let then (V , ψ) be a chart on N such that Φ(p) ∈V . The diffeomorphism Φ induces a new chart of coordinates (Φ−1(V)⊂M, ζ)

(where p∈Φ−1(V)) defined by ζ(q)!

= ψ(Φ(q)) for all q ∈Φ−1(V), so saying thatgeneral relativity is invariant under diffeomorphisms is equivalent to saying thatgeneral relativity is invariant under smooth — but otherwise arbitrary — changesof coordinates (see Appendix C of [71]).

Whenever spacetime is foliated by means of a family of 3D spacelike hypersur-faces, the freedom in choosing coordinates translates into the freedom of choosingthe lapse function and shift vector: this is often referred to as a slicing con-dition. As this choice amounts to a gauge transformation, it does not have anyinfluence on the physics of the spacetime under consideration; nevertheless, it doeshave an influence on the numerical evolution of that spacetime, as the lapse andshift appear in the ADM and BSSN systems (3.1.14) — (3.1.15) and (3.1.41) —(3.1.45). Therefore, the lapse and shift must be chosen wisely, and in particular:

1. singularities (either physical or coordinate-related ones) must be avoided,otherwise infinities are likely to appear in field variables, causing the nume-rical simulation to crash. This typically translates in a clever choice of thelapse function, which must ‘slow down’ observers approaching the singula-rity;

2. strong distorsions in the space coordinates must be counteracted as muchas possible in passing from one spacelike slice to the next one by choosingthe shift vector appropriately, otherwise the spatial derivatives of all metric-related quantities may get very large;

3. the lapse function and shift vector would better satisfy simple algebraic rela-tions or, in case this is not possible, evolution equations which are simple toimplement and, most importantly, to solve. For example, elliptic evolutionPDEs for the lapse and shift must be avoided.

In what follows, some popular slicing conditions are outlined and their pros anscons are discussed.

Geodesic slicing with zero shift

This is probably the simplest possible slicing condition:

α = 1, βi = 0 . (3.1.49)

Recalling (2.3.17), the above choice amounts to let the Eulerian observer be thesame as the coordinate observer,

nµ = (1, 0, 0, 0) ≡ ∂t . (3.1.50)

54


In turn, the coordinate observer follows the spacetime geodesics, as its 4-accelerationis aµ = Dµ(log(α)) = 0 (see (2.3.7)). If the spacetime contains a physical singu-larity (e.g. , a black hole) then geodesics focus on that, making geodesic slicing aninconvenient choice in most situations.

Maximal slicing

Consider an open set Ω in spacetime. A congruence in Ω is a family of curvessuch that through each point p∈Ω there passes exactly one curve in the family;the expansion θ of a congruence of timelike curves whose tangent vector is ξ isdefined as

θ := ∇µξµ (3.1.51)

and measures the tendency of the curves in the congruence to converge to a point(if θ < 0) or diverge from one another (if θ > 0). For example, the expansion of acongruence of Eulerian observers is given by (2.3.10),

θ = ∇µnµ = −K , (3.1.52)

so choosing the foliation in such a way that K ≤ 0 (thus θ ≥ 0) prevents theEulerian observer to fall into physical singularities. A popular choice is maximalslicing, which amounts to have θ = 0 at all times, or

K != 0

!= ∂tK . (3.1.53)

This slicing is ‘maximal’ in the sense that imposing K != 0 on a spacelike hypersur-

face can be shown to maximise the volume of the latter. With the choice (3.1.53),the time evolution equation for K (3.1.27) reduces to the following elliptic PDE inthe unknown α,

−DiDiα + α

[KijK

ij + 4π(S + E)]. (3.1.54)

The maximal slicing condition θ = ∇µnµ = −K = 0 defines an incompressible

flow of Eulerian observers. Relation (2.3.7) gives the 4-acceleration of Eulerianobservers as aµ = Dµ(log(α)) and this quantity must grow faster and faster asthe Eulerian observer itself approaches a physical singularity, so as to balance thefocusing of geodesics which gets stronger and stronger with increasing gravitationalpull; the net effect is then α −−−−→

t→+∞0, which is often referred to as the ‘collapse

of the lapse’ (see [10] and [37] for a more thorough discussion and illuminatingexamples). Recalling (2.3.6), the proper time measured by the Eulerian observeris τ = αt, so α −−−−→

t→+∞0 implies τ −−−−→

t→+∞0 and the Eulerian observer itself is

forbidden from reaching the physical singularity in a finite interval of proper time.

55


Being an elliptic PDE, (3.1.54) is expensive to solve; thus, maximal slicingshould only be imposed approximately (it is, after all, just gauge condition) andin such a way as to generate a governing equation of α which is simple to deal

with. One way to do that is to allow for K != 0 in the initial data and then drive

K back to zero as time flows by imposing

∂tK!

= −CK ⇒ K(t) = K(t0) e−C(t−t0) , (3.1.55)

where C > 0 is some constant and t0 specifies the beginning of the time evolution.Inserting (3.1.55) into (3.1.27) yields

DiDiα + α

(KijK

ij − 4π(S + E))− βi∂iK − CK = 0 (3.1.56)

but, since this is again an elliptic PDE in the unknown α, a time evolution termcan be added to get the parabolic PDE

∂λα = DiDiα + α

(KijK

ij − 4π(S + E))− βi∂iK − CK = 0 , (3.1.57)

where λ is some real parameter. Choosing λ ≡ εt for some fixed value ε∈R, thelast equation reads

∂tα = −ε(∂tK + CK) ≡= −ε

(−DiD

iα + α(KijK

ij + 4π(S + E))

+ βi∂iK + CK)

= 0 ,(3.1.58)

where ε is an effective diffusion constant. Equation (3.1.58) is usually referred toas the (parabolic) K-driver condition. Taking ε→ +∞ in (3.1.58) amounts tosolving (3.1.55), but in numerical simulations ε cannot be taken too large as theCFL condition (see 4.2.1) must be satisfied; alternatively, one can decide to splitthe evolution timestep ∆t into multiple substeps in order to allow for large valuesof ε.

Harmonic coordinates and harmonic slicing

One way to impose slicing conditions is to set the quantities

Γµ := gνρΓµνρ = − 1√− det(g)

∂ν

(gµν√− det(g)

)(3.1.59)

to be equal to some chosen gauge source functions Hµ. The simplest choice istaking harmonic coordinates, that is,

Γµ = 0 . (3.1.60)

56


With this choice, the coordinate functions xµ (they are scalar fields in spacetime;see section 2.1.3) are indeed harmonic functions,

∇µ∇µxρ = gµν∇µ(∂νxρ) = gµν

(∂µ∂νx

ρ − Γσµν∂σxρ)

= −gµνΓρµν ≡ Γρ = 0 .(3.1.61)

Using expression (3.1.59) for Γµ and the inverse ADM metric (2.3.24), the har-monic gauge condition (3.1.60) is shown to be equivalent to the following systemof evolution equations for α and βi,(

∂t − βj∂j)α = −α2K (3.1.62)(

∂t − βj∂j)βi = −α2

(γij∂j log(α) + γjk3Γ

ijk

). (3.1.63)

A more common choice is the harmonic slicing,

Γ0 = 0 , (3.1.64)

which is often coupled to βi = 0. In this case, (3.1.62) reduces to

∂tα = −α2K ⇒ α(t) = α(0)√

det(γ) , (3.1.65)

where (3.1.24) has been used to find the solution.

An even more popular choice is modify a bit the RHS of (3.1.65) to

∂tα = −α2f(α)K , (3.1.66)

where f can be very general; notice that f = 0 means geodesic slicing and f = 1means harmonic slicing. By far one of the most used forms of f (usually used inSpritz as well) is

f(α) =2

α⇒ α = 1 + log(det(γ)) ≡ 1 + Tr(log(γ)) , (3.1.67)

which is known as the 1 + log slicing. To finish with, it is very common to letβi 6= 0 in (3.1.62) and insert the function f(α) = 2

αin the RHS of that equation

to get (∂t − βj∂j

)α = −α2 2

αK = −2αK . (3.1.68)

Minimal distorsion

When adopting the BSSN formalism (see section 3.1.2) to solve Einstein’s equa-tions, it would be desirable to minimise the fluctuations in the conformal 3-metric

57


(3.1.18) in passing from one slice to the next one in the spacetime foliation. Tothis end, define

uij := (det(γ))13 ∂t

((det(γ))

13γij

)⇒ γijuij = 0 . (3.1.69)

Being symmetric and traceless, u can be decomposed into a transverse-tracelesspart uTT with zero divergence and a longitudinal part uL which is equal to thesymmetrised and traceless covariant derivative of some vector X which plays therole of a vector potential; in other words,

uij =: uTTij + uL

ij , (3.1.70)

with

DjuTTij = 0 (3.1.71)

uLij = 2D(iX j) −

2

3γijDkX

k ≡

≡ (det(γ))13 LX(det(γ))−

13 γij ≡ (det(γ))

13 LX γij .

(3.1.72)

Thus, uLij = (det(γ))

13 LX γij means that X generates a change of coordinates which

gives rise to uLij ; the minimal distorsion slicing scheme consists then in setting

uLij

!= 0 ⇒ DjuTT

ij = Djuij = 0 . (3.1.73)

One could elaborate on this result and obtain an elliptic constraint PDE for βi.However, it is more common to impose the minimal distorsion condition onlyapproximately, and a clever way to accomplish this task is to set

∂tΓi !

= 0 , (3.1.74)

where Γi are the conformal connection functions defined by (3.1.35) and (3.1.37);indeed, a straightforward calculation yields

∂tΓi = − ∂juij

uij = (det(γ))−13 uij ≡ ∂t

((det(γ))

13γij

).

(3.1.75)

On the other hand, the time evolution equation of the conformal connection func-tion has already been computed when developing the BSSN formalism. Inserting(3.1.40) into (3.1.74) gives the Gamma-freezing condition

∂tΓi = − 2Aij∂jα + 2α

(ΓijkA

jk − 2


)+


i +2

3Γi∂jβ

j +1

3γij∂j∂kβ

k + γjk∂j∂kβi !

= 0 .

(3.1.76)

58


Still, the last relation is a set of elliptic PDEs in the unknowns βi. As done forthe maximal slicing, a time derivative term may be added to (3.1.76) to get theparabolic gamma-driver condition

∂tβi = κ

(∂tΓ

i + ηΓi)

(3.1.77)

for some κ, η > 0. One further step is the hyperbolic gamma-driver condition[14]

∂tβi =

3

4Bi (3.1.78)

∂tBi = ∂tΓ

i − ηBi , (3.1.79)

where B is an auxiliary vector and typically η∼ 12M

, where M is the total energycontained in the spacetime under investigation. The factor of 3

4in (3.1.78) is

usually adopted because it is found to produce smooth GW waveforms [14].

3.2 3+1 decomposition of the hydrodynamics

equations

3.2.1 Conservation of energy and momentum

While the ADM and BSSN systems (3.1.14) — (3.1.15) and (3.1.41) — (3.1.45)concern the LHS of Einstein’s equations, this section deals with the RHS of thelatter, namely, matter and fields in spacetime as represented in the stress-energytensor T . First of all, as a consequence of the second Bianchi identity (2.1.128),

0 = gµτgνσ∇[µRνρ]στ =1

3(2∇µRν

µ −∇νR)

⇒ 2∇µRµν = ∇νR ;

(3.2.1)

therefore, (3.1.1) implies that

∇µTµν = 0 on shell , (3.2.2)

which expresses the conservation of energy and momentum ‘on shell’, i.e. , forspacetime and matter configurations satisifying Einstein’s equations. Second, T issubject to constraints which hold true independently of Einstein’s equations: forexample, the conservation of the baryon number, or Maxwell’s equations for theelectromagnetic field.

59


Substituting (3.1.2) into (3.2.2) with the help of (2.3.8) gives

0 = ∇µSµν −Kpν + 2n(µ∇µp

ν) − pµKµν −KEnν + EgµνDµ(log(α)) . (3.2.3)

Furthermore, let Σt be any slice in the spacetime foliation and let X∈T Σt ; then,(2.2.51) easily gives

DµXµ = ∇µX

µ −XµDµ(log(α)) . (3.2.4)

Finally, use (2.3.11) and the linearity properties (2.1.118) and (2.1.119) of the Liederivative to write

LnE = nµ∂µE =1

α(∂t − Lβ)E =

1

α

(∂t − βi∂i

)E . (3.2.5)

With these tools at hand, recall that p, K and D(log(α)) are all orthogonal to nand project (3.2.3) along the vector n to get

nν∇µSµν + nµnν∇µp

ν −∇µpµ +KE − nµ∇µE = 0 ; (3.2.6)

however, using (2.3.8) and nµSµν = 0 yields

nν∇µSµν = −Sµν∇nν = −Sµν(Kµν +Dν(log(α))) = KµνS

µν , (3.2.7)

and similarly

nµnν∇µpν = −pν∇µ(nµnν) = −pµDµ(log(α)) . (3.2.8)

Therefore, (3.2.6) reduces to(∂t − βi∂i

)E = α

(KijS

ij +KE −Dipi)− 2piDiα (3.2.9)

and the last result expresses the on-shell conservation of energy.

Now project (3.2.3) onto any spatial hypersurface Σt in the spacetime foliation:the result is

Pνρ∇µSµν −Kpρ + Pνρn

µ∇µpν − pµKµρ + EDρ(log(α)) . (3.2.10)

Besides, (2.2.50) and (2.2.22) give

DµSµρ = Pνρ∇µS

µν − SµρDµ(log(α)) ; (3.2.11)

moreover, using the definition of Lie derivative of a vector field (2.1.116) and therelation m := αn, together with (2.3.9) and (2.3.32), one has

Pνρnµ∇µp

ν =1

αPνρm

µ∇µpν =

1

αPνρ(Lmpν − pµ∇νmµ) =

1

αLmpρ + pµKµρ .

(3.2.12)In the end, combining (3.2.11) and (3.2.12) with (2.3.26), (3.2.10) becomes

(∂t − Lβ) pi = α(Kpi −DjS

ji

)− SijDjα− EDiα (3.2.13)

and the last result expresses the on-shell conservation of momentum.

60


3.2.2 Ideal fluids

Matter sources in astrophysical scenarios can usually be conveniently described asfluids, rather than swarms of individual particles. The fluid model entails think-ing about matter as a continuum of fluid elements which are small enoughwith respect to the characteristic lenght of the problem at hand to be considered‘infinitesmally small’, but still large enough with respect to the atomic scale to con-tain a macroscopic number of particles. The latter assumption makes it possibleto define quantities such as the energy and rest-mass densities, pressure, and tem-perature, which are observable quantities resulting from average properties of verylarge groups of particles, for every fluid element, i.e. , at every point in spacetime.

A particularly simple and yet often satisfying approximation is that of an ideal(or perfect) fluid. In order to define an ideal fluid properly, begin by calling theambient spacetime M and consider a point p∈M and any tangent vector Xp ∈TpM. Let γ : [0, 1]→M be a geodesic and let (U ,Φ) be a chart on M such thatγ(λ)∈U for all λ∈ [0, 1] and call Φ(γ(λ)) =: x(λ) ≡

(x0(λ) , . . . , xD−1(λ)

)∈RD;

then, γ satisfies the geodesic equation (2.1.91) and Xµp = ∂xµ

∂λfor all µ∈0, 1, 2, 3.

Notice that the geodesic equation is a second-order ordinary differential equation,so specifying the initial data as

γ(0) = p ⇒ x(0) ≡ Φ(γ(0)) = Φ(p)

∂xµ

∂λ= Xµ

p

(3.2.14)

uniquely determines a solution to it, which means that the geodesic through pwhose tangent vector is Xp is unique. Call

expp :TpM→M , exp(Xp) := s(1) (3.2.15)

the exponential map at p; this map is one-to-one provided that no other geodesicemanating from p crosses s at point s(1). Consider a point q∈U sufficiently close top such that there is a unique geodesic γ connecting p to q; one can always choosethe parametrisation γ(0) = p, γ(1) = q. The vector Xp tangent to γ at p can

be expanded onto an orthonormal tetradep(µ)

D−1

µ=0⊂ TpM as Xp ≡ Xµ

p ep(µ).

Define the Riemann normal coordinates at p as

R43 x(1) ≡(x0(1) , x1(1) , x2(1) , x3(1)

):= Φ(q) = Φ(γ(1))

xµ(1)!

= Xµp ≡ Xp

(ep(µ)

);

(3.2.16)

in words, the Riemann normal coordinates at point q in a sufficiently small neigh-bourhood of p are the components, with respect to an orthonormal frame at p,

61


of the tangent vector Xp such that expp(Xp) = q. Now, notice that any familyλXp of tangent vectors at p gets mapped to a point on a geodesic through p (atleast for some suitable range of the parameter λ), or, in other words, any curve γthrough p such that

((Φ γ)(λ))µ ≡ xµ(λ)!

= λXµp (3.2.17)

solves the geodesic equation; in fact, any geodesic through p may be expressed as(3.2.17), in Riemann normal coordinates, so that

∂2x

∂λ2= 0 . (3.2.18)

On the other hand, (2.1.91) together with (3.2.18) implies that

Γρµν = 0 (3.2.19)

in Riemann normal coordinates, so that

0 = ∇µgνρ = ∂µgνρ − Γσµν gσρ − Γσµρgνσ . (3.2.20)

To summarise, Riemann normal coordinates at p∈M are characterised by

g = η η ≡ Minkowski metric (3.2.21)

∂µgνρ = 0 (3.2.22)

and, as such, constitute a realisation of a local inertial frame at p, thus real-ising Einstein’s Equivalence Principle : the laws of physics reduce to thoseof Special Relativity in a small enough region of spacetime and no one can detectthe presence of a gravitational field by performing local experiments.

Back to matter sources, an ideal or perfect fluid is a fluid which appears tobe homogeneous and isotropic according to any locally inertial observer comovingwith any of the fluid elements. This means that, according to these observers,the fluid undergoes no net transport and no shear stress, so that the stress-energytensor T in any of these reference frames is diagonal,

Tµν =

e 0 0 00 P 0 00 0 P 00 0 0 P

; (3.2.23)

here, e and P are the total energy density and pressure of the fluid as mea-sured by any local inertial observer attached to the fluid element under consider-ation; e may be expressed in terms of the rest mass density ρ and the specificinternal energy ε (measured by the same observer) as

e = ρc2 + ρεc=1= ρ(1 + ε) . (3.2.24)

62


Now focus on one of these observers and denote its 4-velocity by u = (1, 0, 0, 0).Consider a second local inertial observer placed at the same spacetime point as thefirst one and seeing the fluid element moving along with velocity u: this observermoves with velocity −u according to the comoving observer. Then, any 4-vectorX as seen by the second observer is related to the same 4-vector X as measuredby the first one through

Xµ =Bµν(−u) Xν

Bµν(−u) :=

W Wu1 Wu2 Wu3

Wu1 1 + (W − 1)(u1)

2

‖u‖2 (W − 1) v1v2

‖u‖2 (W − 1) u1u3

‖u‖2

Wu2 (W − 1) u1u2

‖u‖2 1 + (W − 1)(u2)

2

‖u‖2 (W − 1) u2u3

‖u‖2

Wu3 (W − 1) u1u3

‖u‖2 (W − 1) u2u3

‖u‖2 1 + (W − 1)(u3)

2

‖u‖2

,

(3.2.25)

where Bµν(−u) is the Lorenz boost to a frame moving with velocity −u with

respect to the initial one and W := 1√1−‖u‖2

is the Lorentz factor relating the

two frames. Indeed, the 4-velocity of the fluid element according to the secondobserver is easily found to be

u := (W,Wu) ⇒ uµuµ = −1 ; (3.2.26)

conversely, the comoving observer sees the second observer moving with velocity

v := (W,−Wu) ⇒ uµuµ = −1 . (3.2.27)

By the way, one can rewrite W in terms of the 4-velocities of the fluid element asseen by the two observers as

− ηµν uµuν ≡ −uµuµ = W . (3.2.28)

Likewise, the stress-energy tensor (3.2.23) as seen by the second observer is found(after some algebra) to be

Tµν = Bρµ(v)Bσ

ν(v) Tρσ = uµuν(e+ P ) + Pηµν . (3.2.29)

However, this is a completely tensorial relation, so it must be true regardless of thecoordinate system adopted (this is the essence of Einstein’s Equivalence Principle).Thus, the generic expression of the stress-energy tensor of an ideal fluid must be

Tµν = Bρµ(v)Bσ

ν(v) Tρσ = uµuν(e+ P ) + Pgµν , (3.2.30)

63


which indeed reduces to (3.2.29) at any point p ∈ M after choosing Riemannnormal coordinates. Notice, in passing, that the total energy density of the fluidas measured by the comoving observer is

e = Tµνuµuν , (3.2.31)

thus justifying expression (3.1.3) for the same quantity as seen by the Eulerianobserver.

3.2.3 Equations of state (EOS)

An equation of state (EOS) must be imposed as a constitutive relation amongthe fluid’s pressure P , the specific internal energy ε, and the (rest)-mass densityρ. Recall the first law of thermodyamics,

dU = TdS − PdV + µdN , (3.2.32)

where V is the fluid’s volume (in some reference frame), m its rest mass, U := mε =ρV ε its total internal energy, T its temperature, S its entropy, µ its chemicalpotential, and N is the number of particles in the fluid. In the simplest scenario,the fluid may be assumed to behave like an ideal (i.e. , non-interacting), classical(i.e. , non-quantum) gas, whose equation of state is well-known to be

PV = NkBT , (3.2.33)

where kB is Boltzmann’s constant. It is also well-known that Meyer’s relation holdsbetween the specific heats at constant volume and pressure, cV and cP respectively,

cP − cV = NkB ⇒ cV =NkBγ − 1

γ :=cPcV

Adiabatic index .(3.2.34)

For a transformation performed at constant volume and for which the number ofparticles is conserved, (3.2.32) reduces to dU = TdS. The specific heat at constantvolume is defined by

cV :=

(∂ε

∂T

)V

= T

(∂s

∂T

)V

, (3.2.35)

where s =: Sm

is the specific entropy. If cV is assumed to be constant during thetransformation, then cV = ε

Tand (3.2.34) allows to recast (3.2.33) as

P (ρ, ε) = (γ − 1) ρε , (3.2.36)

64


which is known as the ideal-fluid or gamma-law EOS. It is worth stressingthat this EOS holds both in the non-relativisic and in the ultrarelativistic limits,as long as the gas can be assumed to be non-interacting (‘ideal’) and non-quantum(‘classical’).

In this work, neutron stars are involved for the larger part. These are ex-treme objects with radii ∼10 km and masses comparable with the mass of the Sun(M∼ 2 · 1030kg), whose rest mass density is thus ∼ 1018 kg

m3 = 1015 gcm3 . Treating

these objects as ideal, classical gases may sounds a bit weird; a much better ap-proximation treats indeed neutron stars as relativistic, degenerate, quantum Fermigases at zero temperature. The last assumption allows to write the EOS for such anobject in closed form, which is not possible when T > 0, and makes a lot of sense,as neutron stars are typically very cold remnants of supernovae explosions. Forneutron stars in binary systems at times close to merger, which are the maintopic of this thesis, the zero-temperature assumption is particularly well-suited,as the typical timescales for the inspiral phase are of the order of tenth, or evenhundreds, of millions of years and the stars get essentially cold by the time theycollapse onto each other. It can be shown that the EOS of a relativistic, degener-ate, quantum Fermi gas at zero temperature in the Newtonian and ultrarelativisticlimits has the form

P (ρ) = KρΓ , (3.2.37)

with

K =π

43

30

(6

g

) 53 gV ~2

m83 c3

, Γ =5

3Newtonian limit (3.2.38)

K =π

23

24

(6

g

) 43 gV ~m

43 c3

, Γ =4

3Ultrarelativistic limit ; (3.2.39)

here g is the spin degeneracy of the fermion species at hand (for example, g =2 · 1

2+ 1 = 2 for neutrons, protons and electrons) and ~ := h

2πdenotes the reduced

Planck’s constant. Write the first law of thermodynamics (3.2.32) in terms of ρfor the case dN = 0 and dS = 0 (isentropic trnsformation) as

dε =P

ρ2dρ ; (3.2.40)

integrating the last equality and using (3.2.37) then gives

ε(ρ) =KρΓ−1

Γ− 1. (3.2.41)

Together, (3.2.37) and (3.2.41) define the polytropic EOS. It is worth notingthat, for isentropic transformations, the ideal-fluid EOS (3.2.36) is equivalent to

65


a polytropic EOS where Γ = γ, where the last quantity is the adiabatic indexdefined in (3.2.34).

The maximum mass of a star whose EOS is polytropic with values of K and Γgiven by (3.2.38) or (3.2.39) can be shown to be less than 1M ; this is way smallerthan the maximum mass measured for a neutron star up to now, which is about2.14M [5, 23, 25]. Therefore, a large number of numerical simulations of BNSsystems have been performed with polytropic EOS, but using values of K and Γdifferent from (3.2.38) and (3.2.39). However, a polytropic EOS with K = 100(geometrised units) and Γ = 2 can be shown to account for a maximum mass ofonly 1.62M for a star satisfying the Tolman-Oppenheimer-Volkov (TOV) set ofequations (see section 5.1.4), so a polytropic EOS cannot, for sure, describe thenature of neutron stars in full detail, and more realistic EOSs are needed.

In addition to the mass problem, realistic EOSs should be ‘hot’, that is, theyshould take non-zero fluid temperatures into account: indeed, while ‘cold’ EOSs(i.e. , EOSs assuming the fluid’s temperature is always zero) are good enoughat describing the inspiral phase of BNS systems, temperatures of the order ∼10 — 100MeV may be reached during and after merger. Therefore, instead oftreating the core of neutron stars simply as a Fermi gas, realistic EOSs take thestrong interactions among nucleons into account as well; since these interactionsare not yet known exactly, several models and approximations were developedthat produce both ‘hot’ and ‘cold’ tabulated EOSs (see, e.g. , the CompOSE [22]or StellarCollapse [66] databases).

3.2.4 Relations among quantities as measured by the Eu-lerian and comoving observers

Consider now a foliation of spacetime and two spacelike slices Σt and Σt+δt in it.Let the worldline of a fluid element cross Σt at point P and Σt+δt at point Q andlet the worldline of the Eulerian observer cross Σt at point P and Σt+δt at pointQ′. The 4-velocity field of the Eulerian observer is the vector field n orthogonalto the foliation, and let u be the 4-velocity field of the fluid element. Let δτ0 bethe difference in proper times between events P and Q according to the observercomoving with the fluid particle and let δτ be the difference in proper times ofbetween events P and Q′ according the Eulerian observer. Notice that, accordingto the Eulerian observer, Q′ and Q are simultaneous events, as they both lie ontoΣt+δt ; let δl the (spacelike) vector connecting Q′ to Q and lying onto Σt+δt . Thesituation is portrayed in figure 3.1.

The Lorentz factor W relating the two observers can be defined as the propor-tionality factor between the corresponding proper times,

δτ = Wδτ0 ; (3.2.42)

66


Σt

Σt+δt

L

UUWU

δln

Wn

δτ = Wδτ0

u

δτ0

P

Q

Q′

Figure 3.1: Worldline L of a fluid element with 4-velocity u crossing thehypersurfaces Σt and Σt+δt in the spacetime foliation. Figure inspired by figure6.1 of [37].

from here, using the triangle identity

δτ0u = δτn+ δl (3.2.43)

and contracting (3.2.43) with nµ to get

δτ0nµ = δτnµnµ + nµδl

µ = −δτ = −Wδτ0 , (3.2.44)

relation (3.2.28) is recovered,

W = −nµuµ = αu0 , (3.2.45)

where (2.3.18) has been used to write the last identity.

The fluid velocity relative to the Eulerian observer is defined as

U :=δl

δτ(3.2.46)

and is tangent to Σt by construction. Now divide (3.2.43) by δτ and use (3.2.45) toget 3+1 decomposition of the fluid velocity u with respect to the foliation Σtt∈R ,

u = W (n+ U) ; (3.2.47)

the unit vector n orthogonal to the spacetime foliation can be made to disappearfrom the last relation by noticing that the orthogonal projection of u onto Σt (see(2.2.22)) is

P iνu

ν = ui −Wni , (3.2.48)

67


so that

U i =P i

νuν

W. (3.2.49)

Notice that (3.2.47) and nµUµ = 0 allow one to write

− 1 = uµuµ = W 2(nµn

µ + 2nµUµ + UµU

µ) = W 2(−1 + UµUµ) , (3.2.50)

from which the familiar, Minkowski-like expression of the Lorentz factor W isrecovered,

W =1√

1− UµUµ; (3.2.51)

of course, behind the Minkowski appearance, the (curved) 3-metric hides insidethe term UµU

µ.

To finish with, let P and Q be as before and let (U ,Φ) be a chart on M suchthat P,Q∈M with and

Φ(P ) =:(t, x1, x2, x3

)≡ x∈R4

Φ(Q) =:(t+ δt, x1 + δx1, x2 + δx2, x3 + δx3

)≡ x+ δx∈R4 .

(3.2.52)

Then, dx ≡ (δx1, δx2, δx3) is the vector tangent to Σt+δt giving the distance of theworldline of constant spatial coordinates, whose tangent vector is ∂t , from pointQ. The fluid coordinate velocity is defined as

vi :=δxi

δt=ui

u0, (3.2.53)

where the fluid’s 4-velocity has been expressed as

uµ =δxµ

δτ0

(3.2.54)

to get the last identity in (3.2.53). Now recall (2.3.11):

∂t = αn+ β , (3.2.55)

where αn and β are the projections of ∂t onto n and onto Σt . Therefore,

δτ = αδt (3.2.56)

(as already noticed in (2.3.6)) and the vector βδt measures the drift of the coordi-nate line x = Const from the Eulerian observer’s worldline between the coordinatetimes t and t+ δt. Therefore,

δl = δx + βδt (3.2.57)

68


and, dividing (3.2.57) by δτ and making use of (3.2.53), (3.2.46) and (3.2.56)results in

v = αU − β . (3.2.58)

All the identities found up to now in this section hold for generic fluids. Inthe special case of ideal fluids, substitute expression (3.2.29) for the ideal-fluidstress-energy tensor into definition (3.1.3) and make use of (3.2.45) in order touncover the relation between the total energy densities E and e as measured bythe Eulerian and comoving observers, respectively,

E + P = W 2(e+ P ) = W 2[ρ(1 + ε) + P ] , (3.2.59)

where (3.2.24) has been used as well. The momentum density of the ideal fluidas measured by the Eulerian observer is obtained by combination of (3.1.4) with(3.2.29) (again recalling (3.2.45)),

pµ = W 2(e+ P )Uµ = (E + P )Uµ , (3.2.60)

where (3.2.59) has been employed to get the last identity. Finally, the stress tensorof the ideal fluid as measured by the Eulerian observer is obtained by means of(3.1.5), (3.2.29) and (3.2.48),

Sij = (E + P )UiUj + Pγij . (3.2.61)

To summarise, then, one can write the 3+1 decomposition of the stress-energytensor of the ideal fluid as

Tµν = E nµnν + 2p(µnν) + Sµν

pi = (E + P )Ui

Sij = (E + P )UiUj + Pγij .

(3.2.62)

Starting from the general law (3.2.9), the last two relations ((3.2.60) and(3.2.61)) allow to express the on-shell conservation of energy for an ideal fluid,

(∂t − Lβ)E = −αDi

((E + P )U i

)+

+ (E + P )(K +KijU

iU j)− 2(E + P )U iDiα = 0 .

(3.2.63)

Similarly, plugging (3.2.60) and (3.2.61) into (3.2.13), one gets the on-shell con-servation of momentum for an ideal fluid,

(∂t − Lβ) ((E + P )Ui) = −αDj

[Pδji + (E + P )U jUi

]+

− [Pγij + (E + P )UiUj] γjkDkα + αK(E + P )Ui − EDiα = 0 .

(3.2.64)

69


Expanding the products in the last relation and making use of (3.2.63) results in

(∂t − Lβ)Ui =− αU jDjUi + UiUjDjα−Diα− αKjkU

jUkUi+

− 1

E + P[αDiP + Ui(∂t − Lβ)P ] ;

(3.2.65)

moreover, (3.2.58) implies

αU jDjUi = vjDjUi + βjDjUi

⇒ LβUi = αU jDjUi − vjDjUi + UjDiβj ,

(3.2.66)

so that (3.2.64) finally reduces to(∂t + vjDj

)Ui =− 1

E + P

[αDiP + Ui

(∂t − βj∂j

)P]

+

+ UjDiβj −Diα + UiU

j(Djα− αKjkU

k).

(3.2.67)

3.2.5 Total rest mass (or baryon number) conservation andelectron fraction advection

Pick a fluid element at a given spacetime point p and an arbitrarily small fluidregion of rest mass m comoving with it; denote with V the volume of this fluidregion as measured by some local inertial observer O comoving with it. Let ρ := m

V

be the rest mass density associated to this fluid region and define the rest massdensity 4-current measured by O as the 4-vector

J := (ρ, 0, 0, 0) . (3.2.68)

Another local Lorentz observer O at p who sees O moving with velocity u measuresthe 4-current J = B(−u) (see (3.2.25)); in components (see (3.2.26)),

Jµ = Bµν(−u) Jν = ρuµ . (3.2.69)

In absence of high-energy reactions, the total rest mass of the fluid can be safelyassumed to be constant. This requirement can be made precise by imposing that,according to O (as well as to any other local Lorentz observer at p) the rate ofchange of m in the fluid volume under consideration is exactly balanced by theflux of the vector J := ρu through the boundary of that volume. Due to lenghtcontraction, the observer O measures a volume δV

W1, so that

∂m

∂t=

∫VW

d3x∂ρ

∂t!

= −∫∂( VW )

d2σ(x) J · n =

∫VW

d3x∇ · J , (3.2.70)

1Without losing generality, rotate the axes of both O and O so that u is aligned, for example,

with the x-axes; then u =(u0, u1, 0, 0

)and the Lorentz boost from O to O reduces to

(W Wu1

Wu1 W

)70


where ∂(VW

)denotes the boundary of V

Wand the divergence theorem has been

employed in the last step. Locally, and using index notation, (3.2.70) means that

∂µJµ = ∂µ(ρuµ)

!= 0 . (3.2.71)

The correct generalisation of (3.2.71) to arbitrary coordinates at point p is

∇µJµ = ∇µ(ρuµ)

!= 0 : (3.2.72)

indeed, (3.2.72) reduces to (3.2.71) when choosing Riemann normal coordinates atp and is a completely tensorial relation, thus being valid in any reference frame.Relation (3.2.72) constitutes a total rest mass conservation law for the fluidand it can be cast in 3+1 form, after having picked a foliation of spacetime bya family of spacelike hypersurfaces Σtt∈R whose orthogonal timelike unit-vectorfield is denoted by n. To this end, recall (3.2.47) and (2.3.10) to write

0 = ∇µ(ρuµ) = nµ∇µ(ρW ) + ρW∇µnµ +Dµ(ρWUµ) + ρWUµDµ(log(α)) =

= Ln(ρW )−KρW +1

αDµ(αρWUµ) ; (3.2.73)

using (2.3.26), the last relation can be written as

(∂t − Lβ)(ρW ) = αKρW −Dµ(αρWUµ) . (3.2.74)

Furthermore, the 4-current J may be decomposed along the directions parallel andorthogonal to the foliation,

Jµ = (−nνJν)nµ + P µνJ

ν ≡ ρ(n)nµ +(J (n)

)µ, (3.2.75)

with ρ(n) := (−nνJν) and(J (n)

)µ:= P µ

νJν being the rest mass density and the

corresponding 4-current measured by the Eulerian observer.

Denote protons and neutrons collectively as ‘baryons’ and assume their massesare equal. Let then mb denote the rest mass of one baryon and call Nb the totalnumber of baryons in the fluid and nb the baryon number density ; the latteris a scalar function of the spacetime point p and nb(p) is measured by an observercomoving with the fluid element placed at p. Similarly, let me be the electron

and any spatial lenght l in the x1 direction as measured by O gets divided by W when measured

by O to get l := lW . Indeed, according to O, measuring the extrema of l constitutes a couple

of simultaneous events A and B whose(t, x1

)coordinates are

(t, x1A

)and

(t, x1B

); on the other

hand, according to O, the same two events are not simulataneous and their spatial coordinates

are

(x1A=W(x1

A+u1t)x1B=W(x1

B+u1t)

), so that x1B − x1A = W

(x1B − x1A

), or l := l

W .

71


mass and let Ne be the total number of electrons in the fluid and ne the electronnumber density ; again, ne is a scalar function of the spacetime point p andne(p) is measured by an observer comoving with the fluid element placed at p. Inabsence of very high-energy processes, Nb can be safely assumed to be constant,so that, following a line of thought analogous to the one which led to (3.2.72), onecan write a baryon number conservation law,

∇µ(nbuµ)

!= 0 . (3.2.76)

A proper treatment of neutrino production, absorption, and transport in the fluidis of critical importance in order to model many interesting astrophysical events(such as jet emissions from compact binary mergers involving neutron stars, ormatter accretion onto a black hole) in a realistic way: indeed, the newest versionof Spritz is able to take care of processes involving neutrinos via a leakage scheme,implemented in the ZelmaniLeak code (see [20] and references therein for furtherdetails). However, this work will not cover this important topic and neutrinos willbe simply discarded. Because of this, reactions such as electron capture on nucleior electron-positron pair annihilation into a neutrino-antineutrino pair are ruledout. Furthermore, assuming again that high-energy processes do not take place,electron-positron annihilations into a pair of photons or into a couple of exoticbaryons are ruled out as well. To summarise, the only negatively-charged particlesin the fluid are assumed to be electrons, and their number is taken to be conserved;as a result, the following electron number conservation law holds,

∇µ(neuµ)

!= 0 . (3.2.77)

Neglecting the electron contribution to the total rest mass of the fluid, one canwrite ρ = mbnb . Then, define the electron fraction as

Ye :=nenb

(3.2.78)

and notice that (3.2.77) implies

0 = mb∇µ(neuµ) = ∇µ

(mbNb

nenbuµ)

= ∇µ(ρYeuµ) , (3.2.79)

expressing that the elecron fraction is advected along the fluid’s streamlines; (3.2.79)is usually called the electron fraction advection equation. As already donewith the total rest mass conservation law (3.2.72), (3.2.79) can be cast in 3+1 formas well: just pick (3.2.74) and perform the substitution ρ 7→ ρYe to get

(∂t − Lβ)(ρYeW ) = αKρYeW −Dµ(αρYeWUµ) . (3.2.80)

72


3.3 3+1 decomposition of Maxwell’s equations

3.3.1 Some basic facts about the electromagnetic field

Following what was done in section 3.2.5, pick a fluid element at a given spacetimepoint p and an arbitrarily small fluid region comoving with it. Let q be theamount of electric charge in this region and let V be its volume as measured bysome local inertial observer O comoving with it. Let ρe := q

Vbe the electric

charge density associated to this fluid region and define the electromagnetic4-current density measured by O as the 4-vector

Je := (ρe, 0, 0, 0) . (3.3.1)

Another local Lorentz observer O at p who sees O moving with velocity u measuresthe 4-current Je = B(−u) (see (3.2.25)); in components (see (3.2.26)),

Jµe = Bµν(−u) Jνe = ρeu

µ . (3.3.2)

The total electric charge of the fluid must be conserved. This requirement can bemade precise by imposing that, according toO (as well as to any other local Lorentzobserver at p) the rate of change of q in the fluid volume under consideration isexactly balanced by the flux of the vector Je := ρeu through the boundary of thatvolume. Due to lenght contraction, the observer O measures a volume V

W, so that

∂q

∂t=

∫δVW

d3x∂q

∂t!

= −∫∂( δVW )

d2σ(x) Je · n =

∫δVW

d3x∇ · Je , (3.3.3)

where ∂(δVW

)denotes the boundary of δV

Wand the divergence theorem has been

employed in the last step. Locally, and using index notation, (3.3.3) means that

∂µJµe = ∂µ(ρeu

µ)!

= 0 . (3.3.4)

The correct generalisation of (3.3.4) to arbitrary coordinates at point p is

∇µJµe = ∇µ(ρeu

µ)!

= 0 : (3.3.5)

indeed, (3.3.5) reduces to (3.3.4) when choosing Riemann normal coordinates atp and is a completely tensorial relation, thus being valid in any reference frame.Relation (3.3.5) constitutes a total electric charge conservation law for thefluid.

In presence of electric charges, a local inertial observer comoving with somefluid element experiences the presence an electromagnetic field, whose dynamics

73


is regulated by Maxwell’s equations,

∇ · E != ρe (3.3.6)

∇× E!

= −∂B

∂t(3.3.7)

∇ ·B != 0 (3.3.8)

∇×B!

= Je +∂E

∂t(3.3.9)

Relation (3.3.8) implies the existence of a vector potential A such that

B = ∇×A . (3.3.10)

Substituting the last result into (3.3.7) yields

∇×(

E +∂A

∂t

)= 0 , (3.3.11)

which implies the existence of a scalar potential ϕ such that

E +∂A

∂t:= −∇ϕ ⇒ E = −∇ϕ− ∂A

∂t. (3.3.12)

The scalar and vector potential may be combined into the electromagnetic 4-potential

A := (ϕ,A) , (3.3.13)

which, in turn, defines the Faraday 2-form F and its Hodge-dual ∗F , whosecomponents in Riemann normal coordinates are

Fµν := ∂[µAν] (3.3.14)

∗F µν :=1

2εµνρσF

ρσ . (3.3.15)

Maxwell’s equations can then be recast compactly for a local Lorentz observer atpoint p as

∂νFµν !

= Jµe (3.3.16)

∂ν∗F µν !

= 0 , (3.3.17)

where F µν := ηµρηνσFρσ and ∗F µν := ηµρηνσ∗F ρσ. Equation (3.3.17) can be conve-niently expanded to

0!

= 3! ∂[µFνρ] ∝ ∂µFνρ + ∂νFρµ + ∂ρFµν . (3.3.18)

74


The generalisation of (3.3.16) and (3.3.17) to arbitrary coordinate systems is givenby the completely tensorial expressions

∇νFµν = Jµe (3.3.19)

∇ν∗F µν = 0 , (3.3.20)

which reduce to (3.3.16) and (3.3.17) when choosing Riemann normal coordinatesat p; the Faraday tensor is now written in terms of covariant derivatives, but, sinceit is antisymmetric and because Γρµν = Γρνµ, it does not change its form,

Fµν := ∇[µAν] = ∂[µAν] . (3.3.21)

As above, (3.3.20) can be recast more explicitly as

0!

= 3!∇[µFνρ] ∝ ∇µFνρ +∇νFρµ +∇ρFµν . (3.3.22)

Notice that taking a covariant derivative with respect to the free index in Maxwell’sequations with sources (3.3.19) implies the local conservation law of total electriccharge (3.3.5). Therefore, solving (3.3.19) automatically takes (3.3.5) into account,so that the latter equation needs not be solved explicitly; that’s why (3.3.5) hasnot been cast in 3+1 form, even if this could have been done straighforwardly bysimply changing ρ 7→ ρe in (3.2.74).

When deducing Maxwell’s equations from an action principle, the Lagrangiandensity associated to the electromagnetic field turns out to be simply proportionalto FµνF

µν and the corresponding stress-energy tensor is naturally defined as aquantity proportional to the variation of that Lagrangian density with respect tothe (inverse) spacetime metric; that is to say, the stress-energy tensor representsthe source of the electromagnetic field (technical details about this are not requiredhere). Following this approach, the stress-energy tensor of the electromag-netic field is found to be

T (EM)µν = FµρFρ

σ − 1

4gµνFρσF

ρσ . (3.3.23)

3.3.2 3+1 treatment of the electromagnetic field

Choose, as usual, a foliation of spacetime by a family of spacelike hypersurfacesΣtt∈R and denote the timelkie unit-vector field orthogonal to it with n. Using thegeneral rule (2.2.32), the electromagnetic 4-current density Je may be decomposedalong the directions parallel and orthogonal to the foliation,

Jµe = (−nνJνe )nµ + P µνJ

νe ≡ ρ(n)

e nµ +(J (n)e

)µ, (3.3.24)

75


with ρ(n)e := (−nνJνe ) and

(J

(n)e

)µ:= P µ

νJνe playing the roles of electric charge

density and electromagnetic 4-current as measured by the Eulerian observer. Fur-thermore, recalling (2.2.43), the Faraday tensor and its Hodge-dual can be splittedalong the directions parallel and orthogonal to the foliation as

Fµν = 2n[µEν] + εµνρσnρBσ (3.3.25)

∗F µν =1

2εµνρσF

ρσ = −2n[µBν] + εµνρσnρEσ , (3.3.26)

where

Eµ :=F µνnν ⇒ nµEµ = 0 (3.3.27)

Bµ :=− ∗F µνnν = −1

2εµνρσnνFρσ ⇒ nµB

µ = 0 (3.3.28)

are the electric and magnetic fields as measured by the Eulerian observer (ascan be easily verified in a local inertial frame comoving with the Eulerian observeritself, recalling (3.3.21)) and they are tangent to all hypersurfaces in the spacetimefoliation.

With these tools at hand, Maxwell’s equations can now be split into 3+1 form.Start by plugging (3.3.26) into the source-free Maxwell equations (3.3.20) to get

∇ν

(−2n[µB ν] + εµνρσnρEσ

)= 0 . (3.3.29)

First of all, use (2.3.10), (3.2.4) and the definition of Lie derivative (2.1.122) towrite

∇ν

(−2n[µB ν]

)= (LnB)µ − nµ∇νB

ν −KBµ =

=1

α(LmB)µ −KBµ − (∇νB

ν − nµBνDν(log(α))) =

=1

α(LmB)µ −KBµ − nµDνB

ν .

(3.3.30)

Second, because εµνρσ ≡ (− det(g))−12 εµνρσ is compatible with the Levi-Civita

spacetime connection and thanks to (2.3.8),

∇ν(εµνρσnρEσ) = εµνρσnρ∇νEσ + εµνρσEσ∇νnρ =

= εµνρσnρDνEσ − (εµνρσKνρ + εµνρσnνDρ(log(α)))

= −εµνρσnνDρEσ − εµνρσnνDρ(log(α))Eσ =

= − 1

αεµνρσnνDρ(αEσ) ≡ 1

αεµρσDρ(αEσ) ,

(3.3.31)

76


where εµνρσnρ∇µEσ = εµνρσnρDµEσ because any part of ∇µEσ parallel to ngets annihilated when contracted against εµνρσnρ . To summarise, the source-freeMaxwell equations (3.3.20) become

(LmB)µ − αKBµ − αnµDνBν + εµνρDν(αEρ) = 0 ; (3.3.32)

the last relation is easily decomposed along the directions parallel and orthogonalto the foliation,

DiBi = 0 (3.3.33)

(∂t − Lβ)Bi = αKBi − εijkDj(αEk) , (3.3.34)

where LmB = (∂t − Lβ)B and Greek indices have been replaced by Latin oneswhere spatial quantities are involved.

The 3+1 splitting of the Maxwell equations with sources is easily deduced from(3.3.32) by substituting 0 with ρ in the RHS of the latter and by noticing that ∗Fbecomes F when switching B 7→ −E and E 7→ B,

− (LmE)µ + αKEµ + αnµDνEν + εµνρDν(αBρ) =

= αJµe ≡ α(ρen

µ +(J (n)e

)µ);

(3.3.35)

again, (3.3.35) can be easily decomposed along along the directions parallel andorthogonal to the foliation,

DiEi = ρe (3.3.36)

(∂t − Lβ)Ei = αKEi + εijkDj(αBk)− α(J (n)e

)i. (3.3.37)

To finish with, plug expression (3.3.25) for F into (3.3.23). After some tediousalgebra, this results in the 3+1 form (3.1.2) of the stress-energy tensor of theelectromagnetic field,

T (EM)µν = E(EM)nµnν + 2p

(EM)(µ nν) + S(EM)

µν , (3.3.38)

with

E(EM) :=1

2

(EiE

i +BiBi)

(3.3.39)

p(EM)i := εijkE

jBk ≡ (E ∧B)i (3.3.40)

S(EM)ij :=

1

2γij(EiE

i +BiBi)− EiEj −BiBj . (3.3.41)

the quantity (3.3.40) is recognised as the Poynting vector.

77


3.3.3 Ideal MHD limit

Consider a local inertial observer O comoving with an element of electrically con-ducting fluid placed at some spacetime point p. Denote the electric field measuredby O with e and assume the fluid is Ohmic; then, according to O, e induces themotion of electric charges as stated by Ohm’s law,

je = σe , (3.3.42)

where je is the electric current density measured by O and σ is called the electricalconductivity. The 4-velocity of the fluid element as measured by O is uµ =(1, 0, 0, 0) ⇔ uµ = ηµν u

ν = (−1, 0, 0, 0), so the Faraday tensor F in this referenceframe is such that

F i0 = ∂iA0 − ∂0Ai = ∂iφ+ ∂tAi = −ei ; (3.3.43)

this means that F 0ν uν = 0

F iν uν = F i0u0 = ei⇒ F µνuν = (0, e) . (3.3.44)

On the other hand, let jµe ≡ (ρe , je) and notice that

jµe uµ = j0eu0 = −ρe . (3.3.45)

Given (3.3.44) and (3.3.45), Ohm’s law may be rewritten as an identity between4-vectors,

je = σe 7→ (0, je) = (ρe , je) + (−ρe, 0, 0, 0)!

= σ (0, e)

jµe + (jνeuν) uµ !

= σF µνuν .(3.3.46)

Now, (3.3.46) is a completely tensorial relation, so it must be valid for any otherlocal inertial observer at p and, more generally, for any observer at p; therefore,the general-relativistic Ohm law reads

Jµe = σF µνuν − (Jνe uν)uµ . (3.3.47)

The first term in the RHS of (3.3.47) is a conduction current, meaning that itis generated by the presence of a non-zero electric field in the rest-frame of eachfluid element, while the second term is a convection current, in that it stemsfrom the effect of relative motion between the electric charges and the observer.

Usually, most fluids and plasmas found in astrophysical scenarios can be con-sidered as being perfect conductors to extremely good approximation, and the

78


corresponding Maxwell theory describing them is then called the ideal mag-netohydrodynamic (MHD) limit. The ideal MHD limit consists in havinginfinite conductivity ; correspondingly, in order that the conduction term σF µνuνin (3.3.47) remains finite, the electric field in the rest-frame of each fluid elementmust vanish,

F µνuν ≡ (0, e)!

= 0 (3.3.48)

(see (3.3.44)). The Faraday tensor and its Hodge-dual can be split along thedirections parallel and orthogonal to u; just replace n with u in (3.3.25) and(3.3.26). In the ideal MHD limit, taking (3.3.48) into account, these two quantitiesare entirely determined by u and by the magnetic field b measured by thecomoving observer (i.e. , any observer comoving with any fluid element),

Fµν = εµνρσuρbσ (3.3.49)

∗Fµν = −2u[µ bν] , . (3.3.50)

Recalling (3.2.45), the magnetic field measured by the Eulerian observer in theideal MHD limit is obtained by plugging (3.3.50) into (3.3.28),

Bµ = −∗F µνnν = (nνbν)uµ − (nνuν) b

µ = (nνuν)uµ +Wbµ . (3.3.51)

Notice thatuµb

µ = −∗F µνuµuν = 0 , (3.3.52)

so, contracting (3.3.51) with uµ and using (3.2.47) results in

nµbµ = −WUiB

i . (3.3.53)

Therefore, using again (3.2.47) leads to the 3+1 decomposition of b along thedirections parallel and orthogonal to the spacetime foliation,

bµ = W(UiB

i)nµ +

Bµ

W+W

(UiB

i)Uµ . (3.3.54)

Again making use of (3.2.45) and thanks to (3.3.54), the electric field measuredby the Eulerian observer in the ideal MHD limit is obtained by plugging (3.3.49)into (3.3.27),

Eµ = F µνnν = εµνρσnνUρBσ ≡ −3εµρσUρBσ . (3.3.55)

In the last relation, Greek indices may be replaced by Latin ones, as only spatialvector are involved. Therefore,

Ei = −3εijkUjBk = −

3εijk√

det(γ)UjBk ≡ −

1√det(γ)

(U×B)i . (3.3.56)

79


The 3+1 decomposition of the electromagnetic stress-energy tensor along thedirections parallel and orthogonal to the spacetime foliation in the ideal MHD limitis deduced from the more general form (3.3.38) by simply substituting expression(3.3.56) into (3.3.39), (3.3.40) and (3.3.41) to get

T (EM id)µν = E(EM id)nµnν + 2p

(EM id)(µ nν) + S(EM id)

µν , (3.3.57)

with

E(EM id) =1

2

[(1 + UiU

i)BjB

j −(UiB

i)2]

(3.3.58)

p(EM id)i =

(BjB

j)Ui −

(UjB

j)Bi (3.3.59)

S(EM id)ij =

1

2

[1

W 2BkB

k +(UkB

k)2]γij −

1

W 2BiB

j +(BkB

k)UiUj+

−(UkB

k)· 2U[iB j] .

(3.3.60)

3.4 Flux-conservative formulation and vector po-

tential evolution

3.4.1 3+1 ideal-fluid magnetohydrodynamics in the idealMHD limit in flux-conservative form

Begin by using (3.2.58) to put the total rest mass density conservation law (3.2.74)in the alternative form

∂t(ρW ) +Di

(ρWvi

)+ ρW

(Diβ

i − αK)

= 0 . (3.4.1)

Now contract (3.1.13) with γij to get

γij∂tγij!

= −2αγijKij + γij(Diβj −Djβi) ≡ 2(Diβ

i − αK), (3.4.2)

but notice that

1

2γij∂tγij =

1

2Tr(γ−1∂tγ

)=

1

2∂t(Tr(log(γ))) =

1

2∂t(log(det(γ))) =

= ∂t

(log(√

det(γ)))

=1√

det(γ)∂t

(√det(γ)

);

(3.4.3)

therefore, (3.4.2) becomes

Diβi − αK =

1√det(γ)

∂t

(√det(γ)

). (3.4.4)

80


On the other hand, application of (2.1.109) to the 3-metric γ and the Levi-Civitacovariant derivative D gives

Di

(ρWvi

)=

1√det(γ)

∂i

(√det(γ) ρWvi

). (3.4.5)

Together, (3.4.4) and (3.4.5) allow to write (3.4.1) in flux-conservative form,

∂t

(√det(γ) ρW

)+ ∂i

(√det(γ) ρWvi

)= 0 . (3.4.6)

This reformulation plays a critical role when looking for a numerical solution ofthe total rest mass conservation law, since (3.4.6), although being fully general-relativistic in nature, is a standard continuity equation: as will be discussed inchapter 4, partial differential equations cast in this way can be effectively tackled,for example, by finite-volume techniques. The price to pay is losing general co-variance, as (3.4.6) is not a tensorial relation anymore and is therefore dependentupon the choice of coordinates.

The electron fraction advection equation (3.2.80) can be cast in flux-conservativeform as well: it is enough to replace ρ 7→ ρYe in (3.4.6) to get

∂t

(√det(γ) ρYeW

)+ ∂i

(√det(γ) ρYeWv

i)

= 0 . (3.4.7)

Next, focus on the law of on-shell conservation of momentum (3.2.13). Using thedefinition of Lie derivative (2.1.122) to write

(Lβp)i = βjDjpi + pjDiβj , (3.4.8)

(3.2.13) becomes

∂tpi +Dj

(αSj i − βjpi

)+(Djβ

j − αK)pi = pjDiβ

j − EDiα . (3.4.9)

Let now τ be any(

11

)-tensor field tangent to all hypersurfaces in the spacetime

foliation chosen. Then, using (2.1.107),

Djτji = ∂jτ

ji + 3Γ

jjkτ

ki − 3Γ

kjiτ

jk =

= ∂jτji +

1√det(γ)

∂k

(√det(γ)

)τ ki − 3Γ

kjiτ

jk =

=1√

det(γ)∂j

(√det(γ)

)τ jk − 3Γ

kjiτ

jk .

(3.4.10)

Therefore, using again (3.4.4), the on-shell momentum conservation law (3.4.9) influx-conservative form is

∂t

(√det(γ) pi

)+ ∂j

(√det(γ)

(αSj i − βjpi

))=

=√

det(γ)[pjDiβ

j − EDiα + 3Γkji

(αSjk − βjpk

)].

(3.4.11)

81


Finally, consider the equation of on-shell conservation of energy (3.2.9). Writing

LβE = βjDjE = Dj

(Eβj

)− EDjβ

j , (3.4.12)

one has

∂tE +Di

(αpi − Eβi

)+ E

(Diβ

i − αK)

= αKijSij − piDiα ; (3.4.13)

using then (3.4.4) and applying (2.1.109) to the 3-metric γ and the Levi-Civita co-variant derivative D yields the on-shell energy conservation law in flux-conservativeform,

∂t

(√det(γ)E

)+ ∂i

(√det(γ)

(αpi − Eβi

))=

=√

det(γ)(αKijS

ij − piDiα).

(3.4.14)

The Spritz code works with perfect conducting fluids in the ideal MHD limit;in this regime, the total stress-energy tensor is the sum of the stress-energy tensorsfor the perfect fluid and for the electromagnetic field in the ideal MHD approxima-tion. Accordingly, the 3+1 decomposition of the total stress-energy tensor alongthe directions parallel and orthogonal to the spacetime foliation is obtained bysumming the contributions from (3.2.62) and (3.3.57)

T (tot)µν = E(tot)nµnν + 2p

(tot)(µ nν) + S(tot)

µν , (3.4.15)

with

E(tot) = E +1

2

[(1 + UiU

i)BjB

j −(UiB

i)2]

(3.4.16)

p(tot)i =

(E + P +BjB

j)Ui −

(UjB

j)Bi (3.4.17)

S(tot)ij =

1

2

[P +

1

W 2BkB

k +(UkB

k)2]γij −

1

W 2BiBj+

+(E + P +BkB

k)UiUj −

(UkB

k)· 2U[iB j] ;

(3.4.18)

recall that the total energy density of the fluid measured by the Eulerian observer,E, is related to the same quantity measured by the comoving observer, e, and tothe specific internal energy ε by (3.2.59).

To summarise, Spritz evolves the following set of equations in flux-conservativeform,

∂tUa + ∂jFja = Sa a∈ 0, 1, 2, 3, 4, 5 , (3.4.19)

82


with the state vector

U0 =√

det(γ) ρW (3.4.20)

Ui =√

det(γ) p(tot)i ∀i∈ 1, 2, 3 (3.4.21)

U4 =√

det(γ)E(tot) (3.4.22)

U5 =√

det(γ) ρYeW , (3.4.23)

the flux vectors (j∈1, 2, 3)

F j0 =

√det(γ) ρWvj (3.4.24)

F ji =

√det(γ)

(α(S(tot)

)ji− βjpi

)∀i∈ 1, 2, 3 (3.4.25)

F j4 =

√det(γ)

(α(p(tot)

)i − E(tot)βi)

(3.4.26)

F j5 =

√det(γ) ρYeWv

j , (3.4.27)

and the source vector

S0 = 0 (3.4.28)

Si =√

det(γ)[p

(tot)j Diβ

j − EDiα+

+ 3Γkji

(α(S(tot)

)jk− βjp(tot)

k

)]∀i∈ 1, 2, 3

(3.4.29)

S4 =√

det(γ)(αKij

(S(tot)

)ij − (p(tot))iDiα

)(3.4.30)

S0 = 0 , (3.4.31)

where E(tot), p(tot)i and S

(tot)ij are defined in (3.4.16), (3.4.17) and (3.4.18), respec-

tively. Actually, Spritz does not evolve the energy equation ∂tU4 + ∂jFj4 = S4 ,

but rather subtracts (3.4.6) from it and uses (3.2.58) to get rid of β,

∂t

(√det(γ)

(E(tot) − ρW

))+

+ ∂i

(√det(γ)

(α(p(tot)

)i+(E(tot) − ρW

)vi − αE(tot)U i

))=

=√

det(γ)(αKij

(S(tot)

)ij − (p(tot))iDiα

).

(3.4.32)

3.4.2 3+1 Maxwell equations in flux-conservative form

In the ideal MHD limit, the 3+1 Maxwell equations (3.3.34) and (3.3.37) take asimpler form and, analogously to (3.4.19), they are amenable to a flux-conservative

83


formulation. Plug expression (3.3.56) for the electric field in the ideal MHD ap-proximation into the source-free Maxwell’s equations (3.3.34) to get

(∂t − Lβ)Bi = αKBi + 3εijkDj

(αεklmU

lBm)

=

= αKBi + 3εkij 3εklm︸︷︷︸

δilδjm−δimδ

jl

Dj

(αU lBm

)= αKBi +Dj

(α · 2U [iB j]

)(3.4.33)

where 3εklm ≡3εklm√

det(γ)is not affected by covariant differentiation because of the

compatibility between D and γ. Now, using the divergence-free condition (3.3.33)for Bi gives

(LβB)i = βjDjBi −BjDjβ

i = Dj

(βjBi

)−BiDjβ

j −Dj

(βiBj

)+ βi

*0

DjBj =

= Dj

(βjBi

)−BiDjβ

j −Dj

(βiBj

)(3.4.34)

and (3.2.58) gives αU = v + β, where v is the fluid’s coordinate velocity (3.2.53);therefore, (3.4.33) becomes

∂tBi +(Djβ

j − αK)Bi +Dj

(2B[iv j]

)= 0 . (3.4.35)

Finally, (3.4.4) and relation (2.1.109) applied to the vector α · 2v[iB j] yield theflux-conservative form of Maxwell’s equations without sources,

∂t

(√det(γ)Bi

)+ ∂j

(√det(γ) 2v[iB j]

)= 0 . (3.4.36)

Notice that the flux terms in the above equation may be rewritten in a moresuggestive fashion as

∂j


)= ∂j

(√det(γ)

(δilδ

jm − δimδ

jl

)2vlBm

)=

= 3εijk∂j

(3εklm

(√det(γ) vlBm

))=(∇×

(√det(γ) v ×B

))i.

(3.4.37)

In the ideal MHD regime, the electric field measured by the Eulerian observeris deduced from the magnetic field measured by the same observer by means of(3.3.56), so there is no need to write Maxwell’s equations with sources (3.3.36) and(3.3.37) in flux-conservative form. However, the divergence-free constraint (3.3.33)for Bi can be expressed in flux-conservative form, thanks to (2.1.109), as

∂i

(√det(γ)Bi

)= 0 ; (3.4.38)

the last relation needs be always fulfilled when (3.4.36) is tackled numerically, oth-erwise unphysical magnetic monopoles may appear. To this end, some techniqueshave been developed, which are briefly listed (without going into too much detail)below.

84


Flux-interpolated constrained transport

In essence, this method consists in integrating the induction equation (3.4.36)over the surface of each elementary, cubic cell in the spatial numerical domain,expressing the RHS of (3.4.36) itself as the wedge product given (3.4.37) andusing Stokes’ theorem to convert the surface integrals on every face of each cellinto the circulation of the electric field (given as (3.3.56)) around the edges of thegiven face. Because the circulation is computed in the same direction (clockwise oranticlockwise) for each face, the contributions to it coming from each edge of thecell cancel exactly, so that the total magnetic flux through the surface of the cellis exactly zero; using the divergence theorem, this means that (3.4.38) is satisfied(as an integral relation holding true in each elementary cell).

As stated, the constrained transport scheme only prevents the generation ofmagnetic field terms violating (3.4.38), but it cannot cancel any such contributionif some are found at the beginning of the evolution. Moreover, though easily imple-mented on uniform meshes, this technique becomes very challenging to deal withon refined grids (see section 4.3.1), because of the huge number of interpolationsrequired to transfer variables from coarser to finer levels (and viceversa) correctly.More about the constrained transport approach in general relativistic MHD canbe found, for example, in [29] and [8].

Divergence-cleaning techniques

Here, the induction equation (3.4.36) is replaced by the system

∂t

(√det(γ)Bi

)+ ∂j


)+ γij∂jΛ = 0 (3.4.39)

DΛ + ∂j

(√det(γ)Bj

)= 0 , (3.4.40)

where Λ is an auxiliary scalar function of the spacetime point which couples thedivergence-free constraint (3.4.38) to the induction equation and D representssome suitable linear differential operator. To see the purpose of doing so, take thedivergence of (3.4.39) to get

∂t

(∂j

(√det(γ)Bj

))+∇2Λ = 0 ; (3.4.41)

thanks to the last relation, taking the time derivative of (3.4.40) gives

∂t(DΛ)−∇2Λ = 0 ; (3.4.42)

finally, applying D to (3.4.41) and using (3.4.40) yields

∂t

(D(∂j

(√det(γ)Bj

)))−∇2

(D(∂j

(√det(γ)Bj

)))= 0 . (3.4.43)

85


Then, (3.4.42) and (3.4.43) show that Λ and ∂j

(√det(γ)Bj

)satisfy the same

equation, so trying to ‘get rid’ of ∂j

(√det(γ)Bj

)is equivalent to trying to ‘get

rid’ of Λ; the latter is, however, simpler matter than the former.

Some choices for the operator D follow.

• Elliptic cleaning:DΛ = 0 ⇒ ∇2Λ = 0 . (3.4.44)

This is the Poisson equation, which is an elliptic partial differential equation(PDE).

• Parabolic cleaning:

DΛ =1

c2par

Λ ⇒ ∂tΛ− c2par∇2Λ = 0 . (3.4.45)

This is the heat equation, which is a parabolic PDE. The solution is smoothed(i.e. its regularity properties are improved) and spread out as time goes by,

thus non-zero contributions to ∂j

(√det(γ)Bj

)get dissipated.

• Hyperbolic cleaning:

DΛ =1

c2hyp

∂tΛ ⇒ ∂2t Λ− c2

hyp∇2Λ = 0 . (3.4.46)

This is the wave equation, which is a hyperbolic PDE. The solution is ad-vected towards the boundaries of the computational domain, thus non-zero

contributions to ∂j

(√det(γ)Bj

)get transported where they cannot badly

affect the evolution of the system under investigation.

• Mixed (parabolic + hyperbolic) cleaning:

DΛ =1

c2par

Λ +1

c2hyp

∂tΛ ⇒ ∂2t Λ +

c2hyp

c2par

∂tΛ− c2hyp∇2Λ = 0 . (3.4.47)

This is the telegrapher’s equation and combines the effects of the heat andwave equations. The solution is then smoothed and dissipated, while beingadvected towards the boundaries of the computational domain.

See [24] for further details.

86


3.4.3 Evolution of the electromagnetic 4-potential

There is another way of ensuring the divergence-free constraint (3.4.38) is satisfied,which is the one actually implemented by Spritz. Once a foliation of spacetimeby a family of spacelike hypersurfaces is chosen, the electromagnetic 4-potentialdefined in (3.3.13) may be decomposed along the directions parallel and orthogonalto that foliation as

Aµ = (−nνAν)nµ + PµνAν =: ϕnµ + Aµ , (3.4.48)

where ϕ := −nνAν and Aµ = PµνAν are the scalar and vector potential mea-

sured by the Eulerian observer, respectively; A is tangent to every spacelike slicein the spacetime foliation. Then, recalling from (2.3.18) that ni = 0, the 3+1decomposition of the ‘purely spatial’ part of the Faraday tensor (3.3.25) reads

DiAj −DjAi = DiAj −DjAi = Fij = 3εijlBl

⇒ 3εkij

(DiAj −DjAi) = 2 · 3εkijDiAj

!= 3ε

kij3εlijBl = 2δkl B

l = 2Bk ,(3.4.49)

so that (renaming indices)

Bi = 3εijkDjAk ⇒ DiB

i = 3εijkDiDjAk = 0 , (3.4.50)

due to antisymmetry of 3εijk

and symmetry of DiDj under the exchange i ↔ j.However, because Fij = DiAj −DjAi = ∂iAj − ∂jAi , the magnetic field measuredby the Eulerian observer may be written more simply as

Bi = 3εijk∂jAk =

3εijk√

det(γ)∂jAk ≡

1√det(γ)

(∇×A)i . (3.4.51)

To summarise, then, evolving the vector potential Ai instead of Bi directly auto-matically entails the divergence-free constraint (3.4.38); Bi can be then deducedfrom Ai through (3.4.51). In order to find an evolution equation for Ai , noticethat A is orthogonal to n and recall expressions (2.3.17) and (2.3.18) to get

0 = nµAµ =1

α

(At − βjAj

)⇒ At = βjAj . (3.4.52)

Using the last result,

Fti = ∂tAi − ∂iAt = ∂tAi − ∂i(ϕnt + At) = ∂tAi − ∂i(−αϕ+ βjAj

); (3.4.53)

on the other hand,

Fti = ntEi − niEt + εtijknjBk = −αEi − 1

αεtijkβ

jBk =

= −αEi + ntεtijkβjBk ≡ −αEi + 3εijkβ

jBk .(3.4.54)

87


In the ideal MHD regime, the electric field measured by the Eulerian observer isgiven by (3.3.56) and, recalling (3.2.58), one has αU j − βj = vj; therefore, com-bining (3.4.53) with (3.4.54) yields the evolution equation for the vector potentialA measured by the Eulerian observer in the ideal MHD approximation,

∂tAi = −3εijkvjBk + ∂i

(βjAj − αϕ

). (3.4.55)

Let Ψ be a scalar function of the spacetime point. A gauge transformation ofthe form

Aµ 7→ A′µ := Aµ + ∂µΨ , (3.4.56)

also stated in terms of ϕ and A as (recall (2.2.22) and use ni = 0)

ϕ = −nνAν 7→ ϕ′ := −nνA′ν = ϕ− nν∂νΨ (3.4.57)

Ai = PiνAν 7→ A′i := Pi

νA′ν = Ai + ∂iΨ , (3.4.58)

leaves the electromagnetic field (i.e , the Faraday 2-form) unchanged,

Fµν = ∂µAν − ∂νAµ 7→7→ F ′µν = ∂µA′ν − ∂νA′µ = ∂µAν − ∂νAµ +

∂µ∂νΨ−∂ν∂µΨ = Fµν .(3.4.59)

Thus, the gauge freedom in Maxwell’s theory allows one to pick A fulfilling someconstraint; two common choices are described below.

Algebraic condition and algebraic gauge

The algebraic condition amounts to

αϕ− βjAj = 0 (3.4.60)

and is meant to simplify the 4-potential evolution equation (3.4.55), which becomes

∂tAi = −3εijkvjBk . (3.4.61)

Choosing the potentials in such a way as to satisfy (3.4.60) is always possible: ifϕ and Ai do not fulfil it, then perform a gauge transformation (3.4.56) to the newpotentials ϕ′ and A′i such that

0!

= αϕ′ − βjA′j = αϕ− αnν∂νΨ− βjAj + βj∂jΨ =

=(αϕ− βjAj

)− α nt︸︷︷︸

− 1α

∂tΨ + βj∂jΨ =(αϕ− βjAj

)− ∂tΨ + βj∂jΨ .

(3.4.62)

Thus, in order for the new potentials to satisfy the algebraic condition (3.4.60),the gauge function Ψ must fulfill

− ∂tΨ + βj∂jΨ = −(αϕ− βjAj

)(3.4.63)

88


and, in case ϕ and Ai already satisfy (3.4.60), Ψ must be such that

− ∂tΨ + βj∂jΨ = 0 . (3.4.64)

All 4-potentials A satisfying (3.4.60) are said to belong to the algebraic gauge.

Lorenz condition and Lorenz gauge

The Lorenz condition is∇µAµ = 0 (3.4.65)

and is an advection equation for A. Proceeding along the same lines as above,in case A does not fulfill (3.4.65), then the gauge-transformed 4-potential A′µ =Aµ + ∂µΨ will satisfy it provided that

0!

= ∇µA′µ = ∇µA′µ +∇µ∂µΨ = ∇µAµ +∇µ∇µΨ

⇒ ∇µ∇µΨ = −∇µAµ ;(3.4.66)

if, instead, A already satisfies (3.4.65), then Ψ must be such that

∇µ∇µΨ = 0 . (3.4.67)

All 4-potentials A satisfying (3.4.65) are said to belong to the Lorenz gauge.With this choice, together with the evolution equation for the vector potential(3.4.55), the following evolution equation for the scalar potential ϕ holds,

∂t

(√det(γ)ϕ

)+ ∂i

(√det(γ)

(αAi − ϕβi

))= 0 ; (3.4.68)

this is easily deduced from the Lorenz condition (3.4.65), making use of (2.3.10),(2.1.109) and the definition of nµ (2.3.17).

Although providing the same physical fields from an analytical point of view,the algebraic and Lorenz gauge choices can produce quite different results whenthe evolution equations (3.4.55) and (3.4.68) are dealt with numerically [28]. Thisissue is discussed in detail in section 5.1, where it is shown how further numericalimprovement may be achieved using the generalised Lorenz condition

∇µAµ = ξnµAµ ; (3.4.69)

here, ξnµAµ is a damping term for the advection of A whose strenght is controlledby the parameter ξ. With this choice, (3.4.68) is modified to

∂t

(√det(γ)ϕ

)+ ∂i

(√det(γ)

(αAi − ϕβi

))= −ξα

√det(γ)ϕ . (3.4.70)

89

Chapter 4Numerical techniques and the Spritz

code

In the previous chapter, the Einstein and GRMHD equations were recast as a sys-tem of non-linear conservation laws; here, the main techniques used to solve thoseequations numerically are presented. After discussing some of the basic propertiesof non-linear hyperbolic partial differential equations (‘PDEs’) in section 4.1, finite-difference and finite-volume numerical schemes are described in section 4.2, withparticular emphasis on high-resolution-shock-capturing (‘HRSC’) methods and onthe effects of strong and weak hyperbolicity on the numerical solutions. Finally,section 4.3 outlines some of the main features and capabilities of the Spritz code,with particular attention to the time evolution of the electromagnetic variablesand to the conservative-to-primitives (‘C2P’) routines.

The main references for the discussion about numerical techniques are chapter6 of [10], [45], [69], and [56].

4.1 A brief introduction to hyperbolic PDEs

4.1.1 Notions of hyperbolicity

Partial differential equations (PDEs) are equations involving an unknownfunction of two or more variables and some of its partial derivatives and they modela great variety of physical systems. For example, Einstein’s, Maxwell’s, and thehydrodynamics equations are second-order PDE in the space and time variables,since they only involve partial derivatives of the unknown function up to secondorder. The most general form of a second-order PDE in two variables ξ and η is

A(ξ, η) ∂2ξu(ξ, η) + 2B(ξ, η) ∂ξ∂ηu(ξ, η) + C(ξ, η) ∂2

ηu(ξ, η) =

= ρ(ξ, η, u, ∂ξu, ∂ηu) ,(4.1.1)

where u is the unknown function, A, B, C are real, scalar, differentiable functionsof ξ and η which do not vanish simultaneously, and ρ is a source term. Theequation above is called:

• elliptic if AC − B2 > 0. The prototype of elliptic equation is the Poissonequation, which in terms of the two spatial coordinates ξ ≡ x and η ≡ y

90

Chapter 4 Numerical techniques and the Spritz code

reads∇2f ≡ ∂2

xf + ∂2yf = ρ . (4.1.2)

In case ρ > 0, this is called the Laplace equation;

• parabolic if AC − B2 = 0. One famous example is the heat or diffusionequation with diffusion coefficient K,

∂tf − ∂x(K∂xf) = ρ ; (4.1.3)

• hyperbolic if AC −B2 < 0. The wave equation with wave speed c

∂2t f − c2∂2

xf = ρ (4.1.4)

belongs to this class.

Elliptic PDE’s are usually cast into boundary value problems : for example,one needs to know f , or f and ∂xf , or f and ∂yf , or both ∂xf and ∂yf onthe whole boundary of the spatial (x, y) domain to single out a unique solution toPoisson’s equation (4.1.2). On the other hand, parabolic and hyperbolic PDE’s areoften presented as initial value problems, i.e. , space-time evolution equationsfor f ; for example, if a unique solution to the wave equation (4.1.4) is looked for,f(t = 0, x) and ∂tf(t = 0, x) are given as initial conditions, while f(t, x = a) andf(t, x = b) or ∂xf(t, x = a) and ∂xf(t, x = b), with x ∈ [a, b], are the boundaryconditions.

Most of the following developments concern hyperbolic PDE’s. In this regard,defining the auxiliary variables r := −∂tf and s := ∂xf , the wave equation (4.1.4)can be cast as the system

∂tu + A∂xu = S , (4.1.5)

where

u := (f, r, s)T Unknown vector (4.1.6)

A :=

0 0 00 0 c2

0 1 0

Characteristic matrix (4.1.7)

S := (−r,−ρ, 0)T Source vector . (4.1.8)

The matrix A admits the three real, distinct eigenvalues 0, c,−c and can thus bediagonalised to D = R−1AR, where D ≡ diag(c,−c, 0) and R is a matrix whosecolumns are eigenvectors of A (only if all eigenvalues are distinct can R be inverted,as in this case the associated eigenvectors are linearly independent of each other

91


and R has therefore maximum rank). Multiplying (4.1.5) by R−1 to the left anddefining w := R−1u results in the decoupled system

∂tw +D∂xw = R−1S ; (4.1.9)

each equation in (4.1.9) is a one-dimensional, linear balance equation, whosegeneral form with velocity v is

∂tu+ v∂xu = S ; (4.1.10)

in case S = 0, the above is called the advection equation,

∂tu+ v∂xu = 0 , (4.1.11)

and is a linear conservation law in one spatial dimension. Focus on this lastequation and notice that the total rate of change of u along some curve x ≡ x(t)in the (t, x(t)) plane is

du

dt= ∂tu+ ∂xu

dx

dt; (4.1.12)

if dxdt

= v, i.e. , if x(t) = x0 +vt for some constant x0∈R, then (4.1.12) and (4.1.11)give du

dt= 0. The curves x(t) = x0 + vt are called the characteristic curves

(or simply characteristics) of (4.1.11) and the last result shows that u is constantalong characteristics. This means that, if the initial datum u(0, x) ≡ u0(x) is givenat t = 0, then u0(x0) = u0(x− vt): the initial datum is transported (‘advected’)along the characteristics with constant speed v without changing in shape, hencethe name ‘advection equation’ given to (4.1.11). Indeed, it is simple matter tocheck that any function of the form u(x− vt) is a solution to (4.1.11).

To summarise, the solution of the wave equation (4.1.4) is the superposition ofthree waves, moving with velocities c, −c and 0 (the last wave is a mere result ofthe definition r := −∂tf and has no physical relevance) with respect to the positivedirection of the x-axis. In fact, this situation always arises in presence of systems oflinear conservation laws (as (4.1.5) is) whose velocity matrix has real eigenvaluesand can be diagonalised, in which case the system is called strongly hyperbolic.If the velocity matrix is real and symmetric, then the system is called symmetrichyperbolic and, because real, symmetric matrices can always be diagonalised (byan orthogonal matrix, i.e. , a rotation), the system is strongly hyperbolic as well.Finally, if the velocity matrix has real eigenvalues, but cannot be diagonalised,then the system is called weakly hyperbolic.

More general systems of N conservation laws (again limiting the discussion toone spatial dimension) may be non-linear. Given, as above, the unknown vector

92


u∈RN and the source vector S∈RN and given a flux vector F(u)∈RN , theycan be cast either in flux-conservative form,

∂tu + ∂xF(u) = S , (4.1.13)

or, noticing that ∂xFi(u) = ∂Fi∂uj∂xuj ≡ J(u)i

j∂xuj with J being the Jacobian of

the map u 7→ F(u), in quasi-linear form,

∂tu + J∂xu = S . (4.1.14)

The appearance of the quasi-linear form is identical to (4.1.5), thus the same clas-sification of the PDE system in terms of symmetric, strong, or weak hyperbolicityholds; here, however, J ≡ J(u(t, x)), so the hyperbolicity character of the system isitself point-dependent. Furthermore, if the system is strongly hyperbolic at everypoint (t, x) in the domain of interest, then J may be diagonalised at each point andthe solution of (4.1.14) is again a (non-linear) superposition of the N fundamen-tal waves, or modes, of the system, each of which moves with a point-dependentvelocity.

4.1.2 A grasp at non-linear dynamics: shocks and rarefac-tion waves

Although the notions of hyperbolicity can be carried over from a linear to a non-linear system of conservation laws almost straightforwardly, non-linear conserva-tion laws are fundamentally different from their linear counterparts. Consider theone-dimensional initial value problem

∂tu+ ∂xF (u) = ∂tu+ ∂F∂u

(u) ∂xu = 0

u(0, x) = u0(x)(4.1.15)

and call the wave propagation speed

∂F

∂u(u) =: λ(u) . (4.1.16)

To mimic what was done in the previous section, the total rate of change of ualong the characteristic curves

x ≡ x(t) = x0 + λ(u(t, x)) t (4.1.17)

isdu

dt= ∂tu+ ∂xu

dx

dt= ∂tu+ λ(u) ∂xu = 0 , (4.1.18)

93


showing that u is constant along these characteristics and that the latter arestraight lines whose slope λ(u) depends on u. Then, the solution to (4.1.15) is

u(t, x) = u0(x0) = u0(x− λ(u0(x0)) t) , (4.1.19)

so that λ ≡ λ(u0(x0)) only depends on the initial datum. Assume λ is a mono-tonically increasing function of u0; then, the flux F satisfies ∂2F

∂u2= ∂λ

∂u> 0 and is

said to be convex. Thus, points in the initial profile with higher values of u0(x0)will travel faster than points with lower values of u0(x0); for a profile like the onedepicted in the top panel of figure 4.1, the wave steepens in the compressive regionon the right side and flattens in the expansive region on the left side. Thus, at acertain time t, the wave gets folded over itself in the compressive region and herecharacteristic lines in the (t, x(t)) plane having different intercepts x0 cross, as λincreases monotonically with u0 (see the bottom panel of figure 4.1).

x

u0(x)

x1 x2

u0(x1)

u0(x2)

Direction ofpropagation

x

t

x1 x2

Figure 4.1: Steepening of the solution to (4.1.15) with convex flux (top panel).Eventually, the characteristics are doomed to cross each other (bottom panel)and a jump discontinuity must be allowed for in the solution.

94


In order to prevent u from becoming multi-valued, a jump discontinuity must beallowed for at the point where characteristics cross. However, u is certainly notdifferentiable here, so it cannot fulfil the conservation law (4.1.15); the strategy isthen to relax the continuity requirement for u. To this end, call a test functiona function φ ∈ C∞(R× [0,+∞] ,R) whose support is compact. Assume, for themoment, that the partial derivatives ∂tu and ∂xF (u) exist, so that (4.1.15) holds.Then, because φ has compact support, multiplying (4.1.15) by φ and integratingby parts yields ∫ +∞

0

dt

∫ +∞

−∞dx [u∂tφ+ F (u) ∂xφ](t, x) +

+

∫ +∞

−∞dx u0(x)φ(0, x) = 0 ,

(4.1.20)

where u(0, x) = u0(x). In fact, in the last relation, u is not required to be contin-uous anymore, but only bounded. Thus, u can have jump discontinuities and iscalled a weak solution of (4.1.15).

Suppose that, in a region Ω ⊂ (R× [0,+∞]), u is smooth on either side ofsome curve C and has a jump discontinuity across C itself. Let C divide Ω intwo regions ΩL and ΩR and let uL and uR be the limiting values of u as the latterapproaches C from ΩL and ΩR , respectively. Then, the original conservation law(4.1.15) holds in Ω\C. On the other hand, select a test function φ having compactsupport in Ω and use (4.1.20) to write

0 =

∫Ω

dtdx [u∂tφ+ F (u) ∂xφ] =

=

∫ΩL

dtdx [u∂tφ+ F (u) ∂xφ] +

∫ΩR

dtdx [u∂tφ+ F (u) ∂xφ]

⇒∫

ΩL

dtdx [u∂tφ+ F (u) ∂xφ] = −∫

ΩR

dtdx [u∂tφ+ F (u) ∂xφ] .

(4.1.21)

Now, since φ has compact support in Ω and since (4.1.15) holds in ΩL away fromC, Green’s theorem yields∫

Ω1

dtdx [u∂tφ+ F (u) ∂xφ] =

∫C

dl(uLn

2 + F (uL)n1)φ , (4.1.22)

and an analogous result holds in ΩR ; here, n ≡ (n1 , n2) is the unit vector orthog-onal to C pointing towards ΩR . Therefore, (4.1.21) implies∫

C

dl[(F (uL)− F (uL))n2 + (uL − uR)n2

]φ = 0 . (4.1.23)

95


The curve C may be regarded as the set C ≡ (x, t) |x = s(t) for some smoothmap s : [0,+∞) → R. Then, the vector tangent to C is (s, 1) and the orthogonalunit-vector pointing towards ΩR is n = (n1 , n2) = 1√

1+s2(−1, s). Thus, because

(4.1.22) must be valid for any test function φ, the Rankine-Hugoniot jumpcondition

F (uL)− F (uL) = σ(uL − uR) (4.1.24)

must hold across the discontinuity curve C in the (x, t) plane; σ := s is called thespeed of the discontinuity propagating along the curve C.

In order to investigate the possible kinds of discontinuities that may arise inthe solution of (4.1.15), consider the special case of the Riemann problem forthe inviscid Burgers’ equation,

∂tu+ ∂xF (u) = 0 , F (u) = 12u2

u(0, x) = u0(x) ≡

uL if x < 0

uR if x ≥ 0

, (4.1.25)

with uL 6= uR . Two scenarios can be distinguished:

• uL > uR Since the flux F (u) = 12u2 is convex, the initial datum is ex-

tremely compressive in nature and the above discussion shows that the char-acteristics of the solution u already cross at t = 0. The solution is foundfrom (4.1.19) by observing that, for Burgers’ equation,

λ(u) = u ⇒ λ(u0(x0)) = u0(x0) (4.1.26)

holds, so that .

u(t, x) =

uL if x < σt

uR if x ≥ σt, (4.1.27)

where the discontinuity speed σ is found from the Rankine-Hugoniot con-straint (4.1.24) to be

σ =1

2(uL + uR) . (4.1.28)

Notice that, because λ(uL) = uL and λ(uR) = uR , σ satisfies the entropycondition

λ(uL) > σ > λ(uL) , (4.1.29)

which implies that characteristics in the (x, t) plane only cross when movingforward in time. A discontinuity curve for a weak solution u of the initialvalue problem (4.1.15) in the (x, t) plane which satisfies both the Rankine-Hugoniot constraint (4.1.24) and the entropy condition (4.1.29) is called ashock wave.

96


• uL < uR Expression (4.1.27) is again a valid solution to the Riemannproblem (4.1.25) with the discontinuity having the same propagation speedσ = 1

2(uL + uR). As opposed to the previous case, however, the character-

istics in the (x, t) plane now diverge from the discontinuity curve and theentropy condition (4.1.29) is not fulfilled. This solution is called a rarefac-tion (or entropy-violating) shock and does not make any physical sense,as the shock cannot be the result of a compression phenomenon. There isin fact another possible solution to the Riemann problem (4.1.25) for theinviscid Burgers’ equation, namely the rarefaction wave

u(t, x) =

uL if x < 0xt

if tuL < x < tuR

uR if x > tuR

. (4.1.30)

In summary, weak solutions need not be unique, and the entropy criterion (4.1.29)turns out to be a very good tool to rule out unphysical solutions.

When considering Riemann problems for systems of non-linear conservationlaws (i.e. , (4.1.13) or (4.1.14) with jump-discontinuous initial data as in (4.1.25)),a wider range of non-linear effects comes into play. As an important example,contact discontinuities may arise: these are waves across the layer of which onlya fraction of the total number of modes of the system jumps discontinuously. Nofurther details about this will be given here.

4.2 Numerical approaches to hyperbolic PDEs

4.2.1 Finite-difference schemes

The simple examples outlined in the previous section gave some background aboutthe main features of linear and non-linear conservation laws and exact solutionsto those problems could be found quite easily. However, the best one can do whentackling the majority of PDE’s describing non-trivial, interesting physical scenariosis to look for approximate solutions to them; the coupled system of Einstein’s,Maxwell’s, and hydrodynamics equations described in the previous parts of thiswork is no exception to this rule. Plenty of approximating schemes have beendeveloped in order to deal with the great variety of PDE on the market; amongthose, finite-difference techniques play a major role.

Consider, for reference, the linear advection equation (4.1.11). The (x, t) do-main is sampled, or discretised, by means of a grid and the unknown function u

97


and its derivatives are only computed at points (xi , tn) belonging to that grid.

The grid spacings in the spatial and temporal directions, namely, ∆x ≡ xi+1 − xjand ∆t ≡ tn+1 − tn , may be constant, in which case the grid is uniform, or not.

Next, derivatives may be approximated in many ways. Focus, for example, onthe spatial derivative and write the Taylor expansions

u(tn, xi+1) = u(tn, xi) + ∂xu(tn, xi) + o(∆x2

)(4.2.1)

u(tn, xi−1) = u(tn, xi)− ∂xu(tn, xi) + o(∆x2

); (4.2.2)

From these, the forward and backwards Euler schemes are directly deduced,

∂xu(tn, xi) =u(tn, xi+1)− u(tn, xi)

∆x+ o(∆x) (4.2.3)

∂xu(tn, xi) =u(tn, xi)− u(tn, xi−1)

∆x+ o(∆x) ; (4.2.4)

One can do a better job by truncating (4.2.1) and (4.2.2) at o(∆x2),

u(tn, xi+1) = u(tn, xi) + ∂xu(tn, xi) +∆x2

2∂2xu(tn, xi) + o

(∆x3

)(4.2.5)

u(tn, xi−1) = u(tn, xi)− ∂xu(tn, xi) +∆x2

2∂2xu(tn, xi) + o

(∆x3

)(4.2.6)

and subtracting (4.2.6) from (4.2.5) to get a centred difference scheme,

∂xu(tn, xi) =u(tn, xi+1)− u(tn, xi−1)

2∆x+ o(∆x2

). (4.2.7)

Clearly, higher-order approximations of the derivative terms can be computedalong this line.

One could then choose to use a forward-time-centred-space (FTCS) ap-proach to solve (4.1.11),

∂tu(tn, xi) =u(tn+1, xi)− u(tn, xi)

∆t+ o(∆t) (4.2.8)

∂xu(tn, xi) =u(tn, xi+1)− u(tn, xi−1)

2∆x+ o(∆x2

), (4.2.9)

but this scheme turns out to be unstable. To see what this means, a Von Neu-mann stability analysis is in order. Since u is disctretised on a grid, write itas the Fourier series

u(t, x) =∑n,j

ξn(k) eik(j∆x) , (4.2.10)

98


where i is now the imaginary unit. Consider the time evolution of the eigenmodeof u having the maximum amplitude,

ξnmax(k) eik(j∆x) , (4.2.11)

by plugging the last expression into the advection equation (4.1.11), approximatingthe derivatives as prescribed by the FTCS scheme ((4.2.8) and (4.2.9)); after a littlealgebra, this results in

ξn+1max(k) = ξnmax(k)

[1− i v∆t

∆xsin(k∆x)

]. (4.2.12)

Because∣∣1− iv∆t

∆xsin(k∆x)

∣∣ > 1 ∀k ∈ R, then (4.2.12) implies that |ξn+1max(k)| >

|ξnmax(k)| ∀k ∈ R, which means that the eigenmode increases in amplitude withtime, causing the numerical solution to explode. For this reason, the FTCS schemecannot be used to solve the advection equation (4.1.11), although the same analysisshows that it is perfectly well-suited for solving, for example, the heat equation(3.4.45).

A simple, stable numerical approximation to the advection equation (4.1.11)

can be ‘magically’ achieved by simply substituting u(tn, xi) 7→ u(tn,xi+1)+u(tn,xi−1)2

in the forward time derivative (4.2.8) to get

u(tn+1, xi)− u(tn, xi)

∆t+ v

u(tn, xi+1)− u(tn, xi−1)

2∆x=

=∆x2

2∆t

u(tn, xi+1)− 2u(tn, xi) + u(tn, xi−1)

∆x2.

(4.2.13)

This is called the Lax-Friedrichs scheme and the same eigenmode analysis per-formed for the FTCS scheme above now gives

ξn+1max(k) = ξnmax(k)

[cos(k∆x)− i v∆t

∆xsin(k∆x)

], (4.2.14)

so that the scheme is stable provided that∣∣cos(k∆x)− i v∆t

∆xsin(k∆x)

∣∣ < 1, or

|v|∆t∆x

≤ 1 . (4.2.15)

The latter constraint is known as the Courant-Friedrichs-Lewy (CFL) stabil-ity condition and implies that increasing the spatial resolution in the numericalscheme requires the timestep ∆t to be reduced accordingly. A different and illu-minating way of interpreting (4.2.15) is that the ‘numerical’ velocity ∆x

∆tmust not

exceed the physical velocity |v| in order for the numerical scheme to be stable.

99


In fact, stability was not achieved because of some ‘magic trick’: expression(4.2.13) is a finite-difference approximation of

∂tu+ v∂xu =∆x2

2∆t∂2xu , (4.2.16)

where the term ∆x2

2∆tplays the role of an artificial viscosity. The price to pay for

the sought-after stability is then some amount of unphysical dissipation in thesolution of the PDE. Notice that, because it is common practice to fix the CFLnumber |v|∆t

∆xand the spatial resolution ∆x and set ∆t accordingly, the viscosity

coefficient in (4.2.16) behaves like ∆x2

2∆t∼∆x and, as such, ∆x2

2∆t−−−→∆x→0

0 : the un-

physical dissipation decreases with increasing spatial resolution. Finally, a viscousterm may be added intentionally to the equation for the purpose of further stabil-ising the numerical solution: such an approach is the idea behind Kreiss-Oligerdissipation.

A widely used finite-difference method for transport equations such as (4.1.11)is the upwind scheme,

u(tn+1, xi)−u(tn, xi)

∆t+ v u(tn, xi)−u(tn, xi−1)

∆xif v > 0

u(tn+1, xi)−u(tn, xi)

∆t+ v u(tn, xi+1)−u(tn, xi)

∆xif v < 0

, (4.2.17)

here presented with first-order approximation of derivative terms (but the exten-sion to higher-order is straightforward). The upwind technique only uses back-wards differences with respect to the direction from which the information is com-ing and the Von Neumann analysis shows that it is stable provided the CFL condi-tion (4.2.15) holds. On the other hand, the scheme is easily shown to be unstableif the first one of (4.2.17) is used when v < 0 or the second one of (4.2.17) is usedwhen v > 0.

More elaborated finite-difference methods are often used. For example, theleapfrog scheme

u(tn+1, xi)− u(tn−1, xi)

2∆t+ v

u(tn, xi+1)− u(tn, xi−1)

2∆x= 0 (4.2.18)

is stable provided the CFL condition (4.2.15) is fulfilled, in which case all modeshave amplitude |ξmax(k)| = 1, meaning that the numerical solution is not damped.However, this technique suffers from mesh drift instability, in that the point (tn, xi)does not communicate with its four neighbours and there are in fact two subsetsof the numerical grid evolving quite independently of one another. One common

100


solution to avoid this issue is to center the time derivative at n + 12

to get theCrank-Nicolson scheme,

u(tn+1, xi)− u(tn, xi)

∆t+

+ vu(tn+1, xi+1)− u(tn+1, xi−1) + u(tn, xi+1)− u(tn, xi−1)

4∆x= 0 .

(4.2.19)

This is stable even if the CFL condition (4.2.15) does not hold (it is unconditionallystable) and again |ξmax(k)| = 1 ∀k ∈ R, meaning that no spurious dissipation isintroduced. As a drawback, this method is implicit (i.e. , it requires informationboth at time t and t+∆t to solve the equation at time t) and it turns out that, forthis reason, a tridiagonal matrix must be inverted at each timestep, thus typicallyrequiring larger amounts of computational resources compared to explicit methods(i.e. , methods only requiring information at time t to solve the equation at timet).

Finally, the very powerful technique method of lines (MOL) consists indiscretising the spatial domain only, leaving the time variable continuous. Forexample, in the case of the linear advection equation (4.1.11), approximating thespatial derivatives to order o(∆x2) as in (4.2.7), this results in the following systemof ordinary differential equations (ODE ), one for each spatial point xi ,

du

dt(t, xi) = −v u(t, xi+1)− u(t, xi−1)

2∆x. (4.2.20)

Again, the CFL constraint (4.2.15) must be fulfilled in order for the numericalevolution to be stable. Equation (4.2.20) can be solved, for example, by means ofRunge-Kutta methods; usually, a 4th order stencil is used. Runge-Kutta techniquesstem from Taylor expanding the LHS and RHS of expressions like

y(t+ ∆t)− y(t) =

∫ t+∆t

t

ds g(s, y(s)) , (4.2.21)

which is the integral version of the Cauchy problemdydt

(t) = g(t, y(t))

y(t0) = y0 .(4.2.22)

Further details can be found in any standard textbook in numerical analysis andare not given here for the sake of brevity.

101


4.2.2 Finite-volume techniques for conservation laws

Finite-volume numerical techniques for PDE’s share some features with finite-difference methods and are in fact equivalent to them sometimes. To fix ideas,consider the one-dimensional conservation law (4.1.15) and discretise the space andtime variables on a mesh of points (tn, xi), with n∈1, . . . , N , i∈1, . . . ,M.However, think now of the (t, x) domain as being divided into a set of adjacent

cells centred at points(tn+ 1

2 , xi

), whose boundaries are tn and tn+1 in the time

direction and xi− 12

and xi+ 12

in the space direction. Integrate (4.1.15) over the

(n, i)-th elementary cell to get∫ tn+1

tndt

∫ xi+1

2

xi− 1

2

dx [∂tu(t, x) + ∂xF (u(t, x))] =

=

∫ xi+1

2

xi− 1

2

dx[u(tn+1, x

)− u(tn, x)

]+

+

∫ tn+1

tndt[F(u(t, xi+ 1

2

))− F

(u(t, xi− 1

2

))].

(4.2.23)

Defining the cell-averaged quantities

uni :=1

∆x

∫ xi+1

2

xi− 1

2

dx u(tn, xi) (4.2.24)

Fn

i± 12

:=1

∆t

∫ tn+1

tndt F

(u(t, xi± 1

2

)), (4.2.25)

dividing (4.2.23) by ∆x and re-organising terms yields

un+1i = uni −

∆t

∆x

(Fn

i+ 12− F n

i− 12

). (4.2.26)

Thus, the numerical integral of uni is conserved over the full numerical domain,except for flux terms at the boundaries: indeed,

∆xM∑i=1

un+1i = ∆x

M∑i=1

uni −∆tM∑i=1

(Fn

i+ 12− F n

i− 12

)=

= ∆xM∑i=1

uni −∆t(Fn

M+ 12− F n

12

).

(4.2.27)

This is one great advantage of writing hyperbolic PDE’s in flux-conservative form;others will be outlined in the next section.

102


Expression (4.2.26) has one problem: the flux terms Fn

i± 12

are not known until

u(t, x) is known for t∈ [tn, tn+1]. On the other hand, the last quantity can only becomputed starting from (4.2.26), which in turn contains F

n

i± 12. This vicious cycle

can be broken, for example, by approximating the flux-terms as

Fn

i± 12≈ 1

2

[F (uni ) + F

(uni±1

)]. (4.2.28)

In this case, rearranging terms, (4.2.26) is approximated by

un+1i − uni

∆t+F(uni+1

)+ F

(uni−1

)2∆x

= 0 , (4.2.29)

which is in fact an FTCS finite-difference approximation to (4.1.15). The FTCSscheme was shown to be unconditionally unstable when applied to the linear ad-vection equation (see (4.2.12)), and the same can in fact be shown to hold in thenon-linear case as well: this makes (4.2.28) not usable in practice. Other flux ap-proximations may be introduced which give rise to stable finite-difference stencilsfor the conservation law, but another approach is usually adopted.

Instead of approximating the flux terms in (4.2.26), the original problem can berestated upstream. Inside each elementary cell, let the function u(tn, x) evaluatedat time tn be substituted with its spatial average uni , which is constant over thespatial volume ∆x = xi+ 1

2−xi− 1

2. The flux terms in (4.2.25) are thus approximated

as

∆t Fn

i+ 12≡∫ tn+1

tndt F

(u(t, xi+ 1

2

))≈ ∆t F

(uni+1 , u

ni

)∆t F

n

i− 12≡∫ tn+1

tndt F

(u(t, xi− 1

2

))≈ ∆t F

(uni , u

ni−1

),

(4.2.30)

where F(uni+1 , u

ni

)and F

(uni , u

ni−1

)denote the solutions to the Riemann problem

which has been produced at the interfaces between the cells i, i + 1 and i − 1,i, respectively. This is the Godunov method and requires solving Riemannproblems at every cell interface in the spatial domain at every timestep during thenumerical evolution: the next section describes some ways of achieving this goal.

4.2.3 Approximate Riemann solvers

Riemann problems for non-linear conservation laws were outlined in section 4.1.2,where a brief touch was given about shocks and rarefaction waves, mentioningother kinds of waves which can arise when dealing with systems of non-linear

103


conservation laws (among those, contact discontinuities are quite ubiquitous). Theprecise structure of the solution to the Riemann problem for a system of non-linearconservation laws largely depends on the details of the PDEs under investigation.Generally speaking, analytic solutions to such Riemann problems are not known,although numerical solutions may be calculated to arbitrary precision in somecircumstances; one then speaks about exact Riemann solvers. For example,an exact Riemann solver for special relativistic MHD is described in [34] and thecode can be found at [30]. Unfortunately, exact Riemann solvers usually employiterative procedures which make them too slow to be used effectively in GRMHDcodes, where numerical grids typically entail a huge number of elementary cells pertimestep; furthermore, since it is just an approximation of the original problemthat is now being tackled, an exact solution may be a bit of an overkill. For thesereasons, a number of approximate Riemann solvers have been developedwhich make somewhat simplifying assumptions about the solution of the Riemannproblem at hand and nevertheless achieve good results in a much more reasonabletime with respect to exact solvers.

The Harten-Lax-Van Leer-Einfeldt (HLLE) approximate Riemann solveronly assumes the presence of two waves in the solution to the Riemann problem.Though being quite a simplicistic approximation, it works well in many cases ofinterest and it is actually the standard Riemann solver in Spritz.

Consider a control volume [xL, xR]× [0, t] in (t, x) space containing the wholewave structure of the solution of the Riemann problem for the non-linear conser-vation law (4.1.15) with initial data

u(0, x) =

uL if x < 0

uR if x ≥ 0. (4.2.31)

The situation is depicted in figure 4.2: only the fastest waves propagating withvelocities σR and σL in the left and right directions are considered and the controlvolume is thus divided in three distinct regions. Integrating (4.1.15) in the controlvolume gives

0 =

∫ T

0

dt

∫ xR

xL

dx [∂tu(t, x) + ∂xF (u(t, x))] =

=

∫ xR

xL

dx [u(T, x)− u(0, x)] +

∫ T

0

dt [F (u(t, xR))− F (u(t, xL))] =

=

∫ xR

xL

dx u(T, x)− (uRxR − uLxL) + T (F (uR)− F (uL)) ;

(4.2.32)

104


x

t

uL

uHLLE

uR

0xL xRt L t R

T

Figure 4.2: Structure of the solution to the Riemann problem in the HLLEapproximation. Three states are separated by two waves whose velocities are σLand σR .

on the other hand,∫ xR

xL

dx u(T, x) =

∫ TσL

xL

dx u(T, x) +

∫ TσR

TσL

dx u(T, x) +

∫ xR

TσR

dx u(T, x) =

= uL(Tσl − xL) +

∫ TσR

TσL

dx u(T, x) + uR(xR − TσR) . (4.2.33)

Collecting results from (4.2.32) and (4.2.33) and dividing by T (σR − σL) yields

1

T (σR − σL)

∫ TσR

TσL

dx u(T, x) =

=σRuR − σLuL − F (uR) + F (uL)

σR − σL=: uHLLE .

(4.2.34)

105


Given this last result, it is natural to propose the approximate solution

u(t, x) =

uL if x < tσL

uHLLE if tσL ≤ x ≤ tσR

uR if x > tσR

, (4.2.35)

where x = 0 represents the interface between two neighbouring cells. The corre-sponding flux FHLLE at each cell interface is easily found by substituting expression(4.2.34) for uHLLE into the Rankine-Hugoniot constraints (see (4.1.24)) at the linesof discontinuity separating the state uHLLE from uL or uR from uHLLE , namely

FHLLE − F (uL) = σL(uHLLE − uL) (4.2.36)

F (uR)− FHLLE = σR(uR − uHLLE) : (4.2.37)

the result is

FHLLE =σRF (uL)− σLF (uR) + σLσR(uR − uL)

σR − σL. (4.2.38)

In case the two waves are assumed to have equal speeds σL = −σR ≡ σ, the HLLEflux above reduces to the Lax-Friedrichs flux

FLxF :=F (uR) + F (uL)

2+σ(uR − uL)

2. (4.2.39)

Up to now, the wave velocities σL and σR were tacitely assumed to be known,but it is not in fact the case. Broadly, a sensible estimation of them dependslargely on the physical scenario described by the equations. For GRMHD, withoutgetting into details, these quantities are computed using both hydrodynamic andelectromagnetic variables reconstructed at cell interfaces (see [33] and [4]).

Plenty of other approximate Riemann solvers are on the market. Most notably,Roe’s solver finds an exact solution to the linear Riemann problem arising whenwriting the non-linear conservation law in quasi-linear form (recall (4.1.14)) andperforming a suitable approximation of the resulting Jacobian matrix (see, forexample, chapter 11 of [69]). Roe’s solver is however not implemented in Spritz

and further details about it are not given here.

4.2.4 Reconstruction techniques

The Godunov scheme described at the end of section 4.2.2 trades the unknownfunction u for its spatial average inside each elementary cell. This is the sim-plest reconstruction technique ; it is quite a rough approximations and it isinadequate for most applications, thus more sophisticated approaches are needed.

106


A first improvement consists in approximating u with a linear function withineach cell i,

u(tn, xi) ≈ unLIN(x) := uni +mni (x− xi) ∀x∈

xi− 1

2, . . . , xi− 1

2

. (4.2.40)

The spatial average of unLIN(x) over the i-th cell is easily seen to be equal to uni ,the spatial average of u over the same cell. Several choices of the slope mn

i arepossible, for example

mni =

uni+1 − uni−1

2∆xCentred (4.2.41)

mni =

uni − uni−1

∆xUpwind (4.2.42)

mni =

uni+1 − uni∆x

Downwind ; (4.2.43)

then, as with Godunov’s approach, the values of unLIN at cell interfaces are usedas piecewise constant initial data for Riemann problems at those points and so-lutions may be found using the preferred exact or approximate Riemann solver.However, all the above expressions for mn

i may lead to unphysical oscillations ofthe reconstructed function at cell boundaries, especially where large gradients of uexist. Thus, some kind of slope limiting fix needs be introduced. One commonapproach consists in requiring the total variation of unLIN, defined at time t = tn

by

TV (unLIN) :=∑i

|unLIN(xi)− unLIN(xi−1)| , (4.2.44)

not to increase with advancing time, namely

TV(un+1

LIN

) !

≤ TV (unLIN) . (4.2.45)

All techniques satisfying (4.2.45) are called total-variation-diminishing (TVD),and all the following choices of mn

i can be shown to fulfill this condition:

• mni = MINMOD

(uni −uni−1

∆x,uni+1−uni

∆x

), where

MINMOD(a, b) :=

a if |a| < |b| and ab > 0

b if |a| > |b| and ab > 0

0 if ab ≤ 0

. (4.2.46)

Thus, MINMOD takes the minimum between the upwind and downwindslopes ((4.2.42) and (4.2.43)) if both have the same sign and reverts to theoriginal Godunov scheme in case they have opposite signs;

107


• mni = MINMOD

(uni+1−uni−1

∆x, 2

uni −uni−1

∆x, 2

uni+1−uni∆x

), where now MINMOD is ex-

tended in such a way as to return the quantity whose absolute value is min-imum if all the three quantities have the same sign, while zero is returned inany other case. This is somtimes referred to as the monotonised centraldifference method or MC limiter ;

• mni = MAXMOD

(mni

(1),mni

(1)), where

MAXMOD(a, b) :=

a if |a| > |b| and ab > 0

b if |a| < |b| and ab > 0

0 if ab ≤ 0

(4.2.47)

and

mni

(1) = MINMOD

(2uni − uni−1

∆x,uni+1 − uni

∆x

)(4.2.48)

mni

(2) = MINMOD

(uni − uni−1

∆x, 2uni+1 − uni

∆x

)(4.2.49)

This is usually called the superbee method.

Better-than-linear approximations to the function u may be performed as well.Begin by noticing that the primitive U of u is proportional to uni ,

U(tn, xi) :=

∫ xi+1

2

xi− 1

2

dy u(tn, x) ≡ ∆xuni . (4.2.50)

Let P ni be a p-th order polynomial that, in some sense, approximates U in the i-th

cell at time tn; then, (4.2.50) implies

u(tn, xi) = ∂xU(tn, x) ≈ ∂xPni (tn, x) ∀x∈

(xi− 1

2, xi+ 1

2

)(4.2.51)

and so the values of ∂xPni and ∂xP

ni+1 can be used as initial data for a Riemann

problem at the intercell inerface placed at xi+ 12.

The construction of the approximating polynomial P ni is delicate. The function

u must be interpolated at p + 1 points in the i-th cell in order for the p-th orderpolynomial to be completely determined. It is known that high-order interpolatingpolynomials are prone to spurious oscillations (especially in regions where thegradient of u is large), so the p + 1 interpolating points must be chosen so as toreduce such oscillations to be minimum, much in the same way as the linear TVDmethods outlined above do. This approach leads to essentially-non-oscillatory(ENO) reconstruction schemes; many implementations have been developed, but

108


further details are not given here. A simpler (but less accurate) way to avoid wildoscillations in the approximating polynomial is to keep its order low: for example,choosing P n

i to be third order means approximating u with a parabola and thecorresponding reconstruction scheme is called the piecewise parabolic method(PPM).

Employing a higher-than-first-order reconstruction scheme together with an ap-proximate Riemann solver is common practice in most GRMHD applications (andmore). Such approaches are widely known as high-resolution-shock-capturing(HRSC) schemes.

4.2.5 Importance of flux-conservative formulation and stronghyperbolicity

There is a reason why so much attention has been devoted to finite-differenceand finite-volume approaches to hyperbolic PDE’s in flux-conservative form up tonow. For reference, consider once again the one-dimensional conservation law inthe initial value problem (4.1.15) with t, x∈ [0,+∞] × R and take a sequence ofnumerical grids

G(r)

r∈N with mesh parameters ∆x(r) , ∆t(r) → 0 as r → +∞;

denote with u(r) the numerical solution computed on the grid G(r).

Theorem (Lax-Wendroff). Under the hypotheses above, suppose that:

1. there exists a function u such that, for all bounded sets Ω ≡ [a, b] × [0, T ]∈[0,+∞]× R, ∫ T

0

dt

∫ b

a

dx∣∣u(r)(t, x)− u(t, x)

∣∣ −−−−→r→+∞

0 ; (4.2.52)

2. for all T > 0, there exists R > 0 such that the total variation of u(r), namely(see (4.2.44))

TV(u(r)(t, ·)

):=∑i

∣∣∣u(r)i (t, ·)− u(r)

i−1(t, ·)∣∣∣ , (4.2.53)

satisfiesTV(u(r)(t, ·)

)< R ∀t∈ [0, T ] (4.2.54)

for all r∈ N.

Then, u is a weak solution to the initial value problem (4.1.15).

109


See [45] for a proof. Thus, the Lax-Wendroff theorem guarantees that the nu-merical solution to an hyperbolic PDE in flux-conservative form converges to anexact weak solution of the same problem as the spatial grid resolution increases.Recall, however, that weak solutions are not in general unique, and an appropriateentropy condition may need be imposed.

The Lax-Wendroff theorem is the main reason why Einstein’s, Maxwell’s, andthe hydrodynamics equations were written as conservation laws in the previousparts of this work. To be more specific, the GRMHD set of equations (3.4.19)which the Spritz code solves and the BSSN system (3.1.41) — (3.1.45) can beshown to be strongly hyperbolic, whereas the ADM system (3.1.14) — (3.1.15)is only weakly hyperbolic [57, 61]. In fact, strong hyperbolicity is critical whenlooking for numerical solutions of systems of conservation laws like (4.1.13). A PDEsystem is well-posed if there exist constants K and α such that the solution usatisfies

‖ua(t, x)‖ ≤ eαt ‖ua(t, x)‖ ∀t > 0, ∀a∈1, . . . ,m , (4.2.55)

where ‖·‖ denotes the L2 norm,

‖f(t, x)‖2 :=

∫Rdx |f(t, x)|2 . (4.2.56)

Then, following theorem holds (see, for example, [39] for a proof):

Theorem. A systems of conservation laws (like (4.1.13)) is well-posed if and onlyif it is strongly hyperbolic.

Notice, however, that well-posedness is necessary, but not sufficient for thenumerical solution to be well-behaved: for example, if u grows exponentially withtime, the numerical scheme is very likely to run into issues and failures.

4.3 The Spritz code

Spritz solves the system of hydrodynamics conservation laws (3.4.19) adoptinga HRSC approach, while a hybrid HRSC - finite-difference technique is used toevolve electromagnetic variables, as described in section 4.3.3. The evolution of thespacetime metric relies on the Einstein Toolkit, which also provides the driverroutines to handle various kinds of EOSs and much more (see section 4.3.1).

110


4.3.1 The Einstein Toolkit environment framework

The Einstein Toolkit [26] is ‘a community-driven software platform of core com-putational tools to advance and support research in relativistic astrophysics andgravitational physics’ [38, 46, 72]. It is a code framework whose main purposeis to perform numerical relativity simulations and it is based on the Cactus in-frastructure [13, 36], which consists of a core of basic pieces of code (the ‘flesh’ )connecting to application modules (the ‘thorns’ ) — hence the name. Thorns canonly interact with other thorns via the flesh and can specify variables, runtimeparameters, and configuration information to be shared with other thorns; also,every thorn need declare scheduling information in order to interact properly withthe rest of the toolkit and in order for its routines to be executed in the desiredorder.

One special driver thorn called Carpet provides adaptive mesh refinement(‘AMR’) capabilities [11]. This involves adapting the spatial resolution in thenumerical domain dynamically, according to the current state of the simulation.Typically, a finer grid spacing is used over and near stars and black holes, leavingless interesting regions covered by coarser meshes. This makes it possible to resolvethe properties of astrophysical objects to great detail, allowing at the same time forlarge computational domains. The advantage is twofold: first, artifacts producedby domain boundaries are greatly reduced, and second, gravitational signal can beextracted in regions where the spacetime curvature is not tremendously high.

Carpet also provides MPI (i.e. , multi-process) and OpenMP (i.e. , multi-threaded)parallelisation schemes and is able to perform efficient IO and memory manage-ment operations. Together with some auxiliary thorns and tools (for example,Simfactory [65]) this enables the user to run simulations in numerical relativityover large, massively-parallel supercomputing machines without too much effort,and removes the need of implementing infrastructure routines into physics-relatedthorns.

Some thorns are grouped into arrangements if they share somewhat similar pur-poses. One basic arrangment is EinsteinBase, whose main thorns are ADMBase,HydroBase, TmunuBase, providing basic standard variables of numerical relativity.Thus, ADMBase defines the 3-metric γ, the extrinsic curvature K, and the lapse andshift with their time derivatives. HydroBase provides the fluid’s rest mass density,pressure, specific internal energy, temperature, electron fraction, specific entropyper particle, and velocity as measured by the Eulerian observer (see (3.2.46)); inaddition, it defines the Lorentz factor relative to the Eulerian and comoving ob-servers (see (3.2.45)) and the magnetic field measured by the Eulerian observer(see (3.3.28)). Finally, TmunuBase defines all the components of the stress-energytensor.

111


Initial data for simulations are provided by the EinsteinInitial arrangement.For example, the TwoPunctures thorn sets up binary black holes initial data, whileTOVSolver sets up a single TOV star (see section 5.1.4). Most importantly forthis thesis, binary neutron stars initial data are generated by LORENE [47], a set ofC++ classes solving PDEs by means of multi-domain spectral methods (in essence,LORENE expands the sought-after solution onto a basis of Cebysev polynomials).LORENE uses such techniques in order to solve for the Hamiltonian and momentumconstraint equations (3.1.10) and (3.1.11), which are recast beforehand accordingto the conformal thin-sandwich approach (see, for example, chapter 9 of [37]).

Einstein’s equations involve the evolution of both the spacetime metric andmatter fields. The first problem is dealt with by the McLachlan thorn, which em-ploys up to eighth order finite-difference schemes to solve the BSSN system (3.1.41)— (3.1.45) and applies Kreiss-Oliger dissipation to get rid of high-frequency noise.On the other hand, time evolution of matter sources is left to GRMHD codes likeGRHydro [51] and IllinoisGRMHD [27]. The Spritz code is designed to be insertedas a matter evolution thorn into the Einstein Toolkit as well, and the hope isthat the tests performed within the scope of this work can help reach that goal.

4.3.2 How Spritz works

Here is a summary of the operations performed by Spritz in order to solve theconservative GRMHD system of equations (3.4.19) and equations (3.4.55) and(3.4.70) for the electromagnetic quantities.

At the very beginning of the simulation, the fluid’s rest mass ρ and pressureP , the fluid’s velocity U i measured by the Eulerian observer (see (3.2.46)) needbe defined at every grid point; these are commonly referred to as the primitivevariables, as opposed to the conservative variables, i.e. , the ones actuallyevolved by Spritz (the ‘state vector’ variables of (3.4.19)). In addition, the mag-netic field Bi measured by the Eulerian observer (see (3.3.28)) and the scalar andvector electromagnetic potentials ϕ and Ai (see (3.4.48)) are defined everywhereas well.

First, the source terms in (3.4.19) are computed by means of a standard finite-difference approach. Then, the flux terms in (3.4.19) are evaluated via a HRSCscheme (see sections 4.2.3 and 4.2.4); typically, at every interface between nu-merical cells, PPM reconstruction is followed by an HLLE approximate Riemannsolver in the three spatial directions. The flux terms thus computed are subtractedfrom the source terms and the time evolution equations resulting from (3.4.19) aresolved with the the method of lines (see section 4.2.1) — which is implemented bythe MOL thorn within the Einstein toolkit — usually choosing a fourth order

112


Runge-Kutta ODE integrator. This makes the conservative variables step forwardin time.

Critically, the flux terms in (3.4.19) depend on the primitive variables, but onlythe conservatives ones have been updated. Therefore, calculating the fluxes at thenext time iteration needs the primitive variables be recovered from the conservativeones, that is, inversion of the ‘state vector’ relations in (3.4.19) is required at eachtimestep. In general, this conservatives-to-primitives — ‘C2P’ for short —retrieval cannot be done analytically, and a numerical inversion is in order; plentyof articles have been devised to this issue, and the main C2P routines used inSpritz are described in section 4.3.4.

The electromagnetic gauge variables are subject to a more sophisticated hybridHRSC - finite-difference treatment to be discussed in section 4.3.3; how Bi isretrieved from Ai is also described in the same section.

4.3.3 Evolution of electromagnetic variables

In Spritz, all metric-related and hydrodynamics variables are located at the centreof each elementary cell in the numerical domain. On the other hand, the magneticfield Bi stands at cells’ faces, the vector potential Ai at cells’ edges, and thedensitised scalar potential ψMHD :=

√det(γ)ϕ — the variable actually evolved by

Spritz when solving (3.4.70) — at cells’ vertices. The precise location of eachquantity is provided in table 4.1 and clarified by figure 4.3.

Variable(s) Location

Metric-related, primitive,and conservative variables

(i, j, k)

Bx(i+ 1

2, j, k

)By

(i, j + 1

2, k)

Bz(i, j, k + 1

2

)Ax

(i, j + 1

2, k + 1

2

)Ay

(i+ 1

2, j, k + 1

2

)Az

(i+ 1

2, j + 1

2, k)

ψMHD :=√

det(γ)ϕ(i+ 1

2, j + 1

2, k + 1

2

)Table 4.1: Location of the variables used in Spritz within the elementary cell

centred at (i, j, k). Figure 4.3 visualises the situation.

113


Figure 4.3: Location of the variables used in Spritz within the elementary cellcentred at (i, j, k). Figure taken by [19] and kindly provided by the authors ofthat publication.

First, notice that the flux term in the induction equation (3.4.36), namely

E i := −3εijkvjBk = −3ε

ijk(

1

α(vj + βj)

)Bk ≡ αEi + 3ε

ijkβjBk , (4.3.1)

is precisely equal to the first term in the RHS of the vector potential evolutionequation (3.4.55); in (4.3.1), Ei := −3ε

ijkUjBk is the electric field measured by

the Eulerian observer in the ideal MHD limit (see (3.3.56)) and use has been madeof (3.2.58). Therefore, it is natural to compute such a flux term using an HRSCapproach, as done in [35].

Focusing on flux terms only and accounting for staggering, the vector potentialevolution equation (3.4.55) reads (considering the z-direction to fix ideas)

(∂tAz)(i+ 1

2, j+ 1

2, k) = E(i+ 1

2, j+ 1

2, k)

z + Gauge terms ; (4.3.2)

this implies that Ez ≡ −2√

det(γ) vxBy must be computed at edge(i+ 1

2, j + 1

2, k). To this end, Spritz averages out the values of Ez computed on

114


the four cells’ faces sharing the edge(i+ 1

2, j + 1

2, k), namely

E(i+ 12, j+ 1

2, k)

z =1

4

(E(i+ 1

2, j, k)

z + E(i, j+ 12, k)

z + E(i+1, j+ 12, k)

z + E(i+ 12, j+1, k)

z

).

(4.3.3)In turn, Ez at cells’ faces is defined by vx and By at the same locations. By isalready stored correctly, while vx must be reconstructed in advance at cells’ facesstarting from its values cells’ centres; typically, PPM reconstruction (see section4.2.4) is employed. Then, the HLLE approximate Riemann solver (see section(4.2.3)) is usually applied to actually compute Ez . The calculation of Ex and Eyfollows from the above by straightforward permutation of the indices.

At this point, if the algebraic gauge (see section 3.4.3) is selected, no gaugeterms appear in (4.3.2) and the latter equation is solved by means of the methodof lines. Then, the vector potential is interpolated from the edges to the center ofeach cell by the average

A(i, j, k)z =

1

4

[A

(i+ 12, j+ 1

2, k)

z + A(i+ 1

2, j− 1

2, k)

z + A(i− 1

2, j+ 1

2, k)

z + A(i− 1

2, j− 1

2, k)

z

],

(4.3.4)with similar relations holding for Ax and Ay. Finally, as prescripted by (3.4.51),

the magnetic field at each cell’s face is obtained as Bi = (det(γ))−12 (∇×A)i; for

example,

By

(i, j+ 12, k)

=1(√

det(γ))(i, j+ 1

2, k)

[(∂zAx)

(i, j+ 12, k) − (∂xAz)

(i, j+ 12, k)], (4.3.5)

where(√det(γ)

)(i, j+ 12, k)

=1

2

[(√det(γ)

)(i, j, k)

+(√

det(γ))(i, j+1, k)

](4.3.6)

and

(∂zAx)(i, j+ 1

2, k) =

A(i, j+ 1

2, k+ 1

2)x − A(i, j+ 1

2, k− 1

2)x

∆z

(∂xAz)(i, j+ 1

2, k) =

A(i+ 1

2, j+ 1

2, k)

z − A(i− 12, j+ 1

2, k)

z

∆x.

(4.3.7)

Spritz also implements the generalised Lorenz gauge (see section 3.4.3). In casethis is selected, gauge terms appear in the time evolution equation for the vec-

115


tor potential (4.3.2), which becomes (compare with (3.4.55) and consider the z-direction to fix ideas)

(∂tAz)(i+ 1

2, j+ 1

2, k) = E(i+ 1

2, j+ 1

2, k)

z +

[∂z

(βjAj − α

ψMHD√det(γ)

)](i+ 12, j+ 1

2, k)

.

(4.3.8)In the last equation, the scalar-potential-related variable

ψMHD :=√

det(γ)ϕ (4.3.9)

is evolved by Spritz according to (3.4.70) at the cell’s vertex(i+ 1

2, j + 1

2, k + 1

2

)(see table 4.1),

(∂tψMHD)(i+12, j+ 1

2, k+ 1

2) =− ∂i(α√

det(γ)Ai − ψMHDβi)(i+ 1

2, j+ 1

2, k+ 1

2)+

− (ξαψMHD)(i+12, j+ 1

2, k+ 1

2) .

(4.3.10)

Once this is done, (4.3.8) is solved with the method of lines and the magnetic fieldat cells’ faces is computed as with the algebraic gauge.

Spritz does not tackle (4.3.10) by means of a HRSC scheme, as ψMHD isnot expected to undergo strong shocks. Instead, the following operations areperformed:

1. interpolate Ax , Ay , Az from cell’s edges to cell’s centre using (4.3.4) (forAz) or the corresponding relations along the x and y directions (for Ax andAy) and interpolate ψMHD from cell’s vertices to cell’s centre as

ψ(i, j, k)MHD =

1

8

(ψ

(i+ 12, j+ 1

2, k+ 1

2)MHD + ψ

(i− 12, j+ 1

2, k+ 1

2)MHD + ψ

(i+ 12, j− 1

2, k+ 1

2)MHD +

+ ψ(i+ 1

2, j+ 1

2, k− 1

2)MHD + ψ

(i+ 12, j− 1

2, k− 1

2)MHD + ψ

(i− 12, j+ 1

2, k− 1

2)MHD +

+ ψ(i− 1

2, j− 1

2, k+ 1

2)MHD + ψ

(i− 12, j− 1

2, k− 1

2)MHD

);

(4.3.11)

2. Compute the terms G := βjAj − α ψMHD√det(γ)

(for (4.3.8)) and

F i := α√

det(γ)Ai − ψMHDβi (for (4.3.10)) at the centre of the cell (i, j, k)

for every i∈1, 2, 3 (do not confuse the grid indices with the vector indices).Notice that Ai = γijAj , where the inverse 3-metric is stored at (i, j, k);

116


3. interpolate G back to the vertex(i+ 1

2, j + 1

2, k + 1

2

)using the values of G

stored at the centres of the cells sharing that vertex, namely

G(i+ 12, j+ 1

2, k+ 1

2) =1

8

(G(i, j, k) +G(i+1, j, k) +G(i, j+1, k) +

+ G(i, j, k+1) +G(i, j+1, k+1) +G(i+1, j, k+1) +

+ G(i+1, j+1, k) +G(i+1, j+1, k+1)).

(4.3.12)

Also, interpolate F i back to cell’s edges; for example,

F z

(i+ 12, j+ 1

2, k) =

1

4

(F z

(i, j, k) + F z(i+1, j, k) + F z

(i, j+1, k) + F z(i+1, j+1, k)

). (4.3.13)

Analogous formulae for F x

(i+ 12, j+ 1

2, k)

and F y

(i+ 12, j+ 1

2, k)

are easily deduced from

the last relation;

4. take finite-difference derivatives of G and F i at cell’s edges and vertices,respectively, as

(∂zG)(i+12, j+ 1

2, k) =

G(i+ 12, j+ 1

2, k+ 1

2) −G(i+ 12, j+ 1

2, k− 1

2)

∆z(4.3.14)

(∂zFz)(i+

12, j+ 1

2, k) =

F z

(i+ 12, j+ 1

2, k+ 1

2)− F z

(i+ 12, j+ 1

2, k− 1

2)

∆z; (4.3.15)

5. finally, interpolate the lapse function α from cells’ centres to the vertex(i+ 1

2, j + 1

2, k + 1

2

)using the same stencil as in (4.3.12) to get the damping

term − (ξαψMHD)(i+12, j+ 1

2, k+ 1

2) .

4.3.4 Conservatives-to-primitives (‘C2P’) solvers

As already mentioned in section (4.3.2), primitive variables need be retrieved fromconservative ones after each time evolution step of the latter in order to computethe flux terms needed to advance the time evolution of the hydrodynmics variablesto the next step. Therefore, the system of equations defining the ‘state vector’ in(3.4.19) must be solved in terms of the rest mass density ρ, the fluid’s coordinatevelocities vi, the specific internal energy ε, and the electron fraction Ye . While thelatter quantity is easily obtained as Ye = U5

U0(compare with (3.4.20) and (3.4.23)),

ρ, vi and ε must, in general, be solved for numerically. The most naive way to dothat is to invert the whole state vector by means, for example, of a Newton-Raphsonroutine (see, for example, section 9.6 of [56]), as was done in [32]. However, boththe computational cost and accuracy of that technique are greatly overcome both

117


by reducing the dimensionality of the original, 5-dimensional problem, and byusing more robust root-finding techniques [64]. In what follows, two of the mainC2P routines used in Spritz are described; the code is provided with a third C2Psolver (called Noble — see [19]) which has not been used in this work and is notdescribed here.

2D Newton-Raphson C2P scheme (‘3eqs’)

First, it follows from (3.2.46) and (3.2.45) that the fluid’s velocity measured bythe Eulerian observer is

U2 ≡ U iUi =W 2 − 1

W 2, (4.3.16)

where W is the Lorentz factor of the comoving observer relative to the Eulerianobserver. Second, following [35], call

D := ρW (4.3.17)

h := 1 + ε+P

ρ(Specific enthalpy) (4.3.18)

Z := ρhW 2 (4.3.19)

and contract the total (i.e. , fluid’s + electromagnetic) momentum density p(tot)i

measured by the Eulerian observer (see (3.4.17)) by Bi to get

BiUi =Bip

(tot)i

Z. (4.3.20)

With (4.3.16) and (4.3.17) in hand, one can easily recast expression (3.4.16) and(3.4.17) for the total energy and momentum density measured by the Eulerianobserver as(

p(tot))2

=(Z +B2

)2 W 2 − 1

W 2− 2Z +B2

Z2

(p

(tot)i Bi

)2

(4.3.21)

E(tot) = Z − P +

(1− 1

2W 2

)B2 −

(Bip

(tot)i

)2

2Z2. (4.3.22)

The original 5D inversion problem is thus reduced to a 2D one in the unknownsP (fluid’s pressure) and W with ρ deduced straightforwardly from W and theconservative variable (3.4.20). Notice, however, that knowledge of ε(ρ, P ) is re-quired; this is the case, for example, when using an ideal-fluid or a polytropic EOS(see (3.2.36), (3.2.37) and (3.2.41)). In particular, when the polytropic EOS isemployed, ε only depends on ρ (through (3.2.41)) and there is no need to solve(4.3.22) explicitly.

118


The quantities W and P are typically found from (4.3.21) and (4.3.22) bymeans of a Newton-Raphson iterative procedure, and then U i is recovered directlyfrom the momentum definition (3.4.17); this done at each time evolution step andfor all points in the spatial domain of the simulation. In case the Newton-Raphsonloop fails to converge at some particular point, or if superluminal velocities and/ornegative pressure or specific energies are found, that point is ‘set to atmosphere ’:ρ is set to a value ρatm which is typically 7 to 10 orders of magnitude smaller thanthe rest mass density at the centre of the star(s) at time t = 0, P and ε arecomputed with a polytropic EOS and the U i are set to zero. Notice that ρatm

cannot be zero, as the hydrodynamics equations (3.4.19) are not even defined insuch a case; this is why an artificial, low-density atmosphere is needed instead ofpure vacuum. Finally, if the recovered value of ρ lies below ρatm at a certain point,the latter is set to atmosphere as well.

The method described here is implemented in Spritz with the name ‘3eqs’and is the standard one when using ideal-fluid or polytropic EOS’s.

1D C2P scheme by Palenzuela et al.

Spritz supports tabulated EOSs (see section 3.2.3), which are typically given inthe form of 3-dimensional tables providing P as functions of ρ and of the fluid’stemperature T and electron fraction Ye (recall (3.2.78)). The 2D Newton-RaphsonC2P scheme outlined in the previous section is not well-suited to handle such asituation, as finding ε ≡ ε(ρ, T, Ye) requires a preliminary inversion of the EOStable to find T ≡ T (ρ, P, Ye) starting from P ≡ P (ρ, T, Ye), and this can result invery slow execution. One possible way out is to use the EOS-table expression ofP ≡ P (ρ, T, Ye) and solve the system (4.3.21)-(4.3.22) with respect to the inde-pendent variables T and W , as done in [64]. Still, however, the Newton-Raphsonalgorithm requires the evaluation of the Jacobian of the map sending (T,W ) to(4.3.21)-(4.3.22); the Jacobian matrix elements are themselves tabulated, but canbe affected by errors, making the root-finding algorithm inaccurate. Therefore,a non-derivative-dependent C2P scheme is more appropriate to handle tabulatedEOS.

Following [54] and [64], define the quantities

q :=E(tot) − ρW

Dr :=

(p(tot)

)2

D2s :=

B2

Dt :=

Bip(tot)

D32

(4.3.23)

and call x := hW . Then:

1. after some algebra, (4.3.21) can be recast (after subtracting ρW from it) as

1

W 2= 1− rx2 + (2x+ s) t2

x2 (x+ s)2 ; (4.3.24)

119


2. compute ρ = DW

;

3. after a little algebra, (4.3.22) can be recast as

ε = W − 1− W 2 − 1

W 2x+W

(q − s+

s

2W 2+

t2

2x2

); (4.3.25)

4. use the EOS table to find T ≡ T (ρ, ε, Ye), then compute P ≡ P (ρ, T, Ye);

5. compute the specific enthalpy h = 1 + ε+ Pρ

;

6. update the initial guess on x by solving the equation

f(x) := x− hW (4.3.26)

by means of Brent’s method, which combines quadratic interpolation nearthe root when possible, reverting to bisection if necessary (more details, forexample, in section 9.3 of [56]).

As with any one-dimensional numerical root-finding technique, the initial guessfor x must be bracketed in advance in an interval [x1 , x2]. The lower bound comesfrom (4.3.22), which implies the inequality

E(tot) < ρhW 2 +B2 ; (4.3.27)

dividing by D and using (4.3.23) and x := hW gives x > 1 + q − s. On the otherhand, suppose the strong energy condition

Tµνuµuν ≥ T

2(4.3.28)

(which arises from the requirement Rµνuµuν ≥ 0 when Einstein’s equations are

written as (3.1.7)) holds. For an ideal fluid, the stress-energy tensor is (3.2.30)and simple algebra shows that (4.3.28) is equivalent to the constraint P ≤ e. Thus,

P ≤ e ≡ ρ(1 + ε) ⇒ 2P ≤ ρ(1 + ε) + P ≡ ρh ≤ ρhW2 (4.3.29)

(since W 2 ≥ 1) and the Cauchy-Schwarz inequality gives(BiUi

)2+B2

W 2≤ B2U2 +B2

(1− U2

)= B2 , (4.3.30)

where (3.2.51) has been used. Therefore, noticing that ρhW 2 = Dx, (4.3.22) yieldsthe constraint

2(E(tot) − ρW

)= Dx+ ρhW 2 − 2P︸︷︷︸

≥0

+2B2−[(BiUi

)2+B2

W 2

]︸︷︷︸

≥B2

−ρW ≥

≥ Dx+B2 − ρW ⇒ x ≤ 2 + 2q − s .

(4.3.31)

120


To summarise, the initial bracketing of x is

x∈ [1 + q − s, 2 + 2q − s] . (4.3.32)

The procedure outlined above is iterated until x converges to some specified tol-erance and is done at each time evolution step for all points in the spatial domainof the simulation. If convergence is reached, then W is retrieved from (4.3.24),ρ ≡ D

Wis computed, and ε is recovered from (4.3.25). If ε lies within the EOS

table, then T ≡ T (ρ, ε, Ye) and P ≡ P (ρ, T, Ye) are calculated, then the U i arededuced from (4.3.21). In case ε happens to fall outside the table limits, T is setto its atmosphere value, then P ≡ P (ρ, T, Ye) and ε ≡ ε(ρ, T, Ye) are computedand Brent’s method is applied to (4.3.22) to find W ; ρ and U i are then recov-ered as above. Finally, in case, for some particular point, no root x is found forf(x) = x − hW within bounds (4.3.32), or if ρ happens to fall outside the tablelimits, or if W < 1, that point is set to atmosphere.

121

Chapter 5BNS simulations with Spritz

Spritz is designed to be included as a thorn into the Einstein Toolkit. Tothis end, the code needs to be tested carefully in all of its parts, with particularattention to the components developed most recently. These include (but arenot limited to) the evolution of the electromagnetic potentials with generalisedLorenz gauge (see section 3.4.3), which is discussed in section 5.1, and the abilityto handle tabulated EOSs correctly, which is considered in section 5.2 ; the latterpoint involves, in particular, the implementation of the conservatives-to-primitivessolver by Palenzuela et al. (see section 4.3.4).

A number of new features which are not discussed in detail in this work haverecently been introduced (or are now on the verge of being introduced) in Spritz.Most importantly, neutrinos are now included via a leakage scheme using theZelmaniLeak code (see [20] and references therein) and high-order reconstructiontechniques have been implemented; in addition, a new C2P scheme has been de-signed (see [40]) which will soon be the default one when tabulated EOSs areinvolved.

5.1 Choice of electromagnetic gauge

Choosing the electromagnetic gauge properly turns out to be a critical step whenperforming magnetised BNS evolutions. In particular, the algebraic gauge is foundto give very good results on uniform grids, as the simple tests in sections 5.1.1 and5.1.2 show. However, it is definitely not suitable for simulations on refined grids,and that’s why Spritz also implements the generalised Lorenz gauge. The lattergauge choice produces much better results in BNS simulations compared to thealgebraic gauge; however, as described in section 5.1.3, it comes with its owndrawbacks, as well.

5.1.1 Special-relativistic shocktube test with algebraic gauge

As a first, simple test of the behaviour of Spritz, I carried out a number ofshocktube trials on uniform grids. In particular, figure 5.1 shows the resultsfor the magnetised shocktube problem number 4 of the testsuite presented in [8],performed adopting the algebraic gauge (see section 3.4.3). Two streams of fluid

122

Chapter 5 BNS simulations with Spritz

described by the ideal-fluid EOS with adiabatic index γ = 53

are travelling along thex axis in Minkowski spacetime in opposite directions with velocities vL = 0.999and vR = −0.999 (this is a strongly relativistic test) and, at time t = 0, theycollide; as a result, one fast and one slow shock wave propagate along each of thetwo spatial directions — left and right.

At the last iteration, the results provided by Spritz using PPM reconstructionand the HLLE approximate Riemann solver are compared with the ‘exact’ solutionprovided by the exact Riemann solver for special relativistic MHD described in [34](see section 4.2.3). With only 200 points along the x axis (∆x = 0.005) in the range[−0.5, 0.5], results are in good agreement with the exact solution, apart from theoscillations in the rest mass density around x = 0 (panel 5.1a); these oscillationswould require a much higher resolution to be accurately resolved, but I decided tosave computational resources for more interesting simulations. Anyway, this simpletest shows that the algebraic gauge is perfectly suitable to evolve the magnetic fieldon uniform grids.

Notice that panel 5.1b is showing the total pressure, i.e. , the sum of the fluid’spressure P and the magnetic pressure

Pmag :=b2

2≡ bµbµ

2, (5.1.1)

where b is the magnetic field measured by the comoving observer, as defined by(3.3.49).

0.4 0.2 0.0 0.2 0.4x [arb. unit]

0

10

20

30

40

50

60

(x)

t = 0.00e+00 [arb. unit]Spritz solution

0.4 0.2 0.0 0.2 0.4x [arb. unit]

t = 1.60e-02 [arb. unit]Spritz solution

0.4 0.2 0.0 0.2 0.4x [arb. unit]


0.4 0.2 0.0 0.2 0.4x [arb. unit]

t = 4.00e-01 [arb. unit]Spritz solutionExact solution

(a) Rest mass density

0.4 0.2 0.0 0.2 0.4x [arb. unit]

0

200

400

600

800

1000

1200

1400

(P+

P mag

)(x)


0.4 0.2 0.0 0.2 0.4x [arb. unit]


0.4 0.2 0.0 0.2 0.4x [arb. unit]


0.4 0.2 0.0 0.2 0.4x [arb. unit]


(b) Total (fluid + magnetic) pressure

123


0.4 0.2 0.0 0.2 0.4x [arb. unit]

1.00

0.75

0.50

0.25

0.00

0.25

0.50

0.75

1.00

v x(x

)


0.4 0.2 0.0 0.2 0.4x [arb. unit]


0.4 0.2 0.0 0.2 0.4x [arb. unit]


0.4 0.2 0.0 0.2 0.4x [arb. unit]


(c) x component of the fluid’s velocity

0.4 0.2 0.0 0.2 0.4x [arb. unit]

0.30

0.25

0.20

0.15

0.10

0.05

0.00

v y(x

)


0.4 0.2 0.0 0.2 0.4x [arb. unit]


0.4 0.2 0.0 0.2 0.4x [arb. unit]


0.4 0.2 0.0 0.2 0.4x [arb. unit]


(d) y component of the fluid’s velocity

0.4 0.2 0.0 0.2 0.4x [arb. unit]

15

10

5

0

5

10

15

B y(x

)


0.4 0.2 0.0 0.2 0.4x [arb. unit]


0.4 0.2 0.0 0.2 0.4x [arb. unit]


0.4 0.2 0.0 0.2 0.4x [arb. unit]


(e) y component of the magnetic field

0.4 0.2 0.0 0.2 0.4x [arb. unit]

2.5

2.0

1.5

1.0

0.5

0.0

A y(x

)


0.4 0.2 0.0 0.2 0.4x [arb. unit]


0.4 0.2 0.0 0.2 0.4x [arb. unit]


0.4 0.2 0.0 0.2 0.4x [arb. unit]


(f) y component of the vector potential

Figure 5.1: Snapshots of the time evolution of the rest mass density (panel 5.1a),total (fluid’s + magnetic) pressure (panel 5.1b), x and y components of the fluid’svelocity (panels 5.1c and 5.1d), y component of the magnetic field (panel 5.1e), andy component of the vector potential (panel 5.1f) on the x axis for the shocktubeproblem number 4 from [8].

124


5.1.2 Special-relativistic magnetic rotor test with algebraicgauge

Other simple tests of the magnetic field evolution in Spritz were performed adopt-ing uniform grids. For example, the magnetic rotor test (see, e.g. , [19] and [8])in Minkowski spacetime consists of a disk of fluid rotating uniformly in an am-bient atmosphere in presence of a uniform magnetic field. In Spritz, the restmass density of the material inside the disk is ρin = 10 (arbitrary units) whileit is equal to ρout = 1 in the sorrounding fluid. The initial angular velocity isω = 9.95, the disk’s radius being r = 0.1, thus yielding a maximum speed of rota-tion ‖U‖ = ωr = 0.995 c. The initial magnetic field B is directed along the x axis,with Bx = 1, and the initial pressure is the same both inside and outside the disk,Pin = Pout = 1. The ideal-fluid EOS (see (3.2.36)) is adopted with adiabatic indexγ = 5

3and the numerical domain is the square [0, 1]× [0, 1] in the (x, y) plane.

As time goes by, the rotor launches torsional Alfven waves into the ambientmedium, as the snapshots of the evolution of the rest mass density 5.2a and thenorm of the magnetic field 5.2d show. Therefore, the angular momentum of therotor is reduced, as can be seen from the evolution of the Lorentz factor in 5.2b.The behaviour of the rotor is the one expected from the literature [8, 19] and,in particular, this indicates that the algebraic gauge works perfectly in this case.Once again then, the algebraic gauge gives excellent results on uniform grids, andcan thus be regarded as the method of choice in such scenarios.

0.0 0.2 0.4 0.6 0.8 1.0x[arb.units]

0.0

0.2

0.4

0.6

0.8

1.0

y[ar

b.un

its]

t = 0.00e+00 [arb.units]

0.0 0.2 0.4 0.6 0.8 1.0x[arb.units]

0.0

0.2

0.4

0.6

0.8

1.0

y[ar

b.un

its]

t = 3.20e-01 [arb.units]

0.0 0.2 0.4 0.6 0.8 1.0x[arb.units]

0.0

0.2

0.4

0.6

0.8

1.0

y[ar

b.un

its]


100

101

Min: 4.34e-01Max: 1.11e+01


0.0 0.2 0.4 0.6 0.8 1.0x[arb.units]

0.0

0.2

0.4

0.6

0.8

1.0

y[ar

b.un

its]


0.0 0.2 0.4 0.6 0.8 1.0x[arb.units]

0.0

0.2

0.4

0.6

0.8

1.0

y[ar

b.un

its]


0.0 0.2 0.4 0.6 0.8 1.0x[arb.units]

0.0

0.2

0.4

0.6

0.8

1.0

y[ar

b.un

its]


100

101

W

Min: 1.00e+00Max: 1.00e+01

(b) Lorentz factor

125


0.0 0.2 0.4 0.6 0.8 1.0x[arb.units]

0.0

0.2

0.4

0.6

0.8

1.0

y[ar

b.un

its]


0.0 0.2 0.4 0.6 0.8 1.0x[arb.units]

0.0

0.2

0.4

0.6

0.8

1.0

y[ar

b.un

its]


0.0 0.2 0.4 0.6 0.8 1.0x[arb.units]

0.0

0.2

0.4

0.6

0.8

1.0

y[ar

b.un

its]


0.2

0.4

0.6

0.8

A

Min: 1.20e-02Max: 9.88e-01

(c) Norm of the vector potential

0.0 0.2 0.4 0.6 0.8 1.0x[arb.units]

0.0

0.2

0.4

0.6

0.8

1.0

y[ar

b.un

its]


0.0 0.2 0.4 0.6 0.8 1.0x[arb.units]

0.0

0.2

0.4

0.6

0.8

1.0

y[ar

b.un

its]


0.0 0.2 0.4 0.6 0.8 1.0x[arb.units]

0.0

0.2

0.4

0.6

0.8

1.0

y[ar

b.un

its]


10 1

100

B

Min: 4.07e-02Max: 3.99e+00

(d) Norm of the magnetic field

Figure 5.2: Magnetic rotor test with algebraic gauge on uniform grid. Therotor launches torsional Alfven waves into the ambient medium, as can be clearlyseen from the panels for the rest mass density 5.2a and the norm of the magneticfield 5.2d. As a result, the rotor loses angular momentum, as the evolution of theLorentz factor shows in panel 5.2b. Finally, the vector potential does not developany strange artifacts (panel 5.2c), showing that the algebraic gauge worksperfectly fine on uniform grids.

126


5.1.3 Magnetised, equal-mass BNS with polytropic EOS

Algebraic gauge

In the previous two sections, the algebraic gauge was shown to yield an accurateevolution of the magnetic field on uniform grids; however, on refined grids —for example, when evolving BNS systems — it is not appropriate at all. Indeed,following [28], the time evolution equation for the vector potential with algebraicgauge (3.4.61) may be rewritten using Bi = εijk∂jAk (equation (3.4.51)) as

∂tAi + vm∂mAi − vm∂iAm = 0 . (5.1.2)

Applying a spatial Fourier transform to the last relation gives

∂tAi + i(vmkmAi − vmkiAm

)≡ ∂tAi + iFi = 0 , (5.1.3)

where vi are the fluid’s coordinate velocities (3.2.53) and

Ai(t,k) :=

∫R3

d3x

(2π)32

Ai(t,x) e−ikmxm

. (5.1.4)

The characteristic matrix J of the system of PDEs (5.1.3) has elements

Jij :=

∂Fi∂Aj

= vmkmδji − kivj ; (5.1.5)

the eigenvalues λ of J are the velocities at which the fundamental gauge modes Aipropagate in spacetime and are given by

0!

= det(J − λI) = εijk(J1

i − λδi1) (J2

j − λδj2) (J3

k − λδk3)

=

− λ(λ− vmkm)2 ⇒ λ1 = 0, λ2,3 = vmkm .(5.1.6)

In particular, there is one static gauge mode, whose velocity is λ = 0. While notan issue on uniform grids, on refined grids interpolation errors occur where refine-ment boundaries come across regions containing the static gauge mode; where thishappens, non-physical magnetic fields are produced. This behaviour is confirmedby figure 5.3, which depicts the time evolution, provided by Spritz, of a magne-tised, equal-mass BNS system with polytropic EOS (polytropic index Γ = 2 —see (3.2.37) (3.2.41)) adopting the algebraic gauge. The trails in the vector poten-tial left behind by the two stars are the effect of the static gauge mode and thestrenght of the corresponding unphysical magnetic field at refinement boundariesaround the stars is only two orders of magnitude lower than the magnetic field

127


at the centre of the stars themselves, which is not quite acceptable for scientificpurposes.

40 30 20 10 0 10 20 30 40x[km]

40

30

20

10

0

10

20

30

40

y[km

]

t = 0.00e+00 ms

40 30 20 10 0 10 20 30 40x[km]

40

30

20

10

0

10

20

30

40

y[km

]

t = 6.62e-01 ms

40 30 20 10 0 10 20 30 40x[km]

40

30

20

10

0

10

20

30

40

y[km

]

t = 1.22e+00 ms

010 17

10 16

10 15

10 14

10 13

10 12

10 11

10 10

10 9

10 8

10 7

A

Min: 0.00e+00Max: 1.09e-07

(a) Norm of the vector potential

40 30 20 10 0 10 20 30 40x[km]

40

30

20

10

0

10

20

30

40

y[km

]

t = 0.00e+00 ms

40 30 20 10 0 10 20 30 40x[km]

40

30

20

10

0

10

20

30

40

y[km

]

t = 6.62e-01 ms

40 30 20 10 0 10 20 30 40x[km]

40

30

20

10

0

10

20

30

40

y[km

]

t = 1.22e+00 ms

010 18

10 17

10 16

10 15

10 14

10 13

10 12

10 11

10 10

10 9

10 8

B

Min: 0.00e+00Max: 2.38e-08

(b) Norm of the magnetic field

Figure 5.3: Magnetised, equal-mass BNS evolution with polytropic EOS(Γ = 2) evolved by Spritz with algebraic gauge. The trails behind the stars inthe vector potential (panel 5.3a), generated by the static gauge mode, causeinterpolation errors at refinement boundaries around the two stars which giverise to strong unphysical magnetic fields (panel 5.3b).

A few more details about this simulation are in order. First, the initial dataare generated by LORENE, which solves the Hamiltonian and momentum constraintequations (3.1.10) and (3.1.11) adopting multi-domain spectral methods for ellipticPDEs (see [47] and, for example, chapter 9 of [37] for the details). The two starshave a baryonic mass (to distinguish it from the ADM mass) of about 1.625Meach, their diameter along the x axis (they are a bit oblate due to the gravitationalpull onto each other) is about 27.4 km and the initial separation between theircentres of mass is around 45 km.

The spatial domain of the simulation is much bigger than the size of theBNS system under study, so as to minimise artifacts due to the interaction of

128


the stars with the domain boundaries and possibly extract the gravitational wavesignal in a region well far away from the sources of the perturbation (this im-portant topic is however not dicussed in this thesis). Thus, the full domain isa cube extending in the range [−950.4M, 950.4M], or [−1403.4 km, 1403.4 km](1M ' 1.47667 km), in the three spatial directions; however, only the centralregion of the (x, y) plane of it at z = 0 is visualised in figure 5.3.

There are six refinement levels on the grid, the two finest (and smallest) onescovering each star and following the motion of the latter during the inspiral phaseby tracking the position of their centre of mass in time (see figure 5.4). The coarserresolution is ∆x = ∆y = ∆z = 4.8M ' 7.1 km, accounting for a resolution of(

4.826−1

)M = 0.15M ' 221.5m in the finest refinement level (every grid refine-

ment improves resolution by a factor of 2). Thus, the diameter of the stars alongthe x axis at the beginning of the simulation is sampled by approximately 123points.

Finally, the CFL number (see (4.2.15)) is C = 0.35, which is a good tradeoffbetween numerical stability (C → 0) and computational efficiency (C → 1).

Figure 5.4: Central region of the AMR grid for the simulation in figure 5.3.The two finest refinement levels cover the two stars and follow them during theinspiral phase. Snapshots produced with VisIt [70].

Generalised Lorenz gauge

Given the problems outlined above, Spritz implements the generalised Lorenzgauge condition (see 3.4.3) as well. This is a hyperbolic equation for the 4-potentialA with a damping term whose strenght is adjustable by the user through a suitableparameter, usually referred to as the ‘damping factor’. In this case, the principalparts of the evolution equations for the scalar and vector potentials (3.4.70) and(3.4.55) (i.e. , the parts containing only the derivatives of the evolved fields ϕ and

129


Ai) are

∂tϕ+ α∂mAm − βm∂mϕ ' 0 (5.1.7)

∂tAi + vm∂mAi − vm∂iAm + α∂iϕ− βm∂iAm ' 0 , (5.1.8)

where ‘'’ means ‘the principal part equals’. Applying a spatial Fourier transformcorresponds to the transformation ∂i 7→ iki , so

∂tϕ+ i(αkmA

m − βmkmϕ)≡ ∂tϕ+ iF0 ' 0 (5.1.9)

∂tAi + i(vmkmAi − (vm + βm) kiAm + αkiϕ

)≡ ∂tAi + iFi . (5.1.10)

Denoting collectively ψ0 := ϕ and ψi := Ai as ψµ with µ∈0, 1, 2, 3, the charac-teristic matrix J of the above system has elements

J0ν :=

∂F0

∂ψν= αkν − βmkmδν0 (5.1.11)

Jiν :=

∂Fi

∂ψj= vmkmδ

νi − (vν + βν) ki + αkiδ

ν0 . (5.1.12)

As with the algebraic gauge, the eigenvalues λ of J are found (after some tediousalgebra) as

0!

= det(J − λI) = εµνρσ(J0µ − λδµ0 ) (J1

ν − λδν1 ) (J2ρ − λδρ2) (J3

σ − λδσ3 ) =

(λ− vmkm)2(λ+ βmkm + α2kmkm)(λ+ βmkm − α2kmkm

)⇒ λ1,2 = vmkm , λ3,4 = βmkm ± α2kmkm .

(5.1.13)

This time, all eigenvalues are (in general) non-zero, so that all gauge modes movewith non-zero speed and propagate towards the boundaries of the computationaldomain. In this way, no trail in the vector potential is expected to follow the starsand, correspondingly, unphysical magnetic fields are not expected to arise in thesame way as seen with the algebraic gauge.

Figure 5.5 — depicting the same BNS system as figure 5.3, now evolved adopt-ing the generalised Lorenz gauge — shows that this is indeed the case. However,unphysical magnetic fields can still be observed at refinement boundaries from thevery beginning of the simulation, probably due to interpolation errors caused bythe propagation of the vector potential. Furthermore, and most seriously, unphys-ical magnetic fields whose strenght is comparable to that of the magnetic fieldsat the stars’ surfaces are generated around the centre of the computational do-main because of unexpectedly large values of the norm of the vector potential andψMHD :=

√det(γ)ϕ in that region. This bad behaviour can possibly be due to

some numerical instability; indeed, adding some amount of Kreiss-Oliger dissipa-tion (see section 4.2.1) to Ai and ψMHD makes the evolution much cleaner, as thesnapshots in figure 5.6 show (compare with 5.5).

130


100 75 50 25 0 25 50 75 100x[km]

100

75

50

25

0

25

50

75

100y[

km]

t = 0.00e+00 ms

100 75 50 25 0 25 50 75 100x[km]

100

75

50

25

0

25

50

75

100

y[km

]

t = 3.31e-01 ms

100 75 50 25 0 25 50 75 100x[km]

100

75

50

25

0

25

50

75

100

y[km

]

t = 1.09e+00 ms

010 18

10 17

10 16

10 15

10 14

10 13

10 12

10 11

10 10

10 9

10 8

A

Min: 0.00e+00Max: 2.29e-08

100 75 50 25 0 25 50 75 100x[km]

100

75

50

25

0

25

50

75

100

y[km

]

t = 1.66e+00 ms

100 75 50 25 0 25 50 75 100x[km]

100

75

50

25

0

25

50

75

100y[

km]

t = 2.15e+00 ms

100 75 50 25 0 25 50 75 100x[km]

100

75

50

25

0

25

50

75

100

y[km

]

t = 2.68e+00 ms

010 18

10 17

10 16

10 15

10 14

10 13

10 12

10 11

10 10

10 9

10 8

A

Min: 0.00e+00Max: 2.28e-08


100 75 50 25 0 25 50 75 100x[km]

100

75

50

25

0

25

50

75

100

y[km

]

t = 0.00e+00 ms

100 75 50 25 0 25 50 75 100x[km]

100

75

50

25

0

25

50

75

100

y[km

]

t = 3.31e-01 ms

100 75 50 25 0 25 50 75 100x[km]

100

75

50

25

0

25

50

75

100

y[km

]

t = 1.09e+00 ms

010 19

10 18

10 17

10 16

10 15

10 14

10 13

10 12

10 11

10 10

10 9

| MHD|

Min: 0.00e+00Max: 6.44e-09

100 75 50 25 0 25 50 75 100x[km]

100

75

50

25

0

25

50

75

100

y[km

]

t = 1.66e+00 ms

100 75 50 25 0 25 50 75 100x[km]

100

75

50

25

0

25

50

75

100

y[km

]

t = 2.15e+00 ms

100 75 50 25 0 25 50 75 100x[km]

100

75

50

25

0

25

50

75

100

y[km

]

t = 2.68e+00 ms

010 19

10 18

10 17

10 16

10 15

10 14

10 13

10 12

10 11

10 10

10 9

| MHD|

Min: 7.98e-210Max: 1.29e-07

(b) Absolute value of ψMHD :=√

det(γ)ϕ

131


100 75 50 25 0 25 50 75 100x[km]

100

75

50

25

0

25

50

75

100y[

km]

t = 0.00e+00 ms

100 75 50 25 0 25 50 75 100x[km]

100

75

50

25

0

25

50

75

100

y[km

]

t = 3.31e-01 ms

100 75 50 25 0 25 50 75 100x[km]

100

75

50

25

0

25

50

75

100

y[km

]

t = 1.09e+00 ms

010 17

10 16

10 15

10 14

10 13

10 12

10 11

10 10

10 9

10 8

10 7

B

Min: 0.00e+00Max: 2.38e-08

100 75 50 25 0 25 50 75 100x[km]

100

75

50

25

0

25

50

75

100

y[km

]

t = 1.66e+00 ms

100 75 50 25 0 25 50 75 100x[km]

100

75

50

25

0

25

50

75

100y[

km]

t = 2.15e+00 ms

100 75 50 25 0 25 50 75 100x[km]

100

75

50

25

0

25

50

75

100

y[km

]

t = 2.68e+00 ms

010 17

10 16

10 15

10 14

10 13

10 12

10 11

10 10

10 9

10 8

10 7

B

Min: 0.00e+00Max: 1.12e-07

(c) Norm of the magnetic field

Figure 5.5: Same BNS system as figure 5.3, but evolved with generalisedLorenz gauge and damping factor ξ = 0.89 = 1.5

∆t[Coarsest ref. level][27]. Unphysical

magnetic fields (panel 5.5c) are generated both at refinement boundaries — dueto interpolation errors in Ai — and at the centre of the domain — due to anumerical instability plaguing both Ai and ψMHD :=

√det(γ)ϕ (panels 5.5a and

5.5b).

132


100 75 50 25 0 25 50 75 100x[km]

100

75

50

25

0

25

50

75

100y[

km]

t = 0.00e+00 ms

100 75 50 25 0 25 50 75 100x[km]

100

75

50

25

0

25

50

75

100

y[km

]

t = 3.31e-01 ms

100 75 50 25 0 25 50 75 100x[km]

100

75

50

25

0

25

50

75

100

y[km

]

t = 1.09e+00 ms

010 18

10 17

10 16

10 15

10 14

10 13

10 12

10 11

10 10

10 9

10 8

A

Min: 0.00e+00Max: 2.30e-08

100 75 50 25 0 25 50 75 100x[km]

100

75

50

25

0

25

50

75

100

y[km

]

t = 1.66e+00 ms

100 75 50 25 0 25 50 75 100x[km]

100

75

50

25

0

25

50

75

100y[

km]

t = 2.15e+00 ms

100 75 50 25 0 25 50 75 100x[km]

100

75

50

25

0

25

50

75

100

y[km

]

t = 2.68e+00 ms

010 18

10 17

10 16

10 15

10 14

10 13

10 12

10 11

10 10

10 9

10 8

A

Min: 0.00e+00Max: 2.28e-08


100 75 50 25 0 25 50 75 100x[km]

100

75

50

25

0

25

50

75

100

y[km

]

t = 0.00e+00 ms

100 75 50 25 0 25 50 75 100x[km]

100

75

50

25

0

25

50

75

100

y[km

]

t = 3.31e-01 ms

100 75 50 25 0 25 50 75 100x[km]

100

75

50

25

0

25

50

75

100

y[km

]

t = 1.09e+00 ms

010 19

10 18

10 17

10 16

10 15

10 14

10 13

10 12

10 11

10 10

10 9

| MHD|

Min: 0.00e+00Max: 6.70e-09

100 75 50 25 0 25 50 75 100x[km]

100

75

50

25

0

25

50

75

100

y[km

]

t = 1.66e+00 ms

100 75 50 25 0 25 50 75 100x[km]

100

75

50

25

0

25

50

75

100

y[km

]

t = 2.15e+00 ms

100 75 50 25 0 25 50 75 100x[km]

100

75

50

25

0

25

50

75

100

y[km

]

t = 2.68e+00 ms

010 19

10 18

10 17

10 16

10 15

10 14

10 13

10 12

10 11

10 10

10 9

| MHD|

Min: 3.92e-208Max: 7.02e-09

(b) Absolute value of ψMHD :=√

det(γ)ϕ

133


100 75 50 25 0 25 50 75 100x[km]

100

75

50

25

0

25

50

75

100y[

km]

t = 0.00e+00 ms

100 75 50 25 0 25 50 75 100x[km]

100

75

50

25

0

25

50

75

100

y[km

]

t = 3.31e-01 ms

100 75 50 25 0 25 50 75 100x[km]

100

75

50

25

0

25

50

75

100

y[km

]

t = 1.09e+00 ms

010 17

10 16

10 15

10 14

10 13

10 12

10 11

10 10

10 9

10 8

10 7

B

Min: 0.00e+00Max: 2.38e-08

100 75 50 25 0 25 50 75 100x[km]

100

75

50

25

0

25

50

75

100

y[km

]

t = 1.66e+00 ms

100 75 50 25 0 25 50 75 100x[km]

100

75

50

25

0

25

50

75

100y[

km]

t = 2.15e+00 ms

100 75 50 25 0 25 50 75 100x[km]

100

75

50

25

0

25

50

75

100

y[km

]

t = 2.68e+00 ms

010 17

10 16

10 15

10 14

10 13

10 12

10 11

10 10

10 9

10 8

10 7

B

Min: 0.00e+00Max: 2.34e-08

(c) Norm of the magnetic field

Figure 5.6: Same BNS system as figure 5.5 with only some amount ofKreiss-Oliger dissipation applied to Ai and ψMHD . The numerical instabilityaffecting Ai and ψMHD has been fixed (compare panel 5.6a with panel 5.5a andpanel 5.6b with 5.5b), thus removing the unphysical magnetic fields around thecentre of the domain (compare 5.6c with 5.5c).

134


Comparison with IllinoisGRMHD

The behaviour of Spritz was compared with that of IllinoisGRMHD [27], aGRMHD code which is part of the Einstein Toolkit and adopts the same stag-gered vector potential approach as Spritz does when solving Maxwell’s equa-tions, albeit with some differences in implementation details. Figure 5.7 showsthat IllinoisGRMHD does not suffer from the same, severe numerical instabilityas Spritz does, although strange ‘blob-like’ artifacts arise in the z component ofthe vector potential and there are strong unphysical magnetic fields just outsidethe two stars.

Less intense unphysical magnetic fields at refinement boundaries seem to ‘move’towards the centre of the computational domain, while they remain fixed at theirpositions when using Spritz (compare 5.7d to 5.5c). This is probably becauseSpritz sets the fluid’s velocity U i measured by the Eulerian observer to zero inthe atmosphere, while IllinoisGRMHD lets the atmospheric fluid move.

This behaviour can be better appreciated by looking at figure 5.8, which de-picts the time evolution of the rest mass density for Spritz with Kreiss-Oligerdissipation on the vector potential and ψMHD and IllinoisGRMHD without dissi-pation: IllinoisGRMHD tends to accrete matter around the two stars much morethan Spritz does.

Finally, figures 5.9 and 5.10 compare the z component of the vector poten-tial (IllinoisGRMHD does not output the norm of the latter quantity) and thedensitised scalar potential ψMHD for Spritz with dissipation and IllinoisGRMHD

without dissipation. A few observations are in order:

1. while Az is essentially zero (to machine precision) for IllinoisGRMHD in thefirst instants of evolution (leftmost panel in 5.9b), it is not so for Spritz.

2. based on the symmetry of the BNS system, the profiles of Az and ψMHD

should be expected to be symmetric around the origin of the (x, y) plane;while this is indeed true for IllinoisGRMHD, it is not for Spritz (comparethe leftmost panels of 5.9a and 5.9b). This may indicate that some bug isstill present in Spritz in the implementation of the generalised Lorenz gaugecondition — or perhaps somewhere else in the code;

3. as already observed above, IllinoisGRMHD develops ‘bubble-like’ artifactsin Az at some point during the evolution, and this probably a symptom ofa numerical instability (probably different in nature from the one observedwith Spritz in figure 5.5). Furthermore, small oscillations can be observedin the evolution of ψMHD as given by IllinoisGRMHD.

135


100 75 50 25 0 25 50 75 100x[km]

100

75

50

25

0

25

50

75

100y[

km]

t = 0.00e+00 ms

100 75 50 25 0 25 50 75 100x[km]

100

75

50

25

0

25

50

75

100

y[km

]

t = 3.31e-01 ms

100 75 50 25 0 25 50 75 100x[km]

100

75

50

25

0

25

50

75

100

y[km

]

t = 1.09e+00 ms

010 18

10 17

10 16

10 15

10 14

10 13

10 12

10 11

10 10

10 9

10 8

|Ax|

Min: 0.00e+00Max: 3.22e-08

100 75 50 25 0 25 50 75 100x[km]

100

75

50

25

0

25

50

75

100

y[km

]

t = 1.66e+00 ms

100 75 50 25 0 25 50 75 100x[km]

100

75

50

25

0

25

50

75

100y[

km]

t = 2.15e+00 ms

100 75 50 25 0 25 50 75 100x[km]

100

75

50

25

0

25

50

75

100

y[km

]

t = 2.38e+00 ms

010 18

10 17

10 16

10 15

10 14

10 13

10 12

10 11

10 10

10 9

10 8

|Ax|

Min: 0.00e+00Max: 3.33e-08

(a) Absolute value of the x component of the vector potential

100 75 50 25 0 25 50 75 100x[km]

100

75

50

25

0

25

50

75

100

y[km

]

t = 0.00e+00 ms

100 75 50 25 0 25 50 75 100x[km]

100

75

50

25

0

25

50

75

100

y[km

]

t = 2.98e-01 ms

100 75 50 25 0 25 50 75 100x[km]

100

75

50

25

0

25

50

75

100

y[km

]

t = 7.94e-01 ms

010 20

10 19

10 18

10 17

10 16

10 15

10 14

10 13

10 12

10 11

10 10

|Az|

Min: 0.00e+00Max: 5.82e-10

100 75 50 25 0 25 50 75 100x[km]

100

75

50

25

0

25

50

75

100

y[km

]

t = 1.32e+00 ms

100 75 50 25 0 25 50 75 100x[km]

100

75

50

25

0

25

50

75

100

y[km

]

t = 1.69e+00 ms

100 75 50 25 0 25 50 75 100x[km]

100

75

50

25

0

25

50

75

100

y[km

]

t = 1.92e+00 ms

010 20

10 19

10 18

10 17

10 16

10 15

10 14

10 13

10 12

10 11

10 10

|Az|

Min: 0.00e+00Max: 2.38e-10

(b) Absolute value of the z component of the vector potential

136


100 75 50 25 0 25 50 75 100x[km]

100

75

50

25

0

25

50

75

100

y[km

]

t = 0.00e+00 ms

100 75 50 25 0 25 50 75 100x[km]

100

75

50

25

0

25

50

75

100

y[km

]

t = 3.31e-01 ms

100 75 50 25 0 25 50 75 100x[km]

100

75

50

25

0

25

50

75

100

y[km

]

t = 1.09e+00 ms

010 1910 18

10 17

10 16

10 15

10 14

10 13

10 12

10 11

10 10

10 9

| MHD|

Min: 0.00e+00Max: 6.61e-09

100 75 50 25 0 25 50 75 100x[km]

100

75

50

25

0

25

50

75

100

y[km

]

t = 1.66e+00 ms

100 75 50 25 0 25 50 75 100x[km]

100

75

50

25

0

25

50

75

100

y[km

]

t = 2.15e+00 ms

100 75 50 25 0 25 50 75 100x[km]

100

75

50

25

0

25

50

75

100

y[km

]

t = 2.38e+00 ms

010 1910 18

10 17

10 16

10 15

10 14

10 13

10 12

10 11

10 10

10 9

| MHD|

Min: 0.00e+00Max: 7.05e-09

(c) Absolute value of ψMHD :=√

det(γ)ϕ

100 75 50 25 0 25 50 75 100x[km]

100

75

50

25

0

25

50

75

100

y[km

]

t = 0.00e+00 ms

100 75 50 25 0 25 50 75 100x[km]

100

75

50

25

0

25

50

75

100

y[km

]

t = 3.31e-01 ms

100 75 50 25 0 25 50 75 100x[km]

100

75

50

25

0

25

50

75

100

y[km

]

t = 1.09e+00 ms

010 18

10 17

10 16

10 15

10 14

10 13

10 12

10 11

10 10

10 9

10 8

|Bz|

Min: 0.00e+00Max: 1.61e-08

100 75 50 25 0 25 50 75 100x[km]

100

75

50

25

0

25

50

75

100

y[km

]

t = 1.66e+00 ms

100 75 50 25 0 25 50 75 100x[km]

100

75

50

25

0

25

50

75

100

y[km

]

t = 2.15e+00 ms

100 75 50 25 0 25 50 75 100x[km]

100

75

50

25

0

25

50

75

100

y[km

]

t = 2.38e+00 ms

010 18

10 17

10 16

10 15

10 14

10 13

10 12

10 11

10 10

10 9

10 8

|Bz|

Min: 0.00e+00Max: 1.56e-08

(d) Absolute value of the z component of the magnetic field

Figure 5.7: Same BNS system as figure 5.5, now evolved by IllinoisGRMHD.The latter code does not seem to suffer from the numerical instability seen withSpritz in 5.5, although some strange ‘blob-like’ artifacts arise in Az (panel 5.7b)and there are unphysical magnetic fields around the two stars (5.7d). Moreover,as with Spritz, the effect of refinement boundaries is clearly visible in Bz .

137


40 20 0 20 40x[km]

40

20

0

20

40

y[km

]

t = 3.31e-02 ms

40 20 0 20 40x[km]

40

20

0

20

40

y[km

]

t = 7.28e-01 ms

40 20 0 20 40x[km]

40

20

0

20

40

y[km

]

t = 1.56e+00 ms

10 13

10 11

10 9

10 7

10 5

10 3

Min: 1.00e-13Max: 9.57e-04

(a) Spritz with Kreiss-Oliger dissipation on A and ψMHD

40 20 0 20 40x[km]

40

20

0

20

40

y[km

]

t = 3.31e-02 ms

40 20 0 20 40x[km]

40

20

0

20

40

y[km

]

t = 7.28e-01 ms

40 20 0 20 40x[km]

40

20

0

20

40

y[km

]

t = 1.56e+00 ms

10 13

10 11

10 9

10 7

10 5

10 3

Min: 1.00e-13Max: 9.57e-04

(b) IllinoisGRMHD without Kreiss-Oliger dissipation on A and ψMHD

Figure 5.8: Evolution of the rest mass density ρ in Spritz with dissipation(panel 5.8a) and in IllinoisGRMHD without dissipation (panel 5.8b).IllinoisGRMHD accretes matter towards the stars much more than Spritz.

40 20 0 20 40x[km]

40

20

0

20

40

y[km

]

t = 3.31e-02 ms

40 20 0 20 40x[km]

40

20

0

20

40

y[km

]

t = 7.28e-01 ms

40 20 0 20 40x[km]

40

20

0

20

40

y[km

]

t = 1.56e+00 ms

010 20

10 19

10 18

10 17

10 16

10 15

10 14

10 13

10 12

10 11

10 10

|Az|

Min: 0.00e+00Max: 9.92e-11


40 20 0 20 40x[km]

40

20

0

20

40

y[km

]

t = 3.31e-02 ms

40 20 0 20 40x[km]

40

20

0

20

40

y[km

]

t = 7.28e-01 ms

40 20 0 20 40x[km]

40

20

0

20

40

y[km

]

t = 1.56e+00 ms

010 20

10 19

10 18

10 17

10 16

10 15

10 14

10 13

10 12

10 11

10 10

|Az|

Min: 0.00e+00Max: 4.52e-10


Figure 5.9: Evolution of Az in Spritz with dissipation (panel 5.9a) and inIllinoisGRMHD (panel 5.9b). With IllinoisGRMHD, Az is zero (to machineprecision) and symmetric around the origin of the (x, y) plane during the firstinstants of evolution, while it is non-zero and asymmetric with Spritz. Moreover,in IllinoisGRMHD, strange ‘blob-like’ structures arise (central panel of 5.9b).

138


40 20 0 20 40x[km]

40

20

0

20

40

y[km

]

t = 3.31e-02 ms

40 20 0 20 40x[km]

40

20

0

20

40

y[km

]

t = 7.28e-01 ms

40 20 0 20 40x[km]

40

20

0

20

40

y[km

]

t = 1.56e+00 ms

010 1910 18

10 17

10 16

10 15

10 14

10 13

10 12

10 11

10 10

10 9

| MHD|

Min: 0.00e+00Max: 6.84e-09


40 20 0 20 40x[km]

40

20

0

20

40

y[km

]

t = 3.31e-02 ms

40 20 0 20 40x[km]

40

20

0

20

40

y[km

]

t = 7.28e-01 ms

40 20 0 20 40x[km]

40

20

0

20

40

y[km

]

t = 1.56e+00 ms

010 1910 18

10 17

10 16

10 15

10 14

10 13

10 12

10 11

10 10

10 9

| MHD|

Min: 0.00e+00Max: 6.82e-09


Figure 5.10: Comparison between the evolution of ψMHD in Spritz (panel5.10a) and IllinoisGRMHD (panel 5.10b). Similarly to what happens with Az,Spritz produces an asymmetric configuration of ψMHD in the first instants of theevolution. Moreover, a mild instability can be observed in the central panel of5.10b.

5.1.4 Magnetised TOV stars with generalised Lorenz gauge

Despite the problems observed with the magnetic field evolution with generalisedLorenz gauge in Spritz (figure 5.5), the results presented here can be consideredvery satisfactory, as achieving such a performance was a truly hard task. Indeed,together with my working group, I had to solve a number of problems in the im-plementation of the generalised Lorenz gauge condition which caused simulationsto fail very early and/or produce a variety of artifacts in the vector potential andψMHD in several regions of the computational domain (with corresponding unphys-ical magnetic fields populating a large part of the grid); furthermore, I managedto solve an issue preventing the actual damping of the 4-potential advection.

These progresses were largely made by performing simple tests such as the

139


shocktube and the magnetic rotor tests in Minkowski spacetime with generalisedLorenz gauge (see sections 5.1.1 and 5.1.2; however, keep in mind that the re-sults presented there employ the algebraic gauge). In addition to this, I ran anumber of test simulations of single magnetised stars satisfying the well-knownTolman-Oppenheimer-Volkov (TOV) equations (see, for example, [50]). Asa reminder, the TOV equations rule the dynamics of non-magnetised, static (thus,in particular, non-rotating and non-pulsating), spherically-symmetric stars whosepressure P and total energy density e are functions of the rest mass density ρ aloneand they read

dP

dr(r) = − (e+ P )

dφ

dr(r) (5.1.14)

dφ

dr(r) =

m(r) + 4πr3P (r)

r(r − 2m(r))(5.1.15)

dm

dr(r) = 4πr2e(r) . (5.1.16)

The quantity φ is defined by the metric,

ds2 = −e2φ(r)dt2 + e2λ(r)dr2 + r2(dθ2 + sin2θ dϕ2

), (5.1.17)

where λ is related to m by

m(r) :=r

2

(1− e−2λ(r)

). (5.1.18)

The quantity m is the gravitational mass of the star, as (5.1.16) states. In the caseof neutron stars, where the rest mass densities can reach values a high as 1015 g

cm3 ,adding a magnetic field of magnitude 1012 ∼ 1013G (typical of pulsars), or even1014 ∼ 1015G (typical of magnetars) is not enough to alter the stellar structuresignificantly, which is why the TOV equations are a good approximation even forstrongly magnetised stars.

On one hand, tests with TOV stars were very useful to debug the magneticfield evolution in Spritz. On the other hand, in such a scenario, the importance ofthe damping term in the generalised Lorenz gauge condition can be fully appreci-ated, as figure 5.11 shows. Indeed, a non-zero damping factor prevents the vectorpotential from being advected too fast, thus delaying the formation of unphysicalmagnetic fields at refinement boundaries and reducing their magnitude.

140


20 10 0 10 20x[km]

20

10

0

10

20

y[km

]

t = 0.00e+00 ms

20 10 0 10 20x[km]

20

10

0

10

20

y[km

]

t = 5.77e-02 ms

20 10 0 10 20x[km]

20

10

0

10

20

y[km

]

t = 5.00e-01 ms

010 15

10 14

10 13

10 12

10 11

10 10

10 9

10 8

10 7

10 6

10 5

A

Min: 0.00e+00Max: 3.29e-05

(a) Norm of the vector potential with damping factor ξ = 0

20 10 0 10 20x[km]

20

10

0

10

20

y[km

]

t = 0.00e+00 ms

20 10 0 10 20x[km]

20

10

0

10

20y[

km]

t = 5.77e-02 ms

20 10 0 10 20x[km]

20

10

0

10

20

y[km

]

t = 5.00e-01 ms

010 15

10 14

10 13

10 12

10 11

10 10

10 9

10 8

10 7

10 6

10 5

A

Min: 0.00e+00Max: 3.29e-05

(b) Norm of the vector potential with damping factor ξ = 19.2

20 10 0 10 20x[km]

20

10

0

10

20

y[km

]

t = 0.00e+00 ms

20 10 0 10 20x[km]

20

10

0

10

20

y[km

]

t = 5.77e-02 ms

20 10 0 10 20x[km]

20

10

0

10

20

y[km

]

t = 5.00e-01 ms

010 15

10 14

10 13

10 12

10 11

10 10

10 9

10 8

10 7

10 6

10 5

B

Min: 0.00e+00Max: 4.22e-05

(c) Norm of the magnetic field with damping factor ξ = 0

20 10 0 10 20x[km]

20

10

0

10

20

y[km

]

t = 0.00e+00 ms

20 10 0 10 20x[km]

20

10

0

10

20

y[km

]

t = 5.77e-02 ms

20 10 0 10 20x[km]

20

10

0

10

20

y[km

]

t = 5.00e-01 ms

010 15

10 14

10 13

10 12

10 11

10 10

10 9

10 8

10 7

10 6

10 5

B

Min: 0.00e+00Max: 4.22e-05

(d) Norm of the magnetic field with damping factor ξ = 19.2

Figure 5.11: Comparison between stable, magnetised, polytropic TOV starswith generalised Lorenz gauge and damping factors ξ = 0 and ξ = 19.2. A non-zero damping factor ξ prevents the vector potential from being advected too fast(compare 5.11b to 5.11a), thus delaying the formation of unphysical magnetic fieldsat refinement boundaries and reducing their magnitude (compare 5.11d to 5.11d).

141


5.2 Realistic, tabulated EOS handling

A realistic description of the physics of neutron stars going beyond the simpleideal-fluid or polytropic EOSs needs tabulated EOSs for which the pressure andspecific internal energy typically depend on the fluid’s rest mass density, tem-perature, and electron fraction (see sections 3.2.3 and 4.3.4). In this work, theSLy4 EOS [16, 62], which is based on the Skyrme model of the nucleus [48],has been used to evolve non-magnetised BNS systems. The EOS table in HDF5

format can be found at https://stellarcollapse.org/SROEOS (the one namedSLy4 3335 rho391 temp163 ye66.h5.bz2).

5.2.1 Non-magnetised, equal-mass BNS with SLy4 EOS

Figure 5.12 shows the time evolution of the rest mass density, temperature, andelectron fraction for a non-magnetised, equal-mass BNS system with SLy4 EOS. Asfor the BNS simulations in section 5.1.3, the initial data are generated by LORENE;the baryonic mass of each star is around 1.55M, their diameter along the xaxis is around 18.5 km and the initial separation between their centres of massis about 45 km. The spatial domain of the simulation is a cube extending in therange [−952.32M, 952.32M] ' [−1406.3M, 1406.3M]; the coarser resolutionis 7.68M ' 113.4m and there are six refinement levels, the finest ones coveringthe two stars and following them during the inspiral phase. This is an exploratoryrun with very low resolution: the finest grid spacing is equal to 0.24Msun'354m,so that only ∼55 points cover the stars’ diameters. The CFL number is C = 0.35.

At the beginning of the run, the temperature inside the stars should be equalto 10−2MeV ; however, for some reason related to the implementation of Spritz,this value gets overwritten by the atmospheric temperature, which can be set bythe user and is equal to 10−3MeV . In any case, the two stars are very ‘cold’initially with respect to the Fermi energy of a typical neutron star, which is oforder ∼ 102MeV ; indeed, the unexpected behaviour of the initial temperaturedoes not seem to have any serious consequences on the subsequent evolution ofthe system, given also that temperatures as high as ∼200MeV are reached anywayat merger time.

Due to the extremely high rest mass density values, there are only two mainweak nuclear reactions taking place in the interior of neutron stars (if the onlyleptons involved are electrons),

n→ p + e− + νe Neutron decay (5.2.1)

p + e− → n + νe Electron capture ; (5.2.2)

if neutrinos and antineutrinos are not considered (as done here), the reactions

142

https://stellarcollapse.org/SROEOS


40 30 20 10 0 10 20 30 40x[km]

40

30

20

10

0

10

20

30

40y[

km]

t = 0.00e+00 ms

40 30 20 10 0 10 20 30 40x[km]

40

30

20

10

0

10

20

30

40

y[km

]

t = 4.77e+00 ms

40 30 20 10 0 10 20 30 40x[km]

40

30

20

10

0

10

20

30

40

y[km

]

t = 5.61e+00 ms

10 10

10 9

10 8

10 7

10 6

10 5

10 4

10 3

Min: 1.00e-10Max: 4.33e-03


40 30 20 10 0 10 20 30 40x[km]

40

30

20

10

0

10

20

30

40

y[km

]

t = 0.00e+00 ms

40 30 20 10 0 10 20 30 40x[km]

40

30

20

10

0

10

20

30

40

y[km

]

t = 4.77e+00 ms

40 30 20 10 0 10 20 30 40x[km]

40

30

20

10

0

10

20

30

40

y[km

]

t = 5.61e+00 ms

10 3

10 2

10 1

100

101

102

T[MeV]

Min: 1.00e-03Max: 2.00e+02

(b) Temperature

40 30 20 10 0 10 20 30 40x[km]

40

30

20

10

0

10

20

30

40

y[km

]

t = 0.00e+00 ms

40 30 20 10 0 10 20 30 40x[km]

40

30

20

10

0

10

20

30

40

y[km

]

t = 4.77e+00 ms

40 30 20 10 0 10 20 30 40x[km]

40

30

20

10

0

10

20

30

40

y[km

]

t = 5.61e+00 ms

0.0

0.2

0.4

0.6

0.8

1.0

Ye

Min: 2.31e-02Max: 6.55e-01

(c) Electron fraction

Figure 5.12: Time evolution of the rest mass density (panel 5.12a), temperature(panel 5.12b), and electron fraction (panel 5.12c) for a non-magnetised, equal-mass (baryonic masses M1 = M2 = 1.55M) BNS system with SLy4 EOS. Anissue with the initial conditions causes the initial temperature inside the stars to beoverwritten by the atmospheric temperature, but this does not seem to influencethe subsequent evolution substantially. A black hole is formed promptly aftermerger.

143


above simplify to

n→ p + e− (5.2.3)

p + e− → n . (5.2.4)

The two stars are assumed to be in β-equilibrium initially, meaning that (5.2.3)and (5.2.4) balance each other exactly. In turn, this constrains the chemical po-tentials µn, µp and µe of neutrons, protons and electrons, respectively, as

µn!

= µp + µe . (5.2.5)

The latter condition can be used to set the initial electron fraction Ye as a functionof the temperature T and the rest mass density ρ.

A black hole is formed just after merger (central snapshots of 5.12) and remainsfixed at the origin of the computational domain, as expected for an equal-masssystem. However, what is depicted in black in the figure is not the actual eventhorizon of the black hole, since this surface could only be found after the systemhas settled down to a nearly stationary configuration; rather, it is the apparenthorizon. In order to understand what this quantity is, consider a foliation ofspacetime by a family Σtt∈R of 3D spacelike hypersurfaces and pick one elementΣt in that family; let then n be the timelike unit vector field orthogonal thefoliation. Consider a 2D spacelike hypersurface S ⊂ Σt and let s be the outward-pointing unit vector orthogonal to S; then, nµsµ = 0. The projection operatoronto S is

mµν := Pµν + nµnν − sµsν , (5.2.6)

where P is the projector onto Σt defined in (2.2.22) and the minus sign in −sµsνstems from the fact that s is spacelike. Define the outgoing and ingoing nullgeodesics as

kµ :=1√2

(nµ + sµ) sµ :=1√2

(nµ − sµ) (5.2.7)

mµνkµ = mµνl

µ = 0 kµkµ = lµlµ = 0 . (5.2.8)

The expansion of the outgoing null geodesics k is defined as

Θ := mµν∇µkν (5.2.9)

but, noticing thatmµν = gµν + 2k(µ lν) (5.2.10)

and that kµ∇µkν = 0 (k is a geodesic) and kν∇µkν = 0, (5.2.9) can be recast as

Θ = ∇µkµ . (5.2.11)

144


Now, define an outer trapped surface as a 2D hypersurface embedded into Σt

such that Θ < 0 and a trapped region as a region Ω ⊂ Σt enclosing at leastone outer trapped surface; an apparent horizon is the boundary of the trappedregion enclosing the maximum number of outer trapped surfaces in the spacetimeunder consideration or, which is the same, is the 2D hypersurface defined by Θ = 0.The last condition may be recast in a 3+1 fashion by noticing that, using formula(2.2.48) for the extrinsic curvature to write

mµν∇µnν = −mijKij , (5.2.12)

definitions (5.2.9) and (5.2.7) lead to

0!

=√

2 Θ = Disi + sisjKij −K . (5.2.13)

The thorn AHFinderDirect from the Einstein Toolkit is able to solve the ellipticPDE (5.2.13) numerically, provided that a sufficiently good ‘guess’ for the positionof the apparent horizon is given. It can be shown that, if the spacetime contains ablack-hole singularity, the apparent horizon always lies inside the the event horizonand coincides with the latter in stationary spacetimes; the apparent horizon is thusan excellent tracker of the position of the black hole during a numerical simulation.

5.2.2 Non-magnetised, unequal-mass BNS with SLy4 EOS

In addition to the equal-mass run depicted in figure 5.12, I performed another BNSsimulation with SLy4 EOS using the same grid setup as in the previous section, nowevolving an unequal-mass configuration with mass ratio 0.8 and initial separationbetween the centres of mass of the two stars of about 47.5 km; this is depictedin figure 5.13. The baryonic mass of the star initially on the left is 1.69M andits diameter along the x axis is ∼ 18 km; the other star has a baryonic mass of1.32M and its diameter along the x axis is ∼ 19.3 km. In order to circumventthe issue related to the overwriting of the initial temperature inside the stars withthe atmospheric temperature, the latter has been set to 10−2MeV .

There is some unexpected, high-electron-fraction material showing up aftermerger at the end of the simulation (rightmost panel of 5.13c) and this is probablydue to failures in the C2P solver in the region near the black hole, which is formedpromptly at merger time. The apparent horizon moves away from the centre ofthe numerical grid, as expected for an unequal-mass BNS system: one says thatthe black hole has been given a ‘kick’. However, this kick looks quite large andcan hardly be entirely physical in nature; rather, it could be due to the extremelylow resolution adopted.

145


40 30 20 10 0 10 20 30 40x[km]

40

30

20

10

0

10

20

30

40y[

km]

t = 0.00e+00 ms

40 30 20 10 0 10 20 30 40x[km]

40

30

20

10

0

10

20

30

40

y[km

]

t = 4.93e+00 ms

40 30 20 10 0 10 20 30 40x[km]

40

30

20

10

0

10

20

30

40

y[km

]

t = 5.83e+00 ms

10 10

10 9

10 8

10 7

10 6

10 5

10 4

10 3

Min: 1.00e-10Max: 4.57e-03


40 30 20 10 0 10 20 30 40x[km]

40

30

20

10

0

10

20

30

40

y[km

]

t = 0.00e+00 ms

40 30 20 10 0 10 20 30 40x[km]

40

30

20

10

0

10

20

30

40

y[km

]

t = 4.93e+00 ms

40 30 20 10 0 10 20 30 40x[km]

40

30

20

10

0

10

20

30

40

y[km

]

t = 5.83e+00 ms

10 3

10 2

10 1

100

101

102

T[MeV]

Min: 1.00e-03Max: 1.62e+02

(b) Temperature

40 30 20 10 0 10 20 30 40x[km]

40

30

20

10

0

10

20

30

40

y[km

]

t = 0.00e+00 ms

40 30 20 10 0 10 20 30 40x[km]

40

30

20

10

0

10

20

30

40

y[km

]

t = 4.93e+00 ms

40 30 20 10 0 10 20 30 40x[km]

40

30

20

10

0

10

20

30

40

y[km

]

t = 5.83e+00 ms

0.0

0.2

0.4

0.6

0.8

1.0

Ye

Min: 2.31e-02Max: 6.55e-01

(c) Electron fraction

Figure 5.13: Time evolution of the rest mass density (panel 5.13a), temperature(panel 5.12b), and electron fraction (panel 5.12c) for a non-magnetised, unequal-mass (baryonic masses 1.32M and 1.69M , mass ratio 0.8) BNS system withSLy4 EOS. The black hole is kicked away from the centre of the computationaldomain, the large kick probably being due to the very low resolution adopted inthis simulation.

146


In order to shed some light on this, I ran a second simulation with maximumresolution equal to 0.20M ' 295m (∼ 66 points covering the stars’ diameters);figure 5.14 compares the rest mass density distributions for the two resolutionsat nearly equal times after merger. The kick looks a bit reduced with increasingresolution, but is still quite large; in any case, the difference is not easy to spot byeye and should be quantified.

40 30 20 10 0 10 20 30 40

x [km]

40

30

20

10

0

10

20

30

40

y[km

]

Finest_dx = 0.24

10 10

10 9

10 8

10 7

10 6

10 5

10 4

10 3

t = 5.45e+00 ms It = 13184

Min: 1.00e-10

Max: 1.25e-04

40 30 20 10 0 10 20 30 40

x [km]

40

30

20

10

0

10

20

30

40

y[km

]

Finest_dx = 0.20

10 10

10 9

10 8

10 7

10 6

10 5

10 4

10 3

t = 5.43e+00 ms It = 15744

Min: 1.00e-10

Max: 1.55e-03

Figure 5.14: Snapshots of the rest mass density at roughly equal times (well aftermerger) for the run of figure 5.13 with maximum grid resolutions of 0.24Msun (leftpanel) and 0.20Msun (right panel). The kick looks a bit reduced with increasedresolution, but is still quite big.

One simple way to do this is to realise that the position of the black hole isapproximated very well by the position of the minimum of the lapse function; then,the distance r between the latter point and the origin of the (x, y) plane is a goodmeasure of the entity of the kick. In figure 5.15, r is plotted as a function of timefor the two resolutions 0.24M and 0.20M . The merger happens at t∼ 5ms,after which the two black holes move at nearly constant, albeit different, velocitiestowards the boundaries of the numerical domain.

A rough measure of the kick velocities is reported in figure 5.16, where a straightline has been fit (adopting a least-squares technique and assuming the uncertaintieson r(t) are all the same) to the rightmost region of the plot in figure 5.15 for bothresolutions. The velocity of each of the two kicks is the slope of the corresponding

147


0 1 2 3 4 5 6t [ms]

2.5

5.0

7.5

10.0

12.5

15.0

17.5

20.0

r(t)

[km

]

Merger

SLy4 EOS M1 = 1.32M , M2 = 1.69M Kick comparison

dx = 0.24Mdx = 0.20M

Figure 5.15: Distance (Euclidean norm) of the point of minimum lapse from theorigin of the (x, y) plane for the unequal-mass (mass ratio 0.8) BNS runs withSLy4 EOS and finest grid resolutions of 0.24M and 0.20M . Merger happens att'5ms and the difference in the velocities of the subsequant kicks are well visible.

interpolating line and it is found that

v1 ' 16361km

sdx = 0.24M (5.2.14)

v2 ' 12893km

sdx = 0.20M . (5.2.15)

as expected, increasing the resolution reduces the kick velocity, thereby catchingthe physics of the process more realistically. Notice, however, that this is quite arough measure: the distance r has been calculated naively, without caring aboutthe non-Euclidean geometry of each of the spacelike slices in the spacetime folia-tion.

148


5.2 5.3 5.4 5.5 5.6 5.7 5.8t [ms]

4

6

8

10

12

14

r(t)

[km

]

v1 16361 kms

v2 12893 kms

SLy4 EOS M1 = 1.32M , M2 = 1.69M Kick velocities

dx = 0.24M min( )dx = 0.20M min( )dx = 0.24M - Fitdx = 0.20M - Fit

Figure 5.16: Linear least-squares fit of the rightmost part of the plot in figure5.15. Increasing the resolution reduces the kick velocity, as expected.

In the present case, a maximum resolution of 0.20M is still very low andmore reliable results could probably be achieved with a maximum resolution atleast equal to 0.15M . Unfortunately, performing higher-resolution runs was notpossible because of lack of computational resources; anyway, it would not proba-bly be worth doing that because of a major issue with initial data. Specifically,the SLy4 initial data computed with LORENE violate the Hamiltonian constraint(3.1.10) seriously inside the stars, as the left panel of figure 5.17 shows for theequal-mass configuration: the quantity

|H| :=∣∣3R +K2 −KijK

ij − 16πE∣∣ (5.2.16)

(see (3.1.10)) reaches values as high as ∼ 10−3, i.e. , it is of the order of the restmass density inside the stars. This result should be compared to the right panelof the same figure, which shows that the Hamiltonian constraint violation for thepolytropic initial data used for the BNS simulations in section 5.1 reaches at mosta value of ∼10−5 near the stars’ surfaces.

149


150 100 50 0 50 100 150x[km]

10 12

10 10

10 8

10 6

10 4

SLy4 EOS M1 = M2 = 1.55 M Finest dx = 0.15

Star surface| |

150 100 50 0 50 100 150x[km]

Polytropic EOS, = 2 M1 = M2 = 1.625 M Finest dx = 0.15

Star surface| |

Figure 5.17: The absolute value of the Hamiltonian constraint violation (‘ |H|′)is comparable in magnitude to the rest mass density ρ inside the stars when theSLy4 EOS is adopted (left panel), while it is about three orders of magnitudesmaller than that if a polytropic EOS (Γ = 2) is chosen (right panel).

This bad behaviour could be due to the specific internal energy ε in the SLy4EOS table to take on negative values, which in turn can cause the specific enthalpy(4.3.18) to become lower than 1; LORENE does not ‘like’ it and this results in thehigh values of |H| observed in the left panel of (5.17). One possible workarounudfor this problem follows.

The EOS tables found at [66] do not contain the specific internal energy ε, butrather log(ε+ ε0), where the purpose of the ‘energy shift’ ε0 is to make ε+ ε0 > 0in case ε < 0; the goal is then to ensure that ε > 0 over the entire EOS table andset ε0 = 0. In order to do so, pick the baryon mass mb as given in the EOS tableand define

mf := mb(1− ε0) . (5.2.17)

Then, compute the total energy density as e = ρ(1 + ε) (the EOS table containslog(ρ)) and define a new rest mass density ρ as

ρ := nbmf =

(ρ

mb

)mf ≡

(mf

mb

)ρ = ρ(1− ε0) . (5.2.18)

Finally, compute the new specific internal energy ε as

ε :=e

ρ− 1 > 0 , (5.2.19)

where the last inequality holds since

ε+ ε0 ≥ 0 ⇒ ε ≥ −ε0 ⇒ e = ρ(1 + ε) ≥ ρ(1− ε0) ≡ ρ . (5.2.20)

150


To finish with, a new EOS table is built from log(ρ) and log(ε), setting ε0 = 0.Thus, LORENE only reads positive values for the specific internal energy and itsbehaviour is improved.

A script performing the ‘clean-up’ of EOS tables coming from [66] (and otheroperations not described here) was written by Dr. Leonardo Werneck following, inturn, a script by Prof. David Radice. Initial data for the LS220 EOS [44] weregenerated following this approach and results are indeed much better if comparedwith the the SLy4 case (left panel of 5.17): figure (5.18) depicts the situation.Unfortunately, ‘uncleaned’ LS220 initial data were not available to produce a directcomparison. Critically, |H| decreases as the spectral resolutions used in LORENE togenerate the LS220 initial data is increased, and this is very encouraging.

150 100 50 0 50 100 150x[km]

10 10

10 9

10 8

10 7

10 6

10 5

10 4

10 3

||

LS220 EOS M1 = M2 = 1.47 M Finest dx = 0.125

Star surfaceLower LORENE res.Higher LORENE res.

Figure 5.18: Absolute value of the Hamiltonian constraint violation for the‘cleaned-up’ LS220 EOS. Results are much better inside the stars if compared tothe SLy4 EOS of 5.17 and increasing the resolution in LORENE makes |H| smaller.

151

https://gwac.wvu.edu/about/people/leonardo-rosa-werneck

http://personal.psu.edu/dur566/

Chapter 6Conclusions and future work

After a detailed study of the necessary topics in the 3+1 formulation of generalrelativity and some discussion about the most common numerical techniques em-ployed to solve the GRMHD equations, a number of results from simulations ofbinary neutron star mergers — as well as less involved physical scenarios — per-formed with the Spritz code have been presented.

Considerable progress has been made in code performance as regards the imple-mentation of the generalised Lorenz gauge condition by fixing a number of flaws inthe source code, unveiling however new problems related to some kind of numericalinstability. For the time being, adding some amount of Kreiss-Oliger dissipationto the time evolution of the scalar and vector electromagnetic potentials is a per-fectly good solution for most purposes. Furthermore, because the instability isseen to arise at the beginning of BNS simulations, Kreiss-Oliger dissipation maybe applied as an initial ‘safety tool’, but then removed just before merger or evenearlier: this is something that should be tested in the near future.

Progress has been made on the side of tabulated EOS handling, as well. Aftera lot of hard work and thanks to the invaluable help from Prof. Bruno Giaco-mazzo and Dr. Federico Cipolletta, I finally managed to solve a problem in theC2P Palenzuela routine which caused memory corruption errors, thus preventingsimulations with SLy4 EOS from running correctly. Even though the BNS runswith SLy4 EOS presented here were performed at extremely low resolution (tosave computational resources) and suffer from the problem with the Hamiltonianconstraint violation in the initial data (recall section 5.2.2), Spritz was able tobring both of them well beyond merger without any issue. Given that the problemwith the initial data seems now on the verge of being solved, the plan for the nearfuture is to try to ‘clean up’ the SLy4 EOS table as done for the LS220 EOS andthen run a magnetised BNS simulation with the SLy4 EOS.

Overall, Spritz stands as a state-of-the-art GRMHD code and the comparisonwith IllinoisGRMHD confirms this. Spritz is able to simulate a large spectrumof interesting astrophysical scenarios in a highly realistic fashion, taking magneticfields, microphysical propeties of matter in neutron stars, and neutrinos (not dis-cussed here) into account. Therefore, I hope the work done within the scope ofthis thesis will be useful to having Spritz included into the Einstein Toolkit

as a thorn at some point in the near future.

152

https://www.brunogiacomazzo.org/

https://www.brunogiacomazzo.org/

https://www.paradisomontagna.it/federico-cipolletta-matematico-astrofisico.html

Chapter 6 Conclusions and future work

Acknowledgements

I would like to thank my advisor, Prof. Bruno Giacomazzo, for being always sohelpful in guiding my work with attention and expertise and for taking care of myfuture as a PhD student. Also, thanks to all the people in the Spritz developers’group: Dott. Federico Cipolletta, Dr. Riccardo Ciolfi, and Dr. Jay Kalinani.

All the simulations presented here were performed either on the MARCONIA3 machine at CINECA (Casalecchio di Reno (BO), Italy) [18, 49] or on theFrontera machine at the TACC (University of Texas at Austin, USA) [31]. Inparticular, I was given access to Frontera by Prof. Manuela Campanelli, PI of theTheoretical and Computational Astrophysics Network (TCAN) [67],which I have worked with for more than five months now. The TCAN is a collab-oration of experts in computational astrophysics from various institutions (e.g. ,the Rochester Institute of Technology, West Virginia University, Johns HopkinsUniversity, and NASA Goddard’s Space Flight Center) whose purpose is to per-form the most accurate study of binary neutron star mergers ever attempted,from inspiral to post-merger evolution, including gamma-ray bursts and neutrinoemission, by means of a combination of state-of-the-art codes such as Spritz,IllinoisGRMHD, and HARM3D [53]. I would like to thank Prof. Bruno Giacomazzo,which is part of the TCAN together with Dr. Federico Cipolletta and Dr. Ric-cardo Ciolfi, for giving me the opportunity to collaborate with the TCAN, andProf. Manuela Campanelli, for letting me in.

153

https://www.rit.edu/directory/mxcsma-manuela-campanelli

Bibliography

[1] B. P. Abbott et al. , GW170817: Observation of Gravitational Waves froma Binary Neutron Star Inspiral , in: Phys. Rev. Lett. 119 (16 Oct. 2017),p. 161101 , doi: 10.1103/PhysRevLett.119.161101 , url: https://link.aps.org/doi/10.1103/PhysRevLett.119.161101.

[2] B. P. Abbott et al. , Observation of Gravitational Waves from a BinaryBlack Hole Merger , in: Phys. Rev. Lett. 116 (6 Feb. 2016), p. 061102 , doi:10.1103/PhysRevLett.116.061102 , url: https://link.aps.org/doi/10.1103/PhysRevLett.116.061102.

[3] A. Anderson and J. W. York , Hamiltonian Time Evolution for GeneralRelativity , in: Phys. Rev. Lett. 81 (6 Aug. 1998), pp. 1154–1157 , doi: 10.1103/PhysRevLett.81.1154 , url: https://link.aps.org/doi/10.1103/PhysRevLett.81.1154.

[4] A. M. Anile , Relativistic Fluids and Magneto-fluids: With Applications inAstrophysics and Plasma Physics , Cambridge Monographs on MathematicalPhysics , Cambridge University Press, 1990 , doi: 10.1017/CBO9780511564130.

[5] J. Antoniadis et al. , A Massive Pulsar in a Compact Relativistic Binary ,in: Science 340.6131 (2013) , issn: 0036-8075 , doi: 10 . 1126 / science .

1233232 , eprint: https://science.sciencemag.org/content/340/6131/1233232.full.pdf , url: https://science.sciencemag.org/content/340/6131/1233232.

[6] R. Arnowitt, S. Deser, and C. W. Misner , Republication of: The dynam-ics of general relativity , in: General Relativity and Gravitation 40.9 (Sept.2008), pp. 1997–2027 , issn: 1572-9532 , doi: 10.1007/s10714-008-0661-1 ,url: https://doi.org/10.1007/s10714-008-0661-1.

[7] S. A. Balbus and J. F. Hawley , Instability, turbulence, and enhancedtransport in accretion disks , in: Rev. Mod. Phys. 70 (1 Jan. 1998), pp. 1–53 ,doi: 10.1103/RevModPhys.70.1 , url: https://link.aps.org/doi/10.1103/RevModPhys.70.1.

[8] D. S. Balsara and D. S. Spicer , A Staggered Mesh Algorithm Using HighOrder Godunov Fluxes to Ensure Solenoidal Magnetic Fields in Magnetohy-drodynamic Simulations , in: Journal of Computational Physics 149.2 (1999),pp. 270–292 , issn: 0021-9991 , doi: https://doi.org/10.1006/jcph.1998.6153 , url: https://www.sciencedirect.com/science/article/pii/S0021999198961538.

154

https://doi.org/10.1103/PhysRevLett.119.161101

https://link.aps.org/doi/10.1103/PhysRevLett.119.161101









https://doi.org/10.1017/CBO9780511564130

https://doi.org/10.1126/science.1233232

https://doi.org/10.1126/science.1233232

https://science.sciencemag.org/content/340/6131/1233232.full.pdf

https://science.sciencemag.org/content/340/6131/1233232.full.pdf

https://science.sciencemag.org/content/340/6131/1233232

https://science.sciencemag.org/content/340/6131/1233232

https://doi.org/10.1007/s10714-008-0661-1

https://doi.org/10.1007/s10714-008-0661-1

https://doi.org/10.1103/RevModPhys.70.1

https://link.aps.org/doi/10.1103/RevModPhys.70.1


https://doi.org/https://doi.org/10.1006/jcph.1998.6153


https://www.sciencedirect.com/science/article/pii/S0021999198961538

https://www.sciencedirect.com/science/article/pii/S0021999198961538

BIBLIOGRAPHY

[9] T. W. Baumgarte and S. L. Shapiro , Numerical integration of Einstein’sfield equations , in: Phys. Rev. D 59 (2 Dec. 1998), p. 024007 , doi: 10.

1103/PhysRevD.59.024007 , url: https://link.aps.org/doi/10.1103/PhysRevD.59.024007.

[10] T. W. Baumgarte and S. L. Shapiro , Numerical Relativity: Solving Ein-stein’s Equations on the Computer , Cambridge University Press, 2010 , doi:10.1017/CBO9781139193344.

[11] M. J. Berger and J. Oliger , Adaptive mesh refinement for hyperbolic par-tial differential equations , in: Journal of Computational Physics 53.3 (1984),pp. 484–512 , issn: 0021-9991 , doi: https://doi.org/10.1016/0021-9991(84)90073- 1 , url: https://www.sciencedirect.com/science/

article/pii/0021999184900731.

[12] R. D. Blandford and R. L. Znajek , Electromagnetic extraction of energyfrom Kerr black holes , in: Monthly Notices of the Royal Astronomical Society179.3 (July 1977), pp. 433–456 , issn: 0035-8711 , doi: 10.1093/mnras/179.3.433 , eprint: https://academic.oup.com/mnras/article-pdf/179/3/433/9333653/mnras179-0433.pdf , url: https://doi.org/10.1093/mnras/179.3.433.

[13] Cactus , url: https://www.cactuscode.org/.

[14] M. Campanelli et al. , Accurate Evolutions of Orbiting Black-Hole Binarieswithout Excision , in: Phys. Rev. Lett. 96 (11 Mar. 2006), p. 111101 , doi:10.1103/PhysRevLett.96.111101 , url: https://link.aps.org/doi/10.1103/PhysRevLett.96.111101.

[15] S. M. Carroll , Spacetime and Geometry: An Introduction to General Rel-ativity , Cambridge University Press, 2019 , doi: 10.1017/9781108770385.

[16] E. Chabanat et al. , A Skyrme parametrization from subnuclear to neutronstar densities Part II. Nuclei far from stabilities , in: Nuclear Physics A635.1 (1998), pp. 231–256 , issn: 0375-9474 , doi: https://doi.org/10.1016/S0375-9474(98)00180-8 , url: http://www.sciencedirect.com/science/article/pii/S0375947498001808.

[17] Y. Choquet-Bruhat , General Relativity and the Einstein Equations , Ox-ford Mathematical Monographs , United Kingdom: Oxford University Press,2009 , isbn: 978-0-19-923072-3.

[18] CINECA , url: https://www.cineca.it/.

[19] F. Cipolletta et al. , Spritz: a new fully general-relativistic magnetohy-drodynamic code , in: Classical and Quantum Gravity 37.13 (June 2020),p. 135010 , doi: 10.1088/1361-6382/ab8be8 , url: https://doi.org/10.1088%2F1361-6382%2Fab8be8.

155

https://doi.org/10.1103/PhysRevD.59.024007


https://link.aps.org/doi/10.1103/PhysRevD.59.024007


https://doi.org/10.1017/CBO9781139193344

https://doi.org/https://doi.org/10.1016/0021-9991(84)90073-1

https://doi.org/https://doi.org/10.1016/0021-9991(84)90073-1

https://www.sciencedirect.com/science/article/pii/0021999184900731

https://www.sciencedirect.com/science/article/pii/0021999184900731

https://doi.org/10.1093/mnras/179.3.433


https://academic.oup.com/mnras/article-pdf/179/3/433/9333653/mnras179-0433.pdf

https://academic.oup.com/mnras/article-pdf/179/3/433/9333653/mnras179-0433.pdf



https://www.cactuscode.org/




https://doi.org/10.1017/9781108770385

https://doi.org/https://doi.org/10.1016/S0375-9474(98)00180-8

https://doi.org/https://doi.org/10.1016/S0375-9474(98)00180-8

http://www.sciencedirect.com/science/article/pii/S0375947498001808

http://www.sciencedirect.com/science/article/pii/S0375947498001808

https://www.cineca.it/

https://doi.org/10.1088/1361-6382/ab8be8

https://doi.org/10.1088%2F1361-6382%2Fab8be8

https://doi.org/10.1088%2F1361-6382%2Fab8be8

BIBLIOGRAPHY

[20] F. Cipolletta et al. , Spritz: General Relativistic Magnetohydrodynamicswith Neutrinos , 2020 , arXiv: 2012.10174 [astro-ph.HE].

[21] F. Cipolletta et al. , Spritz: General Relativistic Magnetohydrodynamicswith Neutrinos , in: arXiv e-prints, arXiv:2012.10174 (Dec. 2020), arXiv:2012.10174 ,arXiv: 2012.10174 [astro-ph.HE].

[22] CompOSE , url: https://compose.obspm.fr/eos/42.

[23] H. T. Cromartie et al. , Relativistic Shapiro delay measurements of anextremely massive millisecond pulsar , in: Nature Astronomy 4.1 (Jan. 2020),pp. 72–76 , issn: 2397-3366 , doi: 10.1038/s41550- 019- 0880- 2 , url:https://doi.org/10.1038/s41550-019-0880-2.

[24] A. Dedner et al. , Hyperbolic Divergence Cleaning for the MHD Equations ,in: Journal of Computational Physics 175.2 (2002), pp. 645–673 , issn: 0021-9991 , doi: https://doi.org/10.1006/jcph.2001.6961 , url: https://www.sciencedirect.com/science/article/pii/S002199910196961X.

[25] P. B. Demorest et al. , A two-solar-mass neutron star measured usingShapiro delay , in: Nature 467.7319 (Oct. 2010), pp. 1081–1083 , issn: 1476-4687 , doi: 10.1038/nature09466 , url: https://doi.org/10.1038/

nature09466.

[26] Einstein toolkit , url: https://einsteintoolkit.org/.

[27] Z. B. Etienne et al. , IllinoisGRMHD: an open-source, user-friendly GRMHDcode for dynamical spacetimes , in: Classical and Quantum Gravity 32.17(Aug. 2015), p. 175009 , doi: 10.1088/0264-9381/32/17/175009 , url:https://doi.org/10.1088/0264-9381/32/17/175009.

[28] Z. B. Etienne et al. , Relativistic magnetohydrodynamics in dynamical space-times: Improved electromagnetic gauge condition for adaptive mesh refine-ment grids , in: Phys. Rev. D 85 (2 Jan. 2012), p. 024013 , doi: 10.1103/PhysRevD.85.024013 , url: https://link.aps.org/doi/10.1103/

PhysRevD.85.024013.

[29] C. R. Evans and J. F. Hawley , Simulation of Magnetohydrodynamic Flows:A Constrained Transport Model , in: apj 332 (Sept. 1988), p. 659 , doi: 10.1086/166684.

[30] Exact Riemann solver for special-relativistic MHD , url: https://www.

brunogiacomazzo.org/?page_id=395.

[31] Frontera , url: https://frontera-portal.tacc.utexas.edu/.

156

https://arxiv.org/abs/2012.10174


https://compose.obspm.fr/eos/42

https://doi.org/10.1038/s41550-019-0880-2

https://doi.org/10.1038/s41550-019-0880-2


https://www.sciencedirect.com/science/article/pii/S002199910196961X

https://www.sciencedirect.com/science/article/pii/S002199910196961X

https://doi.org/10.1038/nature09466




https://doi.org/10.1088/0264-9381/32/17/175009

https://doi.org/10.1088/0264-9381/32/17/175009





https://doi.org/10.1086/166684

https://doi.org/10.1086/166684

https://www.brunogiacomazzo.org/?page_id=395

https://www.brunogiacomazzo.org/?page_id=395

https://frontera-portal.tacc.utexas.edu/

BIBLIOGRAPHY

[32] C. F. Gammie, J. C. McKinney, and G. Toth , HARM: A NumericalScheme for General Relativistic Magnetohydrodynamics , in: The Astrophys-ical Journal 589.1 (May 2003), pp. 444–457 , doi: 10.1086/374594 , url:https://doi.org/10.1086/374594.

[33] B. Giacomazzo , General Relativistic Magnetohydrodynamics: fundamentalaspects and applications , PhD thesis , SISSA, Oct. 2006 , url: http://hdl.handle.net/20.500.11767/4007.

[34] B. Giacomazzo and L. Rezzolla , The exact solution of the Riemannproblem in relativistic magnetohydrodynamics , in: Journal of Fluid Mechan-ics 562 (2006), pp. 223–259 , doi: 10.1017/S0022112006001145.

[35] B. Giacomazzo and L. Rezzolla , WhiskyMHD: a new numerical codefor general relativistic magnetohydrodynamics , in: Classical and QuantumGravity 24.12 (Jan. 2007), S235–S258 , doi: 10.1088/0264-9381/24/12/s16 , url: https://doi.org/10.1088/0264-9381/24/12/s16.

[36] T. Goodale et al. , The Cactus Framework and Toolkit: Design and Ap-plications , in: Vector and Parallel Processing – VECPAR’2002, 5th Inter-national Conference, Lecture Notes in Computer Science , Berlin: Springer,2003 , url: http://edoc.mpg.de/3341.

[37] E. Gourgoulhon , The Initial Data Problem , in: , 3+1 Formalism in Gen-eral Relativity: Bases of Numerical Relativity , Berlin, Heidelberg: SpringerBerlin Heidelberg, 2012, pp. 185–219 , isbn: 978-3-642-24525-1 , doi: 10.

1007/978-3-642-24525-1_9 , url: https://doi.org/10.1007/978-3-642-24525-1_9.

[38] R. Haas et al. , The Einstein Toolkit , version The ‘DeWitt-Morette’ release,ET 2020 11 , To find out more, visit http://einsteintoolkit.org , Nov. 2020 ,doi: 10.5281/zenodo.4298887 , url: https://doi.org/10.5281/zenodo.4298887.

[39] D. Hilditch , An Introduction to Well-Posedness and Free-Evolution , in:International Journal of Modern Physics A 28.22n23 (2013), p. 1340015 ,doi: 10.1142/S0217751X13400150 , eprint: https://doi.org/10.1142/S0217751X13400150 , url: https://doi.org/10.1142/S0217751X13400150.

[40] W. Kastaun, J. V. Kalinani, and R. Ciolfi , Robust recovery of primitivevariables in relativistic ideal magnetohydrodynamics , in: Phys. Rev. D 103(2 Jan. 2021), p. 023018 , doi: 10 . 1103 / PhysRevD . 103 . 023018 , url:https://link.aps.org/doi/10.1103/PhysRevD.103.023018.

157

https://doi.org/10.1086/374594

https://doi.org/10.1086/374594

http://hdl.handle.net/20.500.11767/4007

http://hdl.handle.net/20.500.11767/4007

https://doi.org/10.1017/S0022112006001145

https://doi.org/10.1088/0264-9381/24/12/s16

https://doi.org/10.1088/0264-9381/24/12/s16

https://doi.org/10.1088/0264-9381/24/12/s16

http://edoc.mpg.de/3341

https://doi.org/10.1007/978-3-642-24525-1_9

https://doi.org/10.1007/978-3-642-24525-1_9

https://doi.org/10.1007/978-3-642-24525-1_9

https://doi.org/10.1007/978-3-642-24525-1_9

https://doi.org/10.5281/zenodo.4298887



https://doi.org/10.1142/S0217751X13400150

https://doi.org/10.1142/S0217751X13400150

https://doi.org/10.1142/S0217751X13400150

https://doi.org/10.1142/S0217751X13400150



BIBLIOGRAPHY

[41] T. Kawamura et al. , Binary neutron star mergers and short gamma-raybursts: Effects of magnetic field orientation, equation of state, and mass ra-tio , in: Phys. Rev. D 94 (6 Sept. 2016), p. 064012 , doi: 10.1103/PhysRevD.94.064012 , url: https://link.aps.org/doi/10.1103/PhysRevD.94.064012.

[42] K. Kiuchi et al. , Efficient magnetic-field amplification due to the Kelvin-Helmholtz instability in binary neutron star mergers , in: Phys. Rev. D 92(12 Dec. 2015), p. 124034 , doi: 10 . 1103 / PhysRevD . 92 . 124034 , url:https://link.aps.org/doi/10.1103/PhysRevD.92.124034.

[43] S. S. Komissarov , Electrodynamics of black hole magnetospheres , in: MonthlyNotices of the Royal Astronomical Society 350.2 (May 2004), pp. 427–448 ,issn: 0035-8711 , doi: 10.1111/j.1365- 2966.2004.07598.x , eprint:https://academic.oup.com/mnras/article-pdf/350/2/427/3884469/

350-2-427.pdf , url: https://doi.org/10.1111/j.1365-2966.2004.07598.x.

[44] J. M. Lattimer and F. D. Swesty , A generalized equation of state forhot, dense matter , in: Nuclear Physics A 535.2 (1991), pp. 331–376 , issn:0375-9474 , doi: https : / / doi . org / 10 . 1016 / 0375 - 9474(91 ) 90452 -

C , url: https : / / www . sciencedirect . com / science / article / pii /

037594749190452C.

[45] R. J. LeVeque , Finite Volume Methods for Hyperbolic Problems , Cam-bridge Texts in Applied Mathematics , Cambridge University Press, 2002 ,doi: 10.1017/CBO9780511791253.

[46] F. Loffler et al. , The Einstein Toolkit: a community computational in-frastructure for relativistic astrophysics , in: Classical and Quantum Gravity29.11 (May 2012), p. 115001 , doi: 10.1088/0264-9381/29/11/115001 ,url: https://doi.org/10.1088/0264-9381/29/11/115001.

[47] LORENE , url: https://lorene.obspm.fr/.

[48] Y.-L. Ma and M. Harada , Lecture notes on the Skyrme model , in: arXiv e-prints, arXiv:1604.04850 (Apr. 2016), arXiv:1604.04850 , arXiv: 1604.04850[hep-ph].

[49] MARCONI A3 , url: https://www.hpc.cineca.it/hardware/marconi.

[50] C. W. Misner, K. S. Thorne, and J. A. Wheeler , Gravitation , SanFrancisco: W. H. Freeman, 1973 , isbn: 978-0-7167-0344-0.

[51] P. Mosta et al. , GRHydro: a new open-source general-relativistic magne-tohydrodynamics code for the Einstein toolkit , in: Classical and QuantumGravity 31.1 (Nov. 2013), p. 015005 , doi: 10.1088/0264- 9381/31/1/

015005 , url: https://doi.org/10.1088/0264-9381/31/1/015005.

158







https://doi.org/10.1111/j.1365-2966.2004.07598.x

https://academic.oup.com/mnras/article-pdf/350/2/427/3884469/350-2-427.pdf

https://academic.oup.com/mnras/article-pdf/350/2/427/3884469/350-2-427.pdf

https://doi.org/10.1111/j.1365-2966.2004.07598.x

https://doi.org/10.1111/j.1365-2966.2004.07598.x

https://doi.org/https://doi.org/10.1016/0375-9474(91)90452-C

https://doi.org/https://doi.org/10.1016/0375-9474(91)90452-C

https://www.sciencedirect.com/science/article/pii/037594749190452C

https://www.sciencedirect.com/science/article/pii/037594749190452C

https://doi.org/10.1017/CBO9780511791253

https://doi.org/10.1088/0264-9381/29/11/115001

https://doi.org/10.1088/0264-9381/29/11/115001

https://lorene.obspm.fr/



https://www.hpc.cineca.it/hardware/marconi

https://doi.org/10.1088/0264-9381/31/1/015005

https://doi.org/10.1088/0264-9381/31/1/015005

https://doi.org/10.1088/0264-9381/31/1/015005

BIBLIOGRAPHY

[52] T. Nakamura, K. Oohara, and Y. Kojima , General Relativistic Collapseto Black Holes and Gravitational Waves from Black Holes , in: Progress ofTheoretical Physics Supplement 90 (Jan. 1987), pp. 1–218 , issn: 0375-9687 ,doi: 10.1143/PTPS.90.1 , eprint: https://academic.oup.com/ptps/article-pdf/doi/10.1143/PTPS.90.1/5201911/90-1.pdf , url: https://doi.org/10.1143/PTPS.90.1.

[53] S. C. Noble, J. H. Krolik, and J. F. Hawley , Direct Calculation of theRadiative Efficiency of an Accretion Disk Around a Black Hole , in: TheAstrophysical Journal 692.1 (Feb. 2009), pp. 411–421 , doi: 10.1088/0004-637x/692/1/411 , url: https://doi.org/10.1088/0004-637x/692/1/411.

[54] C. Palenzuela et al. , Effects of the microphysical equation of state in themergers of magnetized neutron stars with neutrino cooling , in: Phys. Rev.D 92 (4 Aug. 2015), p. 044045 , doi: 10.1103/PhysRevD.92.044045 , url:https://link.aps.org/doi/10.1103/PhysRevD.92.044045.

[55] T. Piran , The physics of gamma-ray bursts , in: Rev. Mod. Phys. 76 (4 Jan.2005), pp. 1143–1210 , doi: 10.1103/RevModPhys.76.1143 , url: https://link.aps.org/doi/10.1103/RevModPhys.76.1143.

[56] W. H. Press et al. , Numerical Recipes 3rd Edition: The Art of Scien-tific Computing , 3rd ed. , USA: Cambridge University Press, 2007 , isbn:0521880688.

[57] O. Reula , Strongly hyperbolic systems in General Relativity , in: arXiv e-prints, gr-qc/0403007 (Mar. 2004), gr–qc/0403007 , arXiv: gr-qc/0403007[gr-qc].

[58] L. Rezzolla et al. , The Missing Link: Merging Neutron Stars NaturallyProduce Jet-Like Structures and Can Power Short Gamma-Ray Bursts , in:The Astrophysical Journal 732.1 (Apr. 2011), p. L6 , doi: 10.1088/2041-8205/732/1/l6 , url: https://doi.org/10.1088%2F2041-8205%2F732%2F1%2Fl6.

[59] S. Rosswog and M. Liebendorfer , High-resolution Calculations of Merg-ing Neutron Stars - II. Neutrino Emission , in: mnras 342.3 (July 2003),pp. 673–689 , doi: 10.1046/j.1365-8711.2003.06579.x , arXiv: astro-ph/0302301 [astro-ph].

[60] M. Ruiz et al. , Binary Neutron Star Mergers: A Jet Engine for ShortGamma-Ray Bursts , in: The Astrophysical Journal 824.1 (June 2016), p. L6 ,doi: 10.3847/2041-8205/824/1/l6 , url: https://doi.org/10.3847%2F2041-8205%2F824%2F1%2Fl6.

159

https://doi.org/10.1143/PTPS.90.1

https://academic.oup.com/ptps/article-pdf/doi/10.1143/PTPS.90.1/5201911/90-1.pdf

https://academic.oup.com/ptps/article-pdf/doi/10.1143/PTPS.90.1/5201911/90-1.pdf



https://doi.org/10.1088/0004-637x/692/1/411

https://doi.org/10.1088/0004-637x/692/1/411

https://doi.org/10.1088/0004-637x/692/1/411

https://doi.org/10.1088/0004-637x/692/1/411



https://doi.org/10.1103/RevModPhys.76.1143



https://arxiv.org/abs/gr-qc/0403007


https://doi.org/10.1088/2041-8205/732/1/l6

https://doi.org/10.1088/2041-8205/732/1/l6

https://doi.org/10.1088%2F2041-8205%2F732%2F1%2Fl6


https://doi.org/10.1046/j.1365-8711.2003.06579.x

https://arxiv.org/abs/astro-ph/0302301

https://arxiv.org/abs/astro-ph/0302301

https://doi.org/10.3847/2041-8205/824/1/l6



BIBLIOGRAPHY

[61] O. Sarbach et al. , Hyperbolicity of the Baumgarte-Shapiro-Shibata-Nakamurasystem of Einstein evolution equations , in: prd 66.6, 064002 (Sept. 2002),p. 064002 , doi: 10.1103/PhysRevD.66.064002 , arXiv: gr-qc/0205064[gr-qc].

[62] A. S. Schneider, L. F. Roberts, and C. D. Ott , Open-source nuclearequation of state framework based on the liquid-drop model with Skyrme in-teraction , in: Phys. Rev. C 96 (6 Dec. 2017), p. 065802 , doi: 10.1103/PhysRevC.96.065802 , url: https://link.aps.org/doi/10.1103/

PhysRevC.96.065802.

[63] M. Shibata and T. Nakamura , Evolution of three-dimensional gravita-tional waves: Harmonic slicing case , in: Phys. Rev. D 52 (10 Nov. 1995),pp. 5428–5444 , doi: 10.1103/PhysRevD.52.5428 , url: https://link.aps.org/doi/10.1103/PhysRevD.52.5428.

[64] D. M. Siegel et al. , Recovery Schemes for Primitive Variables in General-relativistic Magnetohydrodynamics , in: The Astrophysical Journal 859.1 (May2018), p. 71 , doi: 10.3847/1538-4357/aabcc5 , url: https://doi.org/10.3847/1538-4357/aabcc5.

[65] Simfactory , url: http://simfactory.org/.

[66] StellarCollapse , url: https://stellarcollapse.org/.

[67] Theoretical and Computational Astrophysics Network (TCAN) , url: https://compact-binaries.org/.

[68] A. Tomasiello , General Relativity , Lecture notes , url: https://virgilio.mib.infn.it/~atom/teaching.html.

[69] E. F. Toro , Riemann Solvers and Numerical Methods for Fluid Dynamics:A Practical Introduction , Berlin, Heidelberg: Springer Berlin Heidelberg,2009 , isbn: 978-3-540-49834-6 , doi: 10.1007/b79761 , url: https://doi.org/10.1007/b79761.

[70] VisIt , url: https://wci.llnl.gov/simulation/computer- codes/

visit.

[71] R. M. Wald , General Relativity , Chicago, USA: Chicago Univ. Pr., 1984 ,doi: 10.7208/chicago/9780226870373.001.0001.

[72] M. Zilhao and F. Loffler , An Introduction to the Einstein Toolkit , in:International Journal of Modern Physics A 28, 1340014-126 (Sept. 2013),pp. 1340014–126 , doi: 10.1142/S0217751X13400149 , arXiv: 1305.5299[gr-qc].

160




https://doi.org/10.1103/PhysRevC.96.065802

https://doi.org/10.1103/PhysRevC.96.065802

https://link.aps.org/doi/10.1103/PhysRevC.96.065802

https://link.aps.org/doi/10.1103/PhysRevC.96.065802




https://doi.org/10.3847/1538-4357/aabcc5



http://simfactory.org/

https://stellarcollapse.org/

https://compact-binaries.org/

https://compact-binaries.org/

https://virgilio.mib.infn.it/~atom/teaching.html

https://virgilio.mib.infn.it/~atom/teaching.html

https://doi.org/10.1007/b79761



https://wci.llnl.gov/simulation/computer-codes/visit

https://wci.llnl.gov/simulation/computer-codes/visit

https://doi.org/10.7208/chicago/9780226870373.001.0001

https://doi.org/10.1142/S0217751X13400149



BIBLIOGRAPHY

[73] J. Zrake and A. I. MacFadyen , Magnetic Energy Production by Turbu-lence in Binary Neutron Star Mergers , in: The Astrophysical Journal 769.2(May 2013), p. L29 , doi: 10.1088/2041-8205/769/2/l29 , url: https://doi.org/10.1088/2041-8205/769/2/l29.

161

https://doi.org/10.1088/2041-8205/769/2/l29

https://doi.org/10.1088/2041-8205/769/2/l29

https://doi.org/10.1088/2041-8205/769/2/l29

Documents

General Relativistic Simulations of Binary Neutron Star Mergers … Summary Binary neutron star (‘BNS’) mergers are violent cosmic events whose rst detec-tion (on 17th August 2017,