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Ulrich Sperhake

Friedrich-Schiller Universität Jena

Numerical simulations of high-energy collisionsNumerical simulations of high-energy collisionsOf black holeOf black hole

CENTRA IST SeminarLisbon, 29th February 2008

Work supported by the SFB TR7

Work with E.Berti, V.Cardoso, J.A.Gonzalez, F.Pretorius

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OverviewOverview

Introduction

Results

Summary

A brief history of BH simulations Results following the recent breakthrough

Black holes in physics

Ingredients of numerical relativity

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1. Black holes in physics1. Black holes in physics

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Black Holes predicted by GRBlack Holes predicted by GR

valuable insight into theory

Black holes predicted by Einstein’s theory of relativity

Vacuum solutions with a singularity

For a long time: mathematical curiosity

Term “Black hole” by John A. Wheeler 1960s

but no real objects in the universe

That picture has changed dramatically!

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How to characterize a black hole?How to characterize a black hole?

Consider light cones

Outgoing, ingoing light

Calculate surface area

of outgoing light

Expansion:=Rate of

change of that area

Apparent horizon:=

Outermost surface with zero expansion

“Light cones tip over” due to curvature

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Black Holes in astrophysicsBlack Holes in astrophysics

Structure formation in the early universe

Black holes are important in astrophysics

Structure of galaxies

Starbursts

Black holes found at centers of galaxies

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Black Holes in astrophysicsBlack Holes in astrophysics

Black holes are important in astrophysics

Gravitational lenses

Important sources of electromagnetic radiation

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Fundamental physics of black holesFundamental physics of black holes

Allow for unprecedented tests of fundamental physics

Strongest sources of Gravitational Waves (GWs) Test alternative theories of gravity No-hair theorem of GR Production in Accelerators

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Gravitational wave physicsGravitational wave physics

Accelerating bodies produce GWs

Weber 1960s Bar detector Claimed detection probably not real

GWs displace particles

GW observatories: GEO600, LIGO, TAMA, VIRGO Bar detectors

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The big pictureThe big picture

Model GR (NR) PN Perturbation theory Alternative Theories?

External Physics Astrophysics Fundamental Physics Cosmology

DetectorsPhysical system

describes

observe

test Provide info

Help d

etec

tion

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2. General relativity2. General relativity

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The framework: General RelativityThe framework: General Relativity

Curvature generates acceleration

Description of geometry

R

g

Metric

Connection

Riemann Tensor

“geodesic deviation”

No “force” !!

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The metric defines everythingThe metric defines everything

Christoffel connection

Covariant derivative

gggg 2

1

R

TTTT

Riemann Tensor

Geodesic deviation

Parallel transport

…

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How to get the metric?How to get the metric?

The metric must obey the Einstein Equations

Ricci-Tensor, Einstein-Tensor, Matter tensor

T

RgRG

RR

2

1

“Trace-reversed’’ Ricci

Einstein Equations TG 8

Solutions: Easy! Take metric Calculate Use that as matter tensor

G

Physically meaningful solutions: Difficult!

“Matter”

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The Einstein equations in vacuumThe Einstein equations in vacuum

Field equations:

0R

Second order PDEs for the metric components

Analytic solutions: Minkowski, Schwarzschild, Kerr, Robertson-Walker,…

Numerical methods necessary for general scenarios!

System of equations extremely complex: Pile of paper!

“Spacetime tells matter how to move, matter tells spacetime how to curve”

Invariant under coordinate (gauge) transformations

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3. The basics of numerical relativity3. The basics of numerical relativity

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A list of tasksA list of tasks

Target: Predict time evolution of BBH in GR

Einstein equations: Cast as evolution system Choose specific formulation Discretize for Computer

Choose coordinate conditions: Gauge

Fix technical aspects: Mesh-refinement / spectral domains Excision Parallelization Find large computer

Construct realistic initial data

Start evolution and wait…

Extract physics from the dataGourgoulhon gr-qc/0703035

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3.1. The Einstein equations3.1. The Einstein equations

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3+1 decomposition3+1 decomposition

GR: “Space and time exist together as Spacetime’’ Numerical relativity: reverse this process! ADM 3+1 decomposition Arnowitt, Deser, Misner (1962)

York (1979)Choquet-Bruhat, York (1980)

3-metric ijlapse

shift

i

lapse, shift Gauge

Einstein equations 6 Evolution equations

4 Constraints

Constraints preserved under evolution!

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The ADM equationsThe ADM equations

Time projection 0 nnR

02 ijijKKKR

Mixed projection 0

nR

0 ijj

i KDKD

Spatial projection 0

R

]2[)( KKKKRDDKL ijjm

imijjiijt

Hamiltonian constraint

Momentum constraints

Evolution equations

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The structure of the ADM equationsThe structure of the ADM equations

Constraints:

t They do not contain time derivatives

They must be satisfied on each slice

The Bianchi identities propagate the constraints:

If they are satisfied initially, they are always satisfied

Evolution equations:

Commonly written as first order system

ijijt KL 2)(

]2[)( KKKKRDDKL ijjm

imijjiijt

Gauge:

Equations say nothing about lapse and shift !

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The ADM equations as an initial value problemThe ADM equations as an initial value problem

Entwicklungsgleichungen

ijijt KL 2)(

]2[)( KKKKRDDKL ijjm

imijjiijt

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Alternatives to the ADM equationsAlternatives to the ADM equations

Unfortunately the ADM equations do not work in NR !!

Weak hyperbolicity: Nearby initial data can diverge

From each other super-exponentially

Many alternative formulations have been suggested

Two successful families so far

ADM based formulations: BSSN

Generalized harmonic formulations

Shibata & Nakamura ’95, Baumgarte & Shapiro ‘99

Choque-Bruhat ’62, Garfinkle ’04, Pretorius ‘05

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The BSSN equationsThe BSSN equations

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3.2. Gauge choices3.2. Gauge choices

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The gauge freedomThe gauge freedom

Remember: Einstein equations say nothing about i ,

Any choice of lapse and shift gives a solution

This represents the coordinate freedom of GR

Physics do not depend on

So why bother?

i ,

Avoid coordinate singularities!

Stop the code from running into the physical singularity

No full-proof recipe, but

Singularity avoiding slicing

Use shift to avoid coordinate stretching

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3.3. Initial data3.3. Initial data

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Initial data problemInitial data problem

Two problems: Constraints, realistic data

York-Lichnerowicz split

Rearrange degrees of freedom

ijij ~4

KAK ijijij 3

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Conformal transverse traceless Physical transverse traceless Thin sandwich

York, Lichnerowicz

Conformal flatness: Kerr is NOT conformally flat!

Non-physical GWs: problematic for high energy collisions!

Wilson, Mathews; York

O’Murchadha, York

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2 families of initial data2 families of initial data

Generalized analytic solutions:

Time-symmetric, -holes:

Spin, Momenta:

Punctures

Brill-Lindquist, Misner (1960s)

Bowen, York (1980)

Brandt, Brügmann (1997)

Isotropic Schwarzschild:

N

Excision Data: IH boundary conditions on excision surface

Meudon group; Cook, Pfeiffer; Ansorg

Quasi-circular: Effective potential PN parameters helical Killing Vektor

2224

22

21

drdr

r

Mdtds

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4. Extracting physics4. Extracting physics

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Basic assumptionsBasic assumptions

Extracting physics in NR is non-trivial !!

We assume that the ADM variables

ij

ij

i

K

Lapse

Shift

3-metric

Extrinsic curvature

are given on each hypersurface t

Even when using other formulations, the ADM variables

are straightforward to calculate

Newtonian quantities are not always well-defined !!

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Global quantitiesGlobal quantities

ADM mass: Global energy of the spacetime

Total angular momentum of the spacetime

By construction all of these are time-independent !!

rS

lkijjikklij

rdSM lim

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1,,ADM

rS

mim

im

ri dSKKP lim

8

1

rS

nmn

mn

r

mii dSKKxJ lim

8

1

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Local quantitiesLocal quantities

Often impossible to define !!

Isolated horizon framework

Calculate apparent horizon

Ashtekar and coworkers

irreducible mass, momenta associated with horizon

16AH

irr

AM

22irr

22irr

2

4P

M

SMM

Total BH mass Christodoulou

Binding energy of a binary: 21ADMb MMME

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Gravitational wavesGravitational waves

Most important result: Emitted gravitational waves (GWs)

Newman-Penrose scalar

GWs allow us to measure

Radiated energy

Radiated momenta

Angular dependence of radiation

Predicted strain hh ,

radrad , JPradE

mnmnC4

Complex 2 free functions

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Angular dependenceAngular dependence

Waves are normally extracted at fixed radius exr

Decompose angular dependence

,,44 t

m

mm Yt,

24 ,

Spin-weighted spherical harmonics

Modes

, are viewed from the source frame !!

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5. Status of black hole simulations5. Status of black hole simulations

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A very brief historyA very brief history

Pioneers: Hahn and Lindquist 1960s, Eppley and Smarr 1970s

Problems:

Computer resources

Instabilities: Gauge, Equations

Breakthrough: Pretorius 2005, Brownsville, NASA Goddard 2005

AMR

Now about 10 codes

GHG

Moving puncture

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The BBH breakthroughThe BBH breakthrough

Simplest configuration

GWs circularize orbit quasi-circular initial data

Pretorius ‘05

Initial data: scalar field

Radiated energy

]%[

][ ex

ME

MR 25 50 75 100 4.7 3.2 2.7 2.3

Eccentricity

2.0...0e

BBH breakthrough

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Astrophysical binaries: basicsAstrophysical binaries: basics

Free parameters

NR can only simulate the last orbits

1: Total mass (merely a scaling factor)

6: Spin

1: Mass ratio

1: Eccentricity; GWs circularize orbit 0 e

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4 stages of BBH merger: Newtonian inspiral (not GW driven)

Post-Newtonian (PN) inspiral phase (GW driven)

merger (Numerical Relativity)

Ring-down (Perturbation Theory)

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Astrophysical binaries: ResultsAstrophysical binaries: Results

Recoil or kick

Unequal mass and/or spin kick

Equal-mass, non-spinning binaries of the total mass of the system

Good agreement with PN predictions for GW emission

GWs quadrupole dominated

%5rad E%98

Certain spin configurations km/s 4000 v

Enough to eject BHs from galaxies

Observations such large kicks not generic Spin-flip

Merger spin realignment X-shaped radio sources

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Zoom whirl orbitsZoom whirl orbits

1-parameter family of initial data: linear momentum

Pretorius & Khurana ‘07

Fine-tune parameter ”Threshold of

immediate merger”

Analogue in gedodesics !

Reminiscent of

”Critical phenomena”

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6. High energy collision6. High energy collision

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MotivationMotivation

No known mechanism to accelerate BHs to cv

Such collisions unlikely to occur in astrophysical scenarios

But:

Probe GR in the most dynamic range

Near the speed of light, structure is lost

Collisions might describe generic collisionscv

Test cosmic censorship

”Threshold of immediate merger” Pretorius 2006

Indication of unexpected phenomena

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Setup of the problemSetup of the problem

Take two black holes

Total rest mass 0,0,0 BA MMM

Linear momentum P

Initial position 0z

For now: Equal-mass, head-on

Target: GW energy

Mode distribution of GW energy

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Analytic studies of a single boosted BHAnalytic studies of a single boosted BH

Schwarzschild boosted with

Aichelburg-Sexl metric

cv

Planar shock wave travelling along const czt

Minkowskian everywhere else

No event horizon!

Analytically studied case: cv

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Analytic studies of a BH binaryAnalytic studies of a BH binary

Superpose two Aichelburg-Sexl metrics

Shock waves collide Penrose, Eardley & Giddings

Future trapped surface!

22 0

fin

MM

Max. gravitational radiation: Penrose 0 %29 M

Perturb superposed Aichelburg-Sexl metrics

D’Eath & Payne ’90s

First correction term reduces GWs to 0 %16 M

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Analytic studies of a BH binaryAnalytic studies of a BH binary

Instant Collision approximation

Applied to superposed Aichelburg-Sexl metricSmarr; Adler & Zeks

i) Flat at sufficiently low ddE

Energy spectrum

ii) Radiation isotropic in the limit

iii) Functional relation radE

iv) Multipole structure 2

1

lElm

Numerical simulations Grand Challenge ’90s

Limited accuracy, mild boosts

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Numerical simulationsNumerical simulations

LEAN code Sperhake ‘07

0ADM MM

BSSN formulation

Puncture initial data Brandt & Brügmann 1996

Based on the Cactus computational toolkit

Mesh refinement: Carpet Schnetter ‘04

Elliptic solver: TwoPunctures Ansorg 2005

Numerically very challenging!

Length scales:

Horizon Lorentz-contracted “Pancake”

Mergers extremely violent

Substantial amounts of unphysical “junk” radiation

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Case exampleCase example

93.20

M

M Boost

initial separation 0202Md

Total radiated energy ME %6.5rad

Mode energy ME %9.420

Junk radiation radjunk 2EE

ME %49.040

ME %09.060

50

0 0MT

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0 6.29 MT

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0 6.82 MT

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0 9.114 MT

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0 3.137 MT

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0 1.156 MT

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0 9.201 MT

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0 2.252 MT

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0 6.291 MT

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0 6.405 MT

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ModesModes

0 111

3.1

Md

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ModesModes

0 222

3.1

Md

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ModesModes

0 192

0.2

Md

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ModesModes

0 240

0.2

Md

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ModesModes

0 162

9.2

Md

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ModesModes

0 202

9.2

Md

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Energy spectrumEnergy spectrum

0.2

Junk radiation affects spectrum for small separations

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ConclusionsConclusions

Introduction to Numerical Relativity

NR has solved the binary problem

Key results on kicks, spin-flips, PN, data analysis

High energy collisions

ME %29rad Penrose limit

Numerical simulations very difficult (junk radiation!)

Merger radiation for ME %5.5rad 3

Todo: Improve accuracy, larger

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Gravitational recoilGravitational recoil

Anisotropic emission of GWs radiates momentum recoil of remaining system

Leading order: Overlap of Mass-quadrupole with octopole/flux-quadrupole Bonnor & Rotenburg ’61, Peres ’62, Bekenstein ‘73

Merger of galaxies

Merger of BHs

Recoil

BH kicked out?

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Gravitational recoilGravitational recoil

Ejection or displacement of BHs has repercussions on:

Escape velocities

km/s 30Globular clustersdSphdE

Giant galaxies

km/s 10020 km/s 300100

km/s 1000

Structure formation in the universe

BH populations

Growth history of Massive Black Holes

IMBHs via ejection?

Structure of galaxies

Merrit et al ‘04

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Kicks of non-spinning black holesKicks of non-spinning black holes

Parameter study Jena ‘07

4...1/ 21 MM

3/ 21 MMkm/s 178

Target: Maximal Kick

Mass ratio:

150,000 CPU hours

Maximal kick for

Convergence 2nd order

%25 %,3 radrad JE

Spin 7.0...45.0

Simulations PSU ’07, Goddard ‘07

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Super KicksSuper Kicks

Test Rochester-Prediction: Jena ‘07

Two models:LeanBam

Model I:

Lean code Moving puncture method: version Not quasi-circular Resolutions:

-e

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1,

44

1,

36

1h

Model II: Safety check with independent code BAM

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ConvergenceConvergence

Discretization error: km/s 43v