Numerical Relativity: generating space-time on computers

Yuichiro Sekiguchi (Toho University)

MSJ-SI: The Role of Metrics in the Theory of Partial Differential Equations

Hokkaido University, 2018.07.02-13

Notation and Convention

Spacetime 𝑀,𝑔𝑎𝑏 : spacetime manifold 𝑀 and the metric 𝑔𝑎𝑏

The signature of the metric : − + + +

We will use the abstract index notation for tensor fields and Einstein’s summation convention

Index of a tensor on 𝑀 is raised/lowered by 𝑔𝑎𝑏/𝑔𝑎𝑏 as usual

Covariant derivative associated with 𝑔𝑎𝑏 is 𝛻𝑎 (𝛻𝑐𝑔𝑎𝑏 = 0)

Riemann tensor for the spacetime : 4 𝑅𝑎𝑏𝑐 𝑑

Ricci tensor and Ricci scalar : 4𝑅𝑎𝑐 = 4𝑅𝑎𝑏𝑐 𝑏, 4𝑅 = 𝑔𝑎𝑏 4𝑅𝑎𝑏

Symmetric and anti-symmetric notation

𝑇(𝑎1⋯𝑎𝑛) =1

𝑛! 𝑇𝑎𝜋(1)⋯𝑎𝜋(𝑛)

𝑛

, 𝑇[𝑎1⋯𝑎𝑛] =1

𝑛! sgn(𝜋)𝑇𝑎𝜋(1)⋯𝑎𝜋(𝑛)

𝑛

We use the geometrical unit : 𝑐 = 𝐺 = 1

Einstein’s equation

2nd order quasilinear partial differential equations for the spacetime metric 𝑔𝑎𝑏

𝐺𝑎𝑏: Einstein tensor

𝐺𝑎𝑏 =8𝜋𝐺

𝑐4𝑇𝑎𝑏

𝐺𝑎𝑏 = 𝑅𝑎𝑏 −1

2𝑔𝑎𝑏𝑅

𝑅𝛼𝛽 = 𝜕𝛾Γ 𝛼𝛽𝛾

− 𝜕𝛼Γ 𝛾𝛽𝛾

+ Γ 𝛼𝛽𝛿 Γ 𝛿𝛾

𝛾− Γ 𝛾𝛽

𝛿 Γ 𝛿𝛼𝛾

Γ 𝛼𝛽𝛾

=1

2𝑔𝛾𝛿(𝜕𝛼𝑔𝛽𝛿 + 𝜕𝛽𝑔𝛼𝛿 − 𝜕𝛿𝑔𝛼𝛽)

𝑅𝑎𝑏: Ricci tensor

𝑅 = 𝑔𝑎𝑏𝑅𝑎𝑏 ∶ Ricci scalar

Γ 𝛼𝛽𝛾

∶ Christoffel symbol

𝑇𝑎𝑏: stress energy momentum tensor

𝑅𝜇𝜈 = −1

2𝑔𝛼𝛽 𝜕𝛼𝜕𝛽𝑔𝜇𝜈 + 𝜕𝜇𝜕𝜈𝑔𝛼𝛽 − 2𝜕𝛽𝜕(𝜇𝑔𝜈)𝛼 + 𝐹𝜇𝜈(𝑔, 𝜕𝑔)

Numerical Relativity (NR) Solving Einstein’s equation on computer

Einstein’s equation in full covariant form are 2nd order quasilinear partial differential equations

The solution, spacetime metric 𝑔𝑎𝑏 is not a dynamical object in the sense that it represents the full geometry of 𝑀 as the metric of the two-sphere does

To reveal the dynamical nature of Einstein’s equation, we must break the four dimensional general covariance and exploit the special nature of time and space

One method is 3+1 decomposition of spacetime (𝑀, 𝑔𝑎𝑏) into a foliation {Σ}, the induced metric 𝛾𝑎𝑏 on Σ, and the extrinsic curvature 𝐾𝑎𝑏 of Σ

Then Einstein’s equation is posed as a Cauchy problem for initial data (Σ, 𝛾𝑎𝑏 , 𝐾𝑎𝑏), which can be solved numerically on computers

We also need to solve the matter equation 𝛻𝑎𝑇𝑎𝑏 = 0

Initial value formulation of general relativity

In other theories of classical physics, we consider the time evolution of quantities from their initial conditions in the given the spacetime background (metric)

In general relativity, however, we are solving for the spacetime (metric) itself. So, we must view general relativity as describing the time evolution of some quantity (initial value formulation)

The initial value formulation must be well posed:

Small changes in initial data should produce only correspondingly “small changes” in the solution

Changes in the initial data in a region, 𝑆, should not produce any changes in the solution outside the causal future, 𝐽+(𝑆), of this region

Existence and uniqueness of solution

Thanks to the general covariance equipped by general relativity, we can employ any convenient coordinates (gauge degrees of freedom)

The local existence and uniqueness of Einstein’s equation can be proven [1] in harmonic coordinates : 𝐻𝜇 = 𝛻𝑎𝛻

𝑎𝑥𝜇 = 0 𝛻𝑎 is the covariant derivative associated with the spacetime metric 𝑔𝑎𝑏

In the harmonic coordinates, Einstein’s equation in vacuum becomes

Then, the theorem due to Leray [2] can be applied to prove local existence and uniqueness of solution of Einstein’s equation [3].

Global existence and uniqueness were also proved in [4]

𝑔𝛼𝛽𝜕𝛼𝜕𝛽𝑔𝜇𝜈 = 𝐹 𝜇𝜈(𝑔, 𝜕𝑔)

[1] Choquet-Bruhat, Y. (1952) Acta. Math. 88, 141.

[2] Leray, J. (1952) “Hyperbolic Differential Equations”, (Princeton University).

[3] Dionne, P. (1962) J. Anal. Math. Jerusalem 10, 1.

[4] Choquet-Bruhat, Y. and Geroch, R. P. (1969) Commun. Math. Phys. 14, 329.

Theorem (Leray 1952) : Let (𝜙0)1,⋯ , (𝜙0)𝑛 be any solution of the quasilinear hyperbolic system

𝑔𝑎𝑏 𝑥; 𝜙𝑗; 𝛻𝑐𝜙𝑗 𝛻𝑎𝛻𝑏𝜙𝑖 = 𝐹𝑖 𝑥; 𝜙𝑗; 𝛻𝑐𝜙𝑗 (1)

Where 𝛻𝑎 is any derivative operator and 𝐹𝑖 is an analytic function

and let (𝑔0)𝑎𝑏 = 𝑔𝑎𝑏 𝑥; (𝜙0)𝑗; 𝛻𝑐(𝜙0)𝑗 . Suppose (𝑀, (𝑔0)𝑎𝑏) is globally hyperbolic. Let Σ be a smooth spacelike Cauchy surface for (𝑀, (𝑔0)𝑎𝑏).

Then, the initial value formulation is well posed:

For Initial data sufficiently close to the initial data for (𝜙0)1,⋯ , (𝜙0)𝑛, there exist an open neighborhood 𝑂 of Σ such that eq. (1) has a solution 𝜙1, ⋯ , 𝜙𝑛, in 𝑂 and (𝑂, 𝑔𝑎𝑏(𝑥; 𝜙𝑗; 𝛻𝑐𝜙𝑗)) is globally hyperbolic.

The solution is unique in 𝑂 and propagate causally:

If the initial data for 𝜙′1, ⋯ , 𝜙′𝑛 agree with that of 𝜙1,⋯ , 𝜙𝑛 on a subset 𝑆 of Σ, then the solutions agree on 𝑂⋂𝐷+(𝑆), where 𝐷+(𝑆) is the future domain of dependence for 𝑆.

The solutions depend continuously on the initial data

Theorem (summarized in Wald 1984 [5]): Let Σ and 𝛾𝑎𝑏 be a 3-dimensional 𝐶∞ manifold and a smooth metric on Σ and let 𝐾𝑎𝑏 be a smooth symmetric tensor field on Σ.

𝛾𝑎𝑏 and 𝐾𝑎𝑏 satisfy the constraint equations

Then there exist a unique 𝐶∞ spacetime, (𝑀, 𝑔𝑎𝑏), called the maximal Cauchy development of (Σ, 𝛾𝑎𝑏 , 𝐾𝑎𝑏) satisfying

(𝑀, 𝑔𝑎𝑏) is a solution of Einstein’s equation and globally hyperbolic

The induced metric and extrinsic curvature of Σ are 𝛾𝑎𝑏 and 𝐾𝑎𝑏

Every other spacetime can be mapped isometrically into a subset of (𝑀, 𝑔𝑎𝑏)

The solution 𝑔𝑎𝑏 depends continuously on the initial data (𝛾𝑎𝑏 , 𝐾𝑎𝑏) on Σ.

Furthermore, (𝑀, 𝑔𝑎𝑏) satisfies the domain of dependence property desired in general relativity:

Suppose (Σ, 𝛾𝑎𝑏, 𝐾𝑎𝑏) and (Σ′, 𝛾′𝑎𝑏, 𝐾′𝑎𝑏) are initial data sets with maximal

developments (𝑀, 𝑔𝑎𝑏) and 𝑀′, 𝑔′𝑎𝑏

. Suppose there is a diffeomorphism

between 𝑆 ⊂ Σ and 𝑆′ ⊂ Σ′ which carries (𝛾𝑎𝑏, 𝐾𝑎𝑏) on 𝑆 into (𝛾′𝑎𝑏, 𝐾′𝑎𝑏) on 𝑆′. Then 𝐷(𝑆) in (𝑀, 𝑔𝑎𝑏) is isometric to 𝐷(𝑆′) in (𝑀′, 𝑔′𝑎𝑏)

[5] Wald, R. M. (1984), “General Relativity” (Univ. of Chicago press)

3+1 decomposition of spacetime

Foliation {Σ} of 𝑀 is a family of spacelike hypersurfaces (slices) which do not intersect each other and fill completely 𝑀

We assume that spacetime is globally hyperbolic. Then each Σ is a Cauchy surface which is parametrized by a global time function 𝑡 and denoted as Σ𝑡 [1]

Now, foliation is characterized by the gradient 1-form, Ω𝑎 = 𝛻𝑎𝑡

The norm of Ω𝑎 is related to a function called the lapse function, 𝛼:

𝑔𝑎𝑏Ω𝑎Ω𝑏 = −𝛼−2

𝛼 characterize the proper time between Σ

Let us introduce the normalized 1-form

𝑛𝑎 = −𝛼Ω𝑎

　taa

[1] e.g. Wald, R. M. (1984), “General Relativity” (Univ. of Chicago press)

The induced metric on Σ

The spatial metric 𝛾𝑎𝑏 induced onto Σ is defined as

𝛾𝑎𝑏 = 𝑔𝑎𝑏 + 𝑛𝑎𝑛𝑏

Using 𝛾𝑎𝑏, a tensor on 𝑀 is decomposed into components tangent and normal to Σ, using the projection operator

⊥𝑏𝑎 = 𝛾𝑏

𝑎 = 𝛿𝑏𝑎 + 𝑛𝑎𝑛𝑏

The projection of a tensor 𝑇 𝑏1⋯𝑏𝑠

𝑎1⋯𝑎𝑟 on 𝑀 into a tensor on Σ is

⊥ 𝑇 𝑏1⋯𝑏𝑠

𝑎1⋯𝑎𝑟 =⊥𝑐1

𝑎1 ⋯ ⊥𝑐𝑟

𝑎𝑟⊥𝑏1

𝑑1 ⋯ ⊥𝑏𝑠

𝑑𝑠 𝑇 𝑑1⋯𝑑𝑠

𝑐1⋯𝑐𝑟

It is easy to check ⊥ 𝑔𝑎𝑏 = 𝛾𝑎𝑏

The covariant derivative operator 𝐷𝑎 associated with 𝛾𝑎𝑏 acting on tensors on Σ is

𝐷𝑐𝑇 𝑏1⋯𝑏𝑠

𝑎1⋯𝑎𝑟 =⊥ 𝛻𝑐𝑇 𝑏1⋯𝑏𝑠

𝑎1⋯𝑎𝑟

It is easy to check Leibnitz’s rules holes and 𝐷𝑐𝛾𝑎𝑏 = 0

Intrinsic and extrinsic curvature for Σ

Riemann tensor 𝑅𝑎𝑏𝑐 𝑑 and the extrinsic curvature tensor 𝐾𝑎𝑏 for Σ are

defined, respectively, by

𝐷𝑎𝐷𝑏 − 𝐷𝑏𝐷𝑎 𝑤𝑐 = 𝑅𝑎𝑏𝑐 𝑑𝑤𝑐 , 𝐾𝑎𝑏 =−⊥ 𝛻(𝑎𝑛𝑏) = −

1

2ℒ𝑛𝛾𝑎𝑏

ℒ𝑛 is Lie derivative with respect to the unit normal vector 𝑛𝑎 = 𝑔𝑎𝑏𝑛𝑏

𝐾𝑎𝑏 is ‘velocity’ of 𝛾𝑎𝑏

The geometry of Σ is described by 𝛾𝑎𝑏 and 𝐾𝑎𝑏. In order that the foliation fits the spacetime, 𝛾𝑎𝑏 and 𝐾𝑎𝑏 must satisfy Gauss, Codazzii, and Ricci relations, respectively, given by

⊥4 𝑅𝑎𝑏𝑐𝑑 = 𝑅𝑎𝑏𝑐𝑑 + 𝐾𝑎𝑐𝐾𝑏𝑑 − 𝐾𝑎𝑑𝐾𝑏𝑐

⊥4 𝑅𝑎𝑏𝑐 𝑑𝑛𝑑 = 𝐷𝑏𝐾𝑎𝑐 − 𝐷𝑎𝐾𝑏𝑐

⊥4 𝑅𝑎𝑏𝑐𝑑𝑛𝑏𝑛𝑑 =⊥ ℒ𝑛𝐾𝑎𝑐 + 𝐾𝑎𝑏𝐾𝑐

𝑏 + 𝐷𝑐𝑎𝑐 + 𝑎𝑎𝑎𝑐

Where 𝑎𝑏 = 𝑛𝑐𝛻𝑐𝑛𝑏 = 𝐷𝑏ln 𝛼

Geometrical meaning of 𝐾𝑎𝑏

𝐾𝑎𝑏 is associated with the difference between the original and parallel-transported unit normal

vector 𝑛𝑎 on Σ𝑡 For a slice Σ𝑡 embedded in the

spacetime in a ‘flat’ manner, 𝐾𝑎𝑏 = 0

２

How Σ𝑡 is embedded in spacetime （curvature seen from outside） ⇒ extrinsic curvature

bx

bb xx

)( ba xn)(

bba xxn

Parallel transport b

aba xKn

‘Evolution’ vector and general covariance

Any vector 𝑡𝑎 dual to Ω𝑎(𝑡𝑎Ω𝑎 = 1) can be the evolution vector

A simple candidate is 𝑡𝑎 = 𝛼𝑛𝑎

Note that the unit normal vector 𝑛𝑎 can not be an evolution vector because ℒ𝑛 ⊥𝑏

𝑎≠ 0 so that Lie derivative (evolution) with respect to 𝑛𝑎 of a tensor tangent to Σ is not a tensor tangent to Σ

On the other hand, ℒ𝛼𝑛 ⊥𝑏𝑎= 0

𝜮𝒕+𝒅𝒕

𝜮𝒕

𝜷𝒊𝒅𝒕 ∶ shift vector

𝒏𝒂: unit normal

𝜶𝒏𝒂𝒅𝒕 ∶ lapse function

𝒕𝒂 : evolution vector We can add a spatial vector, called shift vector 𝛽𝑎 (𝛽𝑎Ω𝑎 = 0), to 𝑡𝑎

Thus the general evolution vector is 𝑡𝑎 = 𝛼𝑛𝑎 + 𝛽𝑎

The degree of freedom in choosing the evolution vector is originated in the general covariance of general relativity

3+1 decomposition of Einstein’s equation Now we proceeds to 3+1 decomposition of Einstein’s equation

𝐺𝑎𝑏 = 4𝑅𝑎𝑏 −1

2𝑔𝑎𝑏

4𝑅 =8𝜋𝐺

𝑐4𝑇𝑎𝑏 = 8𝜋𝑇𝑎𝑏

The decomposition of the stress-energy-momentum tensor 𝑇𝑎𝑏 is 𝑇𝑎𝑏 = 𝐸𝑛𝑎𝑛𝑏 + 2𝑛(𝑎𝑃𝑏) + 𝑆𝑎𝑏

where 𝐸 = 𝑇𝑎𝑏𝑛𝑎𝑛𝑏, 𝑃𝑎 =−⊥ 𝑇𝑎𝑏𝑛

𝑏 , and 𝑆𝑎𝑏 =⊥ 𝑇𝑎𝑏 are the energy density, momentum density, and stress tensor measured by the Eulerian observer 𝑛𝑎

3+1 decompositions of Einstein’s equation corresponds to 𝐺𝑎𝑏𝑛𝑎𝑛𝑏,

⊥ 𝐺𝑎𝑏𝑛𝑏, and ⊥ 𝐺𝑎𝑏, which respectively give (𝐾 = 𝐾𝑎

𝑎 and 𝑆 = 𝑆𝑎𝑎)

𝑅 + 𝐾2 − 𝐾𝑎𝑏𝐾𝑎𝑏 = 16𝜋𝐸 (1)

𝐷𝑏𝐾𝑎𝑏 − 𝐷𝑎𝐾 = 8π𝑃𝑎 (2)

ℒ𝑡 − ℒ𝛽 𝐾𝑎𝑏 = −𝐷𝑎𝐷𝑏𝛼 + 𝛼 𝑅𝑎𝑏 + 𝐾𝐾𝑎𝑏 − 2𝐾𝑎𝑐𝐾𝑏𝑐

−4π𝛼(2𝑆𝑎𝑏 − 𝛾𝑎𝑏(𝑆 − 𝐸))

Note that Eqs. (1) and (2) do not contain time derivative of 𝛾𝑎𝑏 and 𝐾𝑎𝑏 (constraints)

Introducing coordinates

We choose the evolution vector 𝑡𝑎 as the time basis vector: (𝜕𝑡)𝑎 = 𝑡𝑎

We also introduce the natural spatial basis vectors (𝜕𝑖)𝑎 on Σ

(𝜕𝑖)𝑎 are Lie dragged along 𝑡𝑎 : ℒ𝑡(𝜕𝑖)

𝑎 = 0

Then (𝜕𝑖)𝑎 remains purely spatial because ℒ𝑡(Ω𝑎 𝜕𝑖)

𝑎 = 0

Dual basis (𝑑𝑥𝜇)𝑎 are defined in the standard manner

Components of geometrical quantities are (e.g., 𝑔𝑎𝑏 = 𝑔𝜇𝜈(𝑑𝑥𝜇)𝑎(𝑑𝑥𝜈)𝑏)

It can be shown 𝛾𝑖𝑘𝛾𝑘𝑗 = 𝛿𝑗𝑖 so that indices of spatial tensors can be lowered

and raised by the spatial metric

𝑛𝜇 =1

𝛼(1,−𝛽𝑖)𝑇

𝑛𝜇 = (−𝛼, 0,0,0)

𝑔𝜇𝜈 =1

𝛼2−1 𝛽𝑖

𝛽𝑖 𝛼2 − 𝛽𝑖𝛽𝑗

𝑔𝜇𝜈 =−𝛼2 + 𝛽𝑘𝛽𝑘 𝛽𝑖

𝛽𝑖 𝛾𝑖𝑗

ADM (Arnowitt-Deser-Misner) formulation

The 3+1 decompositions of Einstein’s equation together with the definition of 𝐾𝑎𝑏 (tensor equations) are now a set of partial differential equations for 𝛾𝑎𝑏 and 𝐾𝑎𝑏 (ADM system) [2] :

𝑅 + 𝐾2 − 𝐾𝑖𝑗𝐾𝑖𝑗 = 16𝜋𝐸 (3)

𝐷𝑗𝐾𝑖𝑗− 𝐷𝑖𝐾 = 8π𝑃𝑖 (4)

𝜕𝑡 − ℒ𝛽 𝐾𝑖𝑗 = −𝐷𝑖𝐷𝑗𝛼 + 𝛼 𝑅𝑖𝑗 + 𝐾𝐾𝑖𝑗 − 2𝐾𝑖𝑘𝐾𝑗𝑘

−4π𝛼(2𝑆𝑖𝑗 − 𝛾𝑖𝑗(𝑆 − 𝐸))

𝜕𝑡 − ℒ𝛽 𝛾𝑖𝑗 = −2α𝐾𝑖𝑗

Eqs. (3) and (4) are elliptic constraint equations (Hamiltonian and momentum constraints, respectively) that 𝛾𝑎𝑏 and 𝐾𝑎𝑏 must satisfy on each slice Σ

The remaining equations are the evolution equations for 𝛾𝑖𝑗 and 𝐾𝑖𝑗

Note that Einstein’s equation tells us nothing about how to specify the gauge

degrees of freedom 𝛼 and 𝛽𝑖, as expected from general covariance

[2] York, J. W. (1978), in “Sources of Gravitational Radiation” eds. Smarr, L. L. (Cambridge Univ. press)

Evolution of constraints The “evolution” equations for the Hamiltonian (𝐶𝐻) and Momentum

(𝐶𝑀𝑖 ) constraints are

Where 𝐹𝑖𝑗 is the spatial projection of the evolution equation

The evolution equations show that the constraints are satisfied, if

They are satisfied initially and Einstein’s equation is solved correctly (𝐹𝑎𝑏 = 0)

Numerically, however, constraint violation may develop and simulations may crash in a short time

Constraints are elliptic equation : very time-consuming to solve

numerically (solve them at initial, but not solved in time evolution)

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Generating spacetime

（if 𝜶, 𝜷𝒂 are given） we can construct the future slice Σ𝒕+𝒅𝒕 from Σ𝑡

spatial metric 𝜸𝒂𝒃 on 𝜮𝒕 and its ‘velocity’ 𝑲𝒂𝒃~𝜸 𝒂𝒃 is given as initial data

𝛾𝑎𝑏 and 𝐾𝑎𝑏 must satisfy the constraints (solving the elliptic equations)

3+1 decomposed Einstein’s equation provide their time developments

𝜸𝒂𝒃 𝑲𝒂𝒃~𝜸 𝒂𝒃

𝒈𝒂𝒃

𝚺𝒕

𝚺𝒕+𝒅𝒕

Spacetime is generated if we can construct 𝑔𝑎𝑏 from 𝜸𝒂𝒃, 𝑲𝒂𝒃

We need prescriptions to specify 𝛼 and 𝛽𝑖 (coordinate problem in NR)

𝒅𝒙𝒊

𝜷𝒊𝒅𝒕 + 𝒅𝒙𝒊

𝒅𝒔

𝜷𝒊𝒅𝒕 ∶ 𝐬𝐡𝐢𝐟𝐭 𝐯𝐞𝐜𝐭𝐨𝐫 𝒕𝒂 : 𝐭𝐢𝐦𝐞 𝐚𝐱𝐢𝐬

Lapse function

𝒅𝒔𝟐 = − 𝜶𝒅𝒕 𝟐 + 𝜸𝒊𝒋(𝜷𝒊𝒅𝒕 + 𝒅𝒙𝒊)(𝜷𝒋𝒅𝒕 + 𝒅𝒙𝒋)

Gauge conditions Associated directly with the general covariance in general relativity,

there are degrees of freedom in choosing coordinates (gauge freedom)

Slicing condition is a prescription of choosing the lapse function 𝛼

Shift condition is that of choosing the shift vector 𝛽𝑎

Einstein’s equations say nothing about the gauge conditions

Choosing “good” gauge conditions are very important to achieve stable and robust numerical simulations

An improper slicing conditions in a stellar-

collapse problem will lead to appearance of

(coordinate and physical) singularities

Also, the shift vector is important in resolving

the frame dragging effect in simulations of

compact binary merger

Almost all choices are bad …

Collapsing star

Event horizon

singularity

t

space

𝛼 = 1 slicing

Geodesic slicing =1, i=0

In the geodesic slicing, the evolution equation of the trace of the extrinsic curvature is

For normal matter (which satisfies the strong energy condition), the right-

hand-side is positive

Thus the expansion of time coordinate −𝐾 = 𝛻𝑎𝑛𝑎 decreases

monotonically in time

In terms of the volume element 𝛾, (𝛾 = det 𝛾𝑎𝑏) , this means that the volume element goes to zero, as

This behavior results in a coordinate singularity

As can be seen in this example, slicing condition is closely related to the trace

of the extrinsic curvature 𝐾

SEKKK ij

ijt 34

KDK k

kijt

ij

t 2

1ln

Maximal slicing (Smarr & York 1978) Because the decrease in time of the volume element 𝛾 results in a

coordinate singularity, let us maximize the volume element

We take the volume element of a 3D-domain 𝑆 and consider a variation along

the time vector

If 𝐾 = 0 on the slice, the volume is maximal Time evolution is delayed in strong gravity has strong singularity avoidance property But the normal vector gets focused ⇒ eventually simulation crash

Necessary to use 𝛽𝑎

Maximal slicing condition is elliptic equation for 𝛼

Hyperbolic lapse has been developed

Singularity Event horizon

Choice of the lapse function 𝜶

SxdSV 3][

SS

i

it xdKKxdSV 33 )(][ L

)](4)(0 SEKKDDK ij

ij

i

it LL

Utilizing the shift vector 𝜷𝒂

Distortion of the time vector is problematic Distortion of 𝑛𝑎 due to the black hole formation The dragging of the frame around a rotating object

We can use 𝛽𝑎 to minimize these distortions

Minimal distortion shift (Smarr & York 1978) The covariant derivative of any timelike unit vector

can be decomposed as (Helmholtz’s theorem)

Minimize distortion functional

Gives a condition for shift Vector elliptic equation Hyperbolic shift has been

developed

baabababba zhz 3

1

(shear) ,

(twist) ,

metric) (induced ,

TF

)(

][

baab

baab

baabab

z

z

zzgh

ion)(accelerat ,

)(expansion ,

a

c

ca

c

c

zz

z

3dxI ab

ab TF

)(

TFTF ~

2

1 baababtab nK L

ab

ab

b

ab

ab

bb

ab

c

ca

ac

c PKDDAADRDDDD 1622

Utilizing the shift vector 𝜷𝒂

Distortion of the time vector is problematic Distortion of 𝑛𝑎 due to the black hole formation The dragging of the frame around a rotating object

We can use 𝛽𝑎 to minimize these distortions

Minimal distortion shift (Smarr & York 1978) The covariant derivative of any timelike unit vector

can be decomposed as (Helmholtz’s theorem)

Minimize distortion functional

Gives a condition for shift Vector elliptic equation Hyperbolic shift has been

developed

baabababba zhz 3

1

(shear) ,

(twist) ,

metric) (induced ,

TF

)(

][

baab

baab

baabab

z

z

zzgh

ion)(accelerat ,

)(expansion ,

a

c

ca

c

c

zz

z

3dxI ab

ab TF

)(

TFTF ~

2

1 baababtab nK L

ab

ab

b

ab

ab

bb

ab

c

ca

ac

c PKDDAADRDDDD 1622

𝜷𝒊𝒅𝒕 ∶ shift vector

ADM formulation is unstable ! Numerical relativity simulations based on ADM formulation is unstable

This crucial limitation may be captured in terms of hyperbolicity

Consider a first-order system : 𝜕𝑡𝑢𝑖 + (𝐴𝑖𝑗)𝑐𝜕𝑐𝑢

𝑗 = 0. This system is called

Strongly hyperbolic : if a matrix (representation) of 𝐴 has real eigenvalues and a complete set of eigenvectors

Weakly hyperbolic : if 𝐴 has real eigenvalues but not a complete set of eigenvectors

Hyperbolicity is a key property for the stability

Strongly hyperbolic system is well-posed and only characteristic fields corresponding to negative eigenvalues need boundary conditions

Weakly hyperbolic system is not well-posed and the solution can be unbounded faster than exponential

Note that Einstein’s equation is nonlinear (2nd order quasi-linear) so that the above arguments may not be adopted directly

(a first order formulation version of) the ADM system is weakly hyperbolic

seeking (at least) strongly hyperbolic reformulation is a central issue in NR

We need formulations for the Einstein’s equation which is (at least)

strongly hyperbolic (in a linearized regime) (as ‘wave-like’ as possible)

Caution ! : better hyperbolicity is necessary condition, not sufficient

Let us consider Maxwell’s equation in flat spacetime to capture what we

should do to obtain a more stable system

−𝜕 𝐴𝑖𝑡2 + 𝛻𝑘𝛻𝑘𝐴𝑖 − 𝛻𝑖𝛻𝑘𝐴

𝑘 = 𝛻𝑖𝜕𝑡𝜙

𝜕𝜇𝐴𝜇 = 0 𝐹 = 𝛻𝑘𝐴

𝑘 𝛻𝑘𝐸𝑘 = 4𝜋𝜌𝑒

𝜕𝑡𝐹 = −𝛻𝑘𝐸𝑘 − 𝛻𝑘𝛻𝑘𝜙

𝜕𝑡𝐹 = −4𝜋𝜌𝑒 − 𝛻𝑘𝛻𝑘𝜙

Adopting better gauge Introduce new variables Utilize constraints

𝜕𝜇𝜕𝜇𝐴𝑖 = 𝛻𝑖𝜕𝑡𝜙 + 𝛻𝑖𝐹 𝜕𝜇𝜕𝜇𝐴𝑖 = 0

Lorenz gauge :

𝛻𝑘𝐸𝑘 = 4𝜋𝜌𝑒 𝛻𝑘𝐴

𝑘 = 0 Coulomb gauge :

𝜕𝜇𝜕𝜇𝐴𝑖 = 𝛻𝑖𝜕𝑡𝜙 𝜕𝑡𝐹 = −𝛻𝑘𝐸

𝑘 − 𝛻𝑘𝛻𝑘𝜙

Evolution eq for 𝐹

Divergence terms prevents the system from achieving better hyperbolicity

Three ways to achieve better hyperbolicity

Reformulating Einstein’s equation

Strongly/Symmetric hyperbolic reformulations of Einstein’s equation

Choosing a better gauge

Better hyperbolicity vs. Singularity avoidance/frame dragging

Generalized harmonic gauge ( Pretorius, CQG 22, 425 (2005) )

Z4 formalism ( Bona et al. PRD 67, 104005 (2003) )

Introducing new, independent variables BSSN ( Shibata & Nakamura PRD 52, 5428 (1995);

Baumgarte & Shapiro PRD 59, 024007 (1999) )

Kidder-Scheel-Teukolsky ( Kidder et al. PRD 64, 064017 (2001) ) symmetric hyp.

Bona-Masso ( Bona et al. PRD 56, 3405 (1997) )

Nagy-Ortiz-Reula ( Nagy et al. PRD 70, 044012 (2004) )

Using the constraint equations to improve the hyperbolicity

adjusted ADM/BSSN ( Shinkai & Yoneda, gr-qc/0209111 )

BSSN outperforms ! ( Alcubierre’s text book 2008 )

Exact reason is not clear (better hyperbolicity is necessary condition)

BSSN formulation (Shibata & Nakamura 1995; Baumgarte & Shapiro 1998)

Strategy: as wave-like as possible

Introduce new variables Γ𝑎 = 𝜕𝑏𝛾𝑎𝑏

Extract the ‘true’ degrees of freedom of gravity (GW)

Conformal decomposition by York (PRL 26, 1656 (1971); PRL 28, 1082 (1972))

the two degrees of freedom of the gravitational field are carried by the conformal equivalence classes of 3-metric, which are related each other by the conformal transformation :

𝜸𝒂𝒃 = 𝝍𝟒𝜸 𝒂𝒃

Extrinsic curvature is also conformally decomposed

Trace of K is associated with the lapse function (c.f. maximal slicing) ⇒ split

𝑲𝒂𝒃 = 𝝍𝟒𝑨 𝒂𝒃 +𝟏

𝟑𝜸𝒂𝒃𝑲

Reformulation based on new variables :

𝜓, 𝛾 𝑎𝑏, 𝐴 𝑎𝑏, 𝐾 = tr 𝐾 , Γ𝑎 = 𝜕𝑏𝛾𝑎𝑏

BSSN reformulation

0212

1~~~~~ 2

EKAARDD ij

ij

i

i

ii

j

ijij

j PKDDAAD

8

~ln

~~6

~~

k

kk

k

t K 6

1

6

1ln

k

kij

k

ijk

k

jikijijk

k

t A ~

3

2~~~2~

)](4 SEKKDDK ij

ij

i

ik

k

t

k

kij

k

ijk

k

jik

k

jikij

TF

ijijjiijk

k

t

AAA

AAAKSRDDA

~

3

2~~

~~2

~)8()(

~

TFTF4

Hamiltonian constraint is used

i

kj

jkk

kj

ijj

j

ii

j

ji

j

j

j

ij

j

ij

j

ijjki

jk

ii

k

k

t AKAAP

~~

3

1

3

2

~2~

3

2ln

~6

~~216

Momentum constraint is used

Long-lasting problem until 2005

Treatment of black hole

Excision boundary

Problem: Black holes contain physical singularities

Methods 1: Horizon excision The first breakthrough by

F. Pretorius in 2005

Generalized harmonic formulation

Phys. Rev. Lett. 95, 121101 (2005)

Methods 2: Moving puncture gauge

𝜕𝑡 − 𝛽𝑖𝜕𝑖 𝛼 = −2𝛼𝐾, 𝜕𝑡𝛽𝑖 =

3

4Γ𝑖 − 𝜂𝛽𝑖

These gauge conditions map only the spacetime outside the physical singularity

Campanelli et al. (BSSN)

Phys. Rev. Lett. 96, 111101, (2006)

Baker et al. (BSSN)

Phys. Rev. Lett. 96, 111102, (2006)

The year 2005 : Dawn of new era of BH simulations

Lapse function Wyle scalar

BH excision: Very complicated actually

Excision boundary

Excision boundary

A milestone simulation by SXS collaboration:

Long-term simulation of BH-BH merger

Scheel et. al., Phys. Rev. D 79, 024003 (2009); Cohen et. al., Class. Quantum Grav. 26 035005 (2009)

A milestone simulation by SXS collaboration:

Long-term simulation of BH-BH merger

Scheel et. al., Phys. Rev. D 79, 024003 (2009); Cohen et. al., Class. Quantum Grav. 26 035005 (2009)

Lovelace et al. CQG 29, 045003 (2012)

(SXS collaboration) Almost Exact solution of BH-BH binary with spin 0.97 for the last 25.5 orbits

GWs from BH-BH merger

Orbital separation decreases due to GW emission

Orbital velocity increases basically according to Kepler’s law

Also, GW frequency and, accordingly amplitude, increases

GW Amplitude takes maximum at the moment of the merger

After the merger, BH quickly becomes axisymmetric due to its characteristic property (no hair theorem)

GW amplitude quickly decreases because stationary axisymmetric object does not emit GW

separation

velocity

Evolving spacetime with matters

Shock waves are formed in general : treatment of discontinuities (High-resolution shock-capturing schemes)

General relativistic hydrodynamics

Valencia group: Marti et al., Astron. Astrophys. 235, 535 (1991)

General relativistic megnetohydrodynamics

Valencia group: Anton et al., Astrophys. J. 637, 296 (2006)

Shibata and Sekiguchi, PRD 72, 044014 (2005)

First binary neutron star merger simulation by Shibata & Uryu (2000)

Recent trend : More ‘realistic’ simulations

Microphysical equation of state and neutrinos

Stellar core collapse: Dimmelmeier et al., PRL 98, 251101 (2007) Sekiguchi PTP 124, 331 (with neutrinos)

Binary neutron star merger: Sekiguchi et al. PRL 107, 051102 (2011)

Black hole-neutron star merger: Foucart et al. PRD 90, 024026 (2014); Sekiguchi (2015,2016)

Black hole-neutron star merger

Kiuchi, Sekiguchi et al. (2015)

Summary of Numerical Relativity

Summary and outlook

After long-term efforts (more than 50 years) and thanks to recent development of computer resources, Numerical Relativity has become a mature field

Many observation-motivated simulations are ongoing, together with studies of extending the frontier of numerical-relativity simulations themselves

Numerical relativity will contribute in clarifying unsolved issues in GW physics, astrophysics, and nuclear physics, and gravity in the next decade

Gravitational waves : Towards gravitational astronomy: Electromagnetic counterparts of gravitational waves

High energy astrophysics : Formation processes of black holes and their association with the central engine of short and long gamma-ray bursts

Nuclear physics : exploring dense matter physics using GW from NS binaries

Gravitational physics : Testing general relativity via Numerical gravity theories