Upload
others
View
2
Download
0
Embed Size (px)
Citation preview
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 ๐,๐๐๐ : spacetime manifold ๐ and the metric ๐๐๐
The signature of the metric : โ + + +
We will use the abstract index notation for tensor fields and Einsteinโs summation convention
Index of a tensor on ๐ is raised/lowered by ๐๐๐/๐๐๐ as usual
Covariant derivative associated with ๐๐๐ is ๐ป๐ (๐ป๐๐๐๐ = 0)
Riemann tensor for the spacetime : 4 ๐ ๐๐๐ ๐
Ricci tensor and Ricci scalar : 4๐ ๐๐ = 4๐ ๐๐๐ ๐, 4๐ = ๐๐๐ 4๐ ๐๐
Symmetric and anti-symmetric notation
๐(๐1โฏ๐๐) =1
๐! ๐๐๐(1)โฏ๐๐(๐)
๐
, ๐[๐1โฏ๐๐] =1
๐! sgn(๐)๐๐๐(1)โฏ๐๐(๐)
๐
We use the geometrical unit : ๐ = ๐บ = 1
Einsteinโs equation
2nd order quasilinear partial differential equations for the spacetime metric ๐๐๐
๐บ๐๐: Einstein tensor
๐บ๐๐ =8๐๐บ
๐4๐๐๐
๐บ๐๐ = ๐ ๐๐ โ1
2๐๐๐๐
๐ ๐ผ๐ฝ = ๐๐พฮ ๐ผ๐ฝ๐พ
โ ๐๐ผฮ ๐พ๐ฝ๐พ
+ ฮ ๐ผ๐ฝ๐ฟ ฮ ๐ฟ๐พ
๐พโ ฮ ๐พ๐ฝ
๐ฟ ฮ ๐ฟ๐ผ๐พ
ฮ ๐ผ๐ฝ๐พ
=1
2๐๐พ๐ฟ(๐๐ผ๐๐ฝ๐ฟ + ๐๐ฝ๐๐ผ๐ฟ โ ๐๐ฟ๐๐ผ๐ฝ)
๐ ๐๐: Ricci tensor
๐ = ๐๐๐๐ ๐๐ โถ Ricci scalar
ฮ ๐ผ๐ฝ๐พ
โถ Christoffel symbol
๐๐๐: stress energy momentum tensor
๐ ๐๐ = โ1
2๐๐ผ๐ฝ ๐๐ผ๐๐ฝ๐๐๐ + ๐๐๐๐๐๐ผ๐ฝ โ 2๐๐ฝ๐(๐๐๐)๐ผ + ๐น๐๐(๐, ๐๐)
Numerical Relativity (NR) Solving Einsteinโs equation on computer
Einsteinโs equation in full covariant form are 2nd order quasilinear partial differential equations
The solution, spacetime metric ๐๐๐ is not a dynamical object in the sense that it represents the full geometry of ๐ as the metric of the two-sphere does
To reveal the dynamical nature of Einsteinโs 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 (๐, ๐๐๐) into a foliation {ฮฃ}, the induced metric ๐พ๐๐ on ฮฃ, and the extrinsic curvature ๐พ๐๐ of ฮฃ
Then Einsteinโs equation is posed as a Cauchy problem for initial data (ฮฃ, ๐พ๐๐ , ๐พ๐๐), which can be solved numerically on computers
We also need to solve the matter equation ๐ป๐๐๐๐ = 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 โsmall changesโ in the solution
Changes in the initial data in a region, ๐, should not produce any changes in the solution outside the causal future, ๐ฝ+(๐), 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 Einsteinโs equation can be proven [1] in harmonic coordinates : ๐ป๐ = ๐ป๐๐ป
๐๐ฅ๐ = 0 ๐ป๐ is the covariant derivative associated with the spacetime metric ๐๐๐
In the harmonic coordinates, Einsteinโs equation in vacuum becomes
Then, the theorem due to Leray [2] can be applied to prove local existence and uniqueness of solution of Einsteinโs equation [3].
Global existence and uniqueness were also proved in [4]
๐๐ผ๐ฝ๐๐ผ๐๐ฝ๐๐๐ = ๐น ๐๐(๐, ๐๐)
[1] Choquet-Bruhat, Y. (1952) Acta. Math. 88, 141.
[2] Leray, J. (1952) โHyperbolic Differential Equationsโ, (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 (๐0)1,โฏ , (๐0)๐ be any solution of the quasilinear hyperbolic system
๐๐๐ ๐ฅ; ๐๐; ๐ป๐๐๐ ๐ป๐๐ป๐๐๐ = ๐น๐ ๐ฅ; ๐๐; ๐ป๐๐๐ (1)
Where ๐ป๐ is any derivative operator and ๐น๐ is an analytic function
and let (๐0)๐๐ = ๐๐๐ ๐ฅ; (๐0)๐; ๐ป๐(๐0)๐ . Suppose (๐, (๐0)๐๐) is globally hyperbolic. Let ฮฃ be a smooth spacelike Cauchy surface for (๐, (๐0)๐๐).
Then, the initial value formulation is well posed:
For Initial data sufficiently close to the initial data for (๐0)1,โฏ , (๐0)๐, there exist an open neighborhood ๐ of ฮฃ such that eq. (1) has a solution ๐1, โฏ , ๐๐, in ๐ and (๐, ๐๐๐(๐ฅ; ๐๐; ๐ป๐๐๐)) is globally hyperbolic.
The solution is unique in ๐ and propagate causally:
If the initial data for ๐โฒ1, โฏ , ๐โฒ๐ agree with that of ๐1,โฏ , ๐๐ on a subset ๐ of ฮฃ, then the solutions agree on ๐โ๐ท+(๐), where ๐ท+(๐) is the future domain of dependence for ๐.
The solutions depend continuously on the initial data
Theorem (summarized in Wald 1984 [5]): Let ฮฃ and ๐พ๐๐ be a 3-dimensional ๐ถโ manifold and a smooth metric on ฮฃ and let ๐พ๐๐ be a smooth symmetric tensor field on ฮฃ.
๐พ๐๐ and ๐พ๐๐ satisfy the constraint equations
Then there exist a unique ๐ถโ spacetime, (๐, ๐๐๐), called the maximal Cauchy development of (ฮฃ, ๐พ๐๐ , ๐พ๐๐) satisfying
(๐, ๐๐๐) is a solution of Einsteinโs equation and globally hyperbolic
The induced metric and extrinsic curvature of ฮฃ are ๐พ๐๐ and ๐พ๐๐
Every other spacetime can be mapped isometrically into a subset of (๐, ๐๐๐)
The solution ๐๐๐ depends continuously on the initial data (๐พ๐๐ , ๐พ๐๐) on ฮฃ.
Furthermore, (๐, ๐๐๐) satisfies the domain of dependence property desired in general relativity:
Suppose (ฮฃ, ๐พ๐๐, ๐พ๐๐) and (ฮฃโฒ, ๐พโฒ๐๐, ๐พโฒ๐๐) are initial data sets with maximal
developments (๐, ๐๐๐) and ๐โฒ, ๐โฒ๐๐
. Suppose there is a diffeomorphism
between ๐ โ ฮฃ and ๐โฒ โ ฮฃโฒ which carries (๐พ๐๐, ๐พ๐๐) on ๐ into (๐พโฒ๐๐, ๐พโฒ๐๐) on ๐โฒ. Then ๐ท(๐) in (๐, ๐๐๐) is isometric to ๐ท(๐โฒ) in (๐โฒ, ๐โฒ๐๐)
[5] Wald, R. M. (1984), โGeneral Relativityโ (Univ. of Chicago press)
3+1 decomposition of spacetime
Foliation {ฮฃ} of ๐ is a family of spacelike hypersurfaces (slices) which do not intersect each other and fill completely ๐
We assume that spacetime is globally hyperbolic. Then each ฮฃ is a Cauchy surface which is parametrized by a global time function ๐ก and denoted as ฮฃ๐ก [1]
Now, foliation is characterized by the gradient 1-form, ฮฉ๐ = ๐ป๐๐ก
The norm of ฮฉ๐ is related to a function called the lapse function, ๐ผ:
๐๐๐ฮฉ๐ฮฉ๐ = โ๐ผโ2
๐ผ characterize the proper time between ฮฃ
Let us introduce the normalized 1-form
๐๐ = โ๐ผฮฉ๐
ใtaa
[1] e.g. Wald, R. M. (1984), โGeneral Relativityโ (Univ. of Chicago press)
The induced metric on ฮฃ
The spatial metric ๐พ๐๐ induced onto ฮฃ is defined as
๐พ๐๐ = ๐๐๐ + ๐๐๐๐
Using ๐พ๐๐, a tensor on ๐ is decomposed into components tangent and normal to ฮฃ, using the projection operator
โฅ๐๐ = ๐พ๐
๐ = ๐ฟ๐๐ + ๐๐๐๐
The projection of a tensor ๐ ๐1โฏ๐๐
๐1โฏ๐๐ on ๐ into a tensor on ฮฃ is
โฅ ๐ ๐1โฏ๐๐
๐1โฏ๐๐ =โฅ๐1
๐1 โฏ โฅ๐๐
๐๐โฅ๐1
๐1 โฏ โฅ๐๐
๐๐ ๐ ๐1โฏ๐๐
๐1โฏ๐๐
It is easy to check โฅ ๐๐๐ = ๐พ๐๐
The covariant derivative operator ๐ท๐ associated with ๐พ๐๐ acting on tensors on ฮฃ is
๐ท๐๐ ๐1โฏ๐๐
๐1โฏ๐๐ =โฅ ๐ป๐๐ ๐1โฏ๐๐
๐1โฏ๐๐
It is easy to check Leibnitzโs rules holes and ๐ท๐๐พ๐๐ = 0
Intrinsic and extrinsic curvature for ฮฃ
Riemann tensor ๐ ๐๐๐ ๐ and the extrinsic curvature tensor ๐พ๐๐ for ฮฃ are
defined, respectively, by
๐ท๐๐ท๐ โ ๐ท๐๐ท๐ ๐ค๐ = ๐ ๐๐๐ ๐๐ค๐ , ๐พ๐๐ =โโฅ ๐ป(๐๐๐) = โ
1
2โ๐๐พ๐๐
โ๐ is Lie derivative with respect to the unit normal vector ๐๐ = ๐๐๐๐๐
๐พ๐๐ is โvelocityโ of ๐พ๐๐
The geometry of ฮฃ is described by ๐พ๐๐ and ๐พ๐๐. In order that the foliation fits the spacetime, ๐พ๐๐ and ๐พ๐๐ must satisfy Gauss, Codazzii, and Ricci relations, respectively, given by
โฅ4 ๐ ๐๐๐๐ = ๐ ๐๐๐๐ + ๐พ๐๐๐พ๐๐ โ ๐พ๐๐๐พ๐๐
โฅ4 ๐ ๐๐๐ ๐๐๐ = ๐ท๐๐พ๐๐ โ ๐ท๐๐พ๐๐
โฅ4 ๐ ๐๐๐๐๐๐๐๐ =โฅ โ๐๐พ๐๐ + ๐พ๐๐๐พ๐
๐ + ๐ท๐๐๐ + ๐๐๐๐
Where ๐๐ = ๐๐๐ป๐๐๐ = ๐ท๐ln ๐ผ
Geometrical meaning of ๐พ๐๐
๐พ๐๐ is associated with the difference between the original and parallel-transported unit normal
vector ๐๐ on ฮฃ๐ก For a slice ฮฃ๐ก embedded in the
spacetime in a โflatโ manner, ๐พ๐๐ = 0
๏ผ
How ฮฃ๐ก is embedded in spacetime ๏ผcurvature seen from outside๏ผ โ extrinsic curvature
bx
bb xx
)( ba xn)(
bba xxn
Parallel transport b
aba xKn
โEvolutionโ vector and general covariance
Any vector ๐ก๐ dual to ฮฉ๐(๐ก๐ฮฉ๐ = 1) can be the evolution vector
A simple candidate is ๐ก๐ = ๐ผ๐๐
Note that the unit normal vector ๐๐ can not be an evolution vector because โ๐ โฅ๐
๐โ 0 so that Lie derivative (evolution) with respect to ๐๐ of a tensor tangent to ฮฃ is not a tensor tangent to ฮฃ
On the other hand, โ๐ผ๐ โฅ๐๐= 0
๐ฎ๐+๐ ๐
๐ฎ๐
๐ท๐๐ ๐ โถ shift vector
๐๐: unit normal
๐ถ๐๐๐ ๐ โถ lapse function
๐๐ : evolution vector We can add a spatial vector, called shift vector ๐ฝ๐ (๐ฝ๐ฮฉ๐ = 0), to ๐ก๐
Thus the general evolution vector is ๐ก๐ = ๐ผ๐๐ + ๐ฝ๐
The degree of freedom in choosing the evolution vector is originated in the general covariance of general relativity
3+1 decomposition of Einsteinโs equation Now we proceeds to 3+1 decomposition of Einsteinโs equation
๐บ๐๐ = 4๐ ๐๐ โ1
2๐๐๐
4๐ =8๐๐บ
๐4๐๐๐ = 8๐๐๐๐
The decomposition of the stress-energy-momentum tensor ๐๐๐ is ๐๐๐ = ๐ธ๐๐๐๐ + 2๐(๐๐๐) + ๐๐๐
where ๐ธ = ๐๐๐๐๐๐๐, ๐๐ =โโฅ ๐๐๐๐
๐ , and ๐๐๐ =โฅ ๐๐๐ are the energy density, momentum density, and stress tensor measured by the Eulerian observer ๐๐
3+1 decompositions of Einsteinโs equation corresponds to ๐บ๐๐๐๐๐๐,
โฅ ๐บ๐๐๐๐, and โฅ ๐บ๐๐, which respectively give (๐พ = ๐พ๐
๐ and ๐ = ๐๐๐)
๐ + ๐พ2 โ ๐พ๐๐๐พ๐๐ = 16๐๐ธ (1)
๐ท๐๐พ๐๐ โ ๐ท๐๐พ = 8ฯ๐๐ (2)
โ๐ก โ โ๐ฝ ๐พ๐๐ = โ๐ท๐๐ท๐๐ผ + ๐ผ ๐ ๐๐ + ๐พ๐พ๐๐ โ 2๐พ๐๐๐พ๐๐
โ4ฯ๐ผ(2๐๐๐ โ ๐พ๐๐(๐ โ ๐ธ))
Note that Eqs. (1) and (2) do not contain time derivative of ๐พ๐๐ and ๐พ๐๐ (constraints)
Introducing coordinates
We choose the evolution vector ๐ก๐ as the time basis vector: (๐๐ก)๐ = ๐ก๐
We also introduce the natural spatial basis vectors (๐๐)๐ on ฮฃ
(๐๐)๐ are Lie dragged along ๐ก๐ : โ๐ก(๐๐)
๐ = 0
Then (๐๐)๐ remains purely spatial because โ๐ก(ฮฉ๐ ๐๐)
๐ = 0
Dual basis (๐๐ฅ๐)๐ are defined in the standard manner
Components of geometrical quantities are (e.g., ๐๐๐ = ๐๐๐(๐๐ฅ๐)๐(๐๐ฅ๐)๐)
It can be shown ๐พ๐๐๐พ๐๐ = ๐ฟ๐๐ so that indices of spatial tensors can be lowered
and raised by the spatial metric
๐๐ =1
๐ผ(1,โ๐ฝ๐)๐
๐๐ = (โ๐ผ, 0,0,0)
๐๐๐ =1
๐ผ2โ1 ๐ฝ๐
๐ฝ๐ ๐ผ2 โ ๐ฝ๐๐ฝ๐
๐๐๐ =โ๐ผ2 + ๐ฝ๐๐ฝ๐ ๐ฝ๐
๐ฝ๐ ๐พ๐๐
ADM (Arnowitt-Deser-Misner) formulation
The 3+1 decompositions of Einsteinโs equation together with the definition of ๐พ๐๐ (tensor equations) are now a set of partial differential equations for ๐พ๐๐ and ๐พ๐๐ (ADM system) [2] :
๐ + ๐พ2 โ ๐พ๐๐๐พ๐๐ = 16๐๐ธ (3)
๐ท๐๐พ๐๐โ ๐ท๐๐พ = 8ฯ๐๐ (4)
๐๐ก โ โ๐ฝ ๐พ๐๐ = โ๐ท๐๐ท๐๐ผ + ๐ผ ๐ ๐๐ + ๐พ๐พ๐๐ โ 2๐พ๐๐๐พ๐๐
โ4ฯ๐ผ(2๐๐๐ โ ๐พ๐๐(๐ โ ๐ธ))
๐๐ก โ โ๐ฝ ๐พ๐๐ = โ2ฮฑ๐พ๐๐
Eqs. (3) and (4) are elliptic constraint equations (Hamiltonian and momentum constraints, respectively) that ๐พ๐๐ and ๐พ๐๐ must satisfy on each slice ฮฃ
The remaining equations are the evolution equations for ๐พ๐๐ and ๐พ๐๐
Note that Einsteinโs equation tells us nothing about how to specify the gauge
degrees of freedom ๐ผ and ๐ฝ๐, as expected from general covariance
[2] York, J. W. (1978), in โSources of Gravitational Radiationโ eds. Smarr, L. L. (Cambridge Univ. press)
Evolution of constraints The โevolutionโ equations for the Hamiltonian (๐ถ๐ป) and Momentum
(๐ถ๐๐ ) constraints are
Where ๐น๐๐ is the spatial projection of the evolution equation
The evolution equations show that the constraints are satisfied, if
They are satisfied initially and Einsteinโs equation is solved correctly (๐น๐๐ = 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)
i
H
ki
M
j
M
i
j
ij
j
i
Mt
ij
ij
Hk
k
M
k
MkHt
DHFCFDKCCKFDC
FKFCKDCCDC
)2()(2)(
)2()(
L
L
abababab TgTRF
2
18 4
Generating spacetime
๏ผif ๐ถ, ๐ท๐ are given๏ผ we can construct the future slice ฮฃ๐+๐ ๐ from ฮฃ๐ก
spatial metric ๐ธ๐๐ on ๐ฎ๐ and its โvelocityโ ๐ฒ๐๐~๐ธ ๐๐ is given as initial data
๐พ๐๐ and ๐พ๐๐ must satisfy the constraints (solving the elliptic equations)
3+1 decomposed Einsteinโs equation provide their time developments
๐ธ๐๐ ๐ฒ๐๐~๐ธ ๐๐
๐๐๐
๐บ๐
๐บ๐+๐ ๐
Spacetime is generated if we can construct ๐๐๐ from ๐ธ๐๐, ๐ฒ๐๐
We need prescriptions to specify ๐ผ and ๐ฝ๐ (coordinate problem in NR)
๐ ๐๐
๐ท๐๐ ๐ + ๐ ๐๐
๐ ๐
๐ท๐๐ ๐ โถ ๐ฌ๐ก๐ข๐๐ญ ๐ฏ๐๐๐ญ๐จ๐ซ ๐๐ : ๐ญ๐ข๐ฆ๐ ๐๐ฑ๐ข๐ฌ
Lapse function
๐ ๐๐ = โ ๐ถ๐ ๐ ๐ + ๐ธ๐๐(๐ท๐๐ ๐ + ๐ ๐๐)(๐ท๐๐ ๐ + ๐ ๐๐)
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 ๐ผ
Shift condition is that of choosing the shift vector ๐ฝ๐
Einsteinโs equations say nothing about the gauge conditions
Choosing โgoodโ 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 โฆ
Collapsing star
Event horizon
singularity
t
space
๐ผ = 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 โ๐พ = ๐ป๐๐๐ decreases
monotonically in time
In terms of the volume element ๐พ, (๐พ = det ๐พ๐๐) , 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 ๐พ
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 ๐พ results in a
coordinate singularity, let us maximize the volume element
We take the volume element of a 3D-domain ๐ and consider a variation along
the time vector
If ๐พ = 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 โ eventually simulation crash
Necessary to use ๐ฝ๐
Maximal slicing condition is elliptic equation for ๐ผ
Hyperbolic lapse has been developed
Singularity Event horizon
Choice of the lapse function ๐ถ
SxdSV 3][
SS
i
it xdKKxdSV 33 )(][ L
)](4)(0 SEKKDDK ij
ij
i
it LL
Utilizing the shift vector ๐ท๐
Distortion of the time vector is problematic Distortion of ๐๐ due to the black hole formation The dragging of the frame around a rotating object
We can use ๐ฝ๐ to minimize these distortions
Minimal distortion shift (Smarr & York 1978) The covariant derivative of any timelike unit vector
can be decomposed as (Helmholtzโs 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 ๐ท๐
Distortion of the time vector is problematic Distortion of ๐๐ due to the black hole formation The dragging of the frame around a rotating object
We can use ๐ฝ๐ to minimize these distortions
Minimal distortion shift (Smarr & York 1978) The covariant derivative of any timelike unit vector
can be decomposed as (Helmholtzโs 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
๐ท๐๐ ๐ โถ 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 : ๐๐ก๐ข๐ + (๐ด๐๐)๐๐๐๐ข
๐ = 0. This system is called
Strongly hyperbolic : if a matrix (representation) of ๐ด has real eigenvalues and a complete set of eigenvectors
Weakly hyperbolic : if ๐ด 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 Einsteinโs 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 Einsteinโs equation which is (at least)
strongly hyperbolic (in a linearized regime) (as โwave-likeโ as possible)
Caution ! : better hyperbolicity is necessary condition, not sufficient
Let us consider Maxwellโs equation in flat spacetime to capture what we
should do to obtain a more stable system
โ๐ ๐ด๐๐ก2 + ๐ป๐๐ป๐๐ด๐ โ ๐ป๐๐ป๐๐ด
๐ = ๐ป๐๐๐ก๐
๐๐๐ด๐ = 0 ๐น = ๐ป๐๐ด
๐ ๐ป๐๐ธ๐ = 4๐๐๐
๐๐ก๐น = โ๐ป๐๐ธ๐ โ ๐ป๐๐ป๐๐
๐๐ก๐น = โ4๐๐๐ โ ๐ป๐๐ป๐๐
Adopting better gauge Introduce new variables Utilize constraints
๐๐๐๐๐ด๐ = ๐ป๐๐๐ก๐ + ๐ป๐๐น ๐๐๐๐๐ด๐ = 0
Lorenz gauge :
๐ป๐๐ธ๐ = 4๐๐๐ ๐ป๐๐ด
๐ = 0 Coulomb gauge :
๐๐๐๐๐ด๐ = ๐ป๐๐๐ก๐ ๐๐ก๐น = โ๐ป๐๐ธ
๐ โ ๐ป๐๐ป๐๐
Evolution eq for ๐น
Divergence terms prevents the system from achieving better hyperbolicity
Three ways to achieve better hyperbolicity
Reformulating Einsteinโs equation
Strongly/Symmetric hyperbolic reformulations of Einsteinโs 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 ! ( Alcubierreโs 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 ฮ๐ = ๐๐๐พ๐๐
Extract the โtrueโ 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 :
๐ธ๐๐ = ๐๐๐ธ ๐๐
Extrinsic curvature is also conformally decomposed
Trace of K is associated with the lapse function (c.f. maximal slicing) โ split
๐ฒ๐๐ = ๐๐๐จ ๐๐ +๐
๐๐ธ๐๐๐ฒ
Reformulation based on new variables :
๐, ๐พ ๐๐, ๐ด ๐๐, ๐พ = tr ๐พ , ฮ๐ = ๐๐๐พ๐๐
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
๐๐ก โ ๐ฝ๐๐๐ ๐ผ = โ2๐ผ๐พ, ๐๐ก๐ฝ๐ =
3
4ฮ๐ โ ๐๐ฝ๐
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 Keplerโs 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 โrealisticโ 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