Numerical Relativity in higher dimensional spacetimes

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8, October 2012, CENTRA seminar @ Institute Superior Te´cnico
Numerical Relativity
in higher dimensional spacetimes
Hirotada Okawa
Yukawa Institute for Theoretical Physics
Kyoto University, Japan
Institute Superior Te´cnico
From August
Outline of the talk
• Introduction
• Numerical Relativity in higher dimensional spacetimes
– Decomposition of the Einstein equation
– BSSN formalism
– Gauge equations
• Black hole collision
• Black hole in AdS spacetimes
• Summary
Introduction
Veryfication for higher dimensional gravity
• In principle, BH can be formed when the energy is confined to
the sufficient small region.(Cf. “Hoop-Conjecture” Thorne(1972))
• Planck scale is 1019[GeV] if D = 4.
• Higher dimensional theory, Arkani-Hamed+(98),Randall+(99)
• Current experiments show the inverse square law of the gravity
is valid up to 0.1 ∼ 0.01mm.
• Planck scale could be larger than 10 [TeV].
• Possibility of the BH formation for particle collisions
Dimopoulos and Landsberg(2001), Giddings and Thomas(2002)
High-energy collision BH formation? evapolation
High-velocity Collisions in Higher Dimensions
It is believed that Gravity is dominant below the Planck scale.
−→ We could treat it as classical gravity.
• Test particle collisions
• Shock wave collisions Penrose(1974),Eardley and Giddings(2002)
• High-velocity BH collisions Shibata+(2008),Sperhake+(2009)
γm
m
γmBH
mBH
Impact paramter(BH formation)
M and J(feature of BH)
Dissipation(feature of GWs)
NR in Higher Dimensions
4D Numerical Relativity(Binary BHs)
-4 0 4
-4
0
4
y
/
m
x/m
Boyle,Brown,Kidder,Mroue,Pfeiffer,Scheel,Cook,Teukolsky(2007)
-3
-2
-1
 0
 1
 2
 3
-3 -2 -1
 0 1 2
 3
-0.3
-0.2
-0.1
 0
 0.1
 0.2
 0.3
Campanelli,Lousto,Zlochower,Merritt(2007)
• The evolution of Binary
BHs is calculated quite ac-
curately.
• It’s also popular to study
about the Gravitational
Recoil (Kick).
High-velocity BH Collisions in Numerical Relativity
-25 0 25 50 75 100 125
t / rS
-0.015
-0.01
-0.005
0
0.005
0.01
Φ
,
R
ex
 = 30 rS
R
ex
 = 40 rS
R
ex
 = 50 rS
t
∆
 1
 1.005
 1.01
 1.015
 1.02
 1.025
 0 50 100 150
A
 
/
 
1
6
 
 
m
0
2
t / m
0
-5
 0
 5
-10 -5 0 5 10 15 20 25 30
(
y
2
-
y
1
)
 
/
 
2
m
0
(x
2
-x
1
) / 2m
0
scatter
non-prompt merger
prompt merger
Sperhake,Cardoso,Pretorius,Berti,Hinderer,Yunes(2009)
Shibata, HO, Yamamoto(2008)
Witek, Zilha˜o, Gualtieri,Cardoso,
Herdeiro, Nerozzi, Sperhake(2010)
Orbits of a BH in 4D collision
Spin of the formed BH in 4D collision
Waveform of 5D Headon collision
Numerical Relativity
D-dimensional NR(D-1+1 decomposision)
To solve the Einstein equation
Rab − 12gabR = κTab(= 0).
Spatial metric
γab ≡ gab + nanb
na:Normal vector to the space (na = (1/α, βi/α))
Extrinsic curvature
Kab ≡ − 12α (∂tγab −Dbβa −Daβb)
Decomposition of the Einstein equation
−→ Same as 4D
6 4
;
4
Evolution equation
∂tKab = −DbDaα+ α (Rab − 2KacKcb +KabK)
+βcDcKab +KcbDaβ
c +KcaDbβ
c
Constraints
Hamiltonian constraint
R+K2 −KabKab = 0
Momentum constraints
DbK
b
a −DaK = 0
D-dimensional NR(BSSN formalism)
BSSN formalism in D-dim. Yoshino and Shibata (2009)
We cannot evolve γij and Kij
numerically because of the ex-
istence of violating modes.
γij = χ
−1γ˜ij , detγ˜ij = 1,
Kij = χ
−1A˜ij +
1
D − 2χ
−1γijK,
Γ˜i = −γ˜ij,j .
(
∂t − βi∂i
)
χ =
2
D − 1
χ
(
αK − ∂iβi
)
,(
∂t − βl∂l
)
γ˜ij = −2αA˜ij + γ˜il∂jβl + γ˜jl∂iβl −
2
D − 1
γ˜ij∂lβ
l,(
∂t − βi∂i
)
K = α
[
A˜ijA˜
ij +
1
D − 1
K2
]
−DiDiα,(
∂t − βl∂l
)
A˜ij = χ
[
α
(
R¯ij −
1
D − 1
γijR¯
)
−
(
DiDjα−
1
D − 1
γijD
lDlα
)]
+α
(
KA˜ij − 2A˜ilA˜lj
)
+ A˜lj∂iβ
l + A˜il∂jβ
l − D − 2
D − 1
A˜ij∂lβ
l,(
∂t − βj∂j
)
Γ˜i = γ˜jk∂j∂kβ
i +
D − 3
D − 1
γ˜ij∂j∂kβ
k − Γ˜j∂jβi +
2
D − 1
Γ˜i∂jβ
j
−2A˜ij∂jα+ 2α
(
Γ˜ijkA˜
jk − D − 1
2
∂jχ
χ
A˜ij − D − 2
D − 1
γ˜ij∂jK
)
.
Gauge conditions
Puncture Gauge Conditions Alcubierre+(2003)(
∂t − βi∂i
)
α = −ηααK,(
∂t − βj∂j
)
βi = −ηβBi,(
∂t − βj∂j
)
Bi =
(
∂t − βj∂j
)
Γ˜i − ηBBi.
Dennison,Wendell,Baumgarte(2010)
ηα, ηβ and ηB are arbitrary parameters.
(We choose the parameter carefully by
problems to solve.)
• we can treat the evolution near
the singularity well.
• The result of the evolution of the
Scwarzschild-Tangherlini BH:
In the puncture gauge, there is a
special slice as the attractor.
Hannam+(2009),Nakao+(2009)
SACRA code
Yamamoto,Shibata,Taniguchi(2008)
• How to solve the Einstein eq.
– Space : 4th order
finite differences
– Time : 4th order
Runge-Kutta integration
• BH-BH, NS-NS and BH-NS :
To resolve following scales,
– Size of the star,
– Interval between stars,
– Wavelength of GWs,
we need Adaptive Mesh Re-
finement(AMR).
• Economical:
We can run on a computer,
– CPU :3.4 GHz, 6 cores,
– MEM :32 GB,
– EURO:1,300 euros.
0
1
2 4
3
0
1
2
depth
5D Numerical Relativity(Cartoon method)
w
x
y, z
z
y
Cartoon Method Alcubierre+(1999)
• Originally this is used to solve the axisym-
metric problems.
• Numerical errors tend to grow near the
axis(r ∼ 0) on polar coordinates. (Because
1
r
diverges apparently at r = 0.)
• There isn’t such a problem on Cartesian co-
ordinates.
{
y = a cos b
z = a sin b
We can get any values
by the interpolation if we
have the data on z = 0.
We can take Cartesian coordinates and save the costs by Cartoon method.
Black Hole Collision
Initial condition(a Boost BH)
Schwarzschild-Tangherlini Black Hole in isotropic coordinates
ds2 = −α20dt20 + ψ
4
(D−3)
0
(
dw20 + dx
2
0 +
D−3∑
i=1
dy20i
)
α0 =
(
1− µ
4rD−30
)
ψ−10 , ψ0 = 1 +
µ
4rD−30
, r20 = w
2
0 + x
2
0 +
D−3∑
i=1
y20i
a Boost BH solution
ds2 = −γ2
(
α20 − v2ψ
4
D−3
0
)
dt2 − 2γ2v
(
ψ
4
D−3
0 − α
2
0
)
dtdw + ψ
4
D−3
0
[
B20dw
2 + dx2 +
D−3∑
i=1
dy2i
]
B20 = γ
2
(
1− α20v2ψ
−4
D−3
0
)
, γ ≡ 1√
1− v2
extrinsic curvature
Kxx = Kyiyi =
2
D − 3
γvα0ψ
′
0
w
ψ0B0r0
,
Kww =
γ3vB0w
r0
[
2α
′
0
− B
−2
0
(
2
D − 3
α0ψ
−1
0
ψ
′
0
− α
2
0
ψ
−4
D−3
0
α
′
0
v
2
)]
,
Kwx =
γvB0x
r0
[
α
′
0
− B
−2
0
(
2
D − 3
α0ψ
−1
0
ψ
′
0
− α
2
0
ψ
−4
D−3
0
α
′
0
v
2
)]
,
Kwyi =
γvB0yi
r0
[
α
′
0
− B
−2
0
(
2
D − 3
α0ψ
−1
0
ψ
′
0
− α
2
0
ψ
−4
D−3
0
α
′
0
v
2
)]
.
Then, we consider the
initial data for BHs
with initial velocity by
using Boost BHs.
Initial condition(Superposition of Boost BHs)
Initial condition for two Boost BHs
dℓ2 = ψ
4
D−3
(
B2dw2 + dx2 +
D−3∑
i=1
dy2i
)
conformal factor
ψ = ψmain + δψ,
ψmain ≡ 1 + µ1
4rD−21
++
µ2
4rD−22
rA =
√
γ2(w − wA)
2 + (x− xA)
2 +
∑
D−3
i=1
y2
i
(A = 1, 2)
B2 = γ2
[
1− v2ψ
−4
D−3
main
(
1− µ1
4rD−3
1
− µ2
4rD−3
2
)]
We should get δψ by solving the Hamiltonian constraint.
extrinsic curvature
Kij = K
(1)
ij +K
(2)
ij + δKij
We should also get δKij by solving
the Momentum constraints.
• δψ and δKij are the small
corrections when the BHs
are sufficiently apart from
each other.
Orbits of BH Collisionsin 4D and 5D
4D v = 0.6, Zoom-whirl behavior,impact paramter(x = 5.0, 5.2, 5.4)
-4
-2
 0
 2
 4
 6
-10 -5 0 5 10 15 20 25 30
x
w
5D v = 0.6, No Zoom-whirl, impact paramter(x = 1.65, 1.66, 1.75, 2.0)
-6
-4
-2
 0
 2
 4
-10 -5 0 5 10 15 20 25 30
x
w
Scattering of BHs
 3
 3.1
 3.2
 3.3
 3.4
 3.5
 3.6
 3.7
 0.4 0.5 0.6 0.7 0.8
i
m
p
a
c
t
 
p
a
r
a
m
e
t
e
r
 
b
 
[
R
g
]
v
b
C
b
B
Scattering
Merger
Crash
Vertical axis denotes impact paramter.
Horizontal axis denotes initial velocity of BHs.
bC :Lower limit of b which we can see the scattering.
bB:Upper limit of b which we can see the merging.
Exchange of M and J by Scattering
 0.98
 1
 1.02
 1.04
 1.06
 1.08
 1.1
 0 50 100 150 200
/
i
t [R
g
]
Mass
 0.95
 1
 1.05
 1.1
 1.15
 1.2
 1.25
 1.3
 1.35
 0 50 100 150 200
C
e
/
C
p
t [R
g
]
Deformation of a BH
-0.005
 0
 0.005
 0.01
 0.015
 0.02
 0.025
 0.03
 0 50 100 150 200
S
z
 
[
R
g
2
]
t [R
g
]
Spin of a BH
• Mass and Angular momentum grow after scattering.
• “Tidal Heating effect” could explain this.
(Cf. The membrane paradigm(1986), Poisson(2009,2010))
Kretschmann Invariant Scalar
Let’s see the BH scattering
in terms of
the Gauge invariant quantity.
K2 = RabcdRabcd
(Kretschmann invariant scalar)
Normalized by the value
on the Horizon of SBH
K2 = (D − 2)2 (D − 1) (D − 3)E4P
EP : Planck Energy
Schwarzschild black hole
in 5D isotropic coordinates
K2 = 72E4P
Kretschmann Scalar during scattering
-10 -5 0 5 10
w [R
g
]
-10
-5
 0
 5
 10
x
 
[
R
g
]
 0
 1
 2
 3
 4
• Kretschmann in-
variant K on y =
z = 0 plane.
• Red circles de-
note the shape
of AH.
• Two BHs always exist
through the scattering.
• Kretschmann becomes large
outside the Horizon.
HO,
Nakao,
Shibata(2011)
Kretschmann vs Separation between BHs
 0
 50
 100
 150
 200
 250
 300
 350
 400
 
 3 4
 
 5
 
 6
 
 7
 
K
2
 
[
E
P
2
M
-
2
]
Separation before
scattering [R
g
]
 
 3
 
 4
 
 5
 
 6 7
Separation after
scattering [R
g
]
=0.046875 R
g
=0.037500 R
g
=0.031250 R
g
• Horizontal axis denotes separation between two BHs.
(Time goes from left to right.)
• Vertical axis denotes Kretchmann scalar at the center of mass.
Kretschmann vs Initial velocity of BH
 3
 3.1
 3.2
 3.3
 3.4
 3.5
 3.6
 3.7
 0.4 0.5 0.6 0.7 0.8
i
m
p
a
c
t
 
p
a
r
a
m
e
t
e
r
 
b
 
[
R
g
]
v
b
C
b
B
Scattering
Merger
Kretschmann invariant at the same impact parameter
Maximum of Kretschmann Scalar
 1
 10
 40
 0.6 0.67 0.72 0.76
 1.3 1.4 1.5 1.6
K
 
[
E
P
 
M
-
1
]
v
=0.03750,b=3.50
=0.03125,b=3.50
=0.03750,b=3.54
=0.03125,b=3.54
• Horizontal axis denotes the initial velocity of BHs.
• Vertical axis denotes the Maximum of K
at the center of mass(log scale).
Black Hole in AdS spacetimes
BH in the spacetimes with a brane
Future Prediction?
Looking back at the history briefly,
L. Randall and R. Sundrum (1999)
Warped spacetimes
T. Tanaka(2002), R. Emparan et al.(2002)
BHs larger than bulk scale are not static.
H. Kudoh, T. Tanaka and T. Nakamura(2003)
Method to make the solution with a BH. (Small
BH could be static.)
H. Kudoh(2004), H. Yoshino(2006)
It is difficult to make a large BH.
N. Tanahashi and T. Tanaka(2008)
A larger BH seems unstable.
P. Figueras and T. Wiseman(2011)
Method to make a large BH.
We want to see Dynamical Evolution .
Exact Solution of Thick Brane
T. Takahashi, HO, M.Shibata(2012-13?)
(My) Requirement
• I want to start from SACRA code.
• I like easy boundary conditions.
-0.6
-0.4
-0.2
 0
 0.2
 0.4
 0.6
-10 -5 0 5 10
y
-0.25
-0.2
-0.15
-0.1
-0.05
 0
 0.05
 0.1
-1 -0.5 0 0.5 1
V
We can make the static thick brane with a scalar field.(Cf. Giovannini(2002))
S =
∫
d5x
√−g
[
R
2κ
− 1
2
∇aφ∇aφ− V (φ)
]
(1)
φ(y) =
√
3
κ
arctan [sinh(by)] , z = 2b sinh(by) (2)
V (φ) =
3b2
2κ
[
1− 5 sin2(
√
κ
3
φ)
]
(3)
Metric of the Brane Solution
ds2 =
1
1+ b2z2
[
dz2 + ηµνdx
µdxν
]
(4)
(Cf. RS II model)
ds
2
=
ℓ2
z2 + ℓ2
[
dz
2
+ ηµνdx
µ
dx
ν
]
b = ℓ−1 is the thickness of the brane.
5D NR(4+1 decomposition)
To solve Einstein equation
Rab − 12gabR = κTab
Spatial metric γab ≡ gab + nanb
na:Normal vector(na = (1/α, βi/α))
Extrinsic curvature
Kab ≡ − 12α (∂tγab −Dbβa −Daβb)
Decomposition of the Einstein equation
−→ Same form as usual
time
space
Evolution equation
∂tKab = β
cDcKab + KcbDaβ
c + KcaDbβ
c
+α
(
Rab − 2KacKcb +KabK − κ
[
Sab +
ρ−S
3 γab
])
−DbDaα
Constraints
Hamiltonian constraint
R+K2−KabKab = 2κρ
Momentum constraints
DbK
b
a −DaK = κja
BSSN formalism in AdS spacetimes
ds2 =
1
a2(z)
[
−α2dt2 + γij
(
dxi + βidt
) (
dxj + βjdt
)]
(5)
• We define the BSSN variables without a(z) part.
• We get usual evolution equation with BSSN variables + a(z) terms.
(
∂t − βi∂i
)
χ =
1
2
χ
(
αK − ∂iβi
)
+
2a′
a
βzχ,
(
∂t − βl∂l
)
γ˜ij = −2αA˜ij + γ˜il∂jβl + γ˜jl∂iβl −
1
2
γ˜ij∂lβ
l,
(
∂t − βi∂i
)
K = α
[
A˜ijA˜
ij +
1
4
K2 +
κ
3
(2ρ+ S)
]
−
[
aD¯iD¯i
(
α
a
)]TF
+
a′
a
βzK,(
∂t − βl∂l
)
A˜ij = · · · ,(
∂t − βj∂j
)
Γ˜i = · · · .Evolution equations for the scalar field
(
∂t − βl∂l
)
φ = −αΠ,(
∂t − βl∂l
)
Π = αKΠ+
α
a2
∂V
∂φ
− χα
[
D˜iD˜iφ+
(
∂iα
α
− ∂χ
χ
−3a
′
a
δzi
)
D˜iφ
]
.
Coordinates and Boundary Conditions
Coordinates
• We want higher resolution near
the brane.
• Perhaps outer region could be
coarse.(not sure)
• We use y coordinate.
(The interval of z is larger at
far.)
z = 2b sinh(by)
• But we defined BSSN variables
on z(γzz conformal flat).
• We should pay attention to the
derivatives of z.
∂zα =
1
a(y)∂yα
z
x1, x2, x3
Boundary Conditions
• Z2 symmetry at the brane
We can give the variables at −y
from that at y.
α(−y) = α(y), φ(−y) = −φ(y)
• If outer boundaries are enough far,
we can impose the Neumann con-
dition or Outgoing condition. (It
could be correct unless we evolve
for a long time.)
Hamiltonian Constraint
• We want an initial condition with Apparent Horizon(AH).
• Previous study T. Shiromizu and M. Shibata(2000)
• They construct the solution with AH by introducing a scalar field
as gravitational source.
metric ansatz
dl2 =
1
a2(z)
[
dz2 + (1 + w(r, z))4
(
dr2 + r2dθ2 + r2 sin2 θdφ2
)]
Hamiltonian Constraint
∂
2
rw +
2
r
∂rw +
3
2
[(
∂
2
zw −
3bz2
1 + b2z2
∂zw
)
(1 + w)
4
+ 3 (∂zw)
2
(1 + w)
3
]
= −π (1 + w)
5
(∂zψ)
2
− π (1 + w) (∂rψ)
2
We can solve the nonlinear elliptic equation(in principle if solutions exist)!
Initial Condition
How to give the scalar field:
ψ(r, z) =
f(r, z)
(1 + w)s
f(r, z) = ǫz exp
{
−1
2
(
r2
a2
+
z2
b2
)}
We can choose s, a, b and ǫ.
for example,
a = b = 0.3, s = 2, ǫ = 2.25
 0
 0.5
 1
 1.5
 2
 2.5
 3
 0 2 4 6 8 10 12 14
z
 0
 12
 3
 4
 5
 6
 7
 8
r
Preliminary Result
 6.5
 6.55
 6.6
 6.65
 6.7
 6.75
 6.8
 6.85
 6.9
 0 5 10 15 20 25 30 35 40 45
M
a
s
s
[
l
]
time[l]
AH mass
 4.04
 4.06
 4.08
 4.1
 4.12
 4.14
 4.16
 4.18
 4.2
 0 5 10 15 20 25 30 35 40 45
 2.64
 2.66
 2.68
 2.7
 2.72
 2.74
 2.76
 2.78
C
b
[
l
]
C
z
[
l
]
time
Circumfential length on the brane
Circumfential length in the bulk
• This is an evolution test for previous initial condition(Low res.).
• The values related to the AH are in about 4% error.
• It seems good before t ∼ 20.
• The proper length of the equator is not so changed, but the
position of AH on those coordinates is growing.
Summary
• We are in the stage to apply Numerical relativity to various space-
times.
• We can see the dynamical evolution of the BH(s) in higher di-
mensions.
• Scattering of BHs in higher dimensions might be used to inves-
tigate the limit of classical gravity.
• The code of the AdS numerical relativity should show some in-
teresting results(I hope).
Thank you very much for listening!
	þÿ
	Outline of the talk
	Introduction
	Veryfication for higher dimensional gravity
	High-velocity Collisions in Higher Dimensions
	4D Numerical Relativity(Binary BHs)
	High-velocity BH Collisions in Numerical Relativity
	Numerical Relativity
	D-dimensional NR(D-1+1 decomposision)
	D-dimensional NR(BSSN formalism)
	Gauge conditions
	SACRA code
	5D Numerical Relativity(Cartoon method)
	Black Hole Collision
	Initial condition(a Boost BH)
	Initial condition(Superposition of Boost BHs)
	Orbits of BH Collisions in 4D and 5D
	Scattering of BHs
	Exchange of M and J by Scattering
	Kretschmann Invariant Scalar
	Kretschmann Scalar during scattering
	Kretschmann vs Separation between BHs
	Kretschmann vs Initial velocity of BH
	Maximum of Kretschmann Scalar
	Black Hole in AdS spacetimes
	BH in the spacetimes with a brane
	Exact Solution of Thick Brane
	5D NR(4+1 decomposition)
	BSSN formalism in AdS spacetimes
	Coordinates and Boundary Conditions
	Hamiltonian Constraint
	Initial Condition
	Preliminary Result
	Summary