Revisiting stabilities of 5D
Myers-Perry black holes
in numerical relativity
Masaru Shibata (YITP, Kyoto)
With H. Yoshino (KEK)
Myers-Perry black hole with single spin
2
Gd µ
Σ 2
2
ds = −dt + d−5 dt + asin θ dϕ + dr + Σdθ 2
Δ
r Σ
+ r 2 + a 2 sin 2 θ dϕ 2 + r 2 cos 2 θ dΩ2d−4
2
(
2
(
)
)
%
Gd µ (
2
2
2
2
2
' Σ = r + a cos θ ; Δ = r + a − d−5 *
r )
&
µ : mass parameter, a : spin parameter
d − 2 Ωd−2
2
⇒ M=
µ, J =
Ma
16π Gd
d −2
2 parameters (µ , a) exist; but scale invariance exists
a
⇒ q :=
: nondim. spin
1/(d−3)
Gd µ
(
)
(
)
5D Myers-Perry BH: Reminder
•  Spin parameter q = a/µ1/2 = [0,1)
# For q= 1, a naked singularity appears
A =ΩD-2rh µ à 0 for q à 1
•  For q à 1 , black hole is pancake-like
Cp / Ceq à 0
Cp(φ)
“Axial ratio”
Ceq
1
ηm =
Cp ϕ exp imϕ d ϕ
∫

Cp
“Deformation”
( ) (
)
Previous studies for the stability of
MP BH (1)
•  Emparan & Myers gave a conjecture based on
Thermodynamical argument (2003)
?
M, J fixed
A =ΩD-2rhµ à 0
for high-spin
A = 2Ao >0
Likely to be
more stable
•  High-spin BH seems to be unstable for any
dimension including D=5 case with q ~ 1
Previous studies for the stability of
MP BH (2)
•  Is higher-dimensional spinning BH unstable ?
•  Yes, for more than 6D cases: Consensus
ü For axisymmetric perturbation (Dias et al. 09)
ü For non-axisymmetric case (SY 10, Dias + 14)
•  But, could be “No” for 5D case
ü For axisymmetric perturbation, it is No
ü SY 10 suggested in numerical relativity it could be
“Yes” à However, simulation time was too short
à We have been revisiting since last year
ü Dias + 14 show it “No”.
II
Summary of our previous higherdim numerical-relativity results:
•  Use BSSN formulation with cartoon method:
impose SU(D-3) symmetry for extra-dim space
•  Excise a region deep inside horizon
•  Fixed mesh refinement algorithm
•  “Puncture gauge” with appropriate choice of
coefficients
∂tα = −1.5α K
1 i
i
∂t β = B
3
CB i
i
i
∂t B = Γ −
B : C B = 1− 2
µ
Evolution of deformation of AH: 6D (SY10)
Unstable
Deformation (m=2)
The result does agree with Dias+ 14
q=0.74
Marginal
Evolution of deformation of AH: 7D (SY10)
q=0.96
Unstable
Deformation (m=2)
The result does agree with Dias+ 14
q=0.735
6D Evolution of Cp / Ce for high spin
Axial ratio
SY10
q ~ 0.6
Critical value
=NOT very small
q=0.74
q > 0.74
q~1
Cp
Ce
SY10
Looks
exponential Crash
growth
Stable
Exponential
decay
Unstable?
Deformation (m=2)
Evolution of deformation of AH in 5D
The result does not agree with Dias+ 14
III New numerical simulations
•  Long-term simulations are obviously
required for high-spin case;
this will be the case for studying black ring
•  Employ Z4c formulation in which constraint
violation can propagate
•  For 4D black holes, Z4c has been proven to
be robust for evolving high-spin BHs for a
long term
Z4c formulation (Bernuzzi & Hilditch): 5D version
1
l
l




(∂t − β ∂l )γ ij = −2α Aij + γ il β ,j + γ jl β ,i − γij β l ,l
2
Vacuum
χ
l
l
(∂t − β ∂l ) χ = α K − β ,l
2
#
&
#
&
1
1
l

(∂t − β ∂l ) Aij = αχ % Rij − γ ij R ( − χ % Di D jα − γ ij Δα (
4
4
$
'
$
'
1 l 
l
l
l





+α KAij − 2 Ail A j + Ail∂ j β + A jl∂i β − β ,l Aij
2
#
1 2&
l
ij



(∂t − β ∂l ) K = α % Aij A + K ( − Δα : K = K − 2Θ
4 '
$
# i jk 1 ij
&
k
i
ij



∂t − β ∂k Γ = 2α % Γ jk A − γ 3K − 4Θ − 2 χ , j A (
,j
4
$
'
1 i j
1 ik j
j i
jk i

− Γ β, j + Γ β, j + γ β, jk + γ β, jk
2
2
1
l
(∂t − β ∂l )Θ = α H : H = R + K 2 − Ki j K ij
12
2
l
(
)
(
(
)
)
(
(
)
)
Essence of Z4C formulation (Bernuzzi & Hilditch)
H = 0 : Hamiltonian constraint
H i = 0 : Momentum constraint
%' ADM, BSSN: ∂ H ~ −D H i , ∂ H i ~ 0
t
i
t
⇒&
'( Z4c: ∂t Θ ~ H , H i ~ −∂i Θ
⇒ ∂t H ~ −Di H i , ∂t H i ~ −D i H
Constraint violation obeys wave equations
ADM, BSSN: constraint violation cannot propagate
Z4C: constraint violation can propagate
à “error” is washed out à long-term simulation
13
1 Evolution of bar-mode (m=2):
Add bar perturbation and evolve as SY10
0.1
m=2
Preliminary
0.01
q=0.92
q=0.915
q=0.91
q=0.90
q=0.89
2
(t-1.5)
The growth are not
exponential
q=0.92
r
a
e
n
i
l
.
x
Appro
q=0.915
q=0.91
q=0.90
0.001
q=0.89
Numerical results agree with
Dias+ 14 (black dot-dot)
0.0001
0
0
1/2
50
t/µ
1/2 50
t/µ
1 Study for Bar-mode (m=2 mode)
•  For stable BHs, q < ~0.90, oscillation frequency
and damping rate agree with Dias+ 14
e.g., ωr ~ 1.3, ωi ~ 0.028 for q = 0.90 à  For q < =0.90, 5D BH are stable
(our previous analysis was incorrect)
However, for q > ~0.91, we s2ll find an instability à This is associated with other unstable mode. Axial ratio of horizon
Cp / Ceq
2 Evolution of axial ratio with no initial
perturbation: Instability is not due to m=2
0.38
0.37
0.36
0.35
0.34
0.33
0.32
0.31
0.3
0.29
0.28
q ~ 0.875
Stationary states
q ~ 0.89
Looks
stable
q = 0.90
q = 0.91
q=0.900
q=0.910
q=0.915
q=0.920
q = 0.92
0
Preliminary
50
100
150
1/2
tt // (µ
µ1/2
)
200
250
300
0.3
Evolution of the shape of apparent horizon
t=65.5
t=75.8
q=0.92
0.3
0.2
0.1
0.1
0
0
y
0.2
-0.1
-0.1
-0.2
-0.2
-0.3
-0.3
Preliminary
t=86.0
-0.3 -0.2 -0.1 0 0.1 0.2 0.3
This is not
gauge-invariant.
-0.3 -0.2 -0.1 0 0.1 0.2 0.3
x
But, it may
show something.
x
t=96.3
t=106.5
0.3
0.2
0.2
0.2
0.1
0.1
0.1
0
0
0
y
0.3
y
0.3
-0.1
-0.1
-0.1
-0.2
-0.2
-0.2
-0.3
-0.3
-0.3
-0.3 -0.2 -0.1 0 0.1 0.2 0.3
-0.3 -0.2 -0.1 0 0.1 0.2 0.3
-0.3 -0.2 -0.1 0 0.1 0.2 0.3
x
x
x
m=4 mode perturbation by Cartesian grid
q=0.900
q=0.910
q=0.915
q=0.920
q=0.925
0.1
m=4
Preliminary
Stable
Unstable !?
0.01
Noise
level
0.001
0.0001
0
50
100
Growing mode still exists:
Real frequency is by 10% smaller
than in linear analysis for it
150
1/2
1/2
tt// µµ
200
250
300
Cp / Ceq
Results depend only weakly on resolution
0.38
0.37
0.36
0.35
0.34
0.33
0.32
0.31
0.3
0.29
0.28
q=0.92 à ~ 0.875
Stable
q=0.92:Unstable !?
q=0.90
N=60,q=0.92
N=48,q=0.92
N=40,q=0.92
N=60,q=0.90
N=48,q=0.90
N=40,q=0.90
q=0.92
0
Preliminary
50
100
t/µ
1/2
150
200
r+exact --11
Ar+AH/ /A
exact
Evolution of area of apparent horizon
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
-0.02
q=0.92, Unstable ?
q=0.900
q=0.910
q=0.915
q=0.920
q=0.915, Unstable ?
q=0.91, Unstable ?
q=0.90, Stable
0
Preliminary
50
100
150
1/2
tt // µ1/2
200
250
300
Checking 1st law of BH thermodynamics
(
2
1/2
)
µ−a
rh
κδ A = 8π δ M − ΩHδ J > 0 : κ = =
µ
µ
Assume monotonic wave emission
ω
q
⇒ δ M = σδ J : σ = ; Here, ΩH = 1/2 > σ is needed
m
µ
%
3 δµ " 1
$$
⇒ δq =
− q ''
1/2
2 µ #µ σ
&
1/2
µ
σ − ΩH
δA
δq
δ q σµ1/2 − q
⇒
=
=
2
1/2
A 1− q 1− σµ q
1− q2 1− σµ1/2q
(
)
(
)
I checked this relation between δA & δq
is satisfied for m=4 approximately: σ ~ 0.8
Summary
•  Z4c formalism enables us long-term simulations
•  For q < =0.9, 5D Myers-Perry BH appears stable
(I am very sorry for our previous result):
This new result agrees with Dias+ 14
i.e., our results for ωr & ωi agree with Dias+ 14
•  However, simulations still find a non-axisym.
instability (m=4) for 5D high-spin MP BH !?!?
v  New possible critical value is q ~ 0.91 (a/rh ~ 2.2)
v  The instability is NOT associated with bar-mode
v  Unstable BH evolves to a less spinning BH
Ø  To be honest, I do not still believe this result:
Please do not believe as well …
A concern
•  Theory in Z4c formalism is different from GR in
the presence of a constraint violation
•  If a large constraint violation is continuously
generated (e.g., near the BH horizon), something
wrong such as bypassing might occur
•  We have to be careful and need more studies
Prohibited in GR
Another solu2on
Direc2on of constraint viola2on
A solu2on
Solution space
in GR
A concern in Z4c formulation
1
l
l

(∂t − β ∂l )γij = −2α Aij + γil β ,j + γ jl β ,i − γij β l ,l
2
χ
l
(∂t − β ∂l ) χ = α K − β l ,l
2
#
&
#
&
1
1
l

(∂t − β ∂l ) Aij = αχ % Rij − γ ij R ( − χ % Di D jα − γ ij Δα (
4
4
$
'
$
'
l
(
)
1
Add
+α ( K + nΘ)Aij − 2 Ail A lj + Ail∂ j β l + A jl∂i β l − β l ,l Aij
2
constraint #
1 2&
l
ij



(∂t − β ∂l ) K = α % Aij A + K ( − Δα : K = K − 2Θ
(
$
)
4
'
# i jk 1 ij
&
k
i
ij



∂t − β ∂k Γ = 2α % Γ jk A − γ 3K − 4Θ − 2 χ , j A (
,j
4
$
'
1 i j
1 ik j
j i
jk i
− Γ β, j + Γ β, j + γ β, jk + γ β, jk
2
2
1
l
(∂t − β ∂l )Θ = α H : H = R + K 2 − Ki j K ij
2
(
)
(
(
)
)
24
Cp / Ceq
Final state depends on n for q=0.92
Need more researches
0.44
0.42
0.4
0.38
0.36
0.34
0.32
0.3
0.28
Z4c
n=-1
n=1
n=2
q=0.90
q=0.92
0
Preliminary
50
100
t / µ1/2
150
200
No spin: Check of 4th-order convergence
AAH / Aexact - 1
1/2 / N
Δx=1.2=µ
x=1.2µ1/2/N
0
-0.01
-0.02
-0.03
-0.04
-0.05
-0.06
-0.07
-0.08
N=30
N=24
N=20
4
N=24, (4/5)
N=20, (2/3)4
0
200
400
t/µ
600
1/2
800
1000