Revisiting stabilities of 5D Myers-Perry black holes in numerical relativity

Masaru Shibata (YITP, Kyoto)

With H. Yoshino (KEK)

ds2 = −dt2 + Gdµrd−5Σ

dt + asin2θdϕ( )2+ΣΔdr2 +Σdθ 2

+ r2 + a2( )sin2θdϕ 2 + r2 cos2θdΩd−42

Σ = r2 + a2 cos2θ; Δ = r2 + a2 −Gdµrd−5

%

&'

(

)*

µ : mass parameter, a : spin parameter

⇒ M =d − 2( )Ωd−2

16πGdµ, J = 2

d − 2Ma

2 parameters (µ, a) exist; but scale invariance exists

⇒ q := a

Gdµ( )1/(d−3)

: nondim. spin

Myers-Perry black hole with single spin

5D Myers-Perry BH: Reminder

•  Spin parameter q = a/µ1/2 = [0,1) # For q= 1, a naked singularity appears

•  For q à 1 , black hole is pancake-like

A =ΩD-2rh µ à 0 for q à 1

Cp / Ceq à 0 Cp(φ)

Ceq “Axial ratio”

ηm =1Cp

Cp ϕ( )exp imϕ( )dϕ∫“Deformation”

•  Emparan & Myers gave a conjecture based on Thermodynamical argument (2003)

•  High-spin BH seems to be unstable for any

dimension including D=5 case with q ~ 1

A =ΩD-2rhµ à 0 for high-spin

A = 2Ao >0

Previous studies for the stability of MP BH (1)

Likely to be more stable

M, J fixed

?

•  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”.

Previous studies for the stability of MP BH (2)

II Summary of our previous higher-dim 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 = −1.5αK

∂tβi =

13Bi

∂tBi = Γi −

CBµBi : CB =1− 2

Evolution of deformation of AH: 6D (SY10) U

nstable D

efor

mat

ion

(m=2

)

q=0.74

Marginal

The result does agree with Dias+ 14

Evolution of deformation of AH: 7D (SY10) U

nstable

q=0.96 D

efor

mat

ion

(m=2

)

q=0.735

The result does agree with Dias+ 14

6D Evolution of Cp / Ce for high spin

q > 0.74

q=0.74

q ~ 0.6

q ~ 1 Cp

Ce

Critical value =NOT very small A

xial

rat

io SY10

Evolution of deformation of AH in 5D U

nstable? D

efor

mat

ion

(m=2

) Looks exponential growth

Exponential decay

Stable

Crash

The result does not agree with Dias+ 14

SY10

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

12

Z4c formulation (Bernuzzi & Hilditch): 5D version

(∂t −βl∂l ) γ ij = −2α Aij + γ ilβ ,j

l + γ jlβ ,il −

12γ ijβ ,l

l

(∂t −βl∂l )χ =

χ2αK −β ,l

l( )(∂t −β

l∂l ) Aij =αχ Rij −14γ ijR

#

$%

&

'(− χ DiDjα −

14γ ijΔα

#

$%

&

'(

+α K Aij − 2 Ail Ajl( )+ Ail∂ jβ l + Ajl∂iβ l − 1

2β ,ll Aij

(∂t −βl∂l ) K =α Aij A

ij +14K 2

#

$%

&

'(−Δα : K = K − 2Θ

∂t −βk∂k( )Γi = 2α Γ jk

i A jk − 14γ ij 3K − 4Θ( ), j

− 2χ , jAij

#

$%

&

'(

−Γ jβ, ji +

12Γiβ, j

j + γ jkβ, jki +

12γ ikβ, jk

j

(∂t −βl∂l )Θ =αH : H =

12R+ K 2 − Ki

jK ji( )

Vacuum

13

ADM, BSSN: constraint violation cannot propagate Z4C: constraint violation can propagate à “error” is washed out à long-term simulation

Constraint violation obeys wave equations

Essence of Z4C formulation (Bernuzzi & Hilditch) H = 0 : Hamiltonian constraint Hi = 0 : Momentum constraint

⇒ADM, BSSN: ∂tH ~ −DiH

i , ∂tHi ~ 0

Z4c: ∂tΘ ~ H , Hi ~ −∂iΘ

%&'

('

⇒∂tH ~ −DiHi , ∂tH

i ~ −DiH

0.0001

0.001

0.01

0.1

0 50

dm

=2

t / µ1/2

q=0.92q=0.915q=0.91q=0.90q=0.89(t-1.5)2

1 Evolution of bar-mode (m=2): Add bar perturbation and evolve as SY10

q=0.92

q=0.91

q=0.90

q=0.89 Numerical results agree with Dias+ 14 (black dot-dot)

t / µ1/2

The growth are not exponential

Preliminary q=0.915

50 0

Approx. linear

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.  

0.28 0.29

0.3 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38

0 50 100 150 200 250 300

C p /

C eq

t / (µ1/2)

q=0.900q=0.910q=0.915q=0.920

t / µ1/2

Stationary states q ~ 0.89

q = 0.90

2 Evolution of axial ratio with no initial perturbation: Instability is not due to m=2

q ~ 0.875

Preliminary

q = 0.91

q = 0.92 Axi

al r

atio

of

hori

zon

Looks stable

-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

y

x

t=86.0

-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

y

x

t=96.3

-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

y

x

t=75.8Evolution of the shape of apparent horizon

This is not gauge-invariant. But, it may show something.

q=0.92

-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

y

x

t=65.5

-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

y

x

t=106.5Preliminary

0.0001

0.001

0.01

0.1

0 50 100 150 200 250 300

dm

=4

t / µ1/2

q=0.900q=0.910q=0.915q=0.920q=0.925

m=4 mode perturbation by Cartesian grid

Unstable !?

Stable

Noise level

t / µ1/2

Preliminary

Growing mode still exists: Real frequency is by 10% smaller than in linear analysis for it

0.28 0.29

0.3 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38

0 50 100 150 200

C p /

C eq

t / µ1/2

q=0.90

q=0.92

N=60,q=0.92N=48,q=0.92N=40,q=0.92N=60,q=0.90N=48,q=0.90N=40,q=0.90

Results depend only weakly on resolution

Stable q=0.92:Unstable !?

q=0.92 à ~ 0.875

Preliminary

-0.02 0

0.02 0.04 0.06 0.08

0.1 0.12 0.14

0 50 100 150 200 250 300

r + /

r +ex

act -

1

t / µ1/2

q=0.900q=0.910q=0.915q=0.920

Evolution of area of apparent horizon

Preliminary t / µ1/2

q=0.90, Stable

q=0.91, Unstable ?

q=0.915, Unstable ?

q=0.92, Unstable ? A A

H/A

exac

t - 1

Checking 1st law of BH thermodynamics

κδA= 8π δM −ΩHδJ( ) > 0 : κ = rhµ=µ − a2( )

1/2

µAssume monotonic wave emission

⇒ δM =σδJ : σ =ωm

; Here, ΩH =qµ1/2

>σ is needed

⇒ δq = 32δµµ

1µ1/2σ

− q"

#$$

%

&''

⇒ δAA=

δq1− q2

µ1/2 σ −ΩH( )1−σµ1/2q

=δq

1− q2σµ1/2 − q1−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

Solution space in GR

A    solu2on Another      solu2on

Prohibited    in    GR Direc2on    of  constraint    viola2on

24

A concern in Z4c formulation

(∂t −βl∂l ) γ ij = −2α Aij + γ ilβ ,j

l + γ jlβ ,il −

12γ ijβ ,l

l

(∂t −βl∂l )χ =

χ2αK −β ,l

l( )(∂t −β

l∂l ) Aij =αχ Rij −14γ ijR

#

$%

&

'(− χ DiDjα −

14γ ijΔα

#

$%

&

'(

+α K + nΘ( ) Aij − 2 Ail Ajl( )+ Ail∂ jβ l + Ajl∂iβ l − 1

2β ,ll Aij

(∂t −βl∂l ) K =α Aij A

ij +14K 2

#

$%

&

'(−Δα : K = K − 2Θ

∂t −βk∂k( )Γi = 2α Γ jk

i A jk − 14γ ij 3K − 4Θ( ), j

− 2χ , jAij

#

$%

&

'(

−Γ jβ, ji +

12Γiβ, j

j + γ jkβ, jki +

12γ ikβ, jk

j

(∂t −βl∂l )Θ =αH : H =

12R+ K 2 − Ki

jK ji( )

Add constraint

0.28 0.3

0.32 0.34 0.36 0.38

0.4 0.42 0.44

0 50 100 150 200

C p /

C eq

t / µ1/2

q=0.90

q=0.92

Z4cn=-1n=1n=2

Final state depends on n for q=0.92 Need more researches

Preliminary

No spin: Check of 4th-order convergence

-0.08-0.07-0.06-0.05-0.04-0.03-0.02-0.01

0

0 200 400 600 800 1000

AA

H /

Aex

act -

1

t / µ1/2

6x=1.2µ1/2/N

N=30N=24N=20

N=24, (4/5)4

N=20, (2/3)4

Δx=1.2=µ1/2  /  N