Higher Dimensional Black Higher Dimensional Black HolesHolesTsvi PiranTsvi Piran

Evgeny Sorkin & Barak KolThe Hebrew University,

Jerusalem Israel

E. Sorkin & TP, Phys.Rev.Lett. 90 (2003) 171301 B. Kol, E. Sorkin & TP, Phys.Rev. D69 (2004) 064031 E. Sorkin, B. Kol & TP, Phys.Rev. D69 (2004) 064032

E. Sorkin, Phys.Rev.Lett. (2004) in press

Kaluza-Klein, String Theory and other theories suggest that Space-

time may have more than 4 dimensions.

The simplest example is:R3,1 x S1 - one additional dimension

z

r

Higher dimensions

4d space-time R3,1

d – the number of dimensions is the only free parameter in classical GR. It is interesting, therefore, from a pure theoretical point of view to explore the behavior of the theory when we vary this parameter (Kol, 04).

4d difficult

5d …you ain’tseen anything yet

2d

trivial

3d

almosttrivial

What is the structure of Higher Dimensional Black Holes?

Black String

z

r

Asymptotically flat 4d space-time x S1

z

4d space-time

2222122 )21()21( dzdrdrrMdt

rMds +Ω+−+−−= −Horizon topology

S2 x S1

Or a black hole?Locally:

Asymptotically flat 4d space-time x S1

4d space-time

origin teh from distance 4D theisr origin thefrom distance 5D theis

))3/8(1())3/8(1( )3(22212

22

2

R

dRdRR

MdtR

Mds Ω+−+−−= −ππ

2222122 )21()21( dzdrdrrMdt

rMds +Ω+−+−−= −

z

r

z

Horizon topology S3

Black String

Black Hole

A single dimensionless parameter

3−≡ dN

LmGµ

Black hole

L

µ

A single dimensionless parameter

3−≡ dN

LmGµ

B

L

Black Stringµ

What Happens in the inverse What Happens in the inverse direction direction -- a shrinking String?a shrinking String?

3−≡ dN

LmGµ

r

z

L

Uniform B-Str

µ

??Non-uniform B-Str - ?

Black hole

Gregory & Laflamme (GL) : the string is unstable below a certain mass.

Compare the entropies (in 5d): Sbh~µ3/2 vs. Sbs ~ µ2

Expect a phase tranistion

Horowitz-Maeda ( ‘01): Horizon doesn’t pinch off!The end-state of the GL instability is a stable non-uniform string.

?

Perturbation analysis around the GL point [Gubser 5d ‘01 ] the non-uniform branch emerging from the GL point cannot be the end-state of this instability.

µnon-uniform > µuniform Snon-uniform > Suniform

Dynamical: no signs of stabilization (5d) [CLOPPV ‘03].This branch non-perturbatively (6d) [ Wiseman ‘02 ]

Dynamical Instaibility of a Black String Choptuik, Lehner, Olabarrieta, Petryk,

Pretorius & Villegas 2003

2/)1/( minmax −= rrλ

Non-Uniform Black String Solutions (Wiseman, 02)

Objectives:

• Explore the structure of higher dimensional spherical black holes.

• Establish that a higher dimensional black hole solution exists in the first place.

• Establish the maximal black hole mass.• Explore the nature of the transition

between the black hole and the black string solutions

The Static Black hole Solutions

Equations of Motion

22/)cos( zrz +≡≡ χξ ψψ c−=∆

Poor manPoor man’’s Gravity s Gravity ––The Initial Value Problem The Initial Value Problem (Sorkin & TP 03)(Sorkin & TP 03)

Consider a moment of time symmetry:Consider a moment of time symmetry:

0~)(1)(

82

22

2

222242

=+∂∂

+∂∂

∂∂

Ω++=

−ψρψψ

ψ

zrr

rr

drdzdrds

Higher Dim Black holes exist?

There is a Bh-Bstrtransition

Pro

per d

ista

nce

Log(µ-µc)

A similar behavior is seen in 4DA similar behavior is seen in 4D

Apparent horizons: From a simulation of bh merger Seidel & Brügmann

µ

µ

)1/(2/)1/(

minmax

minmax

−′′=′−=

rrrr

λλ

The Anticipated phase diagramThe Anticipated phase diagram [ Kol ‘02 ]

order param

µmerger point

?

x

GLUniform Bstr

We need a “good” order parameter allowing to put all the phases on the same diagram. [Scalar charge]

Non-uniform Bstr

Gubser 5d ,’01Wiseman 6d ,’02

Black hole

Sorkin & TP 03

Asymptotic behaviorAsymptotic behaviorSorkin ,Kol,TP ‘03Harmark & Obers ‘03[Townsend & Zamaklar ’01,Traschen,’03]

At r→∞: •The metric become z-independent as: [ ~exp(-r/L) ]•Newtonian limit

αβαβ

αβα

αβγ

αβαβ

αβαβαβαβ

π

η

η

γ

TGh

hhhh

hhg

d16

021

1

−=∆

=∂−=

<+=

Asymptotic Charges:Asymptotic Charges:The mass: m ≡ ∫T00dd-1x

The Tension: τ ≡ -∫Tzzdd-1x /L

3

34

34

23

22222222

8)3(1

131

)1(

)1(:alyAsymptotic

)(

−

−−

−−

−

Ω≡⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−−

−−=⎟⎟

⎠

⎞⎜⎜⎝

⎛

+≈

+≈

Ω+++−=

dd

d

dd

dd

dCA

kba

dd

kLm

ro

rb

ro

raA

drdredzedteds

πτ

φ

φ

τ

τ

Ori 04

dd 3

8k π

−

=ΩThe asymptotic coefficient b determines the length along

the z-direction.b=kd[(d-3)m-τL]

The mass opens up the extra dimension, while tension counteracts.

For a uniform Bstr: both effects cancel:

Bstr

But not for a BH:

BH

Archimedes for BHs

Smarr’s formula (Integrated First Law)

Together with

∫∫ −+=−= −∂ ][8

116

1),( 01 KKdV

GRdV

GFLI d

dd

d ππββ

dLLIdIdI∂∂

+∂∂

= ββ TSmF −=

We get (gas )pdVTdSdE −=dLTdSdm τ+=

LTSdmdLArea d

τε

+−=−⇒==+→ −

)2()3(][L[m])L(1L 23-d

aGk

LdAG

TSNdN

)4(8

1 −== κ

πThis formula associate quantities on the horizon (T and S) with asymptotic quantities at infinity (a). It will provide a strong test of the numerics.

Numerical SolutionSorkin Kol & TP

Numerical Convergence I:

Numerical Convergence II:

Numerical Test I (constraints):

Numerical Test II (The BH Area and Smarr’s formula):

Analytic expressions Gorbonos & Kol 04

eccentricity

distance

Ellipticity

“Archimedes”

A Possible Bh BstrTransition?

Anticipated phase diagramAnticipated phase diagram [ Kol ‘02 ]

order param

µmerger point

?

x

GLUniform Bstr

Non-uniform Bstr

Gubser 5d ,’01Wiseman 6d ,’02

d=10 a critical dimension

Black hole

Kudoh & Wiseman ‘03ES,Kol,Piran ‘03

GL

xmerger point

?

µThe phase diagram:

BHs

Scalar charge

Uni

form

Bst

r

Non-un

iform

Bstr

Gubser: First order uniform-nonuniformblack strings phase transition. explosions, cosmic censorship?

Universal? Vary d ! E. Sorkin 2004Motivations: (1) Kol’s critical dimension for the BH-BStr merger (d=10)

(2) Problems in numerics above d=10

For d*>13 a sudden change in the order of the phase transition. It becomes smooth

Scalar charge

µ

BHs

Perturbative Non-uniform Bstr

Uni

form

Bst

r

?

)( 3−≡

∝

dN

dc

LmGµ

γµ with γ= 0.686

The deviations of the calculated points from the linear fit are less than 2.1%

Entropies:

)()()3()3(

161

)3(16

41

)2(16

41

)0()0()0()4)(2(

)3)(3(

42

33)0(

43

33

32

22

)0(

µµπ

µ

ππ

BstrBHdd

dd

dd

dd

dd

dd

dBstr

dd

dd

dBH

SSdd

Ld

mG

Sd

mG

S

=−−

ΩΩ

=

⎥⎦

⎤⎢⎣

⎡Ω−

Ω=⎥⎦

⎤⎢⎣

⎡Ω−

Ω=

−−

−−

−−

−−

−−

−−

−−

−−

corrected BH:Harmark ‘03Kol&Gorbonos

for a given mass the entropy of a caged BH is larger

)()(

)()2(216)3(1

)1()1()1(

2

2

)0()1(

µµ

µπζ

BstrBH

dBHBH

SS

Od

mdSS

=

⎥⎦

⎤⎢⎣

⎡+

Ω−−

+=−

The curves intersect at d~13. This suggests that for d>13 A BH is entropically preferable over the string only for µ<µc . A hint for a “missing link” that interpolates between the phases.

A comparison between a uniform and a non-uniform String: Trends in mass and entropy

For d>13 the non-uniform string is less massive and has a higher entropythan the uniform one. A smooth decay becomes possible.

42

uniform

uniform-non21 1; λσλη

µδµ

+=+=S

SK

Interpretations & implications

Above d*=13 the unstable GL string can decay into a non-uniformstate continuously.

µ a smooth mergerµ

BHs

Perturbative Non-uniform Bstr

Uni

form

Bst

r

?filling the blob µ

discontinuous

bµ

a new phase,disconnected

Outlook

•Continue the non-uniform phase to the non-linear regime.

•Check time-evolution: does the evolution stabilize?

•Is the smooth p.t. general: more extra dimensions, other topologies. Would the

critical dimension become smaller than d=10?

SummarySummary

• We have demonstrated the existence of BH solutions.

• Indication for a BH-Bstr transition.• The global phase diagram depends on the

dimensionality of space-time

Open QuestionsOpen Questions• Non uniquness of higher dim black holes• Topology change at BH-Bstr transition• Cosmic censorship at BH-Bstr transition• Thunderbolt (release of L=c5/G)• Numerical-Gravitostatics• Stability of rotating black strings