Higher Dimensional Higher Dimensional Black HolesBlack Holes

Tsvi PiranTsvi PiranEvgeny Sorkin & Barak Kol

The 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:

4d space-time R3,1Higher dimensions

R3,1 x S1 - one additional dimension

z

r

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).

2d

trivial

3d

almost trivial

4d difficult

5d … you ain’t seen anything yet

What is the structure of Higher Dimensional Black Holes?

Black String

2222122 )2

1()2

1( dzdrdrr

Mdt

r

Mds

4d space-time

zAsymptotically flat 4d space-time x S1

Horizon topology S2 x S1

z

r

Or a black hole?

4d space-time

zAsymptotically flat 4d space-time x S1

Locally:

origin teh from distance 4D theisr

origin thefrom distance 5D theis

))3/8(

1())3/8(

1( )3(22212

22

2

R

dRdRR

Mdt

R

Mds

2222122 )2

1()2

1( dzdrdrr

Mdt

r

Mds

Horizon topology S3

z

r

Black String

Black Hole

A single dimensionless parameter

3dN

L

mG

Black hole

L

Black String

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

3dN

L

mG

r

z

L

Uniform B-Str: Non-uniform B-Str - ?

Black hole

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

Compare the entropies (in 5d): Sbh~3/2 vs. Sbs ~ 2

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.

non-uniform > uniform 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 rr

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 c

Poor man’s Gravity – Poor man’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

zrr

rr

drdzdrds

Higher Dim Black holes exist?

There is a Bh-Bstr transition

Pro

per

dist

ance

Log(-c)

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

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

)1/(

2/)1/(

minmax

minmax

rr

rr

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

order param

merger point

?

x

GL

Uniform 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 behavior Asymptotic behavior

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

Sorkin ,Kol,TP ‘03Harmark & Obers ‘03[Townsend & Zamaklar ’01,Traschen,’03]

TGh

hhhh

hhg

d16

02

1

1

Asymptotic Charges:Asymptotic Charges:

The mass: m T00dd-1x

The Tension: -Tzzdd-1x /L

3

34

34

23

22222222

8

)3(1

131

)1

(

)1

(:alyAsymptotic

)(

dd

d

dd

dd

dCA

kb

a

d

d

kL

mr

or

br

or

aA

drdredzedteds

Ori 04

dd 3

8k

The asymptotic coefficient b determines the length along

the z-direction.b=kd[(d-3)m-L]

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

BH

Bstr

Archimedes for BHs

For a uniform Bstr: both effects cancel:

But not for a BH:

Together with

We get (gas )

Smarr’s formula (Integrated First Law)

][8

1

16

1),( 0

1 KKdVG

RdVG

FLI dd

dd

dLL

Id

IdI

TSmF

dLTdSdm pdVTdSdE

LTSdmd

LArea d

)2()3(

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

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

aGk

LdA

GTS

NdN

)4(

8

1

Numerical Solution Sorkin Kol & TP

Numerical Convergence I:

Numerical Convergence II:

Numerical Test I (constraints):

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

Ellipticity

“Archimedes”

Analytic expressions Gorbonos & Kol 04

eccentricity

distance

A Possible Bh Bstr Transition?

Anticipated phase diagramAnticipated phase diagram [ Kol ‘02 ]

order param

merger point

?

x

GL

Uniform Bstr

Non-uniform Bstr

Gubser 5d ,’01Wiseman 6d ,’02

d=10 a critical dimension

Black hole

Kudoh & Wiseman ‘03ES,Kol,Piran ‘03

Scalar charge

GL

xmerger point

?

The phase diagram:

BHs

Uni

form

Bst

r

Non-u

nifor

m

Bstr

Gubser: First order uniform-nonuniform black 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

BHs

Perturbative Non-uniform Bstr

Uni

form

Bst

r

?

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

)(3

dN

dc

L

mG

with = 0.686

corrected BH:Harmark ‘03Kol&Gorbonos

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

)()()3(

)3(

16

1

)3(

16

4

1

)2(

16

4

1

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

)3)(3(

42

33)0(

4

3

33

3

2

22

)0(

BstrBHdd

dd

dd

dd

d

d

dd

dBstr

d

d

dd

dBH

SSd

d

Ld

m

GS

d

m

GS

)()(

)()2(2

16)3(1

)1()1()1(

2

2

)0()1(

BstrBH

dBHBH

SS

Od

mdSS

Entropies:

The curves intersect at d~13. This suggests that for d>13 A BH is entropically preferable over the string only for c . 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;

S

S

Interpretations & implications

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

BHs

Perturbative Non-uniform Bstr

Uni

form

Bst

r

?

b

filling the blob

a smooth merger

discontinuous

a new phase,

disconnected

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