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