Post on 22-Aug-2020
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