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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
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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
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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
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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
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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
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Black String
Black Hole
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A single dimensionless parameter
3−≡ dN
LmGµ
Black hole
L
µ
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A single dimensionless parameter
3−≡ dN
LmGµ
B
L
Black Stringµ
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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
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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 ]
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Dynamical Instaibility of a Black String Choptuik, Lehner, Olabarrieta, Petryk,
Pretorius & Villegas 2003
2/)1/( minmax −= rrλ
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Non-Uniform Black String Solutions (Wiseman, 02)
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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
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The Static Black hole Solutions
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Equations of Motion
22/)cos( zrz +≡≡ χξ ψψ c−=∆
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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
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Pro
per d
ista
nce
Log(µ-µc)
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A similar behavior is seen in 4DA similar behavior is seen in 4D
Apparent horizons: From a simulation of bh merger Seidel & Brügmann
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µ
µ
)1/(2/)1/(
minmax
minmax
−′′=′−=
rrrr
λλ
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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
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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
−=∆
=∂−=
<+=
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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
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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
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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.
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Numerical SolutionSorkin Kol & TP
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Numerical Convergence I:
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Numerical Convergence II:
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Numerical Test I (constraints):
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Numerical Test II (The BH Area and Smarr’s formula):
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Analytic expressions Gorbonos & Kol 04
eccentricity
distance
Ellipticity
“Archimedes”
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A Possible Bh BstrTransition?
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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
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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
?
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)( 3−≡
∝
dN
dc
LmGµ
γµ with γ= 0.686
The deviations of the calculated points from the linear fit are less than 2.1%
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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−−
+=−
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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.
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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
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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
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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?
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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
![Page 43: Higher Dimensional Black Holesusers.wfu.edu/cookgb/YorkFest/Talks/Piran_YorkFest.pdfHigher Dimensional Black Holes Tsvi Piran Evgeny Sorkin & Barak Kol The Hebrew University, Jerusalem](https://reader035.vdocuments.net/reader035/viewer/2022071015/5fce0d72f2e1032d970f399d/html5/thumbnails/43.jpg)
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