Higher Dimensional Black Holes

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Higher Dimensional Black Holes

• Tsvi Piran
• Evgeny 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 

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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
4d space-time R3,1
Higher dimensions
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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).

2d trivial
3d almost trivial
4d difficult
5d you aint seen anything yet
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What is the structure of Higher Dimensional
Black Holes?
Black String
Asymptotically flat 4d space-time x S1
z
4d space-time
Horizon topology S2 x S1
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Or a black hole?
Locally
Asymptotically flat 4d space-time x S1
z
4d space-time
Horizon topology S3
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Black String
Black Hole
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A single dimensionless parameter
L
Black hole
Black String
m
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What Happens in the inverse direction - a
shrinking String?
m
?
Uniform B-Str
Non-uniform B-Str - ?
z
L
r
Black hole
Gregory Laflamme (GL) the string is unstable
below a certain mass.
Compare the entropies (in 5d) Sbhm3/2
vs. Sbs m2 Expect a phase tranistion
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Horowitz-Maeda ( 01) Horizon doesnt
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.
mnon-uniform gt muniform Snon-uniform gt 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
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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
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Poor mans Gravity The Initial Value Problem
(Sorkin TP 03)Consider a moment of time
symmetry
There is a Bh-Bstr transition
Higher Dim Black holes exist?
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Proper distance
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A similar behavior is seen in 4D
Apparent horizons From a simulation of bh merger
Seidel Brügmann
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The Anticipated phase diagram Kol 02
m
order param
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Asymptotic behavior
Sorkin ,Kol,TP 03 Harmark Obers 03 Townsend
Zamaklar 01,Traschen,03

• At r??
• The metric become z-independent as exp(-r/L)
  
• Newtonian limit

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Asymptotic Charges The mass m ?
?T00dd-1x The Tension t ? -?Tzzdd-1x /L
Ori 04
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The asymptotic coefficient b determines the
length along the z-direction. bkd(d-3)m-tL
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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Smarrs formula (Integrated First Law)
Together with
(gas )
We get
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.
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Numerical Solution Sorkin 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 Smarrs
formula)
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eccentricity
Ellipticity
Analytic expressions Gorbonos Kol 04
distance
Archimedes
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A Possible Bh Bstr Transition?
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Anticipated phase diagram Kol 02
m
order param
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merger point
The phase diagram
x
Uniform Bstr
Non-uniform Bstr
?
GL
BHs
Universal? Vary d ! E. Sorkin 2004 Motivations
(1) Kols critical dimension for the BH-BStr
merger (d10) (2) Problems in numerics
above d10
Scalar charge
For dgt13 a sudden change in the order of the
phase transition. It becomes smooth
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with g 0.686
The deviations of the calculated points from the
linear fit are less than 2.1
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Entropies
corrected BH Harmark 03 KolGorbonos
for a given mass the entropy of a caged BH is
larger
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The curves intersect at d13. This suggests that
for dgt13 A BH is entropically preferable over
the string only for mltmc . 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 dgt13 the non-uniform string is less massive
and has a higher entropy than the uniform one. A
smooth decay becomes possible.
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Interpretations implications
Above d13 the unstable GL string can decay into
a non-uniform state continuously.
b
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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 d10?

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Summary

• 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

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Open Questions

• Non uniquness of higher dim black holes
• Topology change at BH-Bstr transition
• Cosmic censorship at BH-Bstr transition
• Thunderbolt (release of Lc5/G)
• Numerical-Gravitostatics
• Stability of rotating black strings