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Higher Dimensional Black Holes Tsvi Piran Evgeny Sorkin & Barak Kol The Hebrew University, 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. D 69 (2004) 064031 E. Sorkin, B. Kol & TP, Phys. Rev. D 69 (2004) 064032 E. Sorkin, Phys. Rev. Lett. (2004) in press

Kaluza-Klein, String Theory and other theories suggest that Spacetime may have more than 4 Kaluza-Klein, String Theory and other theories suggest that Spacetime may have more than 4 dimensions. The simplest example is: R 3, 1 x S 1 - one additional dimension z r Higher dimensions 4 d space-time R 3, 1

d – the number of dimensions is the only free parameter in classical GR. 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 theory when we vary this parameter (Kol, 04). 2 d 3 d trivial almost trivial 4 d difficult 5 d … you ain’t seen anything yet

What is the structure of Higher Dimensional Black Holes? Black String z Asymptotically flat What is the structure of Higher Dimensional Black Holes? Black String z Asymptotically flat 4 d spacetime x S 1 z r 4 d space-time Horizon topology S 2 x S 1

Or a black hole? Locally: z z Asymptotically flat 4 d spacetime x S Or a black hole? Locally: z z Asymptotically flat 4 d spacetime x S 1 r Horizon topology S 3 4 d space-time

Black String Black Hole Black String Black Hole

A single dimensionless parameter L Black String Black hole m A single dimensionless parameter L Black String Black hole m

What Happens in the inverse direction - a shrinking String? m ? Non-uniform B-Str What Happens in the inverse direction - a shrinking String? m ? Non-uniform B-Str - ? Uniform B-Str: z L r Gregory & Laflamme (GL) : the string is unstable below a certain mass. Black hole Compare the entropies (in 5 d): Sbh~m 3/2 vs. Sbs ~ m 2 Expect a phase tranistion

Horowitz-Maeda ( ‘ 01): Horizon doesn’t pinch off! The end-state of the GL instability 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 5 d ‘ 01 ] the nonuniform branch emerging from the GL point cannot be the endstate of this instability. mnon-uniform > muniform Snon-uniform > Suniform Dynamical: no signs of stabilization (5 d) [CLOPPV ‘ 03]. This branch non-perturbatively (6 d) [ Wiseman ‘ 02 ]

Dynamical Instaibility of a Black String Choptuik, Lehner, Olabarrieta, Petryk, Pretorius & Villegas 2003 Dynamical Instaibility of a Black String Choptuik, Lehner, Olabarrieta, Petryk, Pretorius & Villegas 2003

Non-Uniform Black String Solutions (Wiseman, 02) Non-Uniform Black String Solutions (Wiseman, 02)

Objectives: • Explore the structure of higher dimensional spherical black holes. • Establish that 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 The Static Black hole Solutions

Equations of Motion Equations of Motion

Poor man’s Gravity – The Initial Value Problem (Sorkin & TP 03) Consider a Poor man’s 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?

Log(m-mc) Proper distance Log(m-mc) Proper distance

A similar behavior is seen in 4 D Apparent horizons: From a simulation of A similar behavior is seen in 4 D Apparent horizons: From a simulation of bh merger Seidel & Brügmann

m m m m

The Anticipated phase diagram [ Kol ‘ 02 ] m x Uniform Bstr GL The Anticipated phase diagram [ Kol ‘ 02 ] m x Uniform Bstr GL merger point Sorkin & TP 03 ? Black hole order param Non-uniform Bstr Gubser 5 d , ’ 01 Wiseman 6 d , ’ 02 We need a “good” order parameter allowing to put all the phases on the same diagram. [Scalar charge]

Asymptotic behavior Sorkin , Kol, TP ‘ 03 Harmark & Obers ‘ 03 [Townsend 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

Asymptotic Charges: The mass: m T 00 dd-1 x The Tension: t t - Asymptotic Charges: The mass: m T 00 dd-1 x The Tension: t t - Tzzdd-1 x /L t Ori 04

The asymptotic coefficient b determines the length along the z-direction. b=kd[(d-3)m-t. L] The mass The asymptotic coefficient b determines the length along the z-direction. b=kd[(d-3)m-t. L] 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 We get This formula associate quantities on Smarr’s formula (Integrated First Law) Together with 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. (gas )

Numerical Solution Sorkin Kol & TP Numerical Solution Sorkin Kol & TP

Numerical Convergence I: Numerical Convergence I:

Numerical Convergence II: Numerical Convergence II:

Numerical Test I (constraints): Numerical Test I (constraints):

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

eccentricity Ellipticity Analytic expressions Gorbonos & Kol 04 distance “Archimedes” eccentricity Ellipticity Analytic expressions Gorbonos & Kol 04 distance “Archimedes”

A Possible Bh Bstr Transition? A Possible Bh Bstr Transition?

Anticipated phase diagram [ Kol ‘ 02 ] m x Uniform Bstr GL merger Anticipated phase diagram [ Kol ‘ 02 ] m x Uniform Bstr GL merger point ? Black hole order param Non-uniform Bstr Gubser 5 d , ’ 01 Wiseman 6 d , ’ 02 d=10 a critical dimension Kudoh & Wiseman ‘ 03 ES, Kol, Piran ‘ 03

Uniform Bstr The phase diagram: Uniform Bstr Gubser: First order uniform-nonuniform black strings phase Uniform Bstr The phase diagram: Uniform Bstr Gubser: First order uniform-nonuniform black strings phase transition. explosions, cosmic censorship? merger point rm fo i un tr n- Bs o N GL x ? BHs Scalar charge Universal? Vary d ! E. Sorkin 2004 Motivations: (1) Kol’s critical dimension for the BH-BStr merger (d=10) (2) Problems in numerics above d=10 Perturbative Nonuniform Bstr ? For d*>13 a sudden change in the order of the phase transition. It becomes smooth BHs Scalar charge

with g= 0. 686 The deviations of the calculated points from the linear fit with g= 0. 686 The deviations of the calculated points from the linear fit are less than 2. 1%

Entropies: corrected BH: Harmark ‘ 03 Kol&Gorbonos for a given mass the entropy of Entropies: corrected BH: Harmark ‘ 03 Kol&Gorbonos for a given mass the entropy of a caged BH is larger

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

A comparison between a uniform and a non-uniform String: Trends in mass and entropy 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 entropy than the uniform one. A smooth decay becomes possible.

Interpretations & implications Uniform Bstr Above d*=13 the unstable GL string can decay into Interpretations & implications Uniform Bstr Above d*=13 the unstable GL string can decay into a non-uniform state continuously. a smooth merger Perturbative Nonuniform Bstr filling the blob ? discontinuous BHs b a new phase, disconnected

Summary • We have demonstrated the existence of BH solutions. • Indication for a 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

Open Questions • • • Non uniquness of higher dim black holes Topology change Open Questions • • • Non uniquness of higher dim black holes Topology change at BH-Bstr transition Cosmic censorship at BH-Bstr transition Thunderbolt (release of L=c 5/G) Numerical-Gravitostatics Stability of rotating black strings