tomimatsu-sato geometries and the kerr/cft...

TS & Kerr/CFT arXiv:1010.2803 – 1 / 15

Tomimatsu-Sato geometries and the

Kerr/CFT correspondance

Sanjeev Seahra (with J Gegenberg, H Liu and B Tippett)

Black holes VIII: May 13, 2011

Introduction

TS spacetimes

The “dual CFT”

Summary

TS & Kerr/CFT arXiv:1010.2803 – 2 / 15

� Kerr/CFT: attempt to obtain holographic description of extremalKerr black holes (originally)

� based on structure of near-horizon geometry

� CFT Cardy formula reproduces gravitational entropy

� many properties of CFT remain mysterious

� may be useful to look at to more examples of extremal spinninghorizons in GR

� e.g. extremal Kerr-Bolt (Ghezelbash 2009)

� here we consider generalization to the (admittedlypathological) Tomimatsu-Sato (1972) spacetimes

Introduction

TS spacetimes

The “dual CFT”

Summary

TS & Kerr/CFT arXiv:1010.2803 – 2 / 15

Introduction

TS spacetimes

The “dual CFT”

Summary

TS & Kerr/CFT arXiv:1010.2803 – 2 / 15

Introduction

TS spacetimes

The “dual CFT”

Summary

TS & Kerr/CFT arXiv:1010.2803 – 2 / 15

Introduction

TS spacetimes

The “dual CFT”

Summary

TS & Kerr/CFT arXiv:1010.2803 – 2 / 15

Introduction

TS spacetimes

The “dual CFT”

Summary

TS & Kerr/CFT arXiv:1010.2803 – 2 / 15

Introduction

TS spacetimes

The “dual CFT”

Summary

TS & Kerr/CFT arXiv:1010.2803 – 2 / 15

Tomimatsu-Sato spacetimes

Introduction

TS spacetimes

Basic properties

The metric

Ergoregion and CTCs

Ring singularity

Killing horizons

M and J

The “dual CFT”

Summary

TS & Kerr/CFT arXiv:1010.2803 – 3 / 15

Basic properties of Tomimatsu-Sato spacetimes

Introduction

TS spacetimes

Basic properties

The metric

Ergoregion and CTCs

Ring singularity

Killing horizons

M and J

The “dual CFT”

Summary

TS & Kerr/CFT arXiv:1010.2803 – 4 / 15

4D vacuum solutions of GR: [cf. Hikida and

Kodama 2003]

Introduction

TS spacetimes

Basic properties

The metric

Ergoregion and CTCs

Ring singularity

Killing horizons

M and J

The “dual CFT”

Summary

TS & Kerr/CFT arXiv:1010.2803 – 4 / 15

4D vacuum solutions of GR:

stationary

[cf. Hikida and

Kodama 2003]

Introduction

TS spacetimes

Basic properties

The metric

Ergoregion and CTCs

Ring singularity

Killing horizons

M and J

The “dual CFT”

Summary

TS & Kerr/CFT arXiv:1010.2803 – 4 / 15

stationary

axisymmetric

[cf. Hikida and

Kodama 2003]

Introduction

TS spacetimes

Basic properties

The metric

Ergoregion and CTCs

Ring singularity

Killing horizons

M and J

The “dual CFT”

Summary

TS & Kerr/CFT arXiv:1010.2803 – 4 / 15

asymptotically flat

stationary

axisymmetric

[cf. Hikida and

Kodama 2003]

Introduction

TS spacetimes

Basic properties

The metric

Ergoregion and CTCs

Ring singularity

Killing horizons

M and J

The “dual CFT”

Summary

TS & Kerr/CFT arXiv:1010.2803 – 4 / 15

asymptotically flat

stationary

axisymmetric

(Kerr black holes are special cases of TS solutions)

[cf. Hikida and

Kodama 2003]

The metric

Introduction

TS spacetimes

Basic properties

The metric

Ergoregion and CTCs

Ring singularity

Killing horizons

M and J

The “dual CFT”

Summary

TS & Kerr/CFT arXiv:1010.2803 – 5 / 15

The metric

Introduction

TS spacetimes

Basic properties

The metric

Ergoregion and CTCs

Ring singularity

Killing horizons

M and J

The “dual CFT”

Summary

TS & Kerr/CFT arXiv:1010.2803 – 5 / 15

The metric

Introduction

TS spacetimes

Basic properties

The metric

Ergoregion and CTCs

Ring singularity

Killing horizons

M and J

The “dual CFT”

Summary

TS & Kerr/CFT arXiv:1010.2803 – 5 / 15

fixed length scale

The metric

Introduction

TS spacetimes

Basic properties

The metric

Ergoregion and CTCs

Ring singularity

Killing horizons

M and J

The “dual CFT”

Summary

TS & Kerr/CFT arXiv:1010.2803 – 5 / 15

fixed length scale

The metric

Introduction

TS spacetimes

Basic properties

The metric

Ergoregion and CTCs

Ring singularity

Killing horizons

M and J

The “dual CFT”

Summary

TS & Kerr/CFT arXiv:1010.2803 – 5 / 15

fixed length scale

The metric

Introduction

TS spacetimes

Basic properties

The metric

Ergoregion and CTCs

Ring singularity

Killing horizons

M and J

The “dual CFT”

Summary

TS & Kerr/CFT arXiv:1010.2803 – 5 / 15

fixed length scale

The metric

Introduction

TS spacetimes

Basic properties

The metric

Ergoregion and CTCs

Ring singularity

Killing horizons

M and J

The “dual CFT”

Summary

TS & Kerr/CFT arXiv:1010.2803 – 5 / 15

fixed length scale

The metric

Introduction

TS spacetimes

Basic properties

The metric

Ergoregion and CTCs

Ring singularity

Killing horizons

M and J

The “dual CFT”

Summary

TS & Kerr/CFT arXiv:1010.2803 – 5 / 15

fixed length scale

Ergoregion and closed timelike curves

Introduction

TS spacetimes

Basic properties

The metric

Ergoregion and CTCs

Ring singularity

Killing horizons

M and J

The “dual CFT”

Summary

TS & Kerr/CFT arXiv:1010.2803 – 6 / 15

signature of timelike tα andazimuthal φα Killing vectorsvaries over (ρ, z) plane:

Introduction

TS spacetimes

Basic properties

The metric

Ergoregion and CTCs

Ring singularity

Killing horizons

M and J

The “dual CFT”

Summary

TS & Kerr/CFT arXiv:1010.2803 – 6 / 15

� ergosphere (tαtα > 0)

Introduction

TS spacetimes

Basic properties

The metric

Ergoregion and CTCs

Ring singularity

Killing horizons

M and J

The “dual CFT”

Summary

TS & Kerr/CFT arXiv:1010.2803 – 6 / 15

� closed timelike curves(φαφα < 0)

Introduction

TS spacetimes

Basic properties

The metric

Ergoregion and CTCs

Ring singularity

Killing horizons

M and J

The “dual CFT”

Summary

TS & Kerr/CFT arXiv:1010.2803 – 6 / 15

� closed timelike curves(φαφα < 0)

� conical singularity S withdeficit (excess):2πp2/(1− p2)

Ring singularity

Introduction

TS spacetimes

Basic properties

The metric

Ergoregion and CTCs

Ring singularity

Killing horizons

M and J

The “dual CFT”

Summary

TS & Kerr/CFT arXiv:1010.2803 – 7 / 15

Ring singularity

Introduction

TS spacetimes

Basic properties

The metric

Ergoregion and CTCs

Ring singularity

Killing horizons

M and J

The “dual CFT”

Summary

TS & Kerr/CFT arXiv:1010.2803 – 7 / 15

Ring singularity

Introduction

TS spacetimes

Basic properties

The metric

Ergoregion and CTCs

Ring singularity

Killing horizons

M and J

The “dual CFT”

Summary

TS & Kerr/CFT arXiv:1010.2803 – 7 / 15

Ring singularity

Introduction

TS spacetimes

Basic properties

The metric

Ergoregion and CTCs

Ring singularity

Killing horizons

M and J

The “dual CFT”

Summary

TS & Kerr/CFT arXiv:1010.2803 – 7 / 15

Killing horizons

Introduction

TS spacetimes

Basic properties

The metric

Ergoregion and CTCs

Ring singularity

Killing horizons

M and J

The “dual CFT”

Summary

TS & Kerr/CFT arXiv:1010.2803 – 8 / 15

Killing horizons

Introduction

TS spacetimes

Basic properties

The metric

Ergoregion and CTCs

Ring singularity

Killing horizons

M and J

The “dual CFT”

Summary

TS & Kerr/CFT arXiv:1010.2803 – 8 / 15

Killing horizons

Introduction

TS spacetimes

Basic properties

The metric

Ergoregion and CTCs

Ring singularity

Killing horizons

M and J

The “dual CFT”

Summary

TS & Kerr/CFT arXiv:1010.2803 – 8 / 15

Killing horizons

Introduction

TS spacetimes

Basic properties

The metric

Ergoregion and CTCs

Ring singularity

Killing horizons

M and J

The “dual CFT”

Summary

TS & Kerr/CFT arXiv:1010.2803 – 8 / 15

Killing horizons

Introduction

TS spacetimes

Basic properties

The metric

Ergoregion and CTCs

Ring singularity

Killing horizons

M and J

The “dual CFT”

Summary

TS & Kerr/CFT arXiv:1010.2803 – 8 / 15

Mass and angular momentum

Introduction

TS spacetimes

Basic properties

The metric

Ergoregion and CTCs

Ring singularity

Killing horizons

M and J

The “dual CFT”

Summary

TS & Kerr/CFT arXiv:1010.2803 – 9 / 15

Introduction

TS spacetimes

Basic properties

The metric

Ergoregion and CTCs

Ring singularity

Killing horizons

M and J

The “dual CFT”

Summary

TS & Kerr/CFT arXiv:1010.2803 – 9 / 15

Introduction

TS spacetimes

Basic properties

The metric

Ergoregion and CTCs

Ring singularity

Killing horizons

M and J

The “dual CFT”

Summary

TS & Kerr/CFT arXiv:1010.2803 – 9 / 15

Introduction

TS spacetimes

Basic properties

The metric

Ergoregion and CTCs

Ring singularity

Killing horizons

M and J

The “dual CFT”

Summary

TS & Kerr/CFT arXiv:1010.2803 – 9 / 15

Introduction

TS spacetimes

Basic properties

The metric

Ergoregion and CTCs

Ring singularity

Killing horizons

M and J

The “dual CFT”

Summary

TS & Kerr/CFT arXiv:1010.2803 – 9 / 15

Properties of the “dual CFT”

Introduction

TS spacetimes

The “dual CFT”

NHES metric

Central charge

Temperature

Cardy formula

Summary

TS & Kerr/CFT arXiv:1010.2803 – 10 / 15

Near horizon extremal spinning (NHES) metric

Introduction

TS spacetimes

The “dual CFT”

NHES metric

Central charge

Temperature

Cardy formula

Summary

TS & Kerr/CFT arXiv:1010.2803 – 11 / 15

Introduction

TS spacetimes

The “dual CFT”

NHES metric

Central charge

Temperature

Cardy formula

Summary

TS & Kerr/CFT arXiv:1010.2803 – 11 / 15

Introduction

TS spacetimes

The “dual CFT”

NHES metric

Central charge

Temperature

Cardy formula

Summary

TS & Kerr/CFT arXiv:1010.2803 – 11 / 15

Introduction

TS spacetimes

The “dual CFT”

NHES metric

Central charge

Temperature

Cardy formula

Summary

TS & Kerr/CFT arXiv:1010.2803 – 11 / 15

Introduction

TS spacetimes

The “dual CFT”

NHES metric

Central charge

Temperature

Cardy formula

Summary

TS & Kerr/CFT arXiv:1010.2803 – 11 / 15

Introduction

TS spacetimes

The “dual CFT”

NHES metric

Central charge

Temperature

Cardy formula

Summary

TS & Kerr/CFT arXiv:1010.2803 – 11 / 15

Introduction

TS spacetimes

The “dual CFT”

NHES metric

Central charge

Temperature

Cardy formula

Summary

TS & Kerr/CFT arXiv:1010.2803 – 11 / 15

Introduction

TS spacetimes

The “dual CFT”

NHES metric

Central charge

Temperature

Cardy formula

Summary

TS & Kerr/CFT arXiv:1010.2803 – 11 / 15

Central charge

Introduction

TS spacetimes

The “dual CFT”

NHES metric

Central charge

Temperature

Cardy formula

Summary

TS & Kerr/CFT arXiv:1010.2803 – 12 / 15

Calculation of the central charge of candidate CFT dual to NHESgeometry proceeds as in Kerr/CFT (Strominger et al 2009):

1. find diffeomorphisms ξn preserving selected BCs on metricfluctuations

2. use covariant formalism (Barnich and Compere 2008) to find(Virasoro) algebra of associated charges Qξn under Diracbracket

i{Qζm, Qζn} = (m−n)Qζm+ζn+γ2J∆m

δm+n,0

3. read off the central charge c = 6γr20/G = 12γ2J∆

(recover Kerr/CFT for γ = 1 and J∆ = JADM)

Central charge

Introduction

TS spacetimes

The “dual CFT”

NHES metric

Central charge

Temperature

Cardy formula

Summary

TS & Kerr/CFT arXiv:1010.2803 – 12 / 15

δm+n,0

Central charge

Introduction

TS spacetimes

The “dual CFT”

NHES metric

Central charge

Temperature

Cardy formula

Summary

TS & Kerr/CFT arXiv:1010.2803 – 12 / 15

δm+n,0

Central charge

Introduction

TS spacetimes

The “dual CFT”

NHES metric

Central charge

Temperature

Cardy formula

Summary

TS & Kerr/CFT arXiv:1010.2803 – 12 / 15

δm+n,0

Central charge

Introduction

TS spacetimes

The “dual CFT”

NHES metric

Central charge

Temperature

Cardy formula

Summary

TS & Kerr/CFT arXiv:1010.2803 – 12 / 15

δm+n,0

Temperature

Introduction

TS spacetimes

The “dual CFT”

NHES metric

Central charge

Temperature

Cardy formula

Summary

TS & Kerr/CFT arXiv:1010.2803 – 13 / 15

The temperature of the candidate CFT found from analyzing scalarwave equation �Φ = 0 in NHES geometry

1. change from (t, r, φ) to (w±, y) coordinates such that � isrepresented by quadratic function of SL(2,R) generators

2. conformal symmetry broken by periodic identification ofw− ∼ w−e

2π/γ

3. induces a finite Unruh temperature for theSL(2,R)× SL(2,R) invariant vacuum according toco-rotating (t, r, φ) observers:

TL = (2πγ)−1

(again γ = 1 reproduces Kerr/CFT result)

Temperature

Introduction

TS spacetimes

The “dual CFT”

NHES metric

Central charge

Temperature

Cardy formula

Summary

TS & Kerr/CFT arXiv:1010.2803 – 13 / 15

2π/γ

TL = (2πγ)−1

Temperature

Introduction

TS spacetimes

The “dual CFT”

NHES metric

Central charge

Temperature

Cardy formula

Summary

TS & Kerr/CFT arXiv:1010.2803 – 13 / 15

2π/γ

TL = (2πγ)−1

Temperature

Introduction

TS spacetimes

The “dual CFT”

NHES metric

Central charge

Temperature

Cardy formula

Summary

TS & Kerr/CFT arXiv:1010.2803 – 13 / 15

2π/γ

TL = (2πγ)−1

Temperature

Introduction

TS spacetimes

The “dual CFT”

NHES metric

Central charge

Temperature

Cardy formula

Summary

TS & Kerr/CFT arXiv:1010.2803 – 13 / 15

2π/γ

TL = (2πγ)−1

Cardy formula

Introduction

TS spacetimes

The “dual CFT”

NHES metric

Central charge

Temperature

Cardy formula

Summary

TS & Kerr/CFT arXiv:1010.2803 – 14 / 15

� Cardy formula: SCFT =1

3π2cTL =

πr20G

Cardy formula

Introduction

TS spacetimes

The “dual CFT”

NHES metric

Central charge

Temperature

Cardy formula

Summary

TS & Kerr/CFT arXiv:1010.2803 – 14 / 15

3π2cTL =

πr20G

� Geometric entropy: SBH =A∆

πr20G

Cardy formula

Introduction

TS spacetimes

The “dual CFT”

NHES metric

Central charge

Temperature

Cardy formula

Summary

TS & Kerr/CFT arXiv:1010.2803 – 14 / 15

3π2cTL =

πr20G

� SCFT = SBH for all members NHES class including:

Cardy formula

Introduction

TS spacetimes

The “dual CFT”

NHES metric

Central charge

Temperature

Cardy formula

Summary

TS & Kerr/CFT arXiv:1010.2803 – 14 / 15

3π2cTL =

πr20G

� extremal Kerr

Cardy formula

Introduction

TS spacetimes

The “dual CFT”

NHES metric

Central charge

Temperature

Cardy formula

Summary

TS & Kerr/CFT arXiv:1010.2803 – 14 / 15

3π2cTL =

πr20G

� extremal Kerr

� extremal Kerr-Bolt

Cardy formula

Introduction

TS spacetimes

The “dual CFT”

NHES metric

Central charge

Temperature

Cardy formula

Summary

TS & Kerr/CFT arXiv:1010.2803 – 14 / 15

3π2cTL =

πr20G

� extremal Kerr

� extremal Kerr-Bolt

� Tomimatsu-Sato (sort of . . . )

Cardy formula

Introduction

TS spacetimes

The “dual CFT”

NHES metric

Central charge

Temperature

Cardy formula

Summary

TS & Kerr/CFT arXiv:1010.2803 – 14 / 15

Cardy formula

Introduction

TS spacetimes

The “dual CFT”

NHES metric

Central charge

Temperature

Cardy formula

Summary

TS & Kerr/CFT arXiv:1010.2803 – 14 / 15

Cardy formula

Introduction

TS spacetimes

The “dual CFT”

NHES metric

Central charge

Temperature

Cardy formula

Summary

TS & Kerr/CFT arXiv:1010.2803 – 14 / 15

Summary

Introduction

TS spacetimes

The “dual CFT”

Summary

TS & Kerr/CFT arXiv:1010.2803 – 15 / 15

� TS spacetime is a Kerrgeneralization that’s full of exoticfeatures ⇒ a “lesion study” ofKerr/CFT

� we have derived c and TL for a CFTthat may be dual to some portion ofTS spacetime

� CFT properties we calculated seemto be ignorant of ring singularity,CTC region, etc

� need to ask more probing questionsto determine if CFT is only dual toH±

Summary

Introduction

TS spacetimes

The “dual CFT”

Summary

TS & Kerr/CFT arXiv:1010.2803 – 15 / 15

Summary

Introduction

TS spacetimes

The “dual CFT”

Summary

TS & Kerr/CFT arXiv:1010.2803 – 15 / 15

Summary

Introduction

TS spacetimes

The “dual CFT”

Summary

TS & Kerr/CFT arXiv:1010.2803 – 15 / 15

tomimatsu-sato geometries and the kerr/cft...

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