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Preprint typeset in JHEP style - HYPER VERSION

A Tomimatsu-Sato/CFT correspondence

Jack Gegenberg, Haitao Liu, Sanjeev S. Seahra, Benjamin K. Tippett

Department of Mathematics and Statistics

University of New Brunswick

Fredericton, NB E3B 5A3

Canada

Abstract: We analyze the δ= 2 Tomimatsu-Sato spacetime in the context of the proposed

Kerr/CFT correspondence. This 4-dimensional vacuum spacetime is asymptotically ﬂat

and has a well-deﬁned ADM mass and angular momentum, but also involves several exotic

features including a naked ring singularity, and two disjoint Killing horizons separated by a

region with closed timelike curves and a rod-like conical singularity. We demonstrate that

the near horizon geometry belongs to a general class of Ricci-ﬂat metrics with SL(2,R)×

U(1) symmetry that includes both the extremal Kerr and extremal Kerr-bolt geometries.

We calculate the central charge and temperature for the CFT dual to this spacetime and

conﬁrm the Cardy formula reproduces the Bekenstein-Hawking entropy. We ﬁnd that all

of the basic parameters of the dual CFT are most naturally expressed in terms of charges

deﬁned intrinsically on the horizon, which are distinct from the ADM charges in this

geometry.

arXiv:1010.2803v1 [hep-th] 14 Oct 2010

Contents

1. Introduction 1

2. Classical features of the δ= 2 Tomimatsu-Sato spacetime 3

2.1 Line element and coordinate systems 3

2.2 Far ﬁeld region 5

2.3 Killing vectors and stationary observers 6

2.4 The naked ring singularity 7

2.5 Distribution of mass and angular momentum 8

2.6 Killing horizons and the near horizon metric 9

2.7 Nature of the Shypersurface 11

3. CFT description of the near horizon geometry 12

3.1 Central charge 12

3.2 Temperature 14

3.3 Entropy 15

4. Discussion 16

A. Genericness of the near horizon extremal spinning (NHES) metric 17

1. Introduction

There is a broad consensus in gravitational physics that the endstate of the collapse of

uncharged matter is the Kerr black hole. This belief is implicitly based on the cosmic

censorship conjecture [1], which states that curvature singularities in general relativity are

necessarily hidden behind event horizons. Since the positive-mass Kerr black hole is the

only known solution of the vacuum ﬁeld equations that is stationary, axisymmetric, stable,

asymptotically ﬂat and free from curvature singularities in causal contact with null inﬁnity,

it is the only viable candidate for the ﬁnal conﬁguration of collapsed matter consistent with

the cosmic censorship conjecture.

However, it is worthwhile noting that cosmic censorship has yet to be proved, and

there are inﬁnitely many other axisymmetric, stationary, and asymptotically ﬂat solutions

of the vacuum Einstein ﬁeld equations involving naked singularities. It is interesting to

study these solutions as possible alternatives to the Kerr metric for the description of

the gravitational ﬁeld around a compact astrophysical body with given mass and angular

momentum. It is also interesting to consider these solutions from a completely diﬀerent

perspective: namely, in the context of quantum gravity. In recent years, there has been

– 1 –

much activity in obtaining dual descriptions of classical black hole spacetimes by conformal

ﬁeld theories (CFTs) living on their boundaries. A natural question is: given that the

Einstein ﬁeld equations admit non-black hole solutions describing collapsed objects, is it

possible to describe such spacetimes using a dual CFT?

To attempt to address this question, we will focus on the Tomimatsu-Sato geometries

[2,3], which are a class of vacuum solutions in general relativity labelled by a parameter

δ. For δ= 1, the Tomimatsu-Sato spacetimes reduces down to the Kerr solution, but

for δ6= 1 they involve naked singularities. In all cases, the solutions are asymptotically

ﬂat, axisymmetric and stationary. The algebraic complexity of these solutions increases

rapidly with δ, so for this reason the δ= 2 case (which we denote by TS2) has been most

intensively studied in the literature [4,5,6,7]. For this choice, the spacetime contains

a ring singularity on the boundary of an ergoregion. At the center of the geometry is a

rod-like conical singularity which carries all of the spacetime’s gravitational mass, and is

surrounded by a region containing closed timelike curves. To date, it is not clear whether

or not physical matter can collapse to a TS2 geometry, and if the TS2 geometry is stable.

In this paper, we calculate the properties of a candidate CFT dual to the TS2 geometry

using tools from the recently proposed Kerr/CFT correspondence [8], which is in turn

inspired by the AdS/CFT correspondence [9]. The latter approach has been particularly

successful in giving a quantum mechanical description of the thermodynamics of a large

variety of extremal black holes whose asymptotic region is a space of constant negative

curvature (for which supersymmetry allows for a continuation from high to low energy

scales). The essence of the calculation is the identiﬁcation of a CFT living on the boundary

of the spacetime that is in some sense “dual” to the black hole horizon.

However, the fact that this duality was initially restricted to asymptotically anti-de

Sitter (AdS) black holes was disappointing, given the mounting physical evidence that the

large scale geometry of our universe does not have an overall negative curvature. The

discovery that extremal Kerr black holes are dual to a CFT [8,10,11] partially overcomes

this problem. This Kerr/CFT correspondence depends on the fact that the near horizon

geometry of the extremal Kerr metric (NHEK) is a warped product of two dimensional AdS

and S2[12]. The asymptotic region of the NHEK metric supports a CFT, enough of whose

properties are known to construct thermodynamics which describe the original extremal

Kerr black hole. One problem is that, unlike the original AdS/CFT correspondence, the

boundary CFT lives on a spacetime which cannot be identiﬁed with the boundary of the

asymptotically ﬂat region of the original Kerr black hole; rather the CFT lives on the

boundary of the NHEK geometry, which is not asymptotically ﬂat. In spite of this caveat,

the program of enlarging the class of geometries which have a dual CFT description is

important for our eventual understanding of quantum gravity, and should be pursued.

In this spirit, the original Kerr/CFT correspondence has been extended in a number of

directions: including the CFT description of near-extremal Kerr black holes obtained by

Castro & Larsen [13] and Kerr-Bolt spacetimes obtained by Ghezelbash [14].

In the current work, we obtain the further extension of the Kerr/CFT correspondence

to non-black hole spacetimes, namely the TS2 geometry. Our treatment is based on the

discovery by Kodama and Hikida that, despite the existence of the naked ring singularity,

– 2 –

the TS2 geometry does indeed contain a pair of disjoint Killing horizons located at each tip

of the rod at the center of the spacetime [6,7]. Each horizon has isometry group SL(2,R)×

U(1) and is extremal in the sense that it has zero surface gravity (i.e. it is degenerate).

These properties are shared by the horizons in extremal Kerr and extremal Kerr-Bolt, so it

is reasonable to postulate that the TS2 geometry also admits a dual CFT description. We

demonstrate that this is indeed that case by ﬁrst showing that the near horizon geometry

in the TS2 solution belongs to a general class of Ricci ﬂat solutions we call the near

horizon extremal spinning (NHES) metrics. These represent the most general vacuum

metrics describing the geometry near a non-toriodal, axisymmetric, extremal horizon in

general relativity [15], and include the near horizon Kerr and Kerr-Bolt solutions. We

then proceed to obtain a dual CFT description for entire class of NHES geometries by

computing the central charge from asymptotic symmetries [8], and the temperature from

analyzing the hidden symmetries of the scalar wave equation [13,16]. When inserted into

the Cardy formula, our derived central charge and temperature successfully reproduce the

Bekenstein-Hawking entropy of the horizon.

We ﬁnd that the basic parameters of the dual CFT are most naturally expressed

in terms of the charges deﬁned intrinsically on the horizon, rather than the ADM charges

associated with the symmetries of the parent spacetime. For example, in the TS2 geometry

the central charge is proportional to the angular momentum of one of the horizons as

opposed to the ADM angular momentum of the entire spacetime. This seems to suggest

that the CFT we discuss is really dual to the TS2 Killing horizon(s) and not the entire

spacetime—indeed, the CFT we derive shows no awareness of the closed timelike curve

region, ring singularity, or other pathologies of the full TS2 geometry.

The plan of the paper is as follows: In §2, we give a self-contained analysis of the basic

properties of the TS2 metrics, including the ring curvature singularity, the existence of

stationary observers, and the distribution of mass and angular momentum, the near horizon

geometry and the nature of the rod singularity. In §3, we calculate the central charge and

temperature of the dual CFT and conﬁrm that the macroscopic and microscopic entropies

agree. In §4, we conclude with a discussion of the the relevant classical and quantum

properties of the TS2 geometry. In the appendix §A, we brieﬂy discuss the uniqueness of

the NHES geometries.

2. Classical features of the δ= 2 Tomimatsu-Sato spacetime

In this section we introduce and analyze the δ= 2 Tomimatsu-Sato (TS2) spacetime (M, g)

[2,3]. The discussion parallels and extends several previous studies [4,5,6,7].

2.1 Line element and coordinate systems

In prolate spherical coordinates (x, y), the TS2 line element is

ds2=−f(d˜

t−ωdφ)2+σ2

fEdx2

x2−1+dy2

1−y2+ (x2−1)(1 −y2)dφ2.(2.1)

– 3 –

Here we have

f=f(x, y)≡A(x, y)

B(x, y), E =E(x, y)≡A(x, y)

p4(x2−y2)3,

ω=ω(x, y)≡4σqC(x, y)

pA(x, y)(1 −y2),(2.2)

where

A(x, y)≡p4(x2−1)4+q4(1 −y2)4−2p2q2(x2−1)(1 −y2)

×2(x2−1)2+ 2(1 −y2)2+ 3(x2−1)(1 −y2);

B(x, y)≡p2(x4−1) −q2(1 −y4)+2px(x2−1)2+

4q2y2px(x2−1) + (px + 1)(1 −y2)2;

C(x, y)≡q2(px + 1)(1 −y2)3−p3x(x2−1) 2(x4−1) + (x2+ 3)(1 −y2)

−p2(x2−1) 4x2(x2−1) + (3x2+ 1)(1 −y2).(2.3)

Also, σis a constant with dimensions of length, while pand qare dimensionless constants.

This metric is Ricci-ﬂat (Rαβ = 0) for any σif we impose

p2+q2= 1.(2.4)

The prolate spheroidal coordinates (x, y) appearing in (2.1) are related to the quasi-

cylindrical coordinates (ρ, z) by

ρ2=σ2(x2−1)(1 −y2), z =σxy, (2.5)

where σis a constant length parameter. Below, we will also make use of the bipolar

coordinate system (˜r, θ), deﬁned by

x2=σ

σ−˜rcos2(θ/2) , y2=σ−˜r

σ−˜rcos2(θ/2) .(2.6)

We take the ranges of the various coordinates to be

ρ∈[0,∞), z ∈(−∞,∞), x ∈[1,∞), y ∈[−1,1],

˜r∈[0, σ], θ ∈[0, π], φ ∈[0,2π].(2.7)

The coordinate traces of the (x, y) and (˜r, θ) coordinates in the (ρ, z) plane are illustrated

in Fig. 1. Note that the bipolar coordinates only cover the upper half of the (ρ, z) plane

in we take positive square roots in (2.6). Also on this ﬁgure are shown special locations,

S={p∈ M|x= 1,|y|<1 or ρ= 0,|z|< σ or θ=π},(2.8a)

H±={p∈ M|x= 1, y =±1 or ρ= 0, z =±σor r= 0},(2.8b)

that we will study in detail below.

– 4 –

Figure 1: Coordinate traces of the prolate spheriodal (x, y) (left ) and bipolar (˜r, θ) (right ) coordi-

nates in the cylindrical (ρ, z) plane. Note we have taken positive square roots in (2.6), which means

that the bipolar coordinates only cover the upper-half of the (ρ, z) plane.

2.2 Far ﬁeld region

In the limit x→ ∞, the line element reduces to

ds2=−1−4

px +O1

x2d˜

t2+16σq(1 −y2)

p2xd˜

t dφ

+σ21 + 4

px +O1

x2dx2+x2dy2

1−y2+x2(1 −y2)dφ2.(2.9)

Making the following identiﬁcations:

R=σx, y = cos ϑ, M =2σ

Gp, J =qGM2=4σ2q

Gp2,(2.10)

puts this in the form

ds2≈ −1−2GM

Rd˜

t2+4GJ sin2ϑ

Rd˜

t dφ+

1 + 2GM

RdR2+R2(dϑ2+ sin2ϑ dφ2).(2.11)

This matches what we would expect for an aysmptotically ﬂat metric about a matter

distribution of ADM mass Mand angular momentum J(we use units in which ~=c= 1).

From the above identiﬁcation we can infer that to ensure that Rand Mare positive, we

should enforce

σ > 0,0≤p≤1.(2.12)

– 5 –

Furthermore, we see that the sign of qdetermines whether the angular momentum is

pointing in the positive or negative z-direction. Without loss of generality, we can orient

the z-axis such that the body has positive rotation and hence qis non-negative. Then, we

put

q=p1−p2(2.13)

throughout the following discussion.

2.3 Killing vectors and stationary observers

The metric (2.1) has two obvious Killing vectors:

˜

tα∂α=∂˜

t, φα∂α=∂φ.(2.14)

We have the following:

˜

tα˜

tα=−f(x, y), φαφα=−f(x, y)ω2(x, y) + σ2(x2−1)(1 −y2)

f(x, y),

˜

tαφα=f(x, y)ω(x, y).(2.15)

These Killing vectors can be used to deﬁne the 4-velocity of stationary observers:

uα=κ−1(˜

tα+ Ωφα), uαuα=−1,(2.16)

Here, Ω is a constant representing the observer’s angular velocity and κis a normalization

factor. The local geometry seen by such an observer will be time-independent, hence the

name “stationary”. In the Kerr and Schwarzschild geometries, such observers cannot exist

in the trapped regions bounded by event horizons; i.e., there is no Ω ∈Rsuch that uαis

timelike inside an event horizon.

Generally, stationary observers will exist if we can choose Ω ∈Rsuch that κ2>0,

where

κ2=−˜

tα˜

tα−2Ω˜

tαφα−Ω2φαφα.(2.17)

In the TS2 geometry, this becomes

κ2=f(x, y)[1 −ω(x, y)Ω]2−σ2(x2−1)(1 −y2)Ω2

f(x, y).(2.18)

The righthand side is a quadratic equation in Ω, and it follows that if this equation has

two distinct real roots there will be values of Ω for which κ2>0; i.e., values for which

stationary observers exist. Hence, the condition for the existence of stationary observers

is just that the discriminant of the quadratic be positive, which reduces to

σ2(x2−1)(1 −y2) = ρ2>0.(2.19)

In other words, there is no trapped region anywhere in the portion of the TS2 manifold

covered by the (x, y ) coordinates, excluding Sand H±.

However, there is an ergosphere in the TS2 geometry where one is obliged to have

Ω>0; i.e. stationary observers are forced to rotate. This ergosphere is the portion of the

– 6 –

Figure 2: The ergosphere (tan) and the re-

gion with closed timelike curves (CTCs) (pur-

ple) when p= 0.8. The boundaries of the er-

goshere are lines θ= constant.

Figure 3: Contour plot of the Kretchmann

scalar tanh(σ−4RαβγδRαβ γδ ) in the (ρ, z) plane

for p= 1/3. The centre of the ﬂower-shaped

region is the ring singularity.

manifold where ˜

tαis spacelike. It’s boundary is the hypersurface where ˜

tαis null; i.e., the

locus of points where

f(x, y) = A(x, y)=0.(2.20)

One can show that the boundary of the ergosphere is two θ= constant hypersurfaces in

the bipolar coordinates (2.6). There is also a portion of the manifold where the angular

Killing vector φαbecomes timelike. Since φis a periodic coordinate, this region involves

closed timelike curves (CTCs) and is hence acausal. The shape of these two regions in

illustrated in Fig. 2.

2.4 The naked ring singularity

From examination of the Kretschmann scalar Rαβγδ Rαβγδ (which is plotted in Fig. 3), we

see that there is a curvature singularity in the manifold when:1

p2x4+ 2px3−2px −1=0.(2.21)

One can analytically demonstrate that there is always only one root of this polynomial

xring =xring(p) with xring ∈[1,∞) for p∈[0,1]. The position on this singularity is

coincident with the intersection of the inner boundary of the ergosphere and the x-axis.

It is possible to write a closed form expression for xring , but it is complicated and not

1It is not diﬃcult to conﬁrm that there are no other curvature singularities for x∈[1,∞) and y∈[−1,1].

– 7 –

illuminating, but one can derive the following limits:

xring(p) = 1 + 1−p

4+O[(1 −p)2],

xring(p) = a

p1/3+O(p1/3),(2.22)

where a≈0.79. So for papproaching 1, the ring singularity is on the ρ= 0 line and

for papproaching 0 it is at inﬁnity. We can construct series expansions of various metric

coeﬃcients about the ring singularity. Deﬁning ˜x=x−xring, we ﬁnd

A=a1,0˜x+O(˜x2, y2),(2.23a)

B=b2,0˜x2+b0,2y2+O(˜x3,˜xy2),(2.23b)

C=c1,0˜x+O(˜x2, y2),(2.23c)

where ai,j, etc are complicated nonzero functions of p. Notice that Bapproaches zero faster

than Aand Cnear the singularity. From this it follows that fdiverges while ωremains

ﬁnite on the singularity. Since xring >1, we conclude that stationary observers can exist

arbitrarily close to the ring singularity and it must be a naked. It is also interesting to note

that as stationary observers approach the ring singularity from outside the CTC region,

their angular velocity must approach a ﬁxed value:

Ω→Ωring = Ωring(p) = 1

ω(xring(p),0) .(2.24)

This is analogous to the phenomena of stationary observers forced to co-rotating with the

black hole as they approach the Killing horizon in the Kerr geometry, although here the

observers approach the singularity directly. However, if they approach the ring singularity

from inside the CTC region, their angular velocity is unconstrained.

2.5 Distribution of mass and angular momentum

Extending the results of [7], we now use the Komar formulae to determine where the mass

and angular momentum of the TS2 geometry resides. Generally speaking, the mass or

angular momentum contained within a closed 2-surface ∂Σ is given by

M∂Σ=−1

16πG Z∂Σ

αβγδ∇γ˜

tδdxα∧dxβ,(2.25a)

J∂Σ= + 1

32πG Z∂Σ

αβγδ∇γφδdxα∧dxβ,(2.25b)

where ˜

tαand φαare the time and rotational Killing vectors, respectively. If we take ∂Σ to

be a surface of constant x=x0>1, the Komar integrals reproduce the ADM mass and

angular momentum:

Mx=x0=M, Jx=x0=J. (2.26)

This means that all of the mass and angular momentum of the spacetime is contained

within H±and S; i.e., the ring singularity itself carries no mass or angular momentum.

– 8 –

We can actually explicitly calculate the mass and angular momentum contained within

H±by ﬁrst transforming (2.1) to the bipolar coordinates (2.6). In these coordinates, H±

corresponds to the ˜r= 0 hypersurface, and we obtain

M±= 0, J±=p

4(1 −p)J=σ2

Gpr1 + p

1−p.(2.27)

Interestingly, the H±surfaces contain zero mass but non-zero angular momentum. Finally,

we deﬁne2

MS=M−2M±=M, JS=J−2J±=1−p

2J. (2.28)

That is, all the mass but only a fraction of the total angular momentum is carried within

S.

2.6 Killing horizons and the near horizon metric

To further investigate the geometry near H±, we again change coordinates from (x, y) to

(˜r, θ) in the line element (2.1). For concreteness, we take positive square roots in (2.6)

which means that ˜r= 0 corresponds to H+. The immediate neighbourhood of H+is

deﬁned by ˜rσ, and in this regime the line element reduces to:

ds2=˜

Γ(θ)−˜r2

4r2

0

d˜

t2+r2

0

˜r2d˜r2+r2

0dθ2+sin2θ

˜

Γ(θ)q˜r

2pr0

d˜

t−r0dφ2

,(2.29a)

˜

Γ(θ)≡cos2θ+ 2(2p2−1) cos θ+ 1

4p2,(2.29b)

r2

0≡2σ2(p+ 1)

p2=G2M2+√G4M4−G2J2

2=2GJ±p1−p2

p.(2.29c)

We call (2.29) the “near horizon Tomimatsu-Sato” (NHTS) geometry. If we change coor-

dinates according to

˜r=√2¯r, ˜

t=√2¯

t, (2.30)

we see that the NHTS geometry is a special case of the following spacetime:

ds2= Γ(θ)−¯r2

r2

0

d¯

t2+r2

0

¯r2dr2+r2

0dθ2+sin2θ

Γ(θ)γ¯r

r0

d¯

t−r0dφ2

,(2.31a)

Γ(θ) = 1

2α(cos2θ+ 1) + βcos θ, γ2=α2−β2.(2.31b)

We call this the generic “near-horizon extremal spinning” (NHES) geometry. It is eas-

ily conﬁrmed that this metric has Rαβ = 0 and is free from curvature singularities. As

demonstrated in Appendix A, the above is isometric to the most general vacuum solu-

tion representing the near horizon (non-toriodal) geometry of an axisymmetric extremal

horizon in 4-dimensional general relativity [15]. For various choices of the parameters

(α, β, r0), this metric describes the near-horizon Tomimatsu-Sato, extremal-Kerr [12], or

the extremal Kerr-bolt [14] solutions (the exact parameters corresponding to each of these

parent geometries are given in Table 1).

2Note that calculating MSand JSfrom (2.25) directly is problematic because the x= 1 hypersurface

overlaps with some portion of H±.

– 9 –

Parent solution α β γ r2

0

TS2 1

2p21−1

2p21

p2−11/2

2γGJ∆

Extremal Kerr [12]1012GJ

Extremal Kerr-bolt [14] 1 N

a1−N2

a21/2

2a2

Table 1: NHES metric parameters for the near-horizon limit of the TS2, extremal Kerr, and

extremal Kerr-bolt geometries. Here, Nis the NUT charge of the Kerr-bolt geometry and a=J/M.

Notice that all of the solutions have γ2=α2−β2, which means they are all Ricci ﬂat.

It is obvious that the metric (2.31) has a Killing horizon at ¯r= 0 that we denote by

∆ (and which corresponds to H±in the full TS2 geometry) generated by the null Killing

vector `α∂α=∂¯

t. The surface gravity of the horizon κcan be deduced from the relation

`α∇α`β=κ`β, which gives κ= 0; that is, the horizon is extremal. We can extend the

solution across the horizon by changing to the global coordinates:

¯r=r0(p1 + r2cos t+r),(2.32a)

¯

t=r2

0√1 + r2sin t

¯r,(2.32b)

φ=−γϕ+ ln cos t+rsin t

1 + √1 + r2sin t.(2.32c)

In terms of these dimensionless coordinates (t, r, θ, ϕ), the NHES line element is

ds2=r2

0Γ(θ)−(1 + r2)dt2+dr2

1 + r2+dθ2+γ2sin2θ

Γ2(θ)(dϕ +r dt)2.(2.33)

Notice that the period of the new angular coordinate is not 2π:

φ∼φ+ 2π, ϕ ∼ϕ+2π

γ.(2.34)

Just as for the NHEK and the NHTB solutions, the NHES metric (2.31) has SL(2,R)×U(1)

symmetry. The SL(2,R) isometry is generated by the three Killing vectors

Kα

1∂α= +2r(1 + r2)−1/2sin t ∂t−2(1 + r2)1/2cos t ∂r+ 2(1 + r2)−1/2sin t ∂ϕ,(2.35a)

Kα

2∂α=−2r(1 + r2)−1/2cos t ∂t−2(1 + r2)1/2sin t ∂r−2(1 + r2)−1/2cos t ∂ϕ,(2.35b)

Kα

0∂α= +2 ∂t,(2.35c)

which satisfy the algebra:

[K1, K2]=2K0,[K0, K2]=2K1,[K0, K1] = −2K2.(2.36)

The U(1) rotational isometry is generated by the Killing vector ¯

Kα

0∂α=−∂ϕ.

The metric and area on the horizon are easy to calculate:

ds2

∆=r2

0Γ(θ)dθ2+sin2θ

Γ(θ)dφ2, A∆= 4πr2

0,(2.37)

– 10 –

which implies that the total horizon area in TS2 is

ATS2 = 2 ×A∆= 4π[G2M2+pG4M4−G2J2].(2.38)

We see that the combined area of the TS2 horizons is precisely half that of a Kerr black

hole with the same mass and angular momentum, consistent with the results of Ref. [7].

We now determine the mass and angular momentum of ∆. A well-deﬁned algorithm

for calculating these entirely in terms of quantities deﬁned on ∆ is the isolated horizon

formalism [17]. Since we are dealing with a vacuum spacetime, the mass and angular

momentum are deﬁned as

J∆= + 1

32πG Z∆

αβγδ∇γφδdxα∧dxβ=r2

0

2γG ,(2.39a)

M2

∆=r4

0+ 2G2J2

∆

4G2r2

0

=(1 + γ2)J∆

2γG ,(2.39b)

where φαis the Killing vector generating the rotational isometry, as before. Other than

the fact that the above is to be calculated using the NHES metric, we note that (2.39a) is

identical to the Komar angular momentum (2.25b) we deﬁned above for the H±surfaces

in the TS2 manifold. Hence, it is not surprising that J∆=J±when (2.29c) is enforced.

However, we note that M∆6=M±. Again, this is not surprising: M±can be thought of as

the charge of the horizon conjugate to the generator of asymptotic time translations; i.e.,

the ADM mass. However, as pointed out in [17], there is no guarantee that the isolated

horizon mass agree with the ADM mass since quantities deﬁned on the horizon do not

necessarily have any knowledge of the asymptotically ﬂat portion of the spacetime. For

γ= 1, we recover the Kerr-extremality condition J∆=GM2

∆.

Finally, we note that for generic choices of αand β, the horizons ∆ have conical

singularities on the north (θ= 0) and south (θ=π) poles. The conical deﬁcit (or excess)

about each pole is:

δN,S= 2π1−(α±β)−1.(2.40)

Referring to Table 1, we see that the NHEK metric has no singularities, the NHTS metric

has a defect at the south pole, while the NHKB metric has defects at both poles. Also

note that if γ6= 1, the NHES geometry necessarily involves conical singularities.

2.7 Nature of the Shypersurface

We now attempt to further understand the nature of the Shypersurface. First, we note

that the Killing vectors ˜

tαand φαare both timelike in the neighbourhood of Sand it is

easy to conﬁrm

lim

x→1[(˜

tα˜

tα)(φβφβ)−(˜

tαφα)2] = 0; (2.41)

i.e., the Killing vectors are parallel on S. In fact, it is not hard to ﬁnd a linear combination

that vanishes in the x→1 limit:

lim

x→1(˜

tα+ ΩSφα) = 0,ΩS=p

4σr1−p

1 + p.(2.42)

– 11 –

Next, following [7] let us introduce new coordinates

τ=ΩS˜

t−φ

ΩS

, x = cosh η. (2.43)

In these coordinates, Scorresponds to η= 0. Expanding the TS2 line element in the η1

regime yields

ds2=−(1 −y2)2(1 −p)

g(y)dτ2+(1 −p2)(1 + p)g(y)σ2

p4(1 −y2)dη2+dy2

1−y2

+η2g(y)σ2

(1 −p)(1 −y2)dφ2+4ση2[g(y)−8y2(p+ 1)]

g(y)p1−p2dτ dφ, (2.44)

where

g(y)≡(1 −p)y4+ 2(p+ 3)y2+ 1 −p > 0.(2.45)

Examination of the (t, y) = constant sections reveals there is a conical singularity about

S, and the deﬁcit angle is

δ=2π

1−p2.(2.46)

This is the same deﬁcit angle as we found above in each of the horizons H±. The metric

on Sitself is seen to be

ds2

S=−(1 −y2)2(1 −p)

g(y)dτ2+(1 −p2)(1 + p)g(y)σ2

p4(1 −y2)2dy2.(2.47)

We recognize a Lorentzian 1 + 1 spacetime, which demonstrates Sis a line-like object.

To summarize the results of this section and §2.5 above, we have seen that Sis a one-

dimensional object carrying non-zero mass and angular momentum. It is coincident with

a conical singularity and is surrounded by closed timelike curves.

3. CFT description of the near horizon geometry

In this section, we seek a CFT dual to the generic near horizon geometry (2.31) deﬁned

above. We will ﬁnd that the existence of such a CFT is plausible, and that the basic

parameters of the theory are given most concisely in terms of near horizon quantities.

3.1 Central charge

We ﬁrst ﬁnd the central charge of the candidate dual CFT by analyzing the asymptotic

symmetries of metric ﬂuctuations using the method of Ref. [8] based on the covariant

formalism of Refs. [18,19].3We assume the NHES metric is perturbed according to gαβ →

gαβ +hαβ and assume the same boundary conditions on hαβ as in Ref. [8]. The most

3The CFT description of very similar near horizon geometries arising from extremal black holes of varying

matter content has been been obtained in Refs. [10,11].

– 12 –

general diﬀeomorphism consistent with the assumed boundary conditions has the following

form as r→ ∞

ξα=ζα+χα,(3.1a)

ζα∂α= [(ϕ) + O(r−2)]∂ϕ−[r0(ϕ) + O(r0)]∂r,(3.1b)

χα∂α= [C+O(r−3)]∂τ,(3.1c)

where (ϕ) and Care an arbitrary function and constant, respectively. If we choose

(ϕ) = 1

γeiγmϕ, (ϕ) = (ϕ+ 2π/γ), m = 0,±1,±2, . . . (3.2)

and drop subleading terms, we see that the generators

ζm=γ−1eiγmϕ[∂ϕ−iγmr∂r],(3.3)

satisfy the Virasoro algebra:

i[ζm, ζn]=(m−n)ζm+n.(3.4)

Deﬁning

kζ[h, g] = −1

4αβµν [ζν∇µh−ζν∇σhµσ +1

2h∇νζµ

−hνσ ∇σζµ+1

2hσν(∇µζσ+∇σζµ)]dxα∧dxβ,(3.5)

the charges Qζmassociated with the diﬀeomorphisms ζmsatisfy the algebra

{Qζm, Qζn}=−i(m−n)Qζm+ζn+1

8πG Z∂Σ

kζm[£ζng, g],(3.6)

where {,}is the Dirac bracket. Here, ∂Σisa(t, r) = constant hypersurface in the limit

r→ ∞. Evaluating the integral in the NHES background yields

i{Qζm, Qζn}= (m−n)Qζm+ζn+c

12mm2+2

γ2δm+n,0,(3.7)

where the central charge cis given by

c=6γr2

0

G= 12γ2J∆.(3.8)

Passing over to the quantum theory, we make the switch {,}→−i[,] and deﬁne Lnas the

quantum versions of the Qζn:

Ln=Qζn+c

12 1

γ2−1

2δn,0,(3.9)

which yields the familiar expression

[Lm, Ln]=(m−n)Lm+n+c

12m(m2−1)δm+n,0.(3.10)

– 13 –

Our formula for the central charge (3.8) reproduces the NHEK [8] and NHKB [14]

results with appropriate identiﬁcations listed in Table 1. Indeed, it looks very much like

the original extremal-Kerr result c= 12J, apart from the γfactor and the appearance of

the horizon angular momentum J∆instead of the ADM value J. Of course, in Kerr such

a distinction is irrelevant because J=J∆, but in the TS2 spacetime this is not the case.

This suggests that any CFT description that we deduce for the TS2 geometry is really a

CFT description of one of the horizons.

3.2 Temperature

We now turn our attention to the temperature of the dual CFT. A particularly eﬃcient

way of deriving the left and right moving temperatures involves analysis of the massless

scalar wave equation ∇α∇αΦ = 0 in the near horizon region (2.31) [13,16]. Let us make

a separation of variables ansatz:

Φ(¯

t, ¯r, θ, φ) = e−iω¯

t+imφS(θ)R(¯r).(3.11)

We ﬁnd the wave equation reduces to:

−KS(θ) = ∂2

θ+cos θ

sin θ∂θ+m2γ2−m2Γ(θ)

sin2θS(θ),(3.12a)

KR(¯r) = ¯r2∂2

¯r+ 2¯r∂¯r+r4

0ω2

¯r2−2γmωr2

0

rR(¯r),(3.12b)

where Kis a separation constant. The angular equation is solvable in terms of Huen

functions and yields values for Konce boundary conditions are enforced. Turning our

attention to the radial equation, if we change variables according to

w+=−1

2¯

t

r0

+r0

¯r, w−=1

2eφ/γ, y =rr0

2¯reφ/2γ,(3.13)

then the radial equation becomes (after noting im =∂φand iω =−∂¯

t):

KR =1

4(y2∂2

y−y∂y) + y2∂+∂−R. (3.14)

The diﬀerential operator appearing on the right can be expressed as

1

4(y2∂2

y−y∂y) + y2∂+∂−=H=¯

H,(3.15)

where Hand ¯

Hare the quadratic Casimirs

H=−H2

0+1

2(H+1H−1+H−1H+1 ),¯

H=−¯

H2

0+1

2(¯

H+1 ¯

H−1+¯

H−1¯

H+1),(3.16)

formed by the SL(2,R) generators

H+1 =i∂+, H0=i(w+∂++1

2y∂y), H−1=i(w2

+∂++w+y∂y−y2∂−),(3.17a)

¯

H+1 =i∂−,¯

H0=i(w−∂−+1

2y∂y),¯

H−1=i(w2

−∂−+w−y∂y−y2∂+),(3.17b)

– 14 –

respectively. As argued in detail elsewhere [13], this would seem to imply that the solutions

of the wave equation are representations of SL(2,R). However, this is not really the case

because the coordinate transformation (3.13) implies that the w−coordinate should be

identiﬁed as follows:

w−∼w−e2π/γ .(3.18)

The net eﬀect of this identiﬁcation is to break the conformal symmetry and to ensure

that observers using the original (¯

t, ¯r) coordinates (i.e., co-rotating with the horizon) will

observe a thermal bath of (left-moving) radiation of temperature

TL=1

2πγ ,(3.19)

from the SL(2,R)×SL(2,R) invariant vacuum via the Unruh eﬀect.4This reproduces the

Frolov-Thorne temperature [20] of the extremal Kerr horizon when γ= 1.

3.3 Entropy

We now brieﬂy discuss the thermodynamics of the NHES horizon ∆. The Bekenstein-

Hawking entropy of the horizon is, as usual,

SBH =A∆

4G=πr2

0

G= 2πγJ∆.(3.20)

This expression is highly analogous to to the one derived in [10] for various extremal

spinning black holes with magnetic and electric charges. The form of the entropy suggests

we view J∆and γas the thermodynamic state variables for our system.5Following a

similar procedure to the one in [10], a temperature can be associated with each coordinate

via

dSBH =dJ∆

TJ

+dγ

Tγ

.(3.21)

Now, since the charge J∆generated by the zero mode ζ0of the Virasoro algebra (3.4), TJ

ought to be identiﬁed with the left-moving temperature TLof the dual CFT. This yields

TL=1

2πγ ,(3.22)

consistent with the temperature derived from the broken conformal symmetry of the scalar

wave equation. Putting this temperature into the Cardy formula gives the entropy of the

dual CFT:

SCFT =1

3π2cTL=1

3π26γr2

0

G 1

2πγ =πr2

0

G=SBH.(3.23)

Hence the macroscopic and microscopic entropies agree.

4For the extremal situation we are considering here, the right-moving (Hawking) temperature is zero

(TR= 0) since the w+coordinate does not require an identiﬁcation.

5One could equivalently view the system to be thermodynamically described by the isolated horizon

quantities (J∆, M∆).

– 15 –

4. Discussion

In this paper, we have examined the classical properties of the δ= 2 Tomimatsu-Sato

geometry and then calculated the basic parameters associated with its dual CFT description

using techniques from the recently proposed Kerr/CFT correspondence. In particular, we

have conﬁrmed that the TS2 geometry involves a naked ring-like curvature singularity and

a spinning rod-like conical singularity. The former exists on the boundary of a Kerr-like

ergosphere, while the latter is surrounded by a region containing closed timelike curves.

All the ADM mass and a portion of the angular momentum of the geometry is carried

by the strut. The remaining angular momentum is carried by a pair of Killing horizons

located at each end of the rod, while the ring curvature singularity contains no mass or

angular momentum whatsoever. The two Killing horizons are extremal in the sense that

their surface gravity is zero, but unlike the extremal Kerr solution there exists no special

relationship between the ADM mass and angular momentum at inﬁnity.

As pointed out by Kodama and Hikida [7], there is no completely compelling evidence

that the TS2 spacetime considered here cannot be the endpoint of gravitational collapse.

Indeed, there been recent work examining the prospects for observing the TS2 geometry in

a realistic astrophysical context [21]. There is good motivation for determining whether or

not TS2 objects exist in our universe, because their observation would provide an irrefutable

counterexample to the cosmic censorship conjecture. But it remains to be seen if the TS2

solution is stable, which is a necessary condition for such an object to be seen in nature.

The TS2 solution escapes the black hole uniqueness theorem by the existence of the naked

ring singularity, so it can be viewed as an asymptotically-ﬂat alternative geometry to Kerr.

Given Mand J, we have seen that the total horizon area of a TS2 object be half that of the

matching Kerr black hole. Applying the standard interpretation, this means that the former

has half the entropy of the latter. Thermodynamic lore then implies that the endpoint of

non-equilibrium gravitational processes ought to be the Kerr black hole rather than the

TS2 spacetime. Such arguments have been used in the past to account for the Gregory-

Laﬂamme instability of black strings. However, as pointed out by Reall [22], the existence

of this type of global thermodynamic instability in a system does not imply a classical

instability of the state with lower entropy. In other words, the perturbative stability of

the TS2 geometry is still an open question. Addressing this by direct computation or by

making arguments based on the local thermodynamic stability properties of the solution

is an interesting avenue for future work.6

We have demonstrated that the near horizon geometry of the TS2 model is of a form

that represents the most general extremal horizon in vacuum general relativity, and we

have calculated the central charge and temperature of the CFT dual to this entire class.

In addition to providing the basic parameters for a CFT description of the TS2 horizon(s),

we reproduce known results for extremal Kerr [8] and extremal Kerr-Bolt [14] solutions.

Despite the rather exotic features (such as the ring singularity) of the TS2 geometry away

6An important conceptual diﬃculty with such a study is that the existence of the closed timelike curve

region makes the Cauchy problem for perturbations ill-deﬁned. We thank H. Kodama for bringing this to

our attention.

– 16 –

from the horizons, the dual CFT seems to be sensitive to only properties of the horizons

themselves. For example, the central charge is c= 12γJ∆, where J∆is the angular mo-

mentum of the horizon only, not the entire spacetime. In conventional spacetimes, such

distinctions are irrelevant because the ADM and horizon charges usually agree.

In addition to the dependence on angular momentum, the temperature and central

charge of the CFT depend on another parameter γ. The precise physical or geometric

interpretation of the quantity is unclear, but it appears to be related to the existence of

singularities in the near horizon geometry; i.e., if γ6= 1, the near horizon geometry has

conical singularities and if γ= 1 we recover results for extremal Kerr. We note that the

conical singularities in the near horizon of TS2 or extremal Kerr-Bolt are very similar like

the singularities of 4D static black hole solutions with multiple D6-branes in 4-dimensional

N= 2 supergravity [23]. So, it is natural to speculate about a stringy origin of these

conical singularities, as in Refs. [23,24,25,26]. Furthermore, we know from the examples

of the AdS3/CFT2and Kerr/CFT correspondences, there are alternatives to the covariant

version of the Brown-Henneaux formalism employed here to obtain the central charge. In

particular, it has been shown that the extremal Kerr central charge can be obtained by

dimensionally reducing the action from four to two dimensions and the careful addition of

counterterms [27,28,29]. It would be interesting to re-do these calculations for the near-

horizon extremal spinning metric considered in this work with the goal of gaining greater

insight into the role of the γparameter. Work in this direction is in progress.

Acknowledgments

We thank Viqar Husain and Hideo Kodama for useful conversations. H. Liu thanks Wei

Song for valuable discussion and also appreciates the hospitality and inspiring atmosphere

of the 8th Simons Workshop on Mathematics and Physics. This work is supported by the

Natural Sciences and Engineering Research Council of Canada.

A. Genericness of the near horizon extremal spinning (NHES) metric

Kunduri and Lucetti [15] have derived the most general 4-dimensional vacuum metric

representing the geometry near an axisymmetric, extremal, and non-toriodal horizon:

ds2= Ξ(ζ)−¯r2

r2

0

dv2+ 2 dv d¯r+Ξ(ζ)

Q(ζ)dζ2+Q(ζ)

Ξ(ζ)(dx + ¯r dv)2,(A.1)

with

Ξ(ζ) = 1

b+bζ2

4, Q(ζ) = 1

λb2r2

0

(λbζ + 2) (2λ−bζ ),(A.2)

where (b, r0, λ) are constants. Changing coordinates via

v=¯

t−r2

0

¯r, ζ =λ2−1+(λ2+ 1) cos θ

bλ , x =−r2

0bλφ

λ2+ 1 + ln ¯r

r0,(A.3)

– 17 –

puts the metric in the NHES form

ds2= Γ(θ)−¯r2

r2

0

d¯

t2+r2

0

¯r2dr2+r2

0dθ2+sin2θ

Γ(θ)γ¯r

r0

d¯

t−r0dφ2

,(A.4a)

Γ(θ) = 1

2α(cos2θ+ 1) + βcos θ, (A.4b)

with

α=(λ2+ 1)2

2bλ2, β =1−λ4

2bλ2, γ2=α2−β2.(A.5)

Therefore, the NHES metric used in the main text is just another representation of the most

general metric representing the near geometry around an axisymmetric-extremal horizon in

4-dimension vacuum general relativity. We conclude by noting that the absence of conical

singularities demands α= 1 and β= 0 and hence reproduces the NHEK geometry; i.e., the

only extremal near horizon vacuum geometry with S2topology in 4 dimensions is extremal

Kerr, as ﬁrst pointed out in [15].

References

[1] R. M. Wald, Gravitational collapse and cosmic censorship,gr-qc/9710068.

[2] A. Tomimatsu and H. Sato, New exact solution for the gravitational ﬁeld of a spinning mass,

Phys. Rev. Lett. 29 (Nov, 1972) 1344–1345.

[3] A. Tomimatsu and H. Sato, New series of exact solutions for gravitational ﬁelds of spinning

masses,Progress of Theoretical Physics 50 (1973), no. 1 95–110.

[4] G. W. Gibbons and R. A. Russell-Clark, Note on the Sato-Tomimatsu solution of Einstein’s

equation,Phys. Rev. Lett. 30 (1973) 398–399.

[5] F. J. Ernst, New representation of the Tomimatsu–Sato solution,Journal of Mathematical

Physics 17 (1976), no. 7 1091–1094.

[6] W. Hikida and H. Kodama, An Investigation of the Tomimatsu-Sato Spacetime,

gr-qc/0303094.

[7] H. Kodama and W. Hikida, Global structure of the Weyl and the delta=2 Tomimatsu- Sato

spacetime,Class. Quant. Grav. 20 (2003) 5121–5140, [gr-qc/0304064].

[8] M. Guica, T. Hartman, W. Song, and A. Strominger, The Kerr/CFT Correspondence,Phys.

Rev. D80 (2009) 124008, [arXiv:0809.4266].

[9] O. Aharony, S. S. Gubser, J. M. Maldacena, H. Ooguri, and Y. Oz, Large N ﬁeld theories,

string theory and gravity,Phys. Rept. 323 (2000) 183–386, [hep-th/9905111].

[10] T. Hartman, K. Murata, T. Nishioka, and A. Strominger, CFT Duals for Extreme Black

Holes,JHEP 04 (2009) 019, [arXiv:0811.4393].

[11] G. Compere, K. Murata, and T. Nishioka, Central Charges in Extreme Black Hole/CFT

Correspondence,JHEP 05 (2009) 077, [arXiv:0902.1001].

[12] J. M. Bardeen and G. T. Horowitz, The extreme Kerr throat geometry: A vacuum analog of

AdS(2) ×S(2),Phys. Rev. D60 (1999) 104030, [hep-th/9905099].

[13] A. Castro, A. Maloney, and A. Strominger, Hidden Conformal Symmetry of the Kerr Black

Hole,Phys. Rev. D82 (2010) 024008, [arXiv:1004.0996].

– 18 –

[14] A. M. Ghezelbash, Kerr-Bolt Spacetimes and Kerr/CFT Correspondence,arXiv:0902.4662.

[15] H. K. Kunduri and J. Lucietti, A classiﬁcation of near-horizon geometries of extremal

vacuum black holes,J. Math. Phys. 50 (2009) 082502, [arXiv:0806.2051].

[16] B. Chen, J. Long, and J.-j. Zhang, Hidden Conformal Symmetry of Extremal Black Holes,

arXiv:1007.4269.

[17] A. Ashtekar et al.,Isolated horizons and their applications,Phys. Rev. Lett. 85 (2000)

3564–3567, [gr-qc/0006006].

[18] G. Barnich and F. Brandt, Covariant theory of asymptotic symmetries, conservation laws

and central charges,Nucl. Phys. B633 (2002) 3–82, [hep-th/0111246].

[19] G. Barnich and G. Compere, Surface charge algebra in gauge theories and thermodynamic

integrability,J. Math. Phys. 49 (2008) 042901, [arXiv:0708.2378].

[20] V. P. Frolov and K. S. Thorne, Renormalized stress - energy tensor near the horizon of a

slowly evolving, rotating black hole,Phys. Rev. D39 (1989) 2125–2154.

[21] C. Bambi and N. Yoshida, Shape and position of the shadow in the δ= 2 Tomimatsu-Sato

space-time,Class. Quant. Grav. 27 (2010) 205006, [arXiv:1004.3149].

[22] H. S. Reall, Classical and thermodynamic stability of black branes,Phys. Rev. D64 (2001)

044005, [hep-th/0104071].

[23] D. Gaiotto, A. Strominger, and X. Yin, New Connections Between 4D and 5D Black Holes,

JHEP 02 (2006) 024, [hep-th/0503217].

[24] V. Balasubramanian, J. de Boer, E. Keski-Vakkuri, and S. F. Ross, Supersymmetric conical

defects: Towards a string theoretic description of black hole formation,Phys. Rev. D64

(2001) 064011, [hep-th/0011217].

[25] O. Lunin, J. M. Maldacena, and L. Maoz, Gravity solutions for the D1-D5 system with

angular momentum,hep-th/0212210.

[26] J. M. Maldacena and L. Maoz, De-singularization by rotation,JHEP 12 (2002) 055,

[hep-th/0012025].

[27] A. Castro, D. Grumiller, F. Larsen, and R. McNees, Holographic Description of AdS2Black

Holes,JHEP 11 (2008) 052, [arXiv:0809.4264].

[28] A. Castro and F. Larsen, Near Extremal Kerr Entropy from AdS2Quantum Gravity,JHEP

12 (2009) 037, [arXiv:0908.1121].

[29] A. Castro, C. Keeler, and F. Larsen, Three Dimensional Origin of AdS2Quantum Gravity,

JHEP 07 (2010) 033, [arXiv:1004.0554].

– 19 –