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We analyze the $\delta=2$ 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 $SL(2,\mathbb{R})\times U(1)$ 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.
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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
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
arXiv:1010.2803v1 [hep-th] 14 Oct 2010
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
1y2+ (x21)(1 y2)2.(2.1)
– 3 –
Here we have
f=f(x, y)A(x, y)
B(x, y), E =E(x, y)A(x, y)
ω=ω(x, y)4σqC(x, y)
pA(x, y)(1 y2),(2.2)
A(x, y)p4(x21)4+q4(1 y2)42p2q2(x21)(1 y2)
×2(x21)2+ 2(1 y2)2+ 3(x21)(1 y2);
B(x, y)p2(x41) q2(1 y4)+2px(x21)2+
4q2y2px(x21) + (px + 1)(1 y2)2;
C(x, y)q2(px + 1)(1 y2)3p3x(x21) 2(x41) + (x2+ 3)(1 y2)
p2(x21) 4x2(x21) + (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(x21)(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
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
px +O1
t2+16σq(1 y2)
t dφ
+σ21 + 4
px +O1
1y2+x2(1 y2)2.(2.9)
Making the following identifications:
R=σx, y = cos ϑ, M =2σ
Gp, J =qGM2=4σ2q
puts this in the form
ds2≈ −12GM
t2+4GJ sin2ϑ
t dφ+
1 + 2GM
RdR2+R2(2+ 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 > 0,0p1.(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
throughout the following discussion.
2.3 Killing vectors and stationary observers
The metric (2.1) has two obvious Killing vectors:
t, φαα=φ.(2.14)
We have the following:
tα=f(x, y), φαφα=f(x, y)ω2(x, y) + σ2(x21)(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:
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,
In the TS2 geometry, this becomes
κ2=f(x, y)[1 ω(x, y)Ω]2σ2(x21)(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(x21)(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
>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(σ4RαβγδRαβ 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+ 2px32px 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 + 1p
4+O[(1 p)2],
xring(p) = a
where a0.79. 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=xxring, we find
A=a1,0˜x+Ox2, y2),(2.23a)
C=c1,0˜x+Ox2, 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:
ring = 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
16πG ZΣ
JΣ= + 1
32πG ZΣ
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
Interestingly, the H±surfaces contain zero mass but non-zero angular momentum. Finally,
we define2
MS=M2M±=M, JS=J2J±=1p
2J. (2.28)
That is, all the mass but only a fraction of the total angular momentum is carried within
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:
Γ(θ)cos2θ+ 2(2p21) cos θ+ 1
02σ2(p+ 1)
We call (2.29) the “near horizon Tomimatsu-Sato” (NHTS) geometry. If we change coor-
dinates according to
˜r=r, ˜
t, (2.30)
we see that the NHTS geometry is a special case of the following spacetime:
ds2= Γ(θ)¯r2
Γ(θ) = 1
2α(cos2θ+ 1) + βcos θ, γ2=α2β2.(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
TS2 1
Extremal Kerr [12]1012GJ
Extremal Kerr-bolt [14] 1 N
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=α2β2, 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 `αα=¯
t. The surface gravity of the horizon κcan be deduced from the relation
`αα`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)
01 + r2sin t
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
0Γ(θ)(1 + r2)dt2+dr2
1 + r2+2+γ2sin2θ
Γ2(θ)(+r dt)2.(2.33)
Notice that the period of the new angular coordinate is not 2π:
φφ+ 2π, ϕ ϕ+2π
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
1α= +2r(1 + r2)1/2sin t ∂t2(1 + r2)1/2cos t ∂r+ 2(1 + r2)1/2sin t ∂ϕ,(2.35a)
2α=2r(1 + r2)1/2cos t ∂t2(1 + r2)1/2sin t ∂r2(1 + r2)1/2cos t ∂ϕ,(2.35b)
0α= +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 ¯
The metric and area on the horizon are easy to calculate:
Γ(θ)2, A= 4πr2
– 10 –
which implies that the total horizon area in TS2 is
ATS2 = 2 ×A= 4π[G2M2+pG4M4G2J2].(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
2γG ,(2.39a)
0+ 2G2J2
=(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 M6=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
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 x1 limit:
tα+ ΩSφα) = 0,S=p
1 + p.(2.42)
– 11 –
Next, following [7] let us introduce new coordinates
, 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)2+(1 p2)(1 + p)g(y)σ2
p4(1 y2)2+dy2
(1 p)(1 y2)2+4ση2[g(y)8y2(p+ 1)]
g(y)p1p2dτ dφ, (2.44)
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
This is the same deficit angle as we found above in each of the horizons H±. The metric
on Sitself is seen to be
S=(1 y2)2(1 p)
g(y)2+(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→ ∞
ζαα= [(ϕ) + O(r2)]ϕ[r0(ϕ) + O(r0)]r,(3.1b)
χαα= [C+O(r3)]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
satisfy the Virasoro algebra:
i[ζm, ζn]=(mn)ζm+n.(3.4)
kζ[h, g] = 1
4αβµν [ζνµhζνσhµσ +1
hνσ σζµ+1
the charges Qζmassociated with the diffeomorphisms ζmsatisfy the algebra
{Qζm, Qζn}=i(mn)Qζm+ζn+1
8πG ZΣ
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}= (mn)Qζm+ζn+c
where the central charge cis given by
G= 12γ2J.(3.8)
Passing over to the quantum theory, we make the switch {,}→−i[,] and define Lnas the
quantum versions of the Qζn:
12 1
which yields the familiar expression
[Lm, Ln]=(mn)Lm+n+c
– 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 Jinstead 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 ααΦ = 0 in the near horizon region (2.31) [13,16]. Let us make
a separation of variables ansatz:
t, ¯r, θ, φ) = e¯
We find the wave equation reduces to:
KS(θ) = 2
θ+cos θ
sin θθ+m2γ2m2Γ(θ)
KRr) = ¯r22
¯r+ 2¯r∂¯r+r4
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
¯r, w=1
2eφ/γ, y =rr0
then the radial equation becomes (after noting im =φand =¯
KR =1
yy∂y) + y2+R. (3.14)
The differential operator appearing on the right can be expressed as
yy∂y) + y2+=H=¯
where Hand ¯
Hare the quadratic Casimirs
2(H+1H1+H1H+1 ),¯
H+1 ¯
formed by the SL(2,R) generators
H+1 =i∂+, H0=i(w+++1
2y∂y), H1=i(w2
H+1 =i∂,¯
– 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 wcoordinate should be
identified as follows:
wwe2π/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
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,
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 Jand 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
dSBH =dJ
Now, since the charge Jgenerated 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
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:
2πγ =πr2
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 Jis 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.
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
dv2+ 2 dv d¯r+Ξ(ζ)
Ξ(ζ)(dx + ¯r dv)2,(A.1)
Ξ(ζ) = 1
4, Q(ζ) = 1
(λbζ + 2) (2λ ),(A.2)
where (b, r0, λ) are constants. Changing coordinates via
¯r, ζ =λ21+(λ2+ 1) cos θ
, x =r2
λ2+ 1 + ln ¯r
– 17 –
puts the metric in the NHES form
ds2= Γ(θ)¯r2
Γ(θ) = 1
2α(cos2θ+ 1) + βcos θ, (A.4b)
α=(λ2+ 1)2
22, β =1λ4
22, γ2=α2β2.(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].
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We devise a new representation of the simplest Tomimatsu–Sato solution of Einstein’s vacuum field equations. This permits us to dispose of the previously troublesome ’’directional singularities’’ through the introduction of an advanced (or retarded) time coordinate. In the neighborhood of the locations in question the T–S space is shown to possess a Killing tensor of valence two, which allows us to solve the geodesic problem in this neighborhood completely. Finally, we present for future analysis a plausible toroidal model of the material source for the T–S solution.
Within 5–10 years, very long baseline interferometry facilities will be able to observe the 'shadow' of super-massive black hole candidates. This will allow us, for the first time, to test gravity in the strong field regime. In this paper, we numerically study the photon orbits in the δ = 2 Tomimatsu–Sato spacetime. The δ = 2 Tomimatsu–Sato spacetime is a stationary, axisymmetric and asymptotically flat exact solution of the vacuum Einstein equations. We compare the associated shadow with the one of Kerr black holes. The shape of the shadow in the δ = 2 Tomimatsu–Sato spacetime is oblate and the difference between the two axes can be as high as 6% when viewed on the equatorial plane. We argue that future space sub-mm interferometers (e.g. VSOP-3) may distinguish the two cases, and thus are able to test the cosmic censorship conjecture.