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Master Thesis

Quantifying Near Extremal Black Holes

Archishna Bhattacharyya

12176745

Supervisor:

Dr. Alejandra Castro

Second examiner:

Prof. Daniel Baumann

A thesis submitted in partial fulfilment of the requirements for the degree of

M.Sc. Physics and Astronomy: track Theoretical Physics

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Contents

1 Abstract 5

2 Introduction 5

3 The phase space of solutions 7

3.1 Horizons . . . 7

3.1.1 Event horizon of a black hole . . . 7

3.1.2 Cosmological event horizon . . . 10

3.2 The Kerr solution . . . 14

3.2.1 Geometry and thermodynamics . . . 14

3.2.2 The first law: another perspective . . . 17

3.3 Kerr-Newman . . . 18

3.4 Kerr Newman - Λ . . . 20

3.5 Kerr-AdS5 and its flat limit . . . 21

4 The near extremal limit 22 4.1 Near extremal Kerr (Newman) . . . 22

4.2 Decoupling limit of Kerr-Newman-Λ . . . 26

4.3 Alternative extremal limits of Kerr-dS . . . 27

4.4 Kerr-AdS5 . . . 30

5 Thermodynamics from gravitational perturbations 31 5.1 Near extremal thermodynamic response . . . 31

5.2 Gravitational dynamics: nAdS2 . . . 33

5.3 A consistent truncation of Kerr . . . 39

6 Observing phase transitions 43 6.1 Kerr Newman in flat space . . . 43

6.2 Kerr Newman with a cosmological constant . . . 47

6.2.1 Kerr Newman in AdS . . . 47

6.2.2 Kerr-Newman-dS: r+= r− . . . 52

6.2.3 Kerr-dS: r+ = rc . . . 55

6.3 Rotating solutions in 5D . . . 57

6.3.1 Kerr-AdS5 with distinct angular momenta . . . 57

6.3.2 Kerr-AdS5 with equal angular momenta . . . 62

6.3.3 Myers-Perry with distinct angular momenta . . . 63

6.3.4 Myers-Perry with equal angular momenta . . . 64

7 A cue for generalisation 65 7.1 Kerr-(A)dS: r+= r− . . . 65

7.2 Kerr-dS: Nariai limit . . . 67

7.3 Kerr-dS: Ultracold limit . . . 68

8 Conclusion 69

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Acknowledgement

I am grateful to Alejandra Castro, my supervisor. There are several ways in which my interactions with her have been enriching. What I have admired the most was

her insistence and persistence on my understanding of every aspect. It has made me more curious and confident of my own reasoning. Her feedback on my presentation and writing skills has been more useful than any academic skills

courses I have taken. I sincerely thank her for her patience and enthusiasm towards this project.

I thank Daniel Baumann, my second examiner, for his enthusiasm and involvement in this project. I am also grateful for his guidance and support at several other

instances during my masters.

I appreciate and thank Jay Armas for encouraging my questions, and his patience with the discussions I benefitted from while working on this project.

I thank Victor Godet, for being enthusiastic and available for discussions. His encouragement has been greatly supportive.

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1

Abstract

The dynamics of near extremal (static) black holes and their thermodynamic re-sponse near extremality are well described by the duality known as the nAdS2/nCFT1 correspondence, and believed to be universal. Recently, new features have been ob-served for rotating solutions which are not exhibited by their charged counterparts: in particular, the thermodynamic response of the Kerr-Newman solution in [2], and the gravitational perturbations exploiting the conformal symmetry of the Near Hori-zon Extreme Kerr solution in [1]. Motivated by these results, we extend the anal-ysis to include a cosmological constant and identify key features that distinguish charged solutions from their rotational analogues, and solutions in anti-de-Sitter and de-Sitter spacetimes.

2

Introduction

Black holes in General Relativity appear as solutions to Einstein’s equations. As classical solutions, they admit a causal structure, where they possess a horizon, which shields the interior from all causal influences occurring outside of it. The laws of black hole mechanics are in one-to-one correspondence with the laws of thermodynamics, which was established by the seminal work of Hawking [5], and Bekenstein [7]. Their contributions are described by the well known equations

TH = ℏ 2πκ (2.1) SBH = c3 GNAH 4 (2.2)

These tell us that a black hole near its horizon is characterised by a temperature and large entropy. Accounting for this property using statistical mechanics, offers a microscopic description of this entropy [9], and has been a prolific area of research, of which we refer to [13] for a review. Recent results over the years have unravelled many connections with the problem of the information paradox, leading to novel insights from Quantum Information Theory. We refer to [14] for the most recent review. Finally, black holes exhibit interesting properties in the framework of holog-raphy, where the basic principle is that a gravitational theory in (D + 1) dimensions is equivalent to a quantum field theory in D dimensions, the first instance of which was realised as the AdS/CFT correspondence [11].

A key feature of black holes governing their holographic description is extremality. A black hole is extremal if it admits degenerate horizons. Extremal black holes ad-mit a liad-mit, where the region very near their horizon becomes a solution to Einstein’s equations, thus allowing a dynamical treatment independent of the full spacetime. Dynamics in this region have provided strong insights into the holographic nature of quantum gravity [10, 23]. One feature, shared by several extremal solutions [34] is the appearance of an infinitely long AdS2 spacetime in this region. This property

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also leads to the attractor mechanism [26], which states that the solution in this region, is specified uniquely by the black hole charges, and is independent of any initial data at asymptotic infinity characterising it. This region is also marked by conformal invariance, in addition to the time translation symmetry, naively indica-tive of a duality: AdS2/CF T1. This, however, is a problematic description as we shall explain in this thesis.

The main obstacle arises due to the symmetries of AdS2, which are such that, finite energy configurations cannot be supported [24, 25]. So we are only able to observe the trivial ground state of the black hole which is its extremal configuration. In the recent years, a framework known as nAdS2/nCF T1 addressing these obstacles [27, 29], (review [28]) has been developed by considering a deformation of AdS2 to allow for low energy excitations above the extremal state. The deformation leads to the breaking of the symmetries by inducing a conformal anomaly, which enters the dual theory as a coupling constant of the Schwarzian effective action, and is the source corresponding to the dynamical mode. The theory in the bulk is a modi-fied the gravitational action involving a scalar field corresponding to this dynamical mode, and is given by JT gravity [32, 33]. This duality governs the thermodynamic response of black holes which are near extremal. Near extremality refers to a devia-tion away from the extremal limit, where, the soludevia-tion is no longer fully decoupled, and dynamics in this region is governed by this duality. Interestingly, the near hori-zon of Kerr black holes [46], also face the same complications as AdS2 [47]. However, a description of their near horizon dynamics shares additional new features [1, 3, 48] that are not exhibited by their static counterparts, the Reissner-Nordström solu-tions.

The Kerr black hole, itself, admits a holographic description known as the Kerr/CFT correspondence [45]. Numerous developments have followed after, which are sum-marised in: [50]. The basic principle behind this duality is that the central charge, cL of one copy of the Virasoro algebra formed by the generators of the asymptotic

symmetries of the Near Horizon Extreme Kerr (NHEK) is matched with the that of a 2D CFT. The left moving temperature, TLvia the Cardy formula then reproduces

the exact formula for the Bekenstein-Hawking entropy of the extreme Kerr.

S = π 2 3 cLTL = π2 3 12J ℏ 1 = 2πJ= SBH (2.3)

Other directions which exploit the conformal symmetry of the Kerr solution to study holographic properties of gravity in this regime are [55, 56, 57, 58]. Another interest-ing aspect comes from studyinterest-ing Kerr black holes in de-Sitter where multiple extremal limits exist, and one of them leads to a dS2 geometry in the decoupled region known as the Nariai limit [20]. It is characterised by the confluence of a black hole event horizon with the cosmological horizon of de-Sitter [17, 19, 21]. This limit is interest-ing from the perspective of 2D gravity in de-Sitter [63, 64]. Prior work also provides a description in terms of a Nariai/CFT proposal, considered in [59]. These studies fall under the broader subject of dS/CFT [15]. An interesting problem in de-Sitter

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physics is to compute the Hartle-Hawking wavefunction [18], which provides a way to measure the fluctuations from the early Universe and was considered in [63, 64]. In addition, the Kerr solution is also interesting from the analysis of its gravita-tional perturbations. Results in such directions is astro-physically relevant as near extremal Kerr black holes are observed in the sky, for example the near-extreme GRS 1915 + 105 black hole is dual approximately to a CFT with cL ∼ 2 × 1079. A

comprehensive review of these developments are found in [66]. A direction analysing gravitational perturbations using Kerr/CFT was considered in [61, 67].

In this thesis, we consider stationary (charged/neutral) solutions in D = 4, 5 with a locally AdS2 near horizon geometry. We study independently the thermodynamic response of these solutions, and inspired by [2], extend the analysis by consider-ing; (i) Kerr a cosmological constant, (ii) Kerr in D = 5. We review instances of

nAdS2/nCF T1, and its relation with the near extremal thermodynamic response of both the Reissner-Nordström, and Kerr varieties. For the Kerr solution, we adopt the approach developed in [1]. The initial goal was to extend the analysis to the case of Kerr-Λ and distinguish between the properties that arise due to the presence of a positive, and negative cosmological constant. It is incomplete work, and so we shall conclude by discussing our attempt at the problem.

Outline: In section 3, we review some basic geometric and thermodynamic

prop-erties of the solutions we consider. In section 4, we discuss the near extremal limit of each of these solutions in detail. In section 5, we discuss the correlation between dynamics of the near region and the thermodynamic response of RN and Kerr black holes, while reviewing aspects of nAdS2/nCF T1 where necessary. In section 6, we generalise the thermodynamic analysis of [2] and discuss the phase transitions we observe for specific conditions. In section 7 we discuss our ansatz to generalise the analysis in [1] to the solutions we examine in section 6. Section 6.2, 6.3 and 7 are

presented as original contributions.

3

The phase space of solutions

We refer to the outer event horizon of a black hole and the cosmological event horizon of de-Sitter space several times in the text, and so we begin with a discussion of their basic notions. We then discuss basic geometric and thermodynamic properties of the solutions we analyse further.

3.1

Horizons

3.1.1 Event horizon of a black hole

Let us represent an observer in a spacetime (M, gab) by an inextendible timelike

curve γ. The future horizon, h+, of γ is defined to be the boundary, ˙I(γ) of I(γ). In General Relativity, we have a theorem which states that [73],

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Figure 3.1: The figure depicts an observer inM surrounded by her future horizon h+.

Figure 3.2: An accelerating observer

γ in Minkowski.

Each point p∈ h+ lies on a null geodesic segment contained entirely within h+ that

is future inextendible. Furthermore, the convergence of these null geodesics that generate h+ cannot become infinite at a point on h+.

Let (M, gab) be an asymptotically flat spacetime, where a family observers Γ escape

to arbitrarily large distances at late times. If the past of these observers I−(Γ) fails to be the entire spacetime, then a black holeB ≡ M − I(Γ) is said to be present. The horizon, h+, of these observers is the future event horizon of the black hole.1 An isometry is a diffeomorphism that leaves the metric gab invariant. If, ξa, is the

infinitesimal generator of a one-parameter group of isometries, and it satisfies 0 = Lξgab = 2∇(aξb), (3.1)

then, it is said to be a Killing isometry.

A Killing horizon is a null hypersurface, H, which is invariant under the one-parameter isometry group generated by ξa; and on which ξaξ

a = 0. It is a general

result, by virtue of Hawking’s rigidity theorem [6], that an event horizon is also a Killing horizon. The statement of the theorem is,

Let (M, gab) be a stationary, asymptotically flat solution of Einstein’s equation that

contains a black hole. Then the event horizon, h+, of the black hole is a Killing

horizon.

1We shall be more specific about the event horizon in subsequent sections, when we consider

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If ξa vanishes on a codimension-2 surface, Σ, at which, two null surfaces, H a and

Hb, generated by null geodesics orthogonal to Σ intersect, then Σ is said to be

a bifurcate Killing horizon and the intersection point is called the bifurcation surface. It follows that, ξa is normal to both H

a and Hb.

Let λ denote the affine parametrization of the null geodesic generators ofH, and let

ka denote the corresponding tangent. Let ν denote the Killing parameter along the

null generators ofH such that,

ξa = ∂ν. (3.2)

Since ξa is normal to H, we have,

ξa= f ka, (3.3)

where,

f = ∂λ

∂ν. (3.4)

We define the surface gravity, κ, ofH by

κ = ξa∇alnf =

∂lnf

∂ν . (3.5)

Equivalently, we have onH

ξb∇bξa= κξa. (3.6)

It follows that κ is constant along each generator of H. Consequently, the relation between the affine parameter λ and Killing parameter ν onH is given by

λ = eκν. (3.7)

We see that the Killing vector is a geodesic on a horizon that is not parameterized affinely, and the surface gravity measures the non affine parameterization. Near the bifurcate Killing horizon, Σ, the orbits of ξa look like Lorentz boosts in Minkowski

spacetime which correspond to accelerations in Rindler. The acceleration horizon is then formed by two intersecting lines corresponding to Ha and Hb defined

previ-ously. This brings us to the zeroth law of black hole mechanics which states that,

Let Ha and Hb be two null surfaces comprising a bifurcate Killing horizon. Then

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3.1.2 Cosmological event horizon

The Einstein-Hilbert action endowed with a positive cosmological constant in D = 4 in the absence of matter fields is given by2

SEH = 1 16πM d4x√−g (R − 2Λ) , (3.8)

whereM is the 4-dimensional manifold over which we integrate the action. Varying this action with respect to the metric, we obtain the Einstein equations

Rµν−

1

2gµνR − Λ

8πgµν = 0. (3.9)

Tracing, we get the Ricci scalar,R = 4Λ. The cosmological constant in 4D is Λ = 3

l2.

The geometry of pure de-Sitter is visualised by embedding a d-dimensional hyper-boloid to (d + 1) dimensional Minkowski spacetime. Then, de-Sitter is viewed as the induced metric on the hyperboloid:

− X2 0 + ni=1 Xi2 = l2. (3.10)

The induced metric is given by

ds2 =−X02+

n

i=1

Xi2. (3.11)

Various coordinate systems cover partially, or globally, this n-dimensional hyper-boloid [16]. The global and static patches will be relevant for our purpose, so we restrict our attention there. We use coordinates on the three sphere, S3 (and set

l = 1) with the parametrizations,

ω1 = cos ψ, ω2 = sin ψ cos θ,

ω3 = sin ψ sin θ cos θ, ω4 = sin ψ sin θ sin ϕ, where, θ, ψ ∈ (0, π), ϕ∈ [0, 2π], 4 ∑ i=1 (ωi)2 = 1. The metric becomes,

dΩ23 = 4 ∑ i=1 (ωi)2 = dψ2+ sin2ψdΩ22. (3.12) 2We have G N = c =1

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In global coordinates (t, ψ, θ, ϕ),

X0 = sinh t, Xi = ωicosh t, i∈ (1, 2, 3, 4). (3.13) The metric is given by

ds2 =−dt2+ cosh2t dΩ23. (3.14) These coordinates foliate dS with a S3 (positive curvature). The range of the co-ordinates starts from, radius r → ∞ at t → −∞, shrinks to r = 0 at time t = 0, and finally grows exponentially to infinite radius as t→ ∞. It is a time-dependent background with topology R1× S3. There is no global timelike Killing vector, and we do not have time translation symmetry. The global patch is although descriptive of the full spacetime but there is no observer, who has full causal access to the whole patch so these coordinates are not useful. We now make a conformal transformation,

cosh t = 1

cos T, (3.15)

so that the metric becomes,

ds2 = 1 cos2T(−dT 2+ dΩ2 3), T ∈ ( −π 2 , π 2). (3.16)

This now makes the causal structure of global dS manifest. We now make the fol-lowing coordinate transformation,

X0 =1− r2sinh t, X4 =1− r2cosh t, Xj = rωj, j = 1, 2, 3. (3.17) The metric then takes the form,

ds2 =−(1 − r2)dt2+ (1− r2)−1dr2+ r2dΩ22, |r| ∈ [0, 1]. (3.18)

This metric represents the static patch of dS, as it covers one of the (left or right) wedges, (Fig.3.3) and it is the only region where an observer can send and receive signals at times prior to when future null infinity is reached. There is a time like Killing vector,

∂t, and this patch consists of a time independent background, as

∂tgµν = 0. This is physically useful, as we now have time translation symmetry and

can define Hamiltonian evolution for r < 1. At r2 = 1, the observer is surrounded by a cosmological horizon [17, 19].

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Figure 3.3: The full Penrose diagram depicts global de-Sitter. The shaded wedge is the static patch.

Friedmann-Robertson-Walker or FRW metric, which is an exact solution of the Einstein’s equations and is a model of the expanding Universe given by

ds2 =−dt2+ a2(t) ( dr2 1− kr2 + r 2dΩ 2 ) . (3.19)

Here, a(t) is a dimensionless scale factor whose role is to change distances over time. k is a curvature parameter which specifies the curvature of the (maximally symmetric) 3-space. Time is always measured with the present time, t0, as reference, such that a(t0) = 1. There exists a point in time3, tBB < t0, where the scale factor vanishes, a(tBB) = 0, and the FRW metric has a singularity. The universe is said

to be composed of causally disconnected regions of space which become spatially homogeneous and isotropic as they have nearly the same densities and temperatures over large scales. This is called the horizon problem and is attributed to inflation [77]. These subjects are sufficiently detailed in their own right and we shall not discuss them. We allude to the only related aspect, which is the notion of a particle

horizon, that is relevant for our understanding of the cosmological event horizon.

First, we rewrite (3.19) as

ds2 = dt2− a2(t)[2+ Sk2(χ)dΩ2], (3.20)

where, dχ = dr/√1− kr2. S

k(χ) parametrizes the curvature of the 3-sphere.4 χ

is called the comoving coordinate and physical distances depend only on proper distances, dχ= a(t)χ. It is useful to introduce conformal time by

dη = dt

a(t), (3.21)

3colloquially referred to as the Big Bang 4S kχ = 1ksinh( kχ), k < 0; Skχ = χ, k = 0; Skχ = 1ksin( kχ), k > 0

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so that, (3.20) becomes

ds2 = a2(η)[2 (2+ Sk2(χ)dΩ2)] (3.22) The size of a causally connected patch of space is determined by the greatest distance traversed by light in a finite amount of time. Since spacetime is isotropic, we are able to define coordinates such that, light travels along the radial coordinate, and angular dependence is removed by considering θ = ϕ = 0. Using the parametrization in (3.20), the redefinition of time in (3.21), and the above considerations, we can say that the evolution is determined by the line element

ds2 = a2(η)[dη2− dχ2]. (3.23) The path of photons corresponds to ds2 = 0 and is defined by

∆χ(η) =±∆η, (3.24)

where, the plus and minus signs correspond to outgoing and incoming photons. This equation tells us that, the maximal comoving distance between two points in time is given by ∆η = η2− η1. Hence, if there is a singularity at the initial time, ti = 0,

the greatest comoving distance from which an observer at time t can receive signals at the speef of light (we set c = 1) is given by

χh(η) = η− ηi =

t ti

dt

a(t) (3.25)

This is called a particle horizon and describes an observer trying to access her causal past, but being restricted by a cosmological event horizon. The size of the horizon is observer dependent and is estimated by the intersection of the past light cone of an observer O with the spacelike surface η = ηi. (Fig. 3.4)

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O event horizon

Figure 3.4: The figure illustrates the size of the cosmological event horizon, which is the maximum distance from which the observer receives signals.

3.2 The Kerr solution

3.2.1 Geometry and thermodynamics

The Kerr black hole is a solution to the Einstein-Hilbert action in 4D given by

S = 1

16πG4 ∫

d4x√−gR. (3.26)

In Boyer-Lindquist coordinates, the metric is

ds2 =ρ2 ( dt− a sin2θdϕ)2+sin 2θ ρ2 ( (r2+ a2)dϕ− ad˜t)2+ρ 2 ∆dr 2 + ρ22, (3.27) where, ∆ = r2− 2Mr + a2, ρ2 = r2+ a2cos2θ, a = J M. (3.28)

The quantities M and J are the mass and angular momentum of the black hole, and they are conserved charges corresponding to the time translation symmetry and the rotational symmetry of the solution, associated with the Killing isometries, ξ1 =

∂t

and ξ2 =

∂ϕ. a is a rotation parameter, useful for computational purposes, and is

the angular momentum per unit mass.

The Kerr solution admits an outer, and an inner horizon. The location of these horizons is obtained by solving

∆(r) = 0, (3.29)

which gives,

r± = M±√M2− a2. (3.30)

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event horizon. footnoteWe shall focus on the thermodynamics of the outer horizon for the rest of the discussion.

The Hawking temperature [5] on the outer horizon is given by

TH =

ℏκ

(3.31)

where κ is the surface gravity,

κ = r+− M

2M r+

. (3.32)

There is an angular velocity associated with the outer horizon which is determined by ΩH = gtt gtϕ |r+ = a 2M r+ . (3.33)

The Bekenstein-Hawking entropy [7] associated with the outer horizon is given by5

SBH =

A

4ℏ =

2πM r+

ℏ (3.34)

The horizon is generated by a linear combination of the time translation and axial Killing vectors given by

χ+= ξ1+ Ω2, (3.35)

whose norm is,

χ2 = gµνχµχν = ∆ ρ2. (3.36)

The above thermodynamic quantities satisfy the first law of black hole thermody-namics [5, 35] on the outer horizon which states:

If a stationary black hole of mass M, angular momentum J, future event horizon surface gravity κ, and angular velocity Ω is perturbed such that it settles down to another black hole with mass M + δM, and angular momentum J + δJ then

dM = κ

8πdA + ΩdJ. (3.37) Alternatively,

dM = T dS + ΩdJ. (3.38) The Kerr spacetime is characterised by a real curvature singularity whose location is determined by the condition

ρ = 0, =⇒ r = 0, θ = π

2. (3.39)

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A singularity in spacetime is said to be present if there is an non- analyticity mea-sured by a curvature (scalar) invariant. The Kretschmann scalar,K, is a quadratic scalar invariant defined as

K = RabcdRabcd (3.40)

whereRabcd is the Riemann tensor. For the Kerr solution,

K = 48M2(r2 − a2cos2θ)(ρ2− 16r2 a2 cos2θ)

ρ6 . (3.41)

This singularity occurs in the shape of a ring, manifest through the following coor-dinate transformation

x + iy = (r + ia) sin θ exp{i

(dϕ + adr)}, z = r cos θ, t′ = ∫ (dt +r 2+ a2 ∆ dr)− r. (3.42)

The metric becomes,

ds2 =−dt′2+dx2+dy2+dz2+ 2M r 3

r4+ a2z2 (

r (xdx + ydy)− a (xdy − ydx) r2+ a2 +

zdz r + dt

)2

(3.43) where r is a function of coordinates. Rewriting,

x = (r cos ϕ + a sin ϕ) sin θ, y = (r sin ϕ− a cos ϕ) sin θ, (3.44)

we get x2+ y2 sin2θ z2 cos2θ = a 2. (3.45) Alternatively, x2+ y2 sin2θ − r 2 = a2. (3.46) Setting r = 0, θ = π 2; we obtain x2+ y2 = a2

which is a ring in the x-y plane.6

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3.2.2 The first law: another perspective

The Wald formalism [76] associates the entropy as a Noether charge. However, the Noether procedure is only valid for global symmetries, whereas in gravity the only symmetries in an arbitrary space time are diffeomorphisms. So the notion of an associated conserved (n− 1) form with some suitably defined gauge symmetry ξµ is

not valid.

In this section, we shall see how the mass and angular momentum can be alter-natively associated with surface charges corresponding to Killing vectors ∂t and ∂ϕ

using the covariant phase space formalism.

Let Φ = (Φi) denote a collection of fields which also includes the metric g µν.

Let L denote the Lagrangian density such that

δL = ∂Φi∂L

∂Φi − dΘ [∂Φ; Φ] (3.47)

where Θ [∂Φ; Φ] denotes a pre-symplectic potential associated to the fields Φ and its variation ∂Φ; where Θ is a (n− 1) form.

We can define a (n− 2) form ω

ω[∂Φ, ∂Φ; Φ] = ∂Θ [∂Φ; Φ], (3.48)

and it is also known as the pre-symplectic form. Contracting the pre-symplectic form with a gauge transformation ∂ξΦi, ∃ a (n − 2, 1) form Kξ [∂Φ; Φ], that

satisfies on-shell

ω[∂ξΦ, ∂Φ; Φ] + dKξ [∂Φ; Φ] = 0, (3.49)

where Φi solves the equations of motion and ∂Φi solves the linearised equations of

motion around Φi.

Subject to the conditions that the charge is integrable and conserved (see [36]) the surface charge associated with Kξ on a codimension 2 surface S by

δQξ [∂Φ; Φ] =

I

S

[∂Φ; Φ]. (3.50)

ω ≈ 0, when Qξ is conserved, implying that the difference of charge between two

surfaces vanish. If ξµ is an exact Killing isometry, then

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For ξµ= ∂

t, we have Q∂t = M . Similarly, for ξ = ∂ϕ, Q∂ϕ = J.

We shall now derive the first law of black hole mechanics for the Kerr space time which admits a bifurcate Killing horizon. We integrate dKξ over a space-like,

codimension-2 surface Σ stretching between the bifurcate Killing horizon, B, and spatial infinity denoted by Σ. Staring from

ω[∂ξϕ, ∂ϕ] + dKξ (ϕ) = 0, (3.52) we have, 0 = ∫ Σ dKξ (∂ϕ)−∂Σ∞ (∂ϕ)−B (∂ϕ). (3.53)

Here ϕ denotes the black hole space time and ∂ϕ its linearised perturbation. Since

ξ = ∂t+ Ω∂ϕ, by the definition of a surface charge;

∂Σ

(∂ϕ) = δQ∂t+ ΩH δQ∂ϕ = δM + ΩHδJ. (3.54)

Again we have,

Kξ(∂ϕ) = δQξ(ϕ)− ξΘ(∂ϕ), (3.55)

and ξµ = 0 on the bifurcate surfaceB, so

B (∂ϕ) =B δQξ (ϕ) = δB (ϕ). (3.56)

This identifies a quantity ∫B (ϕ) intrinsic to the horizon whose variation enters

the first law. We know from Wald’s formalism that the entropy is identified as a Wald-Noether charge. We want to identify δB (ϕ) with κ δA4 and this is valid

as the zeroth law holds and thus we have,

κ

δA

4 = δM + ΩHδJ. (3.57)

3.3

Kerr-Newman

The Kerr-Newman solution is a generalisation of Kerr which includes a charge, Q, comprising an electric and a magnetic component. It is a solution of the Einstein-Maxwell theory in 4D given by the action

S = 1 16πG4 ∫ d4x√−g ( R − 1 4F 2 ) . (3.58)

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The metric is given by ds2 =( Σ∆ (˜r2+ a2)2− ∆a2sin2θ)d˜t 2+ Σ ( d˜r2 ∆ + dθ 2 ) + sin 2θ Σ (

r2+ a2)2− ∆a2sin2θ) (d ˜ϕ− a(˜r

2+ a2− ∆) ˜ r2+ a2)2− ∆a2sin2θd˜t )2 , (3.59) where, Σ = ˜r2+ a2cos2θ, ∆ = (˜r− r+) (˜r− r−) , r±= M2±M2− a2− Q2. (3.60)

All quantities have the same definition as that of Kerr. The charge is given by

Q2 = Q2e+ Q2m, (3.61)

where, Qe and Qm denote the electric and magnetic charges. The associated

one-form potential has non vanishing components,

At=

Qer˜− Qma cos θ

Σ2 , =

−Qear sin2θ + Qmr2+ a2) cos θ

Σ2 . (3.62)

The electric potential is defined by

Φ± =−Att± Aϕϕ (3.63)

It shares the same geometric properties as that of Kerr. The thermodynamic quan-tities satisfy the first law the outer horizon given by,

dM = T dS + ΩdJ + ΦdQ, (3.64)

where the entropy, angular velocity, electric potential are given by

S = π(r2++ a2), Ω = a r2 ++ a2 , Φ = Qr+ r2 ++ a2 , (3.65)

and the temperature is

T = r+   1 a2+Q2 r2 + 4π(r2 ++ a2)   (3.66)

The Kerr Newman solution is the most general solution in 4D which allows us to retrieve the Kerr (Q = 0), Reissner-Nordström (J = 0), and the Schwarzschild

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(Q = J = 0) solution from its line element. The black hole uniqueness theorem [4] states that,

The only stationary, asymptotically flat (analytic) Einstein-Maxwell solution is the extremal Kerr-Newman black hole.

In addition, the no-hair theorem [4] establishes that the Bekenstein-Hawking entropy can be uniquely specified by M, Q, J and will be insensitive to any other parame-ter specified at asymptotic infinity to characparame-terise the solution. From the horizon topology theorem of Hawking [6] it is established that, a spatial cross section of the event horizon of the Kerr-Newman solution uniquely admits the topology of S2.

3.4 Kerr Newman - Λ

The black hole uniqueness theorem no longer holds, for solutions to (3.52) with a cosmological constant, Λ. In 4D, Λ = 3 l2. The action is S = 1 16πG4 ∫ d4x√−g ( R + 6 l2 1 4F 2 ) . (3.67)

The solutions to this gives rise to Kerr-Newman in asymptotically AdS and dS spacetimes. The metric given below describes both cases, where l2 > 0 for AdS, and

l2 < 0 for dS. ds2 =r ρ2 ( d˜t− a Ξsin 2θd ˜ϕ)2+ ρ2 ( d˜r2 ∆r + 2 ∆θ ) + ∆θ ρ2 sin 2θ ( ad˜t− r˜ 2+ a2 Ξ d ˜ϕ )2 . (3.68) with ∆r = ( ˜ r2 + a2) (1 + r˜ 2 l2 ) − 2m˜r + Q2, θ = 1 a2 l2 cos 2θ, ρ2 = ˜r2+ a2cos2θ, Ξ = 1−a 2 l2, Q 2 = qe2+ qm2. (3.69) The entropy, angular velocity, and electric potential are given by

S = πr 2 ++ a2 Ξ , Ω = Ξa r2 ++ a2 , Φ = Qr+ r2 ++ a2 . (3.70)

The temperature, mass, angular momentum, and the electric and magnetic charges are T = r+ ( 1 + a2/l2+ 3r2+/l2− (a2+ Q2) /r+2) 4π (r2 ++ a2) , M = m Ξ2, (3.71)

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J = aM, Qe =

qe

Ξ, Qm =

qm

Ξ . (3.72)

These thermodynamic quantities are related by the first law (3.57). m is a mass parameter, similar to the rotation parameter for angular momentum. It is convenient for purposes of locating the horizons, as it is more useful to solve than, ∆r.

3.5 Kerr-AdS

5

and its flat limit

In five and higher dimensions, the uniqueness theorem does not hold and the land-scape of solutions is richer.7 The horizon topology of the solutions we consider are

S3. There is also no analytic solution of the Einstein-Maxwell theory so we do not have a direct generalisation of Kerr Newman in 5D.8 The stationary solution we analyse, is a Kerr black hole spinning in two independent rotational planes, em-bedded in an asymptotically AdS spacetime, which has the rotational symmetry group SO(4). The flat space limit gives rise to the simpler Myers-Perry solution [37]. These are both solutions to Einstein gravity with (and without) a negative cosmological constant in 5D whose action is given by

S = 1 2 5 ∫ d5x−g(5) ( R(5) + 12 l2 5 ) . (3.73)

The Kerr-AdS5 solution is given by

ds2 = −∆r ρ2 ( d˜t− a sin 2θ Ξa d ˜ϕ− b cos 2θ Ξb d ˜ψ )2 +−∆θsin 2θ ρ2 ( ad˜t−r˜ 2+ a2 Ξa d ˜ϕ )2 +∆θcos 2θ ρ2 ( bd˜t−r˜ 2+ b2 Ξa d ˜ψ )2 + ρ 2 ∆r d˜r2 + ρ 2 ∆θ 2 (3.74) +1 + ˜r 2l−2 ˜ r2ρ2 ( abd˜t− b(˜r 2+ a2)sinθ Ξa d ˜ϕ− a(˜r 2+ b2)cos2θ Ξb d ˜ψ )2 ,

where a and b are internal rotation parameters along ˜ϕ and ˜ψ, and

r = 1 ˜ r2(˜r 2+ a2)(˜r2+ b2)(1 + r˜2 l2)− 2M,θ = 1 a2 l2 cos 2θb2 l2 sin 2θ, ρ2 = ˜r2 + a2cos2θ + b2sin2θ, Ξa= 1 a2 l2, Ξb = 1 b2 l2. (3.75)

7[34] provides a classification of the extremal solutions. See also [38].

8There is a supersymmetric solution to 5D supergravity, which is called the BMPV black hole

named after the authors in [39]. The non supersymmetric one is the Calabi-Yau solution found in [43], studied further in[44]. Apart from this, there are the black rings which have S1× S2topology

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The outer horizon corresponds to r+ obtained by solving ∆r = 0. The temperature,

entropy, angular momenta Jϕ and Jψ, angular velocity Ωϕ, and Ωψ are given by

T = r 4 +− a2b2+ r+4l−2(2r+2 + a2+ b2) 2πr+(r2++ a2)(r2++ b2) , (3.76) S = π 2(r2 ++ a2)(r+2 + b2) 2raΞb , (3.77) = πma 2 Ξ2 a Ξb , = πmb 2 Ξa Ξ2b ,ϕ= a Ξa r2 ++ a2 ,ψ = b Ξb r2 ++ b2 . (3.78)

The mass of the solution is given by

M = πm(2Ξa+ 2Ξb− ΞaΞb)

4Ξ2

aΞ2b

(3.79)

4

The near extremal limit

In this section we shall discuss properties of the solutions at extremality. In partic-ular we shall study the near horizon geometries of the all the solutions discussed in section 3. We shall then develop the notion of a near extremal limit for each case. In section 4.1, we discuss the Kerr and Kerr Newman solution in flat space. We then consider these solutions in asymptotically AdS and dS spacetimes in section 4.2. The Kerr-dS solution exhibits more than one extremal limit and we mention the different cases in section 4.3. Finally, in section 4.5 we consider the Kerr solution in five dimensions in an asymptotically AdS spacetime.

4.1

Near extremal Kerr (Newman)

We consider the Kerr metric previously described in (3.27), rewritten in the following form ds2 =( Σ∆ (˜r2+ a2)2− ∆a2sin2θ)d˜t 2+ Σ ( d˜r2 ∆ + dθ 2 ) (4.1) +sin 2θ Σ (

r2+ a2)2 − ∆a2sin2θ) (d ˜ϕ− 2aM ˜r

˜ r2+ a2)2− ∆a2sin2θd˜t )2 , with Σ = ˜r2+ a2cos2θ, ∆ = (˜r− r+) (˜r− r−) , r±= M2± M2 − a2. (4.2)

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The extreme Kerr solution is obtained when the outer and inner horizons are coinci-dent, that is when r+ = r−. This is achieved when the mass and angular momentum of the black are related by

M4 = J2 =⇒ M2 = a2, (4.3)

where,

J = aM.

This corresponds to a maximally spinning Kerr solution. The relation in (4.3) is a bound on the value of the the angular parameter, and is known as the extremal bound. Values of a are restricted by this bound, due to cosmic censorship [74]. We thus have,

r+= r− = M.

We see that, the temperature given by (3.26 - 3.27) goes to zero when the solution is extremal. This is said to be the extremal limit of the solution. Let us denote the extremal mass by M0. Let us now rewrite the coordinates in (4.1) by

˜ r = r++ λr, ˜t = 2M0 t λ, ˜ ϕ = ϕ + M0 t λ. (4.4)

We keep J fixed, and take λ→ 0. The line element is now given by

ds2 = M02(1 + cos2θ) [ −r2dt2+dr2 r2 + dθ 2 ] + M02 4 sin 2θ 1 + cos2θ(dϕ + rdt) 2 . (4.5)

The geometry described by (4.5) is known as the Near Horizon Extremal Kerr or NHEK [46]. This follows from a rescaling of the coordinates and finding a suitable diffeomorphism which allows us to go very near the horizon of an extremal Kerr solution.

The Killing vectors of this geometry are given by

ξ−1 = ∂t, ξ0 = t∂t− r∂r, ξ1 = ( 1 r2 + t 2 ) ∂t− 2rt∂r− 2 r∂ϕ, ξ2 = ∂ϕ. (4.6)

These Killing vectors generate an SL(2,R) × U(1) algebra. If we inspect the line element, we see that the NHEK geometry contains an AdS2 spacetime, where the

SL(2,R) is the conformal symmetry group of AdS2 (Fig. 4.1). Thus we see that, the geometry of the solution in this limit, is itself a solution to Einstein’s equations, and has an enhanced conformal symmetry which was not present in the full Kerr

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Figure 4.1: The figure represents the confluence of the outer and inner horizon of Kerr.

Figure 4.2: The NHEK geometry is depicted here. The red shaded region marks the infinitely long AdS2 throat.

solution. The geometry in this sense, is decoupled from the rest of the black hole spacetime and hence this is known as the decoupling limit. It is illuminating to mention, that this decoupling limit consisting of an AdS2 component exists for all extremal solutions at least in 4D, as well as for the stationary solutions we consider in 5D.

We are interested in the thermodynamics of the rotating solutions described in section 3, very close to, but not at extremality, which means that, we wish to maintain a small but finite temperature, as we approach the near horizon region, from the full solution in flat space. We refer to this as a near extreme limit, which is realised by

r±= M0± ελ +

ε2λ2

4M0

+O(λ3), (4.7)

where, λ and ε are small parameters controlling a deviation away from extremal-ity, and a deviation of the mass above its extremal value, M0. We now want to implement the decoupling limit, for the near extremal solution. We do this by keep-ing the angular momentum fixed, and lowerkeep-ing the temperature, by considerkeep-ing the redefinitions, ˜ r = r++ r− 2 + λ ( r + ϵ 2 4r ) , ˜t = 2M0 t λ, ˜ ϕ = ϕ + M0 t λ (4.8)

Taking λ→ 0 while keeping ε fixed, the line element becomes

ds2 = M02(1 + cos2θ) [ −r2 ( 1 ε 2 4r2 )2 dt2+dr 2 r2 + dθ 2 ]

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+ M02 4 sin 2θ 1 + cos2θ [ dϕ + r ( 1 + ε 2 4r2 ) dt ]2 . (4.9)

This metric represents the near horizon of Kerr which is slightly deviated from its extremal configuration. We retrieve NHEK, if ε = 0.

We can similarly define a near extremal limit for Kerr Newman by considering the solution given in section 3.3. We take the near extremal limit by lowering T , but keeping the other charges, J, Q fixed. To do this we consider the following coordinate transformations, ˜ r = r++ r− 2 + λ ( r + ε 2 4r ) ˜ t = (M2+ a2)t λ ˜ ϕ = ϕ + adt λ (4.10)

and take λ → 0, while keeping ε fixed. This gives the near extreme near horizon Kerr Newman ds2 = (M2+ a2cos2θ) ( −r2 ( {1 + ε2 4r2} 2ε 2 r2 ) dt2+ dr2 1 ε2 4r2 r2{1 + ε2 4r2}2− ε2 + dθ2 ) +(M 2+ a2)2sin2θ M2+ a2cos2θ ( dϕ− 2Ma(r + ε 2 4r)dt )2 . (4.11)

This reduces to the near horizon extremal geometry, (which also has the symmetry group SL(2,R) × U(1) ) when ε → 0.

We can get the Reissner-Nordström solution, which is a charged but static solution in 4D from Kerr Newman by setting a = 0 given by

ds2 =−f(r)dt2+ dr 2 f (r)+ r 2dΩ 22, (4.12) where, f (r) = 1− 2M r + Q2 r2. (4.13)

The solution is extremal when r+ = M = Q. Implementing the decoupling limit by redefining

r = Q(1 + λ

z), t =

λ (4.14)

in the RN metric and taking λ→ 0 to get,

ds2 = Q 2

z2 (−dτ

2 + dz2) + Q2 dΩ

22. (4.15)

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case of the previous extremal geometries where, the solution is no longer rotating and the isometries are SL(2,R) × SO(3). This geometry is known as the Bertotti-Robinson spacetime.

The AdS2 geometry appearing in the near horizon region of these extremal solutions is infinitely long. Consider the distance travelled by an observer at an arbitrary point

r0, along a fixed t-slice, to the horizon r+ of the above, RN solution given by ∆s =

r0

r+

dr(1− Q

r) = (r0− r+)− Q log(r0− r+) (4.16)

We see that as r0 → r+, ∆s→ ∞, i.e. the distance becomes infinitely long.

4.2 Decoupling limit of Kerr-Newman-Λ

We first consider the Kerr Newman-Λ solution given by (3.62), and take the extremal limit by lowering the temperature, keeping other parameters fixed.

At extremality T = 0, the two roots of ∆r= 0, where the solution is degenerate are

equal. This gives us the following conditions on a, and m which shall be useful later on in the analysis of section 6.

a2 = r 2 + ( 1 + 3r2 +/l2 ) − Q2 1− r2 +/l2 , (4.17) m = r+ ( 1 + r2 +/l2 )2 − Q2/l2 1− r2 +/l2 . (4.18)

The near horizon geometry is obtained by the following redefinitions ˜

r = r++ λµ0r, ˜t = tµ0/λ, ϕ = ϕ + Ω˜

0

λ (4.19)

where Ω is the angular velocity of the metric given in (3.64), and

µ20 = (r 2 ++ a2)(1 r2 + l2) 1 + 6r2+ l2 3r4 + l4 Q2 l2 . (4.20)

Taking λ→ 0, we obtain the near horizon geometry

ds2 = Γ(θ) [ −r2dt2+ dr 2 r2 + α(θ)dθ 2 ] + γ (θ)(dϕ + krdt)2. (4.21)

The near horizon gauge field becomes,

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where, Γ(θ) = ρ 2 +µ20 r2 ++ a2 , α(θ) = r 2 ++ a2 ∆θµ20 , γ(θ) =θ(r 2 ++ a2)2sin 2θ ρ2 +Σ2 , (4.23) f (θ) = r 2 ++ a2) [

QeΣ(r+2 − a2cos2θ) + 2QmΣar+cos θ ] 2 +Σar+ , and, ρ2+= r2++ a2cos2θ, k = 2ar+Σµ 2 0 (r2 ++ a2)2 . (4.24)

The isometries are given by

ζ2 = ∂ϕ, ζ−1 = ∂t, ζ0 = t∂t− r∂r, ζ1 = ( 1 2r2 + t2 2 ) ∂t− tr∂r− k r∂ϕ. (4.25)

They generate a SL(2,R) × U(1) algebra. The near horizon gauge field A is also invariant under diffeomorphisms generated by this group of isometries. This is the general near horizon geometry for Kerr-Newman black holes in AdS and dS, with a near-horizon AdS2 factor. However, a Kerr solution embedded in de-Sitter, admits more than extremal limit which we consider next.

4.3

Alternative extremal limits of Kerr-dS

de-Sitter space is characterised by a cosmological event horizon as we saw earlier, so black holes in de-Sitter admit multiple extremal limits. We consider the Kerr-dS solution given by the metric

ds2 =br ρ2(dbt− a Σ sin 2θ d bϕ )2 + ρ2 ∆br der 2+ ρ2 ∆θ 2 + ∆θ ρ2 sin 2 θ (a dbt− br 2+ a2 Σ d bϕ) 2 , (4.26) where ∆br = (br2 + a2)(1 br 2 l2 )− 2M br, ∆θ = 1 + a2 l2 cos 2θ, ρ2 =br2+ a2 cos2θ, Σ = 1 + a 2 l2. (4.27)

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denote the locations of the outer, inner and the cosmological horizon of de-Sitter; with rc ≥ r+ ≥ r−. As a consequence, we can define three different extremal limits.

These correspond to, r+ = r−, which has a fibered product of AdS2 and the two-sphere in the near horizon geometry and is obtained in a similar manner as discussed in the previous subsection. The difference is that, l2 < 0 for dS, which is obtained by taking l→ il from the Kerr-AdS solution.

r+ = rcgives rise to a fibered product of dS2×S2 which is called the rotating Nariai limit [20]. 9. The third case is given by the ultracold limit: r

+ = r− = rc which is

a product of the two sphere with Minkowski, R2. We shall focus on the second and third case in this subsection.10.

First, we consider the near horizon corresponding to the Nariai limit, in which the temperature and angular momentum of the black hole event horizon and cosmolog-ical event horizon are in equilibrium. To arrive at the near horizon geometry, we take r+ → rc, and the near horizon limit simultaneously. Let us define a parameter,

ε = rc− r+ λrc

(4.28)

where ε measures a deviation of rc and r+ away from extremality, and λ denotes a

small rescaling parameter. At extremality, ε = 0. We redefine coordinates by bt= λδt, r = br− r+ λrc , ϕ = bϕ− Ωt (4.29) where, Ω = r2 ++ a2 , δ = rc(−1 + a 2/l2+ 6r2 c/l2) r2 c + a2 . (4.30)

Keeping all other parameters fixed, we take λ→ 0 to get

ds2 = Γ(θ) [ −r(ε − r)dt2 + dr 2 r(ε− r) + α(θ)dθ 2 ] + γ(θ)(dϕ + krdt)2 (4.31) where, Γ(θ) = ρ 2 crc δ(r2 c + a2 , γ(θ) =θ(r 2 c + a2) sin 2 θ Σ2ρ2 c , α(θ) = δ(r 2 c + a2) ∆θrc , (4.32) ρ2c = r2c + a2cos2θ, k = 2aΣr 2 c δ(r2 c+ a2)2 . (4.33) 9See also [21]

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r = 1

Figure 4.3: The figure illustrates the cosmological horizons explicitly in the Penrose diagram of dS2 appearing in the Nariai limit.

We now make a final coordinate transformation to make the dS2spacetime appearing in the near horizon manifest by considering the transformations

r→ ε

2(r + 1), t 2

εt (4.34)

which finally gives us,

ds2 = Γ(θ) (−(1 − r2)dt2+ 1 1− r2 dr

2+ α(θ) dθ2) + γ(θ)(dθ + kr dt)2. (4.35)

The near horizon is a fibered product of dS2 × S2. The horizons are located at

|r| = 1. The isometries of this spacetime are generated by ξ2 = ∂ϕ, ξ−1 = ∂t, ξ0 = r sinh t 1− r2 ∂t+ cosh t 1− r2 r− K sinh t 1− r2 ∂ϕ, ξ1 = r cosh t 1− r2 ∂t+ sinh t 1− r2 r− K cosh t 1− r2 ∂ϕ. (4.36)

The other extremal known as the ultra cold limit is achieved when r = r+ = rc.

We start with the above metric (4.34) and make the coordinate transformation,

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Taking δ → 0, with other coordinates held fixed, we obtain

ds2 = eΓ(θ) [−det2+ der2+eα(θ)dθ2] + γ(θ)(dϕ + ek er det)2 (4.38)

where,

eΓ = δΓ, eα = α

δ, ek = δk. (4.39)

We see that the decoupled geometry is a fibered product ofR2× S2.

4.4 Kerr-AdS

5

We consider the metric described in section 3.5. Extremality occurs when ∆r = 0,

is solved for r+ = r− which sends the temperatue, T = 0. We denote by r+, the location of the extremal horizon. It is more intuitive to solve for the mass parameter,

m11 to determine the location of the horizon at extremality,

m = (r 2 ++ a2)2(r+2 + b2)2 2r4 +(2r+2 + a2+ b2) . (4.40)

We can also express the AdS5 radius, l2 by 1 l2 = (ab− r2 +)(ab + r+2) r4 +(2r+2 + a2+ b2) . (4.41)

With these parameters, we then have

r = (r− r+)2ξ +O((r − r+)3), ξ = 4 + 4 l2(3r 2 ++ a 2+ b2). (4.42)

To obtain the near horizon geometry we make the transformations: ˜ r = r+(1 + λr), ˜t = (r+2 + a2)(r+2 + b2) λr3 +ξ t, ˜ ϕ = ϕ + a Ξa r2++ a2t, ψ = ψ +˜ b Ξb r+2 + b2t. (4.43)

Taking the limit, λ→ 0, the metric becomes

ds2 = ρ 2 + ξ ( −r2dt + dr2 r2 ) + ρ 2 +2 ∆θ + ∆θsin 2θ ρ2 + ( 2ar+2 + b2 r+ξ rdt +r 2 ++ a2 Ξa )2 (4.44)

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+∆θcos 2θ ρ2 + ( 2br2 ++ a2 r+ξ rdt +r 2 ++ b2 Ξb )2 +1 + r 2 +/l2 r2 +ρ2+ ( 2abρ2 + r+ξ rdt + b(r 2 ++ a2) sin 2θ Ξa dϕ + a(r 2 ++ b2) cos2θ Ξb )2 where, ρ2+ = r+2 + a2 cos2θ + b2 sin2θ. (4.45) The resulting near horizon geometry has the familiar AdS2 factor arising in the de-coupled spacetime.

5

Thermodynamics from gravitational

perturba-tions

In this section, we shall discuss the thermodynamic response of black holes in their near extreme limit in 5.1. Our focus is on the Kerr solution. We shall then develop how the thermodynamic response is governed by the nAdS2/nCF T1 correspondence. To do so we shall begin with a review in section 5.2 discussing aspects of it for the Reissner-Nordström solution, which is considered to be the universal behaviour of near extremal solutions. In section 5.3, we shall show how using gravitational pertur-bations of NHEK, we are able to obtain the near extremal entropy which corresponds to the universal feature of the solution and yet observe a novel feature exhibited only by the Kerr solution. This section is based on [1, 27, 28, 29].

5.1

Near extremal thermodynamic response

So far, we have seen that a black hole at extremality is characterised by TH = 0,

and admits an AdS2 geometry in its near horizon. From [22], we know that if IE

denotes the on-shell Euclidean action of the gravitational theory, and β, the inverse temperature, then, as GN → 0, we can write the boundary partition function Z(β)

by

Z(β) = exp(−IE). (5.1)

The gravitational free energy is then computed by

F =−1 βlog Z, βF =−IE.

(5.2)

The thermal energy and and thermal entropy of the black hole are then given by

µ =−∂βlog Z(β),

S = (1− β∂β) log Z(β).

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A nearly extreme black hole has small temperature T << M and corresponds to low energy excitations above the extremal state. Let us consider the thermodynamics of the Kerr solution. The near extreme limit as defined in section 4 is given by

r± = M0± +ελ +ε 2λ2 4M0

+O(λ3) (5.4)

where, ε and λ are parameters controlling deviations of the mass from its extremal value, M0, and deviations away from extremality respectively. In this limit, the solution has a temperature,

T = ελ

4πM3 0

+O(λ2). (5.5)

The solution now has an energy given by

E = M − M0 =

ε2λ2

4M0 +O(λ

3). (5.6)

As we noted earlier in section 4, the near extreme limit is taken by increasing temperature and keeping fixed the angular momentum. The increase in entropy is then computed by a small temperature expansion around the ground state entropy keeping fixed J, as follows

S = S0+ T ( ∂S ∂T ) T→0 +· · · (5.7)

The entropy for the Kerr solution is then given by

S = S0+ 2πM0ελ +O(λ2), (5.8)

where,

S0 = 2πM02 (5.9)

is the extremal entropy.

A property of black holes is that, right above their extremal configuration, their response to a small amount of energy causes the scaling symmetry to break explicitly at an energy scale known as the mass gap, denoted by Mgap. It signifies the smallest

energy configuration that the black hole can have above its extremal state. The definition of the mass gap is equivalent to an entropy near extremality which is linear in temperature

S = S0 + 2

Mgap

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For the Kerr solution, we can calculate it to find Mgap= 1 4πM5 0 . (5.11)

The energy temperature relation in terms of the mass gap is then given by

E = T

2

Mgap

. (5.12)

The thermodynamic response of the Kerr solution we describe is a universal property of all near extremal solutions and can be written in general by

S = S0+ 2CT, (5.13)

where, for the Kerr solution, the specific heat C is given by

C = 4πM03. (5.14)

The energy response is given by the equation

E = CT2. (5.15)

5.2

Gravitational dynamics: nAdS

2

A holographic description of gravity in AdS2should naively be described by AdS2/CF T1, where the gravitational theory in two dimensions, is described by a radial and time coordinate. This description suffers certain limitations. The Einstein-Hilbert action in 2D is given by S2 = 1 16πG2 ∫ d2x√−g(R + 2 l2 2 ) (5.16)

where, l2 is the AdS2 radius. Varying the action with the metric we get,

R = −2 l2 2

(5.17)

which tells us that all on-shell solutions are AdS2. At this point we should note that, from variation of the action, we are only left with a constraint. This is the first shortcoming.

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To understand other limitations, let us consider the Reissner-Nordström solution given by ds2 =−(r− r +)(r− r) r2 dt 2+ r2 (r− r+)(r− r−) dr 2+ r2dΩ 22. (5.18) where, r±= Q Lp+ E Lp2±2QE Lp2+ E2 Lp4, (5.19)

Lp is the Planck length and

E = M Q Lp

. (5.20)

denotes the energy above extremality. The extremal limit corresponds to

E = 0, r+ = QLp, TH = 0.

The entropy and temperature for the non extremal solution are given by:

SBH = π r+2 Lp2 , (5.21) TH = r+− r− 4πr+2 . (5.22)

We want to derive an energy temperature relation near extremality. To do so we expand in the excitation energy given by,

E = M− Q/Lp, (5.23)

where E/Q << 1. To leading order in E,

r±≃ QLp± Lp

2EQ +· · · (5.24)

The entropy and temperature become

S ≃ π(Q2+ 2Q2EQ), (5.25) TH 2EQ 2πQ2L p . (5.26) =⇒ E ∼ 2π2Q3TH2Lp. (5.27)

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We see that the energy of a typical quantum of Hawking emission is of order TH.

The mass gap is given by

Egap∼

1

Q3 L

p

. (5.28)

We saw in section 4, given by (4.15) the appearance of AdS2 × S2 in the near horizon of extremal RN. We now wish to understand using relation (5.24) what happens when we try to allow for a finite temperature and excitation energy in this geometry. Firstly, if we want to retain a finite energy configuration with a fixed charge and take the near horizon limit, Lp → 0, TH → ∞. The converse is also

problematic, where we keep a fixed charge and finite temperature, and then we see that Egap→ ∞, and the only allowed energy configuration is E = 0. Let us define

ζ = r− r+ Lp2

, (5.29)

with Q fixed the metric becomes

ds2 Lp2 =−ζ 2 Q2 dt 2+ Q2 ζ2 2+ Q2 dΩ 22. (5.30)

In null coordinates, u± = arctan(t± Qζ2) becomes

ds2 Lp2 = 4Q 2 du+du sin2(u+− u−) + Q 2 dΩ 22, (5.31)

which is the Bertotti-Robinson geometry AdS2 × S2. We see that one of the time like boundaries of AdS2 : u+ = u− is just outside the horizon while the other

u+= u+ π is just inside. The 2D action describing this geometry is given by

S2 = 1 4 ∫ d2x√−g {e−2ϕ [R− F2+ 2(∇ϕ)2] + 2 Lp2 }. (5.32)

4πe−2ϕ is the volume of S2 and F = dA is the gauge field strength. The constraint for T++

− 2e−ϕ

++e−ϕ = T++≥ 0. (5.33)

Integrating, in the conformal gauge, ds2 = e du+du with measure eϕ−2ρ du+ across AdS2 from 0 to π along u− = 0 gives

e−2ρ∂+e−ϕ|u+=0− e−2ρ∂+e−ϕ|u+ =

1 2

du+eϕ−2ρT++≥ 0. (5.34)

e−2ρ vanishes quadratically near the boundaries of AdS2. If T++ > 0 then (5.34) implies that e−ϕ should diverge linearly at least at one of the two boundaries. Thus the geometry cannot be asymptotic to AdS2 × S2 when T++ is nonzero. Thus we

(36)

see from the equations of motion, that any finite energy configuration destroys the

AdS2 geometry.

The AdS2 spacetime is given by

ds2 =−r2dt2+ dr 2

r2 . (5.35)

The isometries of AdS2 are given by

ξ−1 = ∂t, ξ0 = t∂t− r∂r, ξ1 = 1 2( 1 r2 + t 2)∂ t− tr∂r (5.36)

which generates an SL(2,R) algebra. This is also the conformal group for a CFT1. Now, scale invariance requires the energy-momentum tensor of a CFT to be traceless. In 1D, this amounts to having simply, T00 = t = const. The zero energy configura-tions of AdS2 admit symmetries where, the time coordinate of the boundary metric, can be reparametrized by an arbitrary function f (t), while preserving the asymptotic boundary conditions. Explicitly, they are given by the diffeomorphsims12,

t → f(t) + 2f ′′(t)f(t)2 4r2f(t)2− f′′(t)2, r 4r 2f(t)2− f′′(t)2 4rf′(t)3 . (5.37)

The nAdS2/nCF T1 correspondence [27] gives a prescription to observe finite en-ergy configurations where the boundary metric now has time dependence. This is achieved by modifying the gravity action. The motivation to consider the modified action is that, it is a special case of the dilaton gravity actions which arise as a result of the dimensional reduction of the Reissner-Nordström solution.

The Einstein-Hilbert action in 2D is modified by coupling a matter field Φ. This is implemented through the following action which is known as Jackiw-Teitelboim gravity [32, 33] given by I = 1 16πG2 ∫ d2x√−gΦ(R + 2) + 1 8πG2 ∫ dt√−γΦK. (5.38)

The metric (5.35) transforms under (5.37) as

ds2 =−r2 ( 1 + {f(t), t} 2r2 )2 dt2+ dr 2 r2 , (5.39)

12The construction of these diffeomorphisms assume the Fefferman-Graham gauge so that they

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where, {f(t), t} = ( f′′ f′ ) 1 2 ( f′′ f′ )2 . (5.40)

{f(t), t} is the Schwarzian derivative and it is the lowest derivative local expression

that is invariant under SL(2,R) transformations,

f (t)→ af (t) + b

cf (t) + d, ad− bc = 1. (5.41)

Varying the action with the metric results in the equation of motion given by

∇µ∇νΦ− gµν□2Φ + gµνΦ = 0. (5.42)

Additionally variation with Φ gives (5.17). Solving equation (5.42) for AdS2 given by (5.35) we get Φ(t, r) = α1r + α2rt + α3 ( 1 r + rt 2 ) , (5.43)

where α1, α2, α3 are arbitrary constants.

If we, however, consider the solution for the metric (5.16), then we get Φ = σ(t)r + σ ′′(t)− σ{f(t), t} 2r . (5.44) We additionally get, ( 1 f′ ( (f′σ)′ f′ )) = 0. (5.45)

Here, (5.45) relates σ(t) is to the diffeomorphisms that preserve time reparametriza-tions on the boundary metric.

We derived this relation from the equations of motion but we shall now, alternatively try to obtain it from the effective action for f (t) which we in turn, is obtained from (5.38).

We now consider Euclidean AdS2 in Poincare‘ coordinates given by:

ds2 = dt

2+ dz2

z2 (5.46)

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