Black Holes in Infinite Dimensions

Master Thesis

Black Holes in Infinite Dimensions

Pedro Cervantes Correa

January 2018

Abstract

The description of a black hole in the limit of very large number of spacetime dimensions D simplifies considerably. When D approaches infinity, the gravitational field lines of force are infinitely dispersed among the infite number of spacetime directions. This implies that outside the near-horizon region of the black hole the background spacetime will be flat, while the gravitational field will be strongly localized near the horizon. Thus, we can attempt to replace the black hole by a sphere cut off at the horizon in an empty background.

In this project we attempt to obtain the physical conditions that the sphere has to meet in order to be able to reproduce the dynamics of the black hole when embedded in the empty background. This is described in the effective equation that we derive starting from the Einstein equations.

Finally, we apply our results to take a look at ’black droplets’, black holes localized at the boundary of AdS and extending a finite distance into the bulk.

Sammanfattning

Beskrivningen av ett svart h˚ al i gr¨ ansen f¨ or ett mycket stort antal rumtidsdimensioner D f¨ orenklas avsev¨ art. N¨ ar D n¨ armar sig o¨ andligheten ¨ ar gravitationskraftens f¨ altlinjer o¨ andligt spridda bland det o¨ andliga antalet rumtidsriktningar. Detta inneb¨ ar att rumti- den kommer att vara platt utanf¨ or den omedelbara regionen runt det svart h˚ alets ho- risont, medan gravitationsf¨ altet kommer att vara starkt n¨ ara horisonten. Vi kan d¨ arf¨ or ers¨ atta det svarta h˚ alet med en sf¨ ar som str¨ acker sig ut till horisonten i en annars tom bakgrund.

I det h¨ ar arbetet f¨ ors¨ oker vi finna de fysikaliska villkor som sf¨ aren m˚ aste uppfylla f¨ or att korrekt reproducera ett svart h˚ als dynamik n¨ ar det befinner sig i en tom bakgrund.

Detta beskrivs i den effektiva ekvationen som vi h¨ arleder fr˚ an Einsteins ekvationer.

Slutligen till¨ ampar vi v˚ ara resultat p˚ a s˚ a kallade svarta droppar. Svarta droppar ¨ ar

svarta h˚ al som ¨ ar lokaliserade till AdS-gr¨ ansen och str¨ acker sig ett begr¨ ansat avst˚ and in

i volymen.

Contents

1 Introduction 1

1.1 Geometry at large D . . . . 2

1.2 Absence of interactions . . . . 3

1.3 Objectives . . . . 4

2 Theory Foundations 5 2.1 (D − 1) + 1 Formalism . . . . 5

2.1.1 Foliation of spacetime . . . . 5

2.1.2 Tensor decomposition . . . . 7

2.1.3 Extrinsic curvature . . . . 7

2.1.4 Decomposition of stress-energy tensor . . . . 8

2.1.5 Decomposition of the Riemann tensor/Gauss-Codazzi and Ricci equations . . . . 8

2.1.6 Decomposition of the Einstein equations . . . . 9

2.1.7 Summary . . . 11

3 Effective Theory 12 3.1 Metric Ansatz . . . 12

3.2 Solution of the Einstein equations at leading order . . . 13

3.2.1 Integration of the extrinsic curvature tensor components . . . 14

3.2.2 Integration of the metric components . . . 16

3.2.3 Imposition of the vector constraint equation . . . 16

3.2.4 Surface gravity on the horizon . . . 18

3.2.5 Metric and extrinsic curvature matching . . . 19

4 Black droplets 20 4.1 Original construction . . . 21

4.2 Alternative construction . . . 23

4.3 Observations . . . 24

5 Conclusion 26

Bibliography 28

i

Chapter 1

Introduction

In the large D limit of General Relativity the dynamics of black holes simplify dramat- ically. Due to the nature of the black hole geometry as D → ∞, their gravitational potential vanishes exponentially fast outside the horizon r = r ₀ . Since no gravitational field exists in the exterior of the horizon, they behave as non-interacting particles beyond length scales of order r 0 . Therefore, we can treat them as a spheres cut off at the radius r = r ₀ living in a flat spacetime background.

In this document we will concern ourselves with how can we identify a surface Σ B with a black hole in a given background spacetime. If black holes are solutions to Einstein’s equations, it should be possible to derive from these which conditions does Σ B need to satisfy when embedded in a specific background to be able to be replaced by a certain black hole geometry. Starting with a metric ansatz, we solve the Einstein equations at leading order in 1/D to obtain an effective equation in terms of the metric functions.

This equation encodes the requirements that the embedding of the surface Σ _B in a certain background spacetime needs to fulfill. To conclude, we will use our derivation to replicate a known solution in Anti-de Sitter space, black droplets.

This first chapter will be dedicated to introduce the basic concepts and peculiarities of the geometry of black holes when D → ∞.

1

Chapter 1. Introduction 2

1.1 Geometry at large D

The generalization of the Schwarzschild metric to an arbitrary number of dimensions D is given by

ds ² = −

 1 −

 r 0

r

 D−3  dt ² +

 1 −

 r 0

r

 D−3  −1

dr ² + r ² dΩ D−2 , (1.1) known as the Schwarzschild-Tangherlini metric with a horizon of radius r ₀ . The ex- pression remains very familiar and by setting D = 4 we readily recover the standard Schwarzschild metric. We can see that the number of additional dimensions are all collected in the (D − 2)-dimensional sphere.

The main feature of black holes at large D that we utilize in this document, the lack of gravitational interaction outside the horizon, stems from the geometry of the D- dimensional hypersphere as D → ∞. The area of a unit sphere in D − 2 dimensions is given by the expression

S _D−2 = 2π ^(D−1)/2

Γ( ^D−1 ₂ ) . (1.2)

Using Stirling’s formula for Gamma functions

Γ(x + 1) ' x ^x+1/2 e ^−x √

2π, (1.3)

we obtain the following approximation for the surface area, which vanishes when D  1

S D−2 ' D

√ 2π

 2πe D

 D/2

→ 0. (1.4)

For the remainder of this work, we will be mainly interested in two different length scales, the horizon radius itself r 0 and

` κ = κ ⁻¹ ∼ r 0

D , (1.5)

which is a length scale that arises in connection with the surface gravity κ. Section 3.2.4

explains in detail how the relation is found.

Chapter 1. Introduction 3

1.2 Absence of interactions

As we move outside the horizon r > r 0 the gravitational potential (r 0 /r) ^D−3 vanishes exponentially in D. However there is a small region, on a scale of r 0 /D where the gravitational potential is still appreciable, denominated as the sphere of influence or near-horizon region. This is the region in which the surface Σ B , that will resolve the black hole geometry, is attached to the background spacetime, hence its relevance.

Once we depart from the near-horizon region, as we move radially outwards into infinite- dimensional space, the gravitational field becomes infinitely diffused across the infinite possible directions, effectively rendering the background spacetime flat. Therefore, the black hole metric at large D is described by a flat spacetime with a hypersphere cut off at r = r ₀ . Because of the flat geometry, no other bodies can gravitationally interact with the black hole beyond r = r 0 . In Figure 1.1 we can see a graphical description of how the gravitational field becomes increasingly concentrated near the horizon region as one increases the number of dimensions.

Figure 1.1. Graphical comparison of the length scale associated with the extension

of a black hole gravitational field. From [1].

Chapter 1. Introduction 4

1.3 Objectives

As we have seen, in D = 4 there is not a clear separation between the black hole and the background, since the curvature of the background is characterized by the same length scale as the near-horizon region. As D grows towards infinity, the spacetime around the black hole becomes increasingly flatter. Therefore, in the limit of D → ∞ there is a clear-cut separation between the black hole (and its field) and the background.

This separation is permitted by the appearance of the new scale r 0 /D; since D is a dimensionless parameter, we can generate many different length scales by combining it in different ways with the natural length scale r ₀ , establishing a hierarchy of scales. In our case, we restrict ourselves to the two scales

r 0

D  r ₀ , (1.6)

since the physics that we intend to capture manifests itself at such scales.

These unique features of the large D limit are what permits us to effectively replace the black hole with a surface in an empty background. Obviously, not any arbitrary surface that we can draw on the background will be a black hole. Since the properties of a black hole stem from the Einstein equations, it should be possible to derive an alternative set of equations that the surface has to satisfy in order to adequately resolve the black hole.

These equations embody the effective theory of black holes in the large D limit.

Ultimately, the equations for the embedding of the surface in the background spacetime turn out to be quite simple

√ −g _tt K = 2κ, (1.7)

where g _tt is the redshift factor on the surface, K is the trace of the extrinsic curvature of the surface and the constant κ is the surface gravity of the black hole. The results obtained from this relation provide the near-horizon metric of the black hole that solves the Einstein equations at leading order in the large D limit.

The main objective of this document is to guide the reader through the necessary steps

to understand the concepts on which the effective theory is build upon, as well as the

significance of the effective theory itself and its possible applications.

Chapter 2

Theory Foundations

The large D effective theory of Black Holes is set up on a reformulation of the Einstein equations as a dynamical evolution or Cauchy problem. The gradients in the direction perpendicular to the Black Hole horizon are expected to be of order D, while the gra- dients in all the remaining directions are expected to be very small in comparison. For the purpose of capturing the large gradients, the spacetime in decomposed in such a way that our preferred direction becomes separated from the rest, allowing us to rewrite the Einstein equations in a more suitable form for our problem. In this chapter we will show how do the basic Einstein equations need to be manipulated to derive the formalism that needs to be adopted to develop the effective theory of black holes.

2.1 (D − 1) + 1 Formalism

2.1.1 Foliation of spacetime

Given the D-dimensional smooth manifold M describing a spacetime with metric γ µν

we foliate it into the (D − 1)-dimensional set of spatial hypersurfaces Σ t with metric g _µν and run it along the direction t, orthogonal to all other D − 1 dimensions. The two metrics are related by:

γ _µν = g _µν + n _µ n _ν . (2.1)

The spacetime M can be described in terms of the induced metric γ _µν as:

ds ² = g _µν dx ^µ dx ^ν = −(α ² − β _i β ⁱ )dt ² + 2γ _ij β ⁱ dtdx ^j + γ _ij dx ⁱ dx ^j . (2.2)

The Latin indices i, j... run from 1 to D − 1. The letter α is the lapse function, which measures the proper time between two adjacent hypersurfaces Σ t and Σ _t+δt . The shift

5

Chapter 2. Effective Theory 6

Figure 2.1. Foliation of the spacetime M by the set of hypersurfaces Σ t . From [2].

vector β _i indicates the displacement of a point p ∈ Σ _t w.r.t to the point p ⁰ ∈ Σ _t+δt generated at Σ _t+δt by the normal vector starting at p ∈ Σ t , as shown in Figure 2.2.

We define the normal vector as:

n _µ = −α(1, 0, ..., 0), n ^µ = 1/α(1, −β ^µ ), (2.3) such that

n _µ n ^µ = −1. (2.4)

Figure 2.2. Representation of the lapse functionα and shift vector β ⁱ for two adjacent

hypersurfaces Σ 1 and Σ 2 . From [3].

Chapter 2. Effective Theory 7

2.1.2 Tensor decomposition

We can decompose any tensor in a spatial part and a timelike part. The spatial part can be obtained by making use of the spatial projection operator γ ^µ ν , which is derived from relation (2.1). If we act on the right-hand side of expression (2.1) with the metric:

g ^µα γ _αν = γ ^µ _ν = g ^µ _ν + n ^µ n _ν = δ ^µ _ν + n ^µ n _ν . (2.5) The timelike projection operation is given by:

N ^µ ν = −n ^µ n ν . (2.6)

We can now define the covariant derivative w.r.t. the spatial metric γ _µν for any tensor field tangent to Σ t . The spatial covariant derivative D ρ of a (p, q)-tensor T ^α

^...α

_β

_...β

∈ Σ _t can be expressed in terms of the covariant derivative ∇ _σ related to the spacetime metric g µν as:

D ρ T ^α

^...α

β

...β

= γ ^α

µ

...γ ^α

µ

γ ^ν

β

...γ ^ν

β

γ ^σ ρ ∇ _σ T ^µ

^...µ

ν

...ν

. (2.7)

2.1.3 Extrinsic curvature

The extrinsic curvature is a measure of the ”bending” of Σ t in M. It evaluates how the normal vector n ^µ changes as one moves on Σ _t . It can be formally defined as the projection of the covariant derivate of the normal vector:

K _αβ = −γ ^µ _α γ ^ν _β ∇ _µ n _ν . (2.8) Alternatively, one can define the extrinsic curvature in terms of the acceleration experi- enced by an observer traveling along the normal vector:

K _αβ = −γ ^µ _α γ ^ν _β ∇ _µ n _ν = −(δ ^µ _α + n ^µ n _α )(δ ^ν _β + n ^ν n _β )∇ _µ n _ν

= −(δ ^µ _α + n ^µ n _α )∇ _µ n _β = −∇ _α n _β − n _α n ^µ ∇ _µ n _β

= −∇ α n _β − n _α a _β ,

(2.9)

where a _β = n ^µ ∇ _µ n _β is the aforementioned acceleration and we used the fact that

n ^ν ∇ _µ n ν = 0.

Chapter 2. Effective Theory 8

Finally, we can also relate the extrinsic curvature to the Lie derivative of the spatial metric along the normal vector as:

L _n γ αβ = n ^µ ∇ _µ γ αβ + γ µβ ∇ _α n ^µ + γ αµ ∇ _β n ^µ = n ^µ (∇ µ n α )n β + n α n ^µ (∇ µ n β ) + ∇ α n β + ∇ β n α

= a α n β + n α a β + 2∇ α n β = −2K αβ ,

(2.10)

K _αβ = − 1

2 L _n γ _αβ . (2.11)

A more convenient but equivalent form is given by taking the Lie derivate w.r.t the normal evolution vector n ^µ = N m ^µ .

K _αβ = − 1

2N L _m γ _αβ = − 1

2N (∂ _t − L _β )γ _αβ (2.12)

2.1.4 Decomposition of stress-energy tensor

By making use of the two projection operators previously defined, one can decompose any (p, q)-tensor T ^α

^...α

β

...β

∈ M into a timelike, spacelike and mixed component.

The three projections of the stress-energy tensor are the following:

E = n ^µ n ^ν T µν , j α = −n ^µ γ ^ν α T µν , S _αβ = γ ^µ α γ ^ν _β T µν . (2.13)

where E is the timelike component or the energy density, j _α is the mixed component or momentum density and S αβ is the spacelike component or stress tensor.

2.1.5 Decomposition of the Riemann tensor/Gauss-Codazzi and Ricci equations

The Gauss-Codazzi relations and the Ricci equation will be very useful tools to proceed forward in the derivation of the decomposed Einstein equations. These relations are the result of taking different projections of the D-dimensional spacetime Riemann tensor, breaking it down in terms of the (D − 1)-dimensional Riemann tensor and the extrinsic curvature tensor. For an in-depth derivation of the following relations see [2].

The Gauss relation is given by

γ ^µ α γ ^ν _β γ ^γ ρ γ ^{σ (D)} _δ R ^ρ σµν = ^(D−1) R ^γ _δαβ + K ^γ α K _δβ − K ^γ _β K _αδ . (2.14)

Chapter 2. Effective Theory 9

The contracted version of the Gauss relation reads

γ ^µ α γ ^{ν (D)} _β R µν + γ αµ n ^µ γ ^ρ β n ^{σ (D)} R ^µ νρσ = ^(D−1) R αβ + KK αβ − K _αµ K ^µ β . (2.15) The scalar Gauss relation is

(D) R + 2 ^(D) R _µν n ^µ n ^ν = ^(D−1) R + K ² − K _αβ K ^αβ . (2.16)

The Codazzi relation and its contracted version are respectively

γ ^γ ρ n ^σ γ ^µ α γ ^{ν (D)} _β R ^ρ σµν = D β K ^γ α − D _α K ^γ β , (2.17)

γ ^µ α n ^{ν (D)} R µν = D α K − D µ K ^µ α . (2.18) The Ricci equation is

γ _αµ n ^ρ γ ^ν _β n ^{σ (D)} R ^µ _ρνσ = 1

N L _m K _αβ + 1

N D _α D _β N + K _αµ K ^µ _β . (2.19) Finally, combining the Ricci equation and the Gauss relation we obtain

γ ^µ α γ ^{ν (D)} _β R µν = − 1

N L _m K _αβ − 1

N D α D _β N + ^(D−1) R _αβ + KK _αβ − 2K _αµ K ^µ _β , (2.20) where N is the lapse function and the Lie derivative is taken along the normal evolution vector m ^µ defined by:

m ^µ = N n ^µ . (2.21)

2.1.6 Decomposition of the Einstein equations

For a spacetime M with metric g µν the Einstein equation with cosmological constant is given by:

R µν − 1

2 Rg µν + Λg µν = 8πGT µν . (2.22) Taking the trace with respect to the metric of both sides:

R − D

2 R + DΛ = 8πGT. (2.23)

Dividing this expression by ( ^D ₂ − 1) we get:

− R + DΛ

D

2 − 1 = 8πG T

D

2 − 1 . (2.24)

Chapter 2. Effective Theory 10

Multiplying by − ¹ ₂ g µν and adding the result to equation (2.12) we obtain the ”trace- reversed” form of the Einstein equation:

R µν − Λg µν D

2 − 1 = 8πG(T µν − 1

D − 2 T g µν ). (2.25)

We are now ready to derive the three projections of the Einstein equation:

• Projection onto Σ _t :

γ ^µ α γ ^ν _β R µν − Λγ ^µ α γ ^ν β g µν D

2 − 1 = 8πG(γ ^µ α γ ^ν _β T µν − 1

D − 2 T γ ^µ α γ ^ν _β g µν ), (2.26) γ ^µ α γ ^ν β R µν is given by equation (2.18). Then we get

L m K _αβ = −D α D _β N + N



R _αβ + KK _αβ − 2K αµ K ^µ _β + ^Λγ

2

−1 + 8πG(S _αβ − _D−2 ¹ T γ _αβ )

 . (2.27)

• Projection perpendicular to Σ _t : R µν n ^µ n ^ν − 1

2 Rg µν n ^µ n ^ν + Λg µν n ^µ n ^ν = 8πG(T µν n ^µ n ^ν ). (2.28) Since g µν n ^µ n ^ν = −1 and T µν n ^µ n ^ν = E we are left with

R µν n ^µ n ^ν + 1

2 R − Λ = 8πGE. (2.29)

Using the scalar Gauss relation (2.15) we get

R + K ² − K _αβ K ^αβ = 16πGE + 2Λ. (2.30)

• Mixed projection

R µν n ^µ γ ^ν α − 1

2 Rg µν n ^µ γ ^ν α + Λg µν n ^µ γ ^ν α = 8πG(T µν n ^µ γ ^ν α ). (2.31) Since g µν n ^µ γ ^ν α = 0 and j α = −n ^µ γ ^ν α T µν

R _µν n ^µ γ ^ν _β = −8πGj _α . (2.32)

The expression for R _µν n ^µ γ ^ν _β is given by the contracted Codazzi relation (2.17), which yields

D µ K ^µ α − D _α K = 8πGj α . (2.33)

Chapter 2. Effective Theory 11

2.1.7 Summary

We collect the three projections we obtained and re-express the Einstein equation (2.22) with the system:

R + K ² − K _αβ K ^αβ = 16πGE + 2Λ, (2.34)

D _µ K ^µ _α − D _α K = 8πGj _α , (2.35)

L m K αβ = −D α D β N + N



R αβ + KK αβ − 2K αµ K ^µ β + ^Λγ

2

−1 + 8πG(S αβ − _D−2 ¹ T γ αβ )



, (2.36)

K _αβ = − 1

2N (∂ t − L _β )γ _αβ . (2.37)

Note that equations (2.34) (2.35) and (2.36) contain only spatial information: (2.34) is

a scalar equation, (2.35) and (2.36) are tensorial equations where every term is tangent

to Σ t . To have a complete description of the spacetime, we need information on how

the induced metric and the extrinsic curvature evolve as we move between leaves of the

foliation in t. This data is complemented by equation (2.37).

Chapter 3

Effective Theory

In this chapter we solve the Einstein equations at leading order in 1/D and derive the effective equation that Σ _B needs to satisfy. For consistency purposes, we will now switch all the notation used in the previous chapter to match the one used in [4].

3.1 Metric Ansatz

The metric ansatz proposed by [4] is

ds ² = N ² (ρ, z) dρ ²

(D − 1) ² − V ² (ρ, z)dt ² + g ab (ρ, z)dz ^a dz ^b + R ² (ρ, z)q ij dx ⁱ dx ^j . (3.1) Note that now ρ is the radial direction which is perpendicular to all other (D − 1) dimensions. The remaining dimensions are divided into a time dimension t and a number p of spatial directions z ^a which are perpendicular to a S ⁿ⁺¹ sphere. With

n = D − p − 3. (3.2)

This parameter will be the one used for the large expansion instead of D. N (ρ, z), V(ρ, z), g _ab (ρ, z) and R(ρ, z) are functions that scale with n in some fashion. In order to establish sensible assumptions of how these functions scale with n, we require our metric ansatz for p = 1 to be able to reproduce the leading-order near-horizon geometry of the Schwarzschild black hole as

ds ² = r ₀ ² dρ ²

n ² − r ² ₀ tanh ² (ρ/2)dt ² + r ² ₀

 1 + 4

n ln cosh(ρ/2)



(dz ² + sin z ² dΩ _n+1 ). (3.3)

Consequently, we assume that N , V and R are O(1) in n and g ab and ∂ a R are either O(1) or O(1/n), but because of the large dimensionality are still relevant. We can also

12

Chapter 3. Effective Theory 13

see that at leading order g ab and R only depend on z, therefore we can assume

K ^a b , K ⁱ j ∼ O(1), (3.4)

whereas V at leading order does have ρ-dependence, thus

K ^t _t , K ∼ O(n). (3.5)

3.2 Solution of the Einstein equations at leading order

We are interested in the Einstein equations in vacuum (T µν = 0) with cosmological constant:

Λ = − (D − 1)(D − 2)

2l ² , (3.6)

where l is some parameter with dimension of length. The Einstein equations obtained in the previous chapter become

R + K ² − K _αβ K ^αβ = (D − 1)(D − 2)

l ² , (3.7)

∇ _µ K ^µ _α − ∇ _α K = 0, (3.8)

L _m K _αβ = −∇ _α ∇ _β N + N (R _αβ + KK _αβ − 2K _αµ K ^µ _β + D − 1

l ² g _αβ ), (3.9) K _αβ = − 1

2N (∂ _t − L _β )g _αβ . (3.10)

We can see from the metric ansatz (3.1) that the lapse function is now N new = _D−1 ^N

and the shift vector β = 0. In addition, the foliation is now taken along the radial direction ρ and the induced metric is denoted by g αβ . Thus, the system becomes

K ² − K ^µ _ν K ^ν µ = R + (D − 1)(D − 2)

l ² , (3.11)

∇ _ν K ^ν µ − ∇ _µ K = 0, (3.12)

D − 1

N ∂ρK ^µ ν + KK ^µ ν = R ^µ ν + δ ^µ ν

D − 1 l ² − 1

N ∇ ^µ ∇ _ν N , (3.13) K ^µ _ν = D − 1

2N g ^µσ ∂ _ρ g _σν . (3.14)

Note that we multiplied equations (3.6) and (3.7) by the inverse metric g ^αµ and then

relabeled the necessary indices to match the notation of [4].

Chapter 3. Effective Theory 14

3.2.1 Integration of the extrinsic curvature tensor components

We can now separate the leading-order, ρ-independent terms as follows

N (ρ, z) = N 0 (z) + O(1/n), (3.15)

g ab (ρ, z) = γ ab (z) + O(1/n), (3.16)

R(ρ, z) = R(z) + O(1/n), (3.17)

K(ρ, z) = 1

r ₀ (z) + O(1/n), (3.18)

where K(ρ, z) is given by:

K ² (ρ, z) = 1

l ² + 1

(D − 1) ² (R − 1

N ∇ ² N ), (3.19)

we can use this definition to rewrite the trace of equation 2.10 as D − 1

N ∂ρK + K ² = (D − 1) ² K ² (ρ, z) = n ²

r 0 (z) ² , (3.20) we then perform a radial integration to obtain

K = n

r ₀ (z) coth  N 0 (z)

r ₀ (z) (ρ − ρ 0 (z)



. (3.21)

The pole at ρ = ρ 0 (z) is consistent with the requirement of having infinite extrinsic curvature at the horizon. We can choose ρ ₀ (z) = 0 since the metric is invariant under ρ → ρ + f (z) and we can rescale ρ by a function of z such that

N ₀ (z) = r ₀ (z). (3.22)

To obtain

K = n

r 0 (z) coth ρ. (3.23)

Similarly, the equation for K ^t _t is given by n

r ₀ (z) ∂ρK ^t _t + KK ^t _t = R ^t _t + D − 1

l ² , (3.24)

we can neglect the right hand side since both terms are of lower order. Then, making use of (3.20) we have

∂ρK ^t _t + coth ρK ^t _t = 0, (3.25)

whose solution is

K ^t t = n

r ₀ (z) sinh ρ , (3.26)

Chapter 3. Effective Theory 15

where we used the boundary condition K ^t t → ∞ at ρ = 0 to set the integration function ρ ₀ (z) = 0.

Since K ^a b and K ⁱ j are O(1), we can’t neglect the terms R ^µ ν + _l ⁿ

in (3.13). For a constant-ρ section

g µν dx ^µ dx ^ν = −V ² (ρ, z)dt ² + γ _ab (z)dz ^a dz ^b + R ² (z)q ij dx ⁱ dx ^j , (3.27) the scalar curvature at leading order is given by

R = n ² 1

R ² 1 − (DR) ²
, (3.28)

and the components of the Ricci tensor at leading order are given by R ^a _b = −n D ^a D _b R

R + ^(γ) R ^a _b , (3.29)

R ⁱ j = δ ⁱ j

 n

R ² (1 − (DR) ²



= δ ⁱ j

R

n = nδ ⁱ j

 1 r ₀ ² − 1

l ²



, (3.30)

where for the last equality we used relation (3.19), ^(γ) R ^a _b is the Ricci tensor for the metric γ ab and we used the abbreviation

(DR) ² = γ ^ab ∂ _a R∂ _b R. (3.31)

Finally we can write the differential equations for K ^a _b and K ⁱ _j as n

r ₀ ∂ρK ^a _b + KK ^a _b = R ^a _b + n

l ² δ ^a _b , (3.32)

n r 0

∂ρK ⁱ j + KK ⁱ j = R ⁱ j + n

l ² δ ⁱ j , (3.33)

which are solved respectively by

K ^a _b = r ₀ f ^a _b tanh(ρ/2), (3.34) K ⁱ _j = δ ⁱ _j 1

r ₀ tanh(ρ/2), (3.35)

where the tensor f _ab is defined as

f _ab (z) = γ _ab (z)

l ² − D _a D _b R

R +

(γ) R _ab

n . (3.36)

Chapter 3. Effective Theory 16

3.2.2 Integration of the metric components

Now that we have all the components of K ^µ _ν we can integrate them using equation (3.14) to obtain the metric components

K ^t t : n

r 0 sinh(ρ) = n 2r 0

1

V ² (ρ, z) ∂ ρ V ² (ρ, z), (3.37) K ^a _b : r 0 f ^a _b tanh(ρ/2) = n

2r ₀ g ^ac (ρ, z)∂ ρ g _cb (ρ, z), (3.38) K ⁱ _j : 1

r 0

tanh(ρ/2) = n 2r 0

1

R ² (ρ, z) ∂ _ρ R ² (ρ, z). (3.39) Integrating radially each side and rearranging the integration functions of z, we can write the resulting metric components as

V(ρ, z) = V ₀ (z) tanh(ρ/2), (3.40)

g _ab (ρ, z) = γ _ab (z) + 4

n r ² ₀ (z)f _ab (z) ln cosh(ρ/2), (3.41) R(ρ, z) = R(z)

 1 + 2

n ln cosh(ρ/2)



. (3.42)

3.2.3 Imposition of the vector constraint equation

All that is left now is to make use of equation (3.12). The only non-vanishing component is given by

∇ _ν K ^ν _a − ∇ _a K = ∂ _ν K ^ν _a + Γ ^ν _νd K ^d _a − Γ ^d _aν K ^ν _d

= ∂ _b K ^b _a + Γ ^ν _νb K ^b _a − Γ ^a _ba K ^b _a − Γ ^t _at K ^t _t − Γ ⁱ _aj K ^j _i − ∂ _a K = 0.

(3.43) We can see immediately from (3.38) that the first and third terms in the last equality will be O(1). Using the relation

Γ ^α _αβ = 1

2 g ^αλ ∂ _β g _αλ = 1

2 det g ∂ _β g = ∂ _β ln p

det g, (3.44)

we can write equation (3.43) as

∂ _b ln q

| det g _µν |K ^b _a − ∂ _a ln p| det g tt |K ^t _t − ∂ _a ln q

| det g _ij |K ^j _i − ∂ _a K = 0, (3.45)

Chapter 3. Effective Theory 17

at leading order, this simplifies to

n(∂ _b ln R)r ₀ f ^a _b tanh(ρ/2)−(∂ _a ln V ₀ ) n

r ₀ sinh(ρ) −n(∂ _a ln R) tanh(ρ/2) r ₀ + n

r ² ₀ ∂ _a r ₀ coth ρ = 0, (3.46) multiplying by (sinh(ρ/2) cosh(ρ/2)) and dividing by n gives

(∂ _b ln R)r 0 f ^a _b sinh ² (ρ/2)−(∂ a ln V 0 ) 1

2r ₀ −(∂ _a ln R) sinh ² (ρ/2) r ₀ + 1

2r ² ₀ ∂ a r 0 (1+2 sinh ² (ρ/2)) = 0, (3.47) Before moving forward, note that we can relate R and r ₀ by combining relation (3.19) and (3.28) as follows

1 r ² ₀ = 1

l ² + 1

R ² (1 − (DR) ² ) → 1 = r ₀ ²  1 l ² + 1

R ² (1 − (DR) ² )



. (3.48)

Taking a logarithm then gives

− ln r ² ₀ = ln R ² + l ² (1 − (DR) ² )
− ln(l ² R ² ), (3.49) and differentiating w.r.t to z ^a leads to

2∂ a ln R − 2∂ a ln r 0 = ∂ a R ² + l ² (1 − (DR) ² )

R ² + l ² (1 − (DR) ² ) , (3.50) we then multiply both sides by 1/r ₀ ² and use (3.48) to find

2

r ₀ ² (∂ a ln R − ∂ a ln r 0 ) = ∂ a R ² + l ² (1 − (DR) ² )

l ² R ² , (3.51)

expanding the right hand side derivative we get 2

r ₀ ² (∂ a ln R − ∂ a ln r 0 ) = 2R∂ _a R − l ² ∂ _c γ ^ab ∂ _a R∂ _b R

l ² R ² = 2∂ _a ln R

l ² − 2∂ _b ln R(∂ ^b R∂ _a R)

R ,

(3.52) and then cleaning up the expression we obtain

1

r ² ₀ (∂ _a ln R − ∂ _a ln r ₀ ) = (∂ _b ln R)  δ ^b _a

l ² − ∂ ^b R∂ _a R R



. (3.53)

Chapter 3. Effective Theory 18

Now we can go back to (3.47) and make the substitution 1

r ₀ (∂ _a ln R − ∂ _a ln r ₀ ) sinh ² (ρ/2) + r 0

n (∂ _b ln R) ^(γ) R _ab sinh ² (ρ/2) − (∂ _a ln V ₀ ) 1 2r ₀

− (∂ _a ln R) sinh ² (ρ/2)

r ₀ + 1

2r ² ₀ ∂ _a r ₀ (1 + 2 sinh ² (ρ/2)) = 0.

(3.54) The first two terms cancel out with the fifth and seventh terms respectively and the third term is subleading in n, thus the radial dependence is canceled out and we are simply left with

∂ a ln V 0 (z) = ∂ a ln r 0 (z). (3.55) To understand the meaning of this relation we need to make a small detour.

3.2.4 Surface gravity on the horizon

The surface gravity for a Killing vector ξ _µ whose null surface is at the horizon is given by

κ ² = − 1

2 (∇ _µ ξ _ν )(∇ ^µ ξ ^ν ), (3.56) where the expression is evaluated at the horizon. Since our metric is diagonal and time independent, we can readily obtain a Killing vector as

ξ ^µ = δ ^µ _t , ξ _µ = δ ^µ _t g _tt . (3.57) The only non-vanishing combinations that arise from this Killing vector and our metric are

∇ _ρ ξ t = ∂ ρ ξ t − Γ ^t _ρt ξ t = 1 2

∂g tt

∂x ^ρ ,

∇ _t ξ _ρ = ∂ _t ξ _ρ − Γ ^t _tρ ξ _t = − 1 2

∂g _tt

∂x ^ρ ,

∇ ^ρ ξ ^t = g ^ρρ (∂ _ρ ξ ^t + Γ ^t _ρt ξ _t ) = 1

2 g ^ρρ g ^tt ∂g _tt

∂x ^ρ ,

∇ ^t ξ ^ρ = g ^tt (∂ t ξ ^ρ + Γ ^ρ tt ξ t ) = − 1

2 g ^tt g ^ρρ ∂g tt

∂x ^ρ .

(3.58)

Plugging these results in (3.56) we obtain

κ ² = − 1

4 g ^ρρ g ^tt  ∂g _tt

∂x ^ρ

 2      ρ=0

= n ² 4r ₀ ² (z)

V ₀ ² (z) cosh ⁴ (ρ/2)

     ρ=0

, (3.59)

κ = nV ₀ (z)

2r 0 (z) . (3.60)

Chapter 3. Effective Theory 19

We can see that we arrive at an equivalent relation to (3.55) with some additional information. Using (3.60) we can conclude that the vector constraint equation (3.55) is simply requiring the surface gravity κ to be constant along the horizon.

3.2.5 Metric and extrinsic curvature matching

Using our previous results for every component we can write the full metric as

ds ² = r ₀ ² (z)



−4˜ κ ² tanh ² (ρ/2)dt ² + dρ ² n ²

 +



γ _ab (z) + 4

n r ² ₀ (z)f _ab (z) ln cosh(ρ/2)

 dz ^a dz ^b + R ² (z)

 1 + 4

n ln cosh(ρ/2)



dΩ _n+1 .

(3.61) To be able to resolve the black hole by a specific surface Σ _B in a given background, the metric and the extrinsic curvature need to be the same when approach from either direction. This ensures that the two zones are smoothly ensembled. The overlap between the two zones takes place in the region 1  ρ  n. Therefore, for a constant value of ρ in that range the induced metric on the surface Σ B has to be

ds ² = −V ₀ ² (z)dt ² + γ ab (z)dz ^a dz ^b + R ² (z)dΩ n+1 . (3.62) The extrinsic curvature evaluated at Σ _B is

K    Σ

= n

r ₀ (z) , (3.63)

we can relate it with the surface gravity by using (3.60), combining them to give

√ −g _tt K    Σ

= 2κ. (3.64)

Using (3.48) we can write this equation as V ₀ ² (z)

R ² (z)



1 − (DR) ² + R ² (z) l ²



= 4 κ ²

n ² = 4˜ κ ² , (3.65)

which is the effective equation that we were seeking. We obtained a differential equation

in z which only depends on the functions V ₀ (z) and R(z). These functions are specified

by the embedding into the background. For instance, in Minkowski space we have

V 0 (z) = 1, so R(z) is the only function of z left that needs to satisfy (3.65).

Chapter 4

Black droplets

One of the many immediate interesting applications of the study of black holes in higher dimensions is the possibility of constructing black droplets. Black droplets arise in the context of the AdS/CFT correspondence as one of the possible descriptions of the bulk gravity dual of an AdS boundary black hole. The authors of [8] make use of the correspondence to portray a heuristic picture of Hawking radiation for strongly coupled large N field theories. By examining the size of the black hole R and the Hawking temperature T _H in relation to other various scales of the field theory, they argue that two different behaviors may arise.

At a given T _H , large asymptotically flat field theory black holes couple strongly to the deconfined plasma. In this context, deviations from the Hartle-Hawking state that decay to equilibrium on a timescale set by the black hole size are referred to as strong coupling, while weak coupling refers to deviations that fall off much more slowly. In the case of strongly coupled large field theory black holes, the corresponding bulk duals of the Hartle-Hawking states will be described by a spacetime with a single connected horizon, denominated black funnel. On the other hand, Hartle-Hawking states of smaller asymptotically flat boundary black holes are dual to AdS bulk spacetimes containing two disconnected horizons. One horizon is described by a planar black hole geometry in the bulk, while the other “hangs”from the AdS boundary dipping into the bulk, thus the black droplet name[8].

In this chapter, we will show how black droplets were originally constructed, embedding the black droplet horizon into hyperbolic space. Then, we will use the equations derived in Chapter 3 to show an alternative way of generating black droplet horizons.

20

Chapter 4. Black droplets 21

4.1 Original construction

The AdS/CFT correspondence conjectures the equivalence between a theory of quantum gravity in Anti-de Sitter spacetime to a quantum field theory living in the boundary of that same spacetime. By choosing a fixed, non-dynamical background spacetime for the quantum field theory, we impose a conformal boundary condition on the higher- dimensional bulk AdS spacetime [8].

Consider a strongly coupled large N field theory living in a D −1 dimensional spacetime.

Given that the boundary spacetime is fixed and gravity is non-dynamical, one is free to choose any arbitrary black hole geometry which does not need to satisfy any equations of motion. This means that the size of the black hole R and the Hawking temperature T _H can be considered to be completely independent quantities. These arbitrarily chosen black holes are referred to as boundary or field theory black holes, to differentiate from their bulk AdS dual counterparts, namely bulk black holes.

Let us consider a spacetime with a single droplet horizon given by the metric[9]

ds ² = ` ² 1 − ξ ²



− ρ ²

R ² ₀ f _ρ (ρ)dt ² + dρ ²

ρ ² + dξ ²

1 − ξ ² + ξ ² dΩ ₂



, (4.1)

with

f _ρ (ρ) = (1 − ρ) ²

(1 + ρ) ⁶ , (4.2)

where the droplet horizon is located at ρ = 1.

We can visualize the horizon by embedding it in hyperbolic space. Euclidean hyperbolic space is given by the following line element

ds ² _H = ` ²

z ² dz ² + dr ² + r ² dΩ D−3  . (4.3)

By demanding that the pullback of hyperbolic space to a curve γ(x) = (z(x), r(x)) matches the pullback of the metric (4.1) to the droplet horizon ρ = 1, we obtain a system of two equations in z(x) and r(x). The pullback of (4.1) to γ(x) is

ds ² _γ,H = ` ² z(x) ²

"

 dz dx

 2

+  dr dx

 2 !

dx ² + r(x) ² dΩ _D−3

#

. (4.4)

On the other hand, the pullback of (4.1) to ρ = 1 is

ds ² = ` ² 1 − ξ ²

 dξ ²

1 − ξ ² + ξ ² dΩ ₂



. (4.5)

Chapter 4. Black droplets 22

Identifying x with ξ, and equating both metrics we obtain the equations z ⁰ (x) ² + r ⁰ (x) ²

z(x) ² = 1

(1 − x ² ) ² , (4.6)

 r(x) z(x)

 2

= x ²

1 − x ² . (4.7)

Differentiating (4.7) we obtain

r ⁰ (x) =

 xz(x)

√ 1 − x ²

 0

= z(x)

(1 − x ² ) ^3/2 + xz ⁰ (x)

√ 1 − x ² . (4.8)

Substituting the previous result in (4.6) we can write z ⁰ (x) ²

1 − x ² + 2xz(x)z ⁰ (x)

(1 − x ² ) ² + x ² z(x) ²

(1 − x ² ) ³ = 0. (4.9)

We have obtained an ODE only in z(x) and together with (4.7) completes the system of equations we were looking for. By solving the last equation and plugging the result in (4.7) we can draw the following parametric plot of z(x) and r(x)

Figure 4.1. Single droplet horizon. The droplet starts at the boundary z = 0 and extends into the bulk, capping off at z = 1. The horizon radius at the boundary is normalized to r = 1.

Figure 4.1 depicts the droplet horizon we were looking for. The analytical solution of

the system is simply a circular droplet.

Chapter 4. Black droplets 23

4.2 Alternative construction

Now we require that the surfaces Σ B of the form r = ¯ r(z), when embedded in Poincar´ e AdS

ds ² = ` ²

z ² −dt ² + dz ² + dr ² + r ² dΩ _n+1  , (4.10) yield a pullback that satisfies the effective equation derived in Chapter 3. For p = 1, the equation becomes

V ₀ ² (z) R ² (z)



1 − γ ^zz R ⁰ (z) ² + R ² (z)

` ²



= 4˜ κ ² . (4.11)

Using equation (3.62), we can identify the components of (4.11) as

V ₀ ² (z) = ` ²

z ² , γ ^zz = z ²

` ² 1

1 + ¯ r ² (z) , R ² (z) = ¯ r ² (z) ` <sup>2</sup> z <sup>2</sup> , R <sup>0</sup> (z) <sup>2</sup> = ¯ r <sup>0</sup> (z) <sup>2</sup> ` ²

z ² + ¯ r ² (z) ` ²

z ⁴ − 2¯ r(z)¯ r ⁰ (z) ` ² z ³ .

(4.12)

Substituting in (4.11), after some algebra we eventually arrive at z + ¯ r¯ r ⁰

z ¯ rp1 + (¯r ⁰ ) ² = 2˜ κ. (4.13) Similarly as in 4.1, we can set the size of the radius at the AdS boundary as a boundary condition. However, now we don’t have a single droplet horizon, but rather a spectrum of solutions parametrized by the surface gravity ˜ κ. In principle, one could vary both the radius at the horizon and the surface gravity independently, but according to the metric (3.61) only their product is invariant under scaling of the geometry

λ = 2˜ κr _b . (4.14)

Fixing r _b = 1 and solving (4.13) for ¯ r ⁰ we obtain

¯

r ⁰ = −z ± 2˜ κzpz ² + ¯ r ² (1 − 4˜ κ ² z ² )

¯

r(1 − 4˜ κ ² z ² ) . (4.15)

Integrating numerically for the − sign for different values of ˜ κ produces the expected

droplet horizons. The parameter λ goes from λ → 0, where we recover the same circular

droplet as in section 4.1, to λ → 1 where the solution rapidly approaches the black

string, as seen in Figure 4.2.

Chapter 4. Black droplets 24

Figure 4.2. Family of solutions for the black droplet horizons. The range of λ values is λ = 0, 0.2, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 0.9998, increasing from left to right.

4.3 Observations

For the first way of constructing black droplets, we only obtained a single horizon. We can see that the parameter that distinguishes every curve within the family of solutions in the second case, the surface gravity, is nowhere to be seen. The generation of the complete set of solutions requires advanced numerical methods to solve the PDE systems that arise[9, 10]. However, by fixing ˜ κ we can easily solve the system and still produce an illustrative result.

On the other hand, the large D approach produces a much more simplified problem.

By running the components (4.12) of the induced metric through the effective equation (4.11) we obtain an easily solvable ordinary differential equation.

Although not directly comparable, the large D solution shown in Figure 4.2 agrees with

the conjecture stated in [9]. Despite the fact that their numerical method breaks down

when the droplets extend beyond a certain point in the bulk, they propose that droplet

solutions might continue to exist deep into the bulk until they eventually merge with

the black string solution.

Chapter 4. Black droplets 25

We can also compare Figure 4.2 with the results obtained numerically in [9], shown in Figure 4.3. Despite the fact that their problem is much more algebraically advanced and requires the use of numerical tools, to a qualitative degree we can see that the results share their basic format; a family of droplet horizons that extend into the bulk as the surface gravity is increased.

Figure 4.3. Family of solutions for the black droplet horizons, from [ 9].

Chapter 5

Conclusion

In this thesis we have shown how an effective theory of black holes can be constructed in the large D limit. The key feature that makes this approach possible is the fact that for large D there exists a neat separation between black hole and background. We can exploit this aspect by reducing the description of our black hole to a geometrical problem: what shape does a surface Σ _B need to take when embedded in a background spacetime to be able to reproduce a black hole?

By decomposing the Einstein equations along the radial and perpendicular directions of our manifold, we can obtain a system of equations in terms of the intrinsic R µν and extrinsic K _µν curvature tensors for each (D − 1)-dimensional constant-ρ slice. We can then solve the system of equations for a metric ansatz at leading order in D; imposing regularity at the horizon and integrating along the radial direction we obtain an equation which only depends on functions of the z direction, the collective degree of freedom of the black hole. This is the effective equation we were seeking, which reads

V ₀ ² (z) R ² (z)



1 − (DR) ² + R ² (z) l ²



= 4˜ κ ² . (5.1)

If the metric components of the induced metric on the surface Σ _B can be combined in the above way, such that the result is equal to four times the constant surface gravity squared of a black hole, then we can resolve the black hole with that particular value of surface gravity by the surface Σ _B .

26

Chapter 5. Conclusion 27

We have also shown that the effective theory of black holes in large D offers signif- icant advantages in simplicity as far as calculating solutions goes, in contrast to the full Einstein equations at finite D. This can be attributed to two main reasons: the radial direction has already been integrated out which reduces the dimensionality of the equations to solve and the fact that for a static black hole there is only one degree of freedom (z), which entails only one equations to solve, as opposed to a system of coupled differential equations.

We have seen in this thesis how we can reproduce known solutions in finite D using equation (5.1). Additionally, in [1] many other known exact solutions are obtained from the same relation. In recent years, new black holes solutions in D > 4 have been found, most notably the black Saturn obtained in [11] and the black di-ring found in [12]. The first describes a 5D Myers-Perry black hole with a concentric black ring while the second describes two concentric black rings with a common rotation plane.

Following the path of this thesis, it would be interesting to see in future work if these novel black hole solutions that live in D > 4 can also be reproduced by our approach.

Although these solutions are not static, which would require a generalization of the

effective theory to include the necessary additional collective degrees of freedom, the

method used to find the surface/surfaces that resolve those black holes should be the

same as the one we used.

Bibliography

[1] R. Emparan, R. Suzuki, and K. Tanabe. The large D limit of General Relativity.

Journal of High Energy Physics, 6:9, June 2013.

[2] E. Gourgoulhon. 3+1 Formalism and Bases of Numerical Relativity. ArXiv General Relativity and Quantum Cosmology e-prints, March 2007.

[3] L. Rezzolla. Lecture notes in Introduction to Numerical Relativity, February 2008.

[4] R. Emparan, T. Shiromizu, R. Suzuki, K. Tanabe, and T. Tanaka. Effective theory of black holes in the 1/D expansion. Journal of High Energy Physics, 6:159, June 2015.

[5] R. Suzuki and K. Tanabe. Stationary black holes: large D analysis. Journal of High Energy Physics, 9:193, September 2015.

[6] Robert M Wald. General relativity. Chicago University, Illinois, 1984.

[7] Ø. Grøn and S. Hervik. Einstein’s General Theory of Relativity: With Modern Applications in Cosmology. Springer, New York, 2007.

[8] V. E. Hubeny, D. Marolf, and M. Rangamani. Hawking radiation in large N strongly coupled field theories. Classical and Quantum Gravity, 27(9):095015, May 2010.

[9] J. E. Santos and B. Way. Black droplets. Journal of High Energy Physics, 8:72, August 2014.

[10] J. E. Santos and B. Way. Black funnels. Journal of High Energy Physics, 12:60, December 2012.

[11] H. Elvang and P. Figueras. Black saturn. Journal of High Energy Physics, 5:050, May 2007.

[12] H. Iguchi and T. Mishima. Black diring and infinite nonuniqueness. Physical Review D, 75(6):064018, March 2007.

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