Geometric relations of black hole thermodynamics

(1)

MSc Physics

Theoretical Physics

Geometric relations of black hole

thermodynamics

Author:

Vasilis Lefkos

Supervisor:

Dr. Alejandra Castro

Examiner:

Dr. B.W. Freivogel

October 24, 2015

(2)

Abstract

General relativity was an innovating theory that changed modern physics. However, quantum gravity seems elusive. Black hole thermodynamics paved the way for the understanding of the quantum theory. We explore some ge-ometric relations and their consequences to thermodynamics, with the use of more horizons than the outer event one.

(3)

The law that entropy always increases, holds, I think, the supreme position among the laws of Nature. If someone points out to you that your pet theory of the universe is in disagreement with Maxwell’s equations — then so much the worse for Maxwell’s equations. If it is found to be contradicted by observation — well, these experimentalists do bungle things sometimes. But if your theory is found to be against the second law of thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation.

-Arthur Eddington

1 Introduction

General relativity is a non-linear theory and non renormalizable perturbatively. The fact that spacetime it self, instead of being a fixed background as in normal field theories, is dynamical, makes it really difficult to find a correct method for quantization. One would need to quantize spacetime itself. The main symmetry of quantum gravity, diffeomorphisms, cause problems with locality in contrast with other gauge field theories, since physical observables are non local. Moreover, causality and unitarity problems arise when one attempts to formulate a quantum field theory near black holes - one example is the black hole information paradox, where pure states passing the horizon come out as mixed states through thermal radiation.

Unfortunately or not, the supreme agreement of general relativity with experi-ment renders it the most popular low energy effective theory. Black holes however, belong to the regime of general relativity where one cannot rely solely upon the classical theory, or even semi-classical. They serve as the perfect playground for quantum gravity and it is a rich arena for theoretical physics.

The attempt of quantizing gravity with normal quantum field theory and the discovery of black hole thermodynamics, a celebrated work by Bekenstein, Bardeen, Carter and Hawking [103], was evidence towards a deep connection between ther-modynamics and quantum gravity. Later, the nourishing of string theory and holography with the greatest conjecture of contemporary physics, the AdS/CFT correspondence, established the use of a thermal field theory.

These innovations led to the exploration and better understanding of conformal field theories, as dual theories to a gravitational theory, even for the case of black holes. Hence the thermal behavior of black holes could be studied both macroscop-ically and microscopmacroscop-ically. A very interesting endeavor is examining gravitational behaviors from a conformal field theory point of view and vice versa.

One of the latest research subjects, however, is the role of inner horizons in black hole thermodynamics and the dual holographic description of a black hole.

(7)

Despite the fact that inner horizons can have problems in the classical theory, such as divergences for observers who cross them, they turn out to have an impact on the system. For instance, they seem important for the holographic dual as well as the string theory description. Inner horizons have thermodynamical properties similarly with the outer event horizons. The same applies for multiple horizon black holes, with different asymptotics or dimensionalities.

The aim of this project is to make use of these horizons and their thermal properties, entropies and potentials, in order to have a clearer view of their role in black hole physics, as well as their dual conformal theories. There are two geometrical potential relations reviewed; one is the entropy product, which seems to be mass independent for a large class of black holes and theories, and which was studied at some level in the literature [58,89,174,190,196]. The second type of relation is one that does not necessarily involve entropies, rather than geometric potentials, namely the surface gravity and the angular velocity of the horizon. These relations are merely observed for some cases in the literature [183,184,203] and not deeply analyzed.

The structure of the project is as follows: We first introduce black hole physics in General Relativity in Chapter 2, as well as black hole thermodynamics in Chap-ter 3. We especially review the first law, which we will use extensively in the rest of the project. Then we preview cosmological gravity in Chapter 4, together with gravity in various dimensions. What follows is Chapter 5, the introduction to conformal field theories and their correlation to black hole physics, with many modern theories, such as Kerr/CFT and hidden conformal symmetry. We end up with inner horizons and their thermal properties, as well as the geometric potential relations and the work done on them in Chapter 6. Finally, there is an appendix with a presentation of all the relations, as well as some computations on various subjects used throughout.

(8)

2 Black hole physics preliminaries

2.1 General Relativity

General Relativity (GR), i.e. Einstein gravity, is governed by the Einstein’s field equations:

Rµν−

1

2Rgµν + gµνΛ = 8πTµν or Gµν = 8πTµν . (2.1) Here and from now on we use geometrized units (GN = c = 1) and also the

”mostly plus” convention for the metrics. The beauty of this equation is that it connects geometry to matter. On the left side one has purely geometrically defined quantities and on the right side the energy momentum tensor. Rµν is the Ricci

tensor, R is the Ricci scalar, gµν is the metric we use and Λ is the cosmological

constant, which is the vacuum energy of spacetime (famous for Einstein’s quote ”the biggest blunder of my life”). The first two are functions of the metric and the connections, also known as Christoffel symbols. The left hand side is equivalently denoted as Gµν, the Einstein tensor.

2.2 What is a black hole?

In order to define a black hole, one needs to have some more tools at hand. Let us start again by having a spacetime manifold M . Then the future (past) asymptotic region of this spacetime will be the region where some trajectories end (start) (e.g. if it is a future null infinity all light rays end there) and we can denote it by J+(J−).

Then we have the chronological past (future) I−(I+) of a point P, which is the set of points from which P can be reached with a future (past) directed timelike curve.

Similarly the causal past (future) J−(J+_{) is the set of all points from which P}

can be reached with either a future (past) timelike or null curve.

So we are ready to define the domain of outer communication D of M as the intersection of the past of the future asymptotic region with the future of the past asymptotic region i.e.

D = I−(J+) ∩ I+(J−) , (2.2) a definition due to Penrose in 1967 [5]. It is the set of points from which there exist future and past directed timelike curves to arbitrarily large distances.

A black hole is now defined as

(9)

The region M − D is called eternal black hole and it is the maximal analytical extension of the black hole (see fig.2.2 for example). This region has of course a boundary H which can be thought as a union of two subsets H+∪H−_{, where H}+_is

the boundary of I−(J+_{), and similarly for H}−_{. There is an equivalent definition.}

Let us have a set of points U ⊂ M . Then the topological closure ¯J−(U )of J−(U ) is the causal past of U but including the limit points (infinities). The boundary of this closure will be naturally

˙

J−(U ) = ¯J−(U ) − J−(U ) . (2.4) Then the future horizon(future boundary of D will be the boundary of the closure of the causal past of J+ [6]

H+= ˙J−(J+) (2.5) and similarly for H−. So we have finally a region defined as a black hole. It should be clear from the previous that no light can escape from this region to an external observer, which serves as a reason for naming it black. For a black object that was formed from a gravitational collapse, H− would not exist. Theoretically, the region behind this subset cannot be reached by any signal and it is called a white hole, but it will not be relevant for this thesis. Similarly to the black hole, it is defined as W = M − I+(J−).

2.3 Penrose diagrams

A very important and strong toolbox for General relativity are Penrose di-agrams(or conformal diagrams). In order to draw one, we must first introduce conformal compactification of a spacetime [7], which is a way of including ”infinity” (the asymptotic region) in our metric. This is the transformation

ds2 → Ω2_ds2 _, _(2.6)

where Ω depends generally on the coordinates and it is a scaling transformation parameter. What is crucial to know is that the causal structure of the spacetime does not change under this transformation, since we are only rescaling. That means we can safely draw conclusions about causal effects from this new metric as well. Let us have a spacial coordinate vector ~r and time t so that Ω2 _{= Ω}2_(~_{r, t). Then we}

can choose this parameter so that this new metric has ”infinity” in finite distance points, by means of

(10)

for an asymptotically flat spacetime. These points, forming the conformal boundary of the spacetime, also include infinity. An easy illustration is the flat 4d Minkowski spacetime conformal diagram. In order to obtain the diagram, we start with the 4d Minkowski metric in polar coordinates

ds2_{M ink} = −dt2+ dr2+ r2dΩ2 , (2.8) with dΩ2 _{being the unit two sphere metric and −∞ < t < ∞ , 0 ≤ r < ∞. We}

then transform into null coordinates u = t − r , v = t + r with −∞ < u ≤ v < ∞. The element becomes

ds2 = −1 2(dudv + dvdu) + 1 4(u − v) 2 dΩ2 . (2.9) The next transformation is needed so that we can have infinity at a finite coordi-nate. We use U = arctan(u) , V = arctan(v) which means −π/2 < U ≤ V < π/2 leading to the metric

ds2 = 1 4cos2_{U cos}2_V − 2(dV dU + dU dV ) + sin2_{(V − U )dΩ}2 . (2.10)

The final step is to go back to time/radial coordinates (T, R) which now have different ranges. If we set R = V − U , T = V + U then the metric becomes

ds2 = 1

(cosT + cosR)2(−dT 2

+ dR2+ sin2RdΩ2) = Ω−2(T, R)ds2_{M ink} . (2.11) with ranges 0 ≤ R < π , |T | + R < π. It is obvious now that the two metrics are related with the scaling parameter. As we will see in the diagram, there are some special points that need to be defined:

- i+ _{is the future timelike infinity with T = π , R = 0}

- i− is the past timelike infinity with T = −π , R = 0 - i0 is the spatial infinity with T = 0 , R = π

- J+ _{is the future null infinity with T = π − R , 0 < R < π}

(11)

Figure 2.1: Minkowski Penrose diagram [9]

The first three are represented as points in the diagram , which means they are S2 _{spheres. The latter are null hypersurfaces R × S}2 _{. Null radial geodesics are at}

45o _{in these diagrams.}

2.4 Black holes as solutions of Einstein’s equations

Einstein’s field equations, admit unique solutions identified as black holes. For now we will specialize in zero cosmological constant spacetimes, or asymptotically flat. It will be easy to generalize for non-zero later (Chapter 4). A spacetime M is asymptotically flat if it is asymptotically simple and there is an open neighborhood of ∂M where Rµν = 0.

2.4.1 Schwarzschild

The first exact vacuum solution to Einstein’s equations was found in 1916 by Karl Schwarzschild. It is a spherically symmetric solution in four dimensional asymptotically flat spacetime. The metric reads (in coordinates (t, r, θ, φ)

ds2 = − 1 − 2M r dt2+ 1 −2M r −1 dr2+ r2dΩ2 , (2.12) where M will be the mass of the black hole. We will discuss what this mass is and how it is computed later, in section 2.6.1. One can already observe that there seems to be a problem at r = 2M , what is called the Schwarzschild radius. It is also the radius at which the boundary of the black hole is formed, known as

(12)

the event horizon. The metric there is singular; however, one can get rid of this singularity with a change in coordinates. It is just a coordinate singularity. On the other hand, the r = 0 point cannot be remedied. It is a curvature singularity and these kind of singularities appear in most black hole solutions. In order to see if a singularity is physical, one needs to check if the quantity

RµνρσRµνρσ ,

which is called the Kretschmann scalar, diverges- this is at least a sufficient con-dition, and usually the case. Another way to define a curvature singularity is geodesic incompleteness; if there is a singularity of this type, geodesics cannot be extended to all values of their affine parameters. That would mean there is a geodesic congruence that converges to that very point.

As promised, a coordinate transformation should make the r = 2M singularity disappear. For, instance, one could do the following: Define a coordinate

v = t + r∗ where r∗ = r + 2M ln r − 2M 2M is the Regge-Wheeler coordinate. Then the metric becomes

ds2 = − 1 − 2M r dv2+ 2drdv + r2dΩ2 , (2.13) which is clearly non singular at r = 2M . These coordinates are called ingoing Eddington-Finkelstein (EF) . The truth is we cheated a bit; the coordinates are initially defined for r > 2M , but we can analytically continue it to all r > 0. We also define the vector fields

`a= ∂ ∂v a +1 2 1 − 2M r ∂ ∂r a , na= − ∂ ∂r a , (2.14)

which are both null (`2 _{= n}2 _{= 0) and they are normalized as ` · n = −1.}

Assum-ing r → ∞ is outwards, then `a _{is outward pointing and n}a _{is inward pointing.}

Similarly there are the outgoing coordinates with v = t − r∗ .

Combining the ingoing with the outgoing coordinates two we get ds2 = − 1 −2M r dudv + r2dΩ2 , (2.15)

(13)

very useful Kruskal-Szekeres coordinates. With a final transformation U = −e−u/4M V = ev/4M with U V = −r − 2M

2M e r/2M _(2.16) we get ds2 = −32M 2 r e −r/2M dU dV + r2dΩ2 , (2.17) which has the privilege of analytic continuation to all values of U, V ∈ R. So we are ready to have a diagrammatic picture of the Schwarzschild solution:

Figure 2.2: The eternal Schwarzschild [10]

where the axes are the U, V coordinates (V = 0 and U = 0) and we can also see the Finkelstein vectors we defined (showing outgoing and ingoing radial null geodesics). The black hole horizon is at r = 2M . The middle curved line is a spacelike trajectory and the vertical curved line is a timelike trajectory for r > 2M . This is a maximal analytic extension of a Schwarzschild black hole spacetime. Of course we have chosen dΩ = 0. Region I is our universe, region II is the black hole. It also contains the white hole (region IV), and a ”parallel universe” in region III. The wiggly lines are singularities, corresponding to U V = 1.

The two square regions on the right and left of the diagram are the domain of outer communications that we defined in section 2.2. We can see that the black hole is the region from which no null rays can escape and hence our definition of excluding the past of the asymptotic future. Similarly with the white hole, but now null rays cannot go in instead of out.

(14)

2.4.2 Reissner-Nordstr¨om

A little later a charged electrovacuum solution was discovered independently by Reissner (1916) and Nordstr¨om (1918). One can obtain the solution starting from the Einstein-Maxwell action

S = 1 16π

Z

d4x√−g R − FµνFµν , (2.18)

which essentially means we have a non zero energy momentum tensor in the Ein-stein equations; more specifically

Rµν− 1 2Rgµν = 2 FµλFνλ− 1 4gµνFλρF λρ . (2.19) The equation of motion for the strength field is ∇µFµν = 0. The solution is also

spherically symmetric like Schwarzschild and its metric reads ds2 = − 1 − 2M r + Q2 r2 dt2+ dr 2 1 −2M_r +Q_r22 + r2dΩ2 . (2.20)

The Maxwell one-form is A = Q_rdt. We can observe that now there is an ex-tra quantity in the metric, namely the electric charge Q. One could also have a generalization of this [206], the dyonic black hole with e2 _{= Q}2 _{+ P}2 _{the dyonic}

charge, where Q, P would be the electric and magnetic charge respectfully, assum-ing that magnetic monopoles exist. The Maxwell one-form would get an extra term A = Q_rdt + P cosθdφ.

The horizons of the Reissner-Nordstr¨om black hole can be found by solving the equation

∆ = 1 −2M r +

Q2

r2 = 0 . (2.21)

We will later see (section 2.5) that this is related to the Killing vector being null on that surface. Hence, the Reissner-Nordstr¨om metric admits two horizons sitting at

r± = M ±

p

M2_{− Q}2 _.

The outer most one is always the event horizon and we name the other one inner horizon. The radicand will determine the reality of the event horizon radius and hence existence of the horizon. The limit where M = Q is called extremal. An extremal black hole is a black hole with the minimum possible mass in relation with the rest of its charges. At that limit the two horizons also coincide. The cosmic censorship conjecture [13] prevents M acquiring values below Q.

(15)

first take ingoing EF coordinates leading to ds2 = −∆

r2dv

2_{+ 2dvdr + r}2_dΩ2 _. _(2.22)

We can see now that the singularities of r± in the previous metric were just a

coordinate singularities, justifying their identifications as horizons. There is only one singularity in these coordinates, at r = 0.

We do similarly for outgoing EF coordinates. The difference comes at the Kruskal type coordinates. We have two sets, since we have two horizons. Specifi-cally the KS coordinates will be

V± = eκ±v _U± _{= −e}−κ±u _. _(2.23)

The parameter κ±is called surface gravity and we will explain its role and meaning

in chapter 2.5. The metric for the plus sign takes the form

ds2 = −r+r− κ2 + e−2κ+r r2 r− r − r− κ+_κ−−1 dV+dU++ r2dΩ2 (2.24) with V+U+ = −e2κ+r r − r+ r+ r − r− r− κ+_κ− . (2.25) This way we get regions I-IV as in the Schwarzschild. These coordinates are sin-gular for r = r− and we need different ones to cover the inner region. That is why

we defined V−, U−. The metric there is

ds2 = −r+r− κ2 − e−2κ−r r2 r+ r+− r κ−_κ+−1 dV−dU−+ r2dΩ2 (2.26) with V−U− = −e2κ−r r−− r r− r+− r r+ κ−_κ+ . (2.27) This gives us another four regions, with one being common with region II of the previous coordinates (see fig 2.3). Similarly we can patch a different set of ”plus” coordinates on top of these four regions, leading to another exterior spacetime, another black hole etc. Following this method the diagram can be extended both ways infinitely.

(16)

Figure 2.3: The RN Penrose diagram [6]

As we can see in the diagram there is a fundamental difference from the Schwarzschild black hole; the curvature singularity at r = 0 is timelike and hidden behind the inner horizon. As for the ”internal infinities” shown in the diagram, what is demonstrated is that if an observer is behind the horizon, these points are infinitely away from them.

2.4.3 Kerr and Kerr-Newman

Roy P.Kerr found a rotating solution to Einstein’s vacuum field equations, general-ized in 1965 by Ezra Newman to include electric/magnetic charge as well, solving the Einstein-Maxwell field equations. The metric is (in Boyer-Lindquist coordi-nates): ds2 = −∆ − a 2_sin2_θ ρ2 dt 2_{− 2asin}2_θr2+ a2− ∆ ρ2 dtdφ+ (r2 _{+ a}2₎2_{− ∆a}2_sin2_θ sin2θdφ2+ρ 2 dr2+ ρ2dθ2 , (2.28)

(17)

with

ρ2 = r2+ a2cos2θ ∆ = r2− 2M r + a2+ e2 a = J/M (2.29) and the new parameter a is essentially the rotating parameter, containing the angular momentum J . The Maxwell one-form is

A = Qr(dt − asin

2_{θdφ) − P cosθ[adt − (r}2_{+ a}2_)dφ]

ρ2 , (2.30)

where Q, P are the electric, magnetic charges respectfully and the dyon charge is e2 _{= Q}2 _{+ P}2_{. Taking various limits leads to the RN (a → 0), Kerr (e → 0) or}

Schwarzschild (a, e → 0) solutions. Like the RN black hole, the Kerr-Newman black hole mass has to obey an inequality

M ≥ e2+ a2 . (2.31) Again, the equal sign corresponds to the extremal limit.

The Kerr-Newman black hole has two horizons at r± = M ± √ M2_{− e}2 _{− a}2 _. The quantity Ω±= − gtt gtφ _H± = a r2 ±+ a2

is called the angular velocity of the horizon. That is because if we consider an observer with zero angular momentum (also known as ZAMO- zero angular mo-mentum observer), then if they try to stay at the horizon, they will rotate with this velocity.

The horizons are generated by the Killing vectors (they become null on the horizons):

ξ±µ = kµ+ Ω±mµ,

where kµ _{= ∂}

t is is timelike at infinity and exists because of time translation

symmetry. The other Killing vector, mµ _{= ∂}

φ is spacelike at infinity and exists

because of rotational symmetry around the φ axis. For Schwarzschild and RN, we had Ω± = 0, so the Killing vector was just ξµ= kµ.

The norm of ξµ _is

ξ2 = gµνξµξν = ∆(r2+ a2cos2θ)2 , (2.32)

which is obviously spacelike from infinity till the horizon.

In order to obtain a Penrose diagram, we follow the same procedure as the RN case. Indeed, we end up with a very similar diagram

(18)

Figure 2.4: The Kerr Penrose diagram [6]

As shown in the diagram (figure 2.4), there is a ring singularity instead of a point for the Kerr black hole. Let us see how this takes shape. The singularity is at the values of r = 0, θ = π/2, where ρ2 _{= 0. Transforming to another set of}

coordinates defined as x + iy = (r + ia)sinθ exp i Z dφ + a ∆dr z = rcosθ t0 = Z dt + r 2_{+ a}2 ∆ dr − r . (2.33) Then the metric becomes

ds2 = −dt02+dx2+dy2+dz2+ 2M r

3

r4_{+ a}2_z2

r(xdx + ydy) − a(xdy − ydx) r2_{+ a}2 + zdz r +dt 0 2 , (2.34)

where r is now a function of the coordinates. We can rewrite the coordinate transformation as

(19)

which gives us x2_{+ y}2 sin2_θ − z2 cos2_θ = a 2 _(2.36) or x2_{+ y}2 sin2_θ − r 2 = a2 . (2.37) Taking now the singular point of our initial metric r = 0, θ = π/2, we end up with the equation

x2+ y2 = a2 , (2.38) which is a ”ring” on the x-y plane.

The other new feature of the Kerr spacetime are closed timelike curves (CTC’s), a time machine inside the Kerr black hole, very interesting yet irrelevant topic. They exist because the spacelike Killing vector mµ _{= ∂}

φ becomes timelike there.

This diagram can also be extended infinitely to both directions and repeats itself.

Ergosurface

We will digress here to elaborate on this special feature of the Kerr geometry due to its rotation. In the Schwarzschild solution, as we mentioned there is only the time translation Killing vector that becomes null on the horizon. That is essentially when the component gtt of the metric vanishes. Specifically looking at the metric

2.12 we can see that it happens for r = 2M .

This Killing vector exists in the Kerr solution as well; but it becomes null on a surface outside the event horizon, called the ergosurface. The region between this surface and the horizon is called ergosphere and no observer can be stationary in this region. They will experience the effect of frame dragging, which means the black hole forces everything beyond this limit, the ergosurface (also called the stationary limit), to rotate with it. This can be seen by requiring constant r, then geodesics move along φ. Suppose there is an observer that wants to stay still in a coordinate sense. The tangent vector to the observer’s world line will be

Pµ = dx

µ

dt = (1, 0, 0, 0) . (2.39) In order for his trajectory to be timelike, one also needs

gµνPµPν < 0 =⇒ gtt < 0 (2.40)

to be satisfied. In Boyer-Lindquist coordinates, this means r2− 2M r + a2_{− a}2_sin2_{θ > 0 =⇒ r > r}+

E or r < r −

(20)

where we have defined the ergosurfaces as

r±_E = M ±√M2_{− a}2_cos2_{θ .} _(2.42)

So if an observer wants to ”stand still” in these coordinates, they have to be outside this region. Equivalently, the Killing vector ∂t is spacelike in this region. In order

to find the radius of this surface we solve

kµkµ= 0 =⇒ gttktkt= 0 ,

which leads to the same result. There is actually an ergosurface inside the inner horizon, but it is never considered due to the instability of r−. The outer one is

important due to the fact that we can extract energy from the black hole ergoregion. This is called Penrose process, discovered as a gedanken experiment from Penrose [156]. This is because the time translation Killing vector is spacelike, so one can have a particle with negative energy

E = Pt = −Pµkµ= −gtt < 0 (2.43)

as we saw earlier. Let us elaborate a little bit more on this process. Suppose there is an observer (or particle) that follows a geodesic towards the ergosphere. Suppose also that this observer is a system consisting of two parts (1 and 2). The total momentum and energy of the system will be

pµ_tot = pµ₁ + pµ₂ =⇒ Etot = E1+ E2 (2.44)

as we saw though in the previous relation, it is possible in the ergosphere that the energy is negative. That means that if one part of the observer has negative energy, lets say E2 < 0, then from (2.44) we have E1 > Etot! So the other part

of the observer emerges with more energy than it initially had. That energy of course is extracted from the rotational energy of the black hole. There is a limit to this extraction. Requiring the particle (part 2 of the observer) to follow a timelike trajectory we have

pµ₂ξµ < 0 , (2.45)

since ξµ _{is the Killing vector of a rotating black hole. Then (2.45) becomes, if L} 2

is the angular momentum of the particle (we will assume this happens outside the outer event horizon for simplicity)

pµ₂kµ+ Ω+p µ

2mµ< 0 =⇒ −E2+ Ω+L2 < 0 =⇒ E2 > L2ΩH , (2.46)

which means the particle must have opposite sign angular momentum than the black hole (E is negative). In terms of the black hole charges, since they changed

(21)

by

δM = E2 δJ = L2 , (2.47)

the limit of possible extraction is given by

Ω+δJ ≤ δM , (2.48)

where we added an equal sign as the ideal process. This process is also called superradiance when referring to waves with frequency ω and momentum m, that get amplified when they enter this region in a band

0 < ω < mΩ±. (2.49)

All these generalize to the Kerr-Newman case, for instance the superradiant band becomes

0 < ω < mΩ±+ qΦ± , (2.50)

where q is the charge of the wave or particle and Φ± is the electric potential of the

horizon, defined as Φ± = −Aµξµ r± . (2.51)

As argued by Bekenstein [186], superradiance follows from the second law of black hole mechanics, which we will present in the next chapter.

2.5 Killing and Cauchy horizons

A Killing horizon is a null hypersurface with a normal vector that is also a Killing vector. Equivalently, we can define it as a null hypersurface which is invariant under the one-parameter isometry group generated by the Killing vector ξµ and on which ξµ_ξ

µ=0 [112]. In order to understand the Cauchy horizon, one needs to elaborate

on some definitions [6,12].

- An inextensible curve is one that has no past or future endpoint on M. - The future(past) domain of dependence D±(Σ) is the set for which every past(future) inextensible curve intersects Σ.

- A Cauchy surface is a subset Σ ⊂ M which is intersected exactly once by each inextensible timelike curve of M. A Cauchy surface satisfies D+(Σ)∪D−(Σ) = M. - An asymptotically flat spacetime is called globally hyperbolic if it admits a Cauchy surface [12]. If the spacetime is not globally hyperbolic, then the domain of dependence of Σ has a boundary on M, which is the Cauchy horizon. The inner horizons of the black holes we mentioned are Cauchy horizons, in addition to being Killing horizons. We will see in chapter 6 the importance of inner horizons.

(22)

Since the Killing vector ξµ is normal to the the Killing horizon H, it will be proportional to its unit normal aµ. Then we have

aν∇νaµ _H = 0 ξµ = f aµ =⇒ ξν∇νξµ _H = κξµ, (2.52) where f is a function and κ = ξµ_∇

µlnf is what we will call the surface gravity.

Using Frobenius’ theorem (B.7) and Killing’s equation (B.2), we can get another expression for κ: κ2 = −1 2(∇ µ_ξν_)(∇ µξν) _N . (2.53)

It turns out that κ is constant on orbits of ξ. The importance of surface gravity will be demonstrated in the next chapter. In terms of geodesics, eq. (2.52) means that the Killing vector is a geodesic on a horizon that is not parameterized affinely, and that is measured by the surface gravity. In order to demonstrate that, suppose we have an affine parameter λ and a non affine parameter ν with which we can express the Killing vector as ξµ _{= ∂}

ν. Then on an orbit of the Killing vector, we

can parameterize ν as ν(λ), so

ξµ = dλ dν

d

dλ . (2.54)

This gives us the expression for f , being f = dλ_dν. Using then its relation with the surface gravity

∂νlnf = κ . (2.55)

we can see that the surface gravity indeed measures the non affine parameterization. Suppose now ξµ vanishes on a spacelike surface Σ. Then near Σ the orbits of ξµ will look like Lorentz boosts in Minkowski or accelerations in Rindler. The acceleration horizon is formed by two intersecting lines, which are null surfaces generated by null geodesics normal to Σ. These are called bifurcate Killing horizons and the intersecting point is called bifurcation surface. The Killing vector is of course null on the bifurcation surface (in four dimensions, it is a two sphere) S. Conversely, if the group of isometries generated by the Killing vector leaves a surface S fixed, then the null geodesics orthogonal to S comprise a bifurcate Killing horizon [112]. In the Penrose diagrams we illustrated before in Kruskal coordinates, the bifurcation sphere is at the point U = V = 0.

2.6 Hamiltonian formulation

Like any field theory, General relativity can be described by a Lagrangian and hence an action, whose Euler-Lagrange equations will yield Einstein’s equations.

(23)

The vacuum action for GR is the Einstein-Hilbert action, namely IEH = 1 16π Z V R√−g d4_{x ,}

where g is the determinant of the metric, R is the Ricci scalar and V is the volume of the spacetime. Adding matter fields φ and some extra terms results in the full action: I[g; φ] = 1 16π Z V R + 16πLM _√ −g d4_{x +} 1 8π I ∂V ε (K − K0)|h|1/2d3y , (2.56)

where LM(φ, φ,µ; gµν, gµν,ρ...) is the matter Lagrangian density and hab is the

in-duced metric on ∂V, the boundary of V, with normal nµ. The action is a scalar with respect to spacetime coordinate transformations, but behaves as a tensor for the hypersurface coordinate transformations.

K is the extrinsic curvature of the submanifold ∂V, a geometrical quantity that depends on the embedding, and is symmetric. We will see below how it is defined in terms of vectors. This term is added so that after varying the action with respect to the metric, we get the correct equations of motion, Einstein’s equations. It cancels out the boundary term that arises when one varies the Einstein Hilbert term. The reason it is needed is that the Ricci tensor depends on higher (second) derivatives of the field (the metric). The factor ε = nµnµ.

K0 is the constant extrinsic curvature of ∂V embedded in flat spacetime, and is

added to cancel the divergence of the action for an asymptotically flat spacetime. This is a term that is non-dynamical and does not affect the equations of motion. Specifically, it is equal to the gravitational action of flat spacetime

If lat= 1 8π I ∂V εK|h|1/2d3y . (2.57) Subtracting this term gives a zero gravitational action for flat spacetime and reg-ularizes the gravitational action for curved asymptotically flat spacetime.

Defining a stress-energy tensor as Tµν = −2 δLM δgµν + LMgµν , we have : δ(IG+ IM) δgµν = 0 =⇒ Gµν = 8πTµν , (2.58)

with Gµν being of course the Einstein tensor.

Now, if one wants to quantize a theory, one needs to proceed to the equivalent Hamiltonian formulation, find canonical commutation relations, so that he can promote fields to operators etc. The most famous work done in the Hamiltonian

(24)

formulation of General relativity was by Arnowitt, Deser and Misner (ADM) [23]. They introduced a 3 + 1 decomposition of spacetime, foliating it with spacelike hypersurfaces Σt. A scalar field t(xµ) is introduced so that constant t describes a

family of hypersurfaces Σt [15]. Each hypersurface has a coordinate set ya, with

the Roman indices taking values excluding time. The normal to the hypersur-faces nµ _{will be parallel to the derivative of t. Let us have a congruence of curves}

parametrized by t, intersecting non-orthogonally the hypersurfaces Σt, with

tan-gent vectors υµ_{= ∂}

txµ. The metric takes the form

ds2 = −N2dt2+ hab(dya+ Nadt)(dyb+ Nbdt) , (2.59)

where N (t, ya) is called the lapse funtion, a normalization of the normal on the hypersurfaces, i.e. nµ = −N ∂µt, and Na(t, yb) is the shift three-vector, which

measures how the normal is not parallel to the vector υµ_{, i.e. υ}µ _{= N n}µ_{+ N}a_eµ a,

where eµ

a = ∂axµ are tangent vectors on Σt. The extrinsic curvature can now be

defined as Kab = n(µ;ν)eµae ν b K = h ab_K ab = nµ;µ (2.60)

The relations between the full metric and the three metric are:

hab = gab N = (−gtt)−1/2 Na = gta gtt = −(N2− NaNa) Na = habNb

gta = Na/N2 gtt = −1/N2 gab= hab− (Na_Nb_/N2₎ √_{g = N}√_{h .} _(2.61)

The Lagrangian becomes

L =√−g4R = −hab∂tpab− N R0− NaRa −2(pab_N b− 1 2πN a_{+ N}|a√_h) ,a , (2.62)

or in terms of the extrinsic curvature L = 3_{R + K}ab_K ab− K2 N√h − 2(nµ_;νnν − nµ_nν ν);µ , (2.63)

where pab _{is the conjugate momentum to the three metric field p}ab ₌ ∂L

∂£thab, then p = pi i = habpab, the quantities R0 ≡ −√h3R + h−1(1 2p 2_{− p}ab pab) Ra≡ −2pab _|b

and the semicolon ( | ) denotes covariant differentation with respect to the three metric. We also have the relations between the fields and the extrinsic curvature

Kab = 1 2N(£thab− Na|b− Nb|a) p ab = √ h 16π(K ab_{− Kh}ab ) . (2.64)

(25)

The shift and the lapse work as Lagrange multipliers. The graviational Hamiltonian takes the form

HG = 1 16π Z Σt N h−1/2(pabpab− 1 2p 2_{) − N h}1/2 3_{R − 2N} ah1/2(h−1/2pab)|b d3y − 1 8π I St N (k − k0) − Nah−1/2pabrb _√ σd2θ (2.65) or in terms of extrinsic curvature

HG = 1 16π Z Σt N (KabKab− K2−3R) − 2Na(Kab− Khab)|b √ hd3y − 1 8π I St N (k − k0) − Na(Kab− Khab)rb _√ σd2θ , (2.66) where σ is the determinant of the induced metric of the boundary Stof Σtand rais

the normal, with the surface one form being dSa = ra

√

σd2_{θ. The dot corresponds}

to the Lie differentation £thab = ˙hab. We have also defined k = σIJkIJ as the trace

of the extrinsic curvature of Stembedded in Σt, which is defined as kIJ = rµ;νe µ Ie

ν J.

The capital roman indices are for the coordinates in the two surface St, similarly

with the lower case indices for the three surfaces Σt. The normal to the two surfaces

is rµ and the tangent vectors are the eµI = ∂xµ

∂ya

∂ya ∂zI.

We have also added a boundary term like before, that does not affect the equations of motion, namely the term with the constant k0. This is the extrinsic

curvature of St in flat space and serves the same purpose: it ensures the action is

zero for flat space and regularizes the action for asymptotically flat space. The variation of the action in Hamiltonian form will be

δIG= Z dt Z Σt ( ˙hab− Hab)δpab− ( ˙pab+ Pab)δhab+ R0δN + 2RaδNa d3y , (2.67)

where Hab, Pab are given below. Imposing stationarity gives Hamilton’s equations

for GR, ˙hab = Hab = 2N h−1/2(pab− 1 2habp) + Na|b+ Nb|a ˙ pab = −Pab= −N√h3Gab+ 1 2N h ab_(pcd_p cd− 1 2π 2_{) − 2N h}−1/2 (pacp_cb− 1 2pp ab₎₊ √ h(N|ab− hab_N|m |m) + (p ab_Nc₎ |c− Na|cpcb− Nb|cpca R0 = 0 Ra = 0 . (2.68)

(26)

2.6.1 ADM charges

The first two of equations (2.68) are called evolution equations and the latter two are the constraint equations, Hamiltonian and momentum respectively. When the equations of motion are satisfied, we are only left with the boundary term. This serves as a definition for the mass or the angular momentum of an asymptotically flat stationary spacetime, quantities that will be named ADM charges [23,207].

The asymptotic behavior of the spacetime will play a big role in this definition. Because the variation of the Hamiltonian must vanish when the equations of motion are satisfied, only the boundary terms will survive the variation. The gravitational Hamiltonian on shell is H_Gos= − 1 8π I St N (k − k0) − Na(Kab− Khab)rb _√ σd2θ . (2.69)

Then the choices of our lapse and shift functions will yield different values. In particular, the ADM mass is defined as

MADM = − 1 8πSlimt→∞ I St (k − k0) √ σd2θ , (2.70) which came about with the choice of an asymptotic flow vector υµ _{= ∂}

txµ, or the

choice of N = 1, Na _{= 0, with S}

t taken to (spatial) infinity as well. The choice

of this vector is of course an asymptotic time translation, so we have a connec-tion between time translaconnec-tion and a conserved charge - the total energy. Another equivalent definition of the ADM energy for asymptotically flat spacetimes, given in the original paper [23] using the metric is

EADM = 1 16π I S∞ (hab_,b− h,a_)dS a, (2.71)

with hab being the three metric on the spacelike hypersurfaces and h = haa. We

can prove that these definitions are equivalent by using the fact that k = σIJrµ;νeµ_IeνJ = h

ab_eµ ae

ν

brµ;ν . (2.72)

Then substituting in 2.70 and integrating by parts, we get 2.71. Taking a choice for asymptotic rotations, one would expect to get the total angular momentum. Suppose there is a rotational symmetry around the φ axis. Indeed, with N = 0, Na _{= ∂}

φya= ma, we get contribution from the other term

JADM = − 1 8πSlimt→∞ I St (Kab− Khab)marb √ σd2θ , (2.73)

(27)

form. Using equation 2.64, we can express this in terms of the conjugate momenta JADM = − 1 8π I S∞ pabmbdSa. (2.74)

An equivalent and handy way of defining mass and angular momentum for axially symmetric stationary spacetimes is using the Komar integrals:

MKomar = − 1 8πSlimt→∞ I St ∇ν_kµ_dS µν = − 1 8π I S∞ kµ;νdSµν = − 1 8π I S∞ ∗dk JKomar = 1 16πSlimt→∞ I St ∇ν_mµ_dS µν = 1 16π I S∞ mµ;νdSµν = 1 16π I S∞ ∗dm (2.75) where dSµν = 2n[µrν] √

σd2θ is the surface element and kµ, mµ are the Killing vectors generating the time translations and rotations respectively. Using Killing’s equation it is easily proven they are equivalent to the ADM definitions. Suppose we have an axially symmetric stationary spacetime in four dimensions (for simplicity lets consider the Kerr metric 2.28). Then taking the asymptotic limit (for r M ) the terms become (keeping leading and second to leading order terms)

gtt ' − r2_{− 2M r} r2 = −1 + 2M r (2.76) gtφ ' −2asin2θ 2M r r2 = − 4J sin2_θ r (2.77) gφφ ' r2sin2θ (2.78) grr ' 1 − 2M r −1 ' 1 + 2M r gθθ ' r 2 _. _(2.79)

So we can rewrite the metric (for convenience we add a small term in the angular part) ds2_asympt ' − 1 −2M r dt2+ 1 + 2M r (dr2+ r2dΩ2) −4J sin 2_θ r dtdφ . (2.80) We will now prove that the Komar expressions are equal to the (M, J ) quantities of the metric. But first lets see how they are equal to the ADM expressions. In order to do that, we will need the inverse asymptotic metric components. These will be (see appendix F for the original ones):

gtt ' − r 2 r2_{− 2M r} = − 1 + 2M r −1 ' −1 − 2M r (2.81) gtφ' −2J r3 g rr _' r2− 2M r r2 = 1 − 2M r (2.82) gθθ ' 1 −2M r 1 r2 g φφ _' 1 − 2M r 1 r2_sin2_θ . (2.83)

(28)

We will also need the explicit expression for the normals. We foliate with spacelike hypsersurfaces Σt, which have a normal

nµ= −∂µt |gtt_|1/2 ' − 1 + 2M r 1/2 ∂µt ' − 1 + M r ∂µt . (2.84)

Then we can also find the normal to the two sphere St, boundary of Σt:

ra = ∂ar |grr_|1/2 ' 1 −2M r −1/2 ∂ar ' 1 + M r ∂ar . (2.85)

Now we are ready to calculate the extrinsic curvatures. For the St sphere, it is

k = ra_|a = habra|b = habra,b− habΓcabrc. (2.86)

For this we need the following quantities Γr_rr = 1 2g rr_g rr,r ' (1 − 2M r )(− 2M r2 ) ' − M r2 (2.87) Γr_θθ = −1 2 g rr_g θθ,r' (r + M )(−1 + 2M r ) = M (1 + 2M r ) − r (2.88) Γr_φφ = −1 2 g rr_g φφ,r 'M(1 + 2M r ) − rsin 2_{θ .} _(2.89) We have then habra,b = hrrrr,r ' (1 − 2M r )(1 + M r ) 0 = −(1 − 2M r ) M r2 (2.90) and hrrΓr_rrrr ' (1 − 2M r ) M r2(1 + M r ) ' M r2(1 − M r ) (2.91) hθθΓr_θθrr ' 1 r2(1 − 2M r )M(1 + 2M r ) − r(1 + M r ) ' − 1 r(1 − 2M r ) (2.92) hφφΓr_φφrr ' − 1 r(1 − 2M r ) . (2.93) Adding them all together with the proper signs we end up with

k = 2 r − 4M r2 = 2 r(1 − 2M r ) . (2.94) Following the same procedure for the same surface embedded in flat space, we would have a 2-metric element

ds2₂ = ˜r2dΩ2 , (2.95) where ˜r = r(1 + M/r) so that ˜r2 _{' r}2_{(1 + 2M/r). Our new extrinsic curvature is}

now

k0 =

2

(29)

Their difference will be

k − k0 = −

2M

r2 . (2.97)

We also have the two-metric determinant

σ = r4sin2θ . (2.98) It is straightforward to show that the mass parameter coincides with the ADM mass: MADM = − 1 8π I S∞ (k − k0) √ σd2θ = − 1 8π Z π 0 Z 2π 0 (−2M r2 )r 2 sin2θdφdθ = M . (2.99) Now we can prove that the Komar expression is also equal to this quantity. We have

∇ν_kµ_n

µrν = kµ;νnµrν = kµ,νnµrν − Γµνρk ρ_n

µrσgνσ . (2.100)

The first term will vanish since

kµ_νnµrν = ktνntgνσrσ = ktrg rr_r

rnt= 0 (2.101)

and the second term becomes

Γµ_νρkρnµrσgνσ = Γttrntrrgrr . (2.102)

We need to calculate the connection Γt_tr = 1

2g

µt_g

µt,r+ gµr,t− grt,µ . (2.103)

The third term vanishes (no r − t component), as well as the second (stationarity). We are left with

Γt_tr = 1 2(g tφ_g tφ,r+ gttgtt,r) ' 1 2[− 2J sin2_θ r3 4J sin2_θ r2 + (1 + 2M r ) 2M r2 ] ' M r2 (2.104)

and all together (the factor of two comes from the surface element) − 2Γt trntrrgrr ' 2M r2 (−1 + M r )(1 + M r )(1 − 2M r ) ' − 2M r2 = k − k0 . (2.105)

So we have proven that the ADM and Komar definitions are equivalent for the mass.

Let us see the angular momentum. The second term in 2.73 vanishes since there is no φ − r component in the metric. As for the first term

Kabmarb = Kφrrr = Kφrgrrrr ' Kφr(1 −

M

(30)

We need to calculate Kφr. Using the definition of the extrinsic curvature we have Kφr = nφ;r = nφ,r− Γµ_φrnµ ' Γtφr(1 − M r ) . (2.107) But Γt_φr = 1 2g tµ_(g µφ,r + gµr,φ− gφr,µ) . (2.108)

The last term vanishes because there is no φ − r component. The second term vanishes because there is no r − t component as well. We are left with

Γt_φr = 1 2(g tt_g tφ,r+ gtφgφφ,r) ' − 1 2(1 + 2M r 4J sin2_θ r2 + 2J sin2_θ r3 r(1 + 2M r )) ' −3J sin 2_θ r2 =⇒ Kφr ' −3Jsin2_θ r2 . (2.109)

Plugging this in the ADM definition 2.73 we get

JADM = − 1 8π Z π 0 Z 2π 0 (−3J sin3θ)dθdφ = J , (2.110)

where we used the fact that R sin3_{xdx = −cosx +}1 3cos

3_{x + C. What is finally left}

is to prove that the Komar definition of angular momentum is also equal to this parameter. Once again, we start with the Komar integrand

∇ν_mµ_n µrν = mµ;νnµrν = mµnµrν ;ν − mµ_n µrν;ν − m µ_n µ;νrν . (2.111)

The first two terms vanish since mµ_n

µ = 0. So

∇ν_mµ_n

µrν = −mµrνnµ;ν = −marbKab (2.112)

Putting again the factor of two from the surface element, we end up with the ADM expression 2.73.

The nice property of the Komar integrals is that they contain Killing vectors. This way if we use Stokes and eq.(B.2-B.6) we get expressions involving the energy tensor. Explicitly I S ∇νξµdSµν = 2 Z Σ (∇νξµ);νdΣµ = 2 Z Σ ξµdΣµ= −2 Z Σ Rµ_νξνdΣµ. (2.113)

For spacelike hypersurfaces Σ we have dΣµ= −nµ

√

hd3y. So by virtue of Einstein’s equations we finally arrive to the results for the charges

MKomar = −2 Z Σ Tµν− 1 2T gµν nµkν√hd3y

(31)

JKomar = Z Σ Tµν− 1 2T gµν nµmν√hd3y . (2.114) Again, these hold for stationary spacetimes. We therefore have an expression for conserved charges involving geometrical quantities and the stress energy tensor.

As for the electric charge, suppose we have a charge distribution on Σ, with current density jµ_{. Then the charge will be}

Q = Z Σ jµdΣµ= 1 4π Z Σ Fµν_;νdΣµ= 1 8π I S FµνdSµν , (2.115)

(32)

3 Black hole thermodynamics

The renowned work done by Hawking, Bardeen and Carter [103] linking the usual thermodynamics to black holes, formulating the ”Black hole mechanics” seems to be strong indication for the deep connection between quantum physics and gravity [4].In this project, the first law will be of outmost importance, but we will also make a sketch for the other laws.

It began when Christodoulou and Ruffini in 1971 [32], when examining the gedanken Penrose process of extracting energy from black holes, found a mass formula for the Kerr Newman black hole

M2 = Mirr+ Q2 4Mirr 2 + J 2 4M2 irr , (3.1)

where Mirr = 1₂pr2++ a2 is an integration constant called irreducible mass and it

turns out it obeys the inequality

δMirr ≥ 0 , (3.2)

with the equality holding for reversible transformations (changes in the black hole charges through processes like superradiance etc) and the inequality for irreversible ones. In order to prove this relation we solve 3.1 with respect to Mirr

Mirr = 1 2 q M +pM2_{− Q}2_{− a}22 + a2 ₌ 1 2 q 2M2_{− Q}2_{+ 2M}p_M2_{− Q}2_{− a}2 (3.3) and then we vary. We will prove the relation for the Kerr black hole for simplicity (non-charged), so that we can also do the parallelism with the Penrose process we showed in chapter 2. So we have (r+ = M + √ M2_{− a}2 _{= M +} 1 M √ M4_{− J}2₎ 4M_irr2 = r2₊+ a2 = 2M2 +√M4_{− J}2 _=⇒ 4MirrδMirr = 2M δM + 1 2 1 √ M4_{− J}2(4M 3_{δM − 2J δJ ) =} δM M√M2_{− a}2(2M 2√_M2_{− a}2_{+ 2M}3_{) −} J δJ M√M2_{− a}2 = δM (r2 ++ a2) − aδJ √ M2_{− a}2 =⇒ δMirr = (δM − Ω+δJ )(r+2 + a2) 4Mirr √ M2_{− a}2 , (3.4)

where we used the facts that J/M = a and Ω+= _r2a ++a2

for a Kerr black hole. We can see now where the Penrose process comes in play; we saw in (2.48) that

(33)

there is a limit in energy extraction from a black hole. The constraint was

δJ Ω+ ≤ δM , (3.5)

which leads to (3.1).

The free energy of the black hole is then M − Mirr and one cannot extract more

energy than this. That led Hawking to postulate the area law δA ≥ 0, for the area of a black hole, since it is related to the irreducible mass as A = 16πM2

irr for

the Kerr-Newman.

Then Hawking showed [208] that black holes are radiating, a peculiar property since they are black - one would not expect them to radiate. Basically he showed that for black holes that are being formed there is a flux of particles due to the time dilation at the horizon, and this flux can be observed at late times to have a Planck distribution of a black body radiation. This was named Hawking radiation after him, and the temperature at which the hole will radiate is named Hawking temperature and it is proven to be

TH =

κ 2π ,

where we set ~ = 1, kB = 1. So this means that black holes evaporate. Their

lifetime can be found by Stefan’s law dE dt ' −σAT 4 H , (3.6) where σ = π 2_k4 B 60~3_c2 (3.7)

is Stefan-Boltzmann’s constant (we restored the units just to demonstrate what this constant depends on) and A is the black hole area. Plugging then E = M, A ∼ M2_{, T}

H ∼ 1 we get the lifetime

τ ∼ M3 . (3.8)

For solar mass black holes, it is much bigger than the life of the universe.

3.1 The laws

Since black holes have a temperature, they have an entropy; as well as a free energy and thermodynamic properties. The laws of black hole mechanics are (in comparison with the usual thermodynamics) :

- Zeroth law: If hAand hB are two null surfaces comprising the bifurcate Killing

horizon, then the surface gravity κ is constant on both surfaces.

Equivalently, if H is a Killing horizon in a spacetime which is a solution of general relativity with the dominant EC, then κ is constant over H.

(34)

It was proved in [112] that there is a one to one correspondence between the constancy of the surface gravity and the Killing horizon being part of a bifurcate Killing horizon, under the condition that all orbits of ξµ on the horizon H diffeo-morphic to R and H has a cross section intersected only once by these orbits. - First law: Any perturbation to a stationary black hole parametrized by (M, J, Q), with horizon surface gravity κ, horizon angular velocity ΩH, electric potential ΦH,

and horizon surface A, will obey the equation dM = κ

8πdA + ΩHdJ + ΦHdQ , (3.9) where we put the subscript H implying this holds for any horizon, as we will see in chapter 6. Equation (3.9) is reminiscent of the first law of thermodynamics, with energy on the left side, work terms on the right and the entropy being proportional to the area, something proposed by Bekenstein and Hawking (BH) and called the BH entropy SBH = A₄. This was motivated by Hawking’s area law (1971), which

is the next one.

- Second law: If the null EC is satisfied, the surface area of the black hole cannot decrease dA ≥ 0. It is a fact that this area theorem is strikingly similar to the entropy law in thermodynamics. It was generalized by Bekenstein, what is known as the Generalized second law (GSL): the entropy of the black hole and the surrounding matter can never decrease.

- Third law: If the weak EC is satisfied, one cannot reduce the surface of a black hole to zero in finite steps.

3.2 Smarr’s formula

Larry Smarr found a mass formula for charged rotating black holes [22]. It was derived by applying Euler’s theorem for homogeneous functions to M. Using the expression for the area of the Kerr-Newman black hole

A = 4π

2M2+ 2 M4− J2− M2Q21/2− Q2

. (3.10) Solving this equation with respect to the mass yields

A2+ 16π2Q4+ 64π2M4+ 8πAQ2− 16πAM2_{− 64π}2_M2_Q2 ₌ = 64π2M4− 64π2J2− 64π2M2Q2 =⇒ M2 = A 16π − 4π J2 A + π Q4 A + Q2 2 . (3.11) A function is homogeneous of order n in a variable x if

(35)

Then deriving with respect to k and using the chain rule nkn−1f (x) = ∂f (kx) ∂(kx) ∂(kx) ∂k =⇒ nf (x) = x ∂f ∂x , (3.13) where in the last step we set the parameter k = 1. This is easily generalized for an arbitrary number of variables

nf (x0, x1, ...xr) = xi∂f

∂xi , (3.14)

where summation is implied. We can see that the mass is homogeneous of degree

1

2 in (J, A, Q

2_{) so one gets the simple bilinear form (we will see how the factors}

come about below):

M = 2 κ

8πA + 2ΩHJ + ΦHQ . (3.15) Note that this is not the integration of the differential formula (3.9). In fact there is a more general expression of the Smarr formula. Let us start from the Komar expression of the angular momentum. Since we have two boundaries (one at infinity and one at the horizon) Gauss’ law gives two terms, so we have

1 8π Z Σ mµ;ν_;ν dΣµ = 1 16π I S∞ mµ;νdSµν− 1 16π I H mµ;νdSµν . (3.16)

Hence the total angular momentum is (the integral calculated at the infinity boundary, i.e. the first term in the right hand side of eq 3.16)

J = 1 8π Z Σ mµ;ν_;ν dΣµ+ 1 16π I H mµ;νdSµν = − 1 8π Z Σ Rµ_νmνdΣµ+ JH , (3.17)

where we let JHbe the angular momentum evaluated at the horizon. Then

manip-ulating the Komar mass expression, essentially following the exact same procedure we have M = − 1 4π Z Σ k_;νµ;νdΣµ+ MH= 1 4π Z Σ Rµ_νkνdΣµ− 1 8π I H kµ;νdSµν = = 1 4π Z Σ Rµ_νξνdΣµ− 1 8π I H ξµ;νdSµν+ 2ΩHJ , (3.18)

where we used the fact that kµ = ξµ − Ωmµ _{as well as the previous equation.}

Now, we said before that dSµν = 2n[µrν]

√

σd2θ. On the horizon the normal nµ can be identified with the Killing vector that generates the horizon, ξµ, and the normalization is such that nµrµ = −1. Then the second term becomes (we will

denote √σd2_{θ = dA since it is the area element):}

− 1 4π I H ξν;µξ[µrν]dA = − 1 4π I H ξµξν;µrνdA = − 1 4π I H κξνrνdA = 1 4πκA , (3.19)

(36)

where on the second step we used Killing’s equation antisymmetry, on the third step the expression for the surface gravity on the horizon ,and on the last step the fact that it is constant over the horizon (zeroth law). Then the first term on eq.(3.18) can be transformed using Einstein’s equation as

1 4π Z Σ Rµ_νξνdΣµ= 2 Z Σ (Tµ_ν− 1 2T δ µ ν)ξνdΣµ . (3.20)

So the generalized Smarr formula becomes M = κA 4π + 2ΩHJ + 2 Z Σ (Tµ_ν − 1 2T δ µ ν)ξ ν_dΣ µ . (3.21)

Of course, for an isolated black hole, one has M = MH, J = JH since the total

charges of the spacetime will be equal to the black hole charges, so the Smarr formula quantities are those of the black hole. In order to obtain (3.15) in four dimensional electrovacuum, one takes the energy momentum tensor of the Maxwell theory, namely FµρFν_ρ− 1 4g µν_F ρσFρσ , (3.22) which is traceless 4πT = 4πgµνTµν = FµρFµρ− 1 44FρσF ρσ _{= 0 ,} _(3.23)

where the strength of the field Fµν = ∂µA¯ν − ∂νA¯µ = ¯Aν,µ − ¯Aµ,ν (or F = d ¯A

in differential geometry language) is antisymmetric and traceless and obeys the Bianchi identity. The quantity ¯Aµ is the gauge 1-form of electromagnetism (we

are using a bar to distinguish it from the horizon area A).

Now using the fact that the Lie derivative of the gauge potential is

£ξA¯µ= ξν∂νA¯µ+ ¯Aν∂µξν = ξνFνµ+ ξν∂µA¯ν+ ¯Aν∂µξν = ξνFνµ+ (ξνA¯ν),µ . (3.24)

Since the Killing vector field is also a symmetry for the electromagnetic field we have

£ξA¯µ = 0 (3.25)

There is also a quantity that is proven to be constant over the horizon, like the surface gravity, associated with the electromagnetic field, namely

ΦH = −( ¯Aµξµ) _H , (3.26)

which is the electric potential on the horizon. The potential is of course defined everywhere, and we will assume it is zero at infinity, as well as the gauge potential

(37)

¯

Aµ. Hence, equation (3.24) becomes

ξνFνµ= Φ,µ = Φ;µ , (3.27)

since Φ is a scalar function.

A way to prove that the potential is constant on a horizon is assuming we have a bifurcate Killing horizon. Then, on the bifurcation sphere S we have ξµ S_{= 0.}

That means also

Eµ= ξνFνµ S

= 0 =⇒ dΦ = E = 0 =⇒ ΦS = const.S (3.28) Then since Φ is constant on orbits of ξ, £ξΦ = 0 , it is constant all over the horizon.

A more general proof without the need of a bifurcate Killing horizon, needs the use of a property of Killing horizons

Rµνξµξν H= 0 =⇒ Tµνξµξν H= 0 =⇒ EµEµ H= 0 . (3.29)

Then using the fact that Eµ_{is proportional to ξ}µ_{on the horizon (since ξ}µ_ξν_F νµ= 0

due to the strength’s antisymmetry) with a proportionality σE, we can attempt to

Lie transport the electric potential along the horizon, with a tangent vector field tµ as £tΦ = tµΦ,µ = tµEµ H = σEtµξµ H = 0 =⇒ Φ= const.H (3.30) We also have (denoting Fµν_F

µν = F2 for simplicity):

FρνA¯ν;ρ = FρνFρν + FρνA¯ρ;ν =⇒ F2 = 2FρνA¯ν;ρ , (3.31)

since Fµν _{is antisymmetric. But since the Killing vector generates a symmetry it}

is also true that £ξ FµνA¯ν = 0. That means

£ξ FµνA¯ν = ξρ FµνA¯ν ;ρ− ξ µ ;ρ F ρν _¯ Aν = (ξρFµνA¯ν − ξµFρνA¯ν);ρ+ ξµFρνA¯ν;ρ = 0 (3.31) =⇒ ξµF2 = −2(ξρFµνA¯ν − ξµFρνA¯ν);ρ, (3.32)

where we used Killing’s equation and Maxwell’s equations Fµν

;µ = 0. They are

source free since the charge is in the singularity, while we integrate from the horizon to infinity. Then substituting the energy tensor we have

2 Z Σ Tµ_νξνdΣµ = 1 2π Z Σ FµρFνρ+ 1 4δ µ νF 2 ξνdΣµ= 1 2π Z Σ FµρΦ;ρdΣµ− 1 8π Z Σ ξµF2dΣµ =

(38)

− 1 2π Z Σ FµρΦ_;ρdΣµ+ 1 4π Z Σ (ξρFµνA¯ν − ξµFρνA¯ν);ρdΣµ. (3.33)

Then we can use Gauss law with two boundaries, since both tensors in the integrals are antisymmetric. The first term will be

1 2π Z Σ FµρΦ ;ρdΣµ= Φ∞ 4π I S∞ FµρdSµρ− ΦH 4π I H FµρdSµρ = −ΦHQ , (3.34)

since the electric potentials are constant at infinity (zero) and at the horizon. As for the second term, we use the convention that the gauge potential vanishes at infinity so only the horizon boundary contributes

1 4π Z Σ (ξρFµνA¯ν − ξµFρνA¯ν);ρdΣµ= 1 4π I H ξ[ρFµ]νA¯νdSµρ = 2ΦHQ , (3.35)

for which we used the fact that Eµ _{= F}µν_ξ

ν is proportional to ξµon the horizon and

the null condition ξ2 _{= 0. So for the Einstein Maxwell theory, one finally arrives to}

the required result (3.15). If there is external matter or current contribution [105] one obtains

M − 2ΩHJ − ΦHQ −

κA

4π = Mmatter− 2ΩHJmatter− ΦHQcurrent . (3.36)

3.3 The first law

3.3.1 Equilibrium state version

The original first law by Bardeen, Carter and Hawking [103] was a slightly dif-ferent statement: two neighboring solutions were considered, both stationary and axisymmetric, and the variation formula was describing the difference of these two solutions. Specifically, the way it is stated is that if there are two stationary solu-tions (M, g) and (M+δM, g+δg) then the difference in masses, angular momenta, areas and charges is eq. (3.9).

The proof begins by assuming that the Killing fields are conserved, they remain unchanged from one solution to the other i.e.

δkµ= δmµ= 0 . (3.37) Then denoting the metric variation δgµν = hµν, the Killing vector generating the

horizon will obey

δξµ = mµδΩH =⇒ δξµ= δ(gµνξν) =

(39)

Varying the formula (3.21) yields δM = 1 4π(δκA + κδA) + 2(ΩHδJ + δΩHJ ) + 1 4πδ Z Σ Rµ_νξνdΣµ, (3.39)

where we reinserted the Ricci tensor for simplicity. Let’s start by manipulating the first term. Following Hawking-Carter-Bardeen’s paper [103], we have the Lie derivatives vanishing

£ξδξν = ξµδξν;µ+ δξµξµ;ν = 0 (3.40)

£δξξν = δξµξν;µ+ ξµδξµ;ν = 0 (3.41)

since as we said the Killing fields remain unchanged, and hence these are essen-tially Lie transporting the Killing vectors along their orbits. Afterwards, using the definition for the surface gravity κ = 1₂(ξ2),µnµ and varying (we are dropping the

indication that equations hold on the horizon but it is implied) we get δκ = 1 2(ξ 2₎ ,νδnν + 1 2n ν_(ξµ_δξ µ+ ξµδξµ),ν = δnνξµ_;νξµ+ nνξµ;νδξµ+ 1 2(n µ ξν + nνξµ)δξµ;ν + δξµ;νξµnν +1 2n ν_(ξ µ;νδξµ− ξµδξµ;ν− ξ µ ;νδξµ− ξµδξν;µ) . (3.42)

The first two terms in the last line vanish because of (3.41) and Killing’s equation (in order to write ξµ;νδξµ = −ξν;µδξµ) and the last two terms vanish because of

(3.40). For the first two terms in the second line, we use a null tetrad [103],also known as the Newman-Penrose formalism, on the horizon to express the metric as gµν = −(nµξν + ξµnν) + ηµη¯ν + ηνη¯µ, (3.43)

where ηµ, ¯ηµ are complex conjugate null vectors lying on the horizon, and they

are orthogonal to the two normals. The convergence and shear of the horizon generators vanish [103], i.e.

ξµ;νηµη¯ν = ξµ;νηµην = 0 (3.44)

and the same will apply for the new Killing vector δξµ_{, which on the horizon is}

proportional to the old one. The first one of (3.44) vanishes since if there was a convergence, there would be a caustic (singular point) at the horizon in finite affine distance for the null generator. The second one vanishes due to the fact that ξµ;ν

is antisymmetric. Furthermore, on the horizon we can decompose the derivative of the Killing vector as [106]

(40)

which comes after doing a decomposition of the derivative on the horizon with respect to the null tetrad, and taking into account Frobenius’ theorem, Killing’s equation and the surface gravity expression. Then we can express these two terms as

ξµ_;ν(δξµnν + ξµδnν) = ξµ;νξµδην , (3.46)

which vanishes because of (3.45). So we end up with δκ = 1

2(n

µ

ξν + nνξµ)δξµ;ν + δΩH , mµ;νξµnν , (3.47)

where we used (3.38). Then taking advantage of (3.43) and (3.44) first term can be rewritten as 1 2(n µ_ξν _{+ n}ν_ξµ_)δξ µ;ν = 1 2(−g µν_{+ η}µ_η_¯ν _{+ η}ν_η_¯µ_)δξ µ;ν = − δξµ_;µ = −1 2(h µ νξ ν + mµδΩH);µ = − 1 2h µ ν;µξ ν , (3.48) because m ;µ

µ = 0 (Killing’s equation), δΩH ;µ = 0 and hµνξµ;ν = h(µν)ξ(µ;ν) = 0

by virtue of Killing’s equation and symmetry of the metric. Since we chose such notation, our surface 2-form now is dSµν = (nµξν − nνξµ)dA. So the second term

of (3.47) is just δΩHmµ;νξµnν = 1 2δΩH(m µ;ν ξµnν + mν;µξνnµ) = − 1 2δΩHm µ;νdSµν dA . (3.49) As for the other term, exploiting the orthogonality and normalization conditions ξ2 _{= 0, ξ} µnν = −1, one has −1 2h ;µ µν ξ ν = −1 2ξµh µ ;λ λ = 1 2ξµnνξ ν hµ ;λ_λ −1 2ξνξ ν nµhµ ;λ_λ = − 1 2(nµξν−nνξµ)ξ ν hµ ;λ_λ = 1 2(nµξν− nνξµ)ξ µ_hν ;λ λ − 1 2(nµξν − nνξµ)ξ µ_hλ ;ν λ , (3.50)

where we added a term that is identically zero due to £ξh = ξνh;ν = 0 and ξ2 = 0.

Moving on this is equal to

(nµξν − nνξµ)ξµ∇[λh ν] λ = ξ µ_∇[λ_hν] λ dSµν dA . (3.51) Plugging these back to eq (3.47)

δκdA = 1 2dSµν(2ξ µ_∇[λ_hν] λ− m µ;ν_δΩ H) . (3.52)

(41)

out of the integral we get Aδκ = −8πJHδΩH + I H ξµ∇[λ_hν] λdSµν . (3.53)

Then using the fact that the asymptotic metric variation is related to the mass variation by [105] (see 2.80) hµν = 2 δM r δµν + O(1/r 2 ) , (3.54) we have I S∞ dSµνkµ∇[λh ν] λ = −δM I S∞ kµ(nνr,νrµ+ 2nµ)dA = 4πδM . (3.55)

Then, postulating that the horizon surface is invariant under rotations, i.e. mνdSµν =

0, we have using Gauss’ law I H ξµ∇[λ_hν] λdSµν = I H kµ∇[λ_hν] λdSµν = Z Σ kµ(∇[ρhν]_ν);ρdΣµ− 4πδM . (3.56)

Combining all the results, we end up with Aδκ + 8πJHδΩH + 4πδM =

Z

Σ

kµ(∇[ρhν]_ν);ρdΣµ (3.57)

or in the language of differential geometry Aδκ + 8πJHδΩH + 4πδM =

Z

Σ

h[µ;ν]_µ _;ν∗ k . (3.58) Using the Einstein-Hilbert action we can express the metric variation as

− 2h[µ;ν] µ ;ν = − 1 √ −gδ(R √ −g) + Gµν_h µν . (3.59)

Before proving this, lets see what the left side is exactly: 2h[µ;ν]_µ _;ν = hµ;ν_µ _;ν− hν;µ

µ ;ν = gµνδgµν− ∇µ∇νδgµν , (3.60)

where = ∇µ∇µ is the box operator. So equation (3.59) can be rewritten as

δ(R√−g) =√−gGµνδgµν+

√

−g[gµν(δgµν) − ∇ν∇µδgµν] , (3.61)

where we can pull out the metric tensor on the first term of the last step since it is compatible. We can already see where this is going; the left side of eq.(3.61) is the variation of the Einstein Hilbert action and the right side is the Einstein tensor, which should vanish when the equations of motion are satisfied (looking at the asymptotic metric we can see that it is a vacuum solution). We will prove that their sum is equal to the variation of the Einstein Hilbert action.

Geometric relations of black hole thermodynamics

MSc Physics

Theoretical Physics