Classical Principles of Holography A report presented in partial fulfilment of the requirements for the degree of Bachelor of Physics Juan José Segura Flórez Degree Candidate in Physics Faculty Advisor: Diego Mauricio Gallego Mahecha Universidad Pedagógica y Tecnológica de Colombia Tunja-Boyacá April 7, 2021 Contents 1 From Kerr Black Hole to Hawking’s Theorem 1.1 2 Kerr Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.1 Event Horizon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Stationary Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 The Penrose Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.4 Irreducible Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.5 Hawking’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.5.1 15 The Area Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Entropy and Information Theory 2.1 21 Information Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.1.1 Definition of Information . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.1.2 Unit Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.1.3 Unknown Information and Known Information . . . . . . . . . . . . . 24 2.1.4 Shannon’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.2 Thermodynamics and Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.3 Boltzmann Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.3.1 Statistical Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.3.2 Boltzmann Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.3.3 Connection between Statistical Mechanics and Thermodynamics . . . 38 i 3 Black Hole Thermodynamics 40 3.1 Bekenstein Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.2 Semi-Classical Estimation of the Constant η . . . . . . . . . . . . . . . . . . 42 3.3 The Four Laws of Black Hole Mechanics . . . . . . . . . . . . . . . . . . . . 44 4 Holographic Principle 4.1 4.2 49 Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.1.1 Hawking Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.1.2 Bekenstein Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.1.3 Susskind Process and Spherical Entropy Bound . . . . . . . . . . . . 53 4.1.4 Bekenstein Bound and Spherical Bound . . . . . . . . . . . . . . . . . 54 A Holographic Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.2.1 Degrees of Freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.2.2 Fundamental System . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4.2.3 Degrees of Freedom According to Local Field Theory . . . . . . . . . 56 4.2.4 Degrees of Freedom According to Spherical Entropy Bound . . . . . . 57 4.2.5 Local Field Theory Estimation is Wrong? . . . . . . . . . . . . . . . . 59 4.2.6 Unitarity and Holographic Principle . . . . . . . . . . . . . . . . . . . 60 4.2.7 Unitarity and Black Hole Complementarity . . . . . . . . . . . . . . . 61 5 Discussion and Conclusions 64 A Foundations of General Relativity 68 A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 A.2 Causal Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 A.3 Future and Past . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 A.4 Conjugate Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 A.5 Congruence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 ii Bibliography 93 iii Abstract A detailed review is made on the classical concepts of General Relativity that allow us to heuristically infer a first formulation of the so-called Holographic Principle. With the study of rotating black holes and their properties related to the area of the Event Horizon, the Hawking Area Theorem is formally stated. We also review of the concept of Entropy to establish the theoretical framework that justifies the connection between the horizon area and entropy through Information Theory on what is known as the Bekenstein Entropy. Including black holes in the laws of thermodynamics imply a generalization of the Second Law. This postulated generalized law has as a consequence a limit in the entropy of the Universe. Finally, under a weak interaction and spherical symmetry considerations it is found a that the entropy of a system is bounded from above by one-fourth of the area (in natural units) of the minimum sphere containing the system. iv Acknowledgements To my thesis supervisor, Dr. Diego Gallego Mahecha, who during this long process has supported me in my growth, not only professionally but also personally. Thank you for having believed in me, for your infinite patience and for always pushing me to be better and better and to reach my full potential. Now I am sure from everything I have lived so far that physics is my life, thank you my friend. To my parents José Gabriel Segura and Selene Flórez Gutierrez, without them I would not have been able to finish this. I love you unconditionally, there is nothing that I cannot do in this world and in this life with you by my side. If I am a physicist today it is because of you and those countless math books that are still in the old library, lined with my childish handwriting. And finally I dedicate this work to people who suffer from depression: every time you do something (no matter how small) is a victory for you, believe in yourself and you will be unstoppable; at the end of the day, there is no smile more genuine than yours. v Introduction Black holes are regions of the spacetime in which the gravity is so intense that if anything falls into it, it cannot leave again. Currently, black holes are a common concept but it took more than two hundred years to become part of the popular culture. John Wheeler was the person who gave its name, however, the first person who talked about black holes was the astronomer John Michell. At Michell’s time it was know that the speed of light was finite, so it was natural to think that gravity could affect light. In 1783, Michell wrote an article in the Philosophical Transactions of the Royal Society of London explaining that if a star is massive and compact enough it will have a gravitational field so strong that the light cannot escape [1]. Theoretically, black holes are solutions to the Einstein’s field equations. And it is known that there are only four types of black holes described by three physical observables: the mass, charge and angular momentum. The first discovered black hole solution to the Einstein equations was the Schwarzschild Black Hole which has neither charge nor angular momentum [2] [3] [4]. A general solution of the Schwarzschild solution is the so-called Kerr black hole, characterized by its mass and angular momentum. The importance of this type of black hole is in the possibility of studying properties and characteristics that are not found in Schwarzschild solutions. For example, in Kerr-type black holes there is an region where energy can be extracted known as the Ergosphere [3] [4] [5] [6]. A general study of the causal structure of black holes shows that the area of the event horizon cannot decrease, a result known as Hawking Area Theorem. This allows to establish vi a starting point to define the entropy in black holes. According to the Bekenstein-Hawking Entropy, the entropy of a black hole is a quarter of the area of the event horizon; something that leads to the formulation of a natural extension of the Second Law of Thermodynamics known as the Generalized Second Law. When we apply the Generalized Second Law to certain scenarios an Entropy bound is found for the contribution of matter in the Universe. In situations of high symmetry, the entropy is bounded by the area of the region (spherical) that contains an arbitrary matter system. That is, the maximum information that describes the system within a spherical region is on its surface, this principle is known as Susskind Holographic Principle [7]. One of the first robust results in quantum gravity came with it a paradox: the loss of Hawking information. This affirms that the evaporation of a black hole is non-unitary and irreversible generating a debate about the approach of the problem and its possible solution [8]. The loss of information arises as a consequence of studying the thermodynamic properties of black holes [9] and all this study has its common origin in the work carried out by Bekeinstein on a formal definition of the entropy of black holes, generalizing the second law of thermodynamics to a wide range of scenarios [10]. On October 25, 1983, Gerard t’Hooft wrote an article in honor of Abdus Salam and on it he stated the general notions of what would later be called the Holographic Principle [11]. t’Hooft said that to reconcile the physics associated with the phenomenon of collapse of bodies with the postulates of quantum mechanics, it would be necessary to consider the possibility that, according to quantum mechanics at Planck scales, spacetime is not a 3 + 1 dimensional manifold but rather is a two-dimensional Boolean network; in this way, it returns the unitarity to the Hawking evaporation process and consequently restricts possible models of quantum supergravity [11]. Taking into account the previous comment, the present monographic work aims to spin and exploit the theoretical apparatus of Classical General Relativity in strong regime, to propose infer of the holographic principle through the deduction of the thermodynamic properties of vii black holes. Currently it is important to be able to study the thermodynamics of different types of black holes or different spacetimes, since it allows establishing phase transitions and in many cases making a holographic study of entropy [12, 13, 14, 15, 16, 17, 18, 19]. Indeed, with the rise of the AdS/CF T conjecture [20], the fact that the holographic principle is a fundamental tool in the compression of quantum gravity has become more evident. In this report a detailed development of the classical principles that motivate the Holographic Principle is made. That is to say, every result that gives rise to Thermodynamics in black holes is studied and developed; specifically, the approach used by Bekenstein to define the concept of entropy and its Universal Limit for a given spacetime [10] [21] [22]. Our goal is heuristically motivated a first approximation of the Holographic Principle [7]. The report is structured as follows: in chapter 1 a study is made of the properties concerning rotating black holes, from which a rigorous proof of the so called area theorem is done. In the second chapter, a rather complete review of the concept of Entropy and its possible interpretations is made. Here special emphasis is placed on the modern interpretation given by means of Information Theory [23]. In the third chapter we use of the theoretical tools of the previous ones to closely follow Bekenstein’s calculations and developments on the subject of entropy in black holes [10]; where we finally state the four dynamic laws of black holes [12]. Finally, in chapter 4 the consequences of considering the Generalized Second Law as a law of the Universe are developed, through which we arrive (under a series of restrictions) at a preliminary formulation of the Holographic Principle [7]. In the fifth chapter a discussion is made about the validity of the results given by General Relativity and those obtained through Quantum Fields Theory and we conclude. Appendix A presents a summary of the modern topological tools currently used to treat issues related to the causal structure of black holes. We use Natural Units c = G = ~ = 1 unless otherwise noted. 1 Chapter 1 From Kerr Black Hole to Hawking’s Theorem 1.1 Kerr Metric In a set of appropriate coordinates, known as Boyer-Lindquist coordinates, the metric that describes space for a rotating star that has collapsed, also known as a Kerr black hole, is given by [5, 3, 24]:  2M ar sin2 θ 2M r ρ2 2 2 dt − ds = − 1 − 2 (dtdφ + dφdt) + dr ρ ρ2 ∆ i
sin2 θ h 2 2 2 2 2 − a ∆ sin θ dφ2 , r + a + ρ2 dθ2 + ρ2 2  (1.1) where, ∆(r) = r2 − 2M r + a2 , (1.2) ρ2 (r, θ) = r2 + a2 cos2 θ. (1.3) and 2 The Kerr metric has two free parameters: M and a which denote the mass and the angular momentum per unit mass respectively: a≡ J , M (1.4) where J is the black hole angular momentum [4]. These set of coordinates are such that when the angular momentum vanishes, i.e., if a → 0, the metric reduces to the Schwarzschild metric in its own coordinates. Indeed, the Kerr metric is a generalization of the Schwarzschild metric. Analysing the geodesic equations extracted from this metric we can infer the isometries of the spacetime. These equations are obtained by minimizing the length between two points in the spacetime, a procedure that closely follows the principle of least action, ending in a set of Euler-Lagrange equations: ∂L d ∂L − = 0, µ ∂x dλ ∂ ẋµ (1.5) With λ a parameter for the path, the dot denoting derivatives with respect to this parameter and L the line element which is given by: L= r gµν dxµ dxν dλ dλ (1.6) and therefore: d dλ  2M r 1− 2 ρ  dt 2M ar sin2 θ dφ + dλ ρ2 dλ  = 0, (1.7) d dλ 
dφ 2M ar sin2 θ dt sin2 θ 2 2 2 2 2 (r + a ) − a ∆ sin θ − + ρ2 dλ ρ2 dλ 3  = 0. The first equation of (1.7) indicates that there exists a conserved quantity in the direction of the t coordinate since complies: ∂L = 0, ∂t (1.8) while the second equation of (1.7) tells us about another quantity that is conserved in the direction of the coordinate φ because: ∂L = 0. ∂φ (1.9) These isometries are described by the Killing vectors [5, 25]: K = (1, 0, 0, 0), (1.10) R = (0, 0, 0, 1). The existence of the Killing vectors K and R shows that the Kerr spacetime is according to [5]: 1. Stationary: The line element ds does not depend explicitly of t. As consequence there are three Killing vector fields on the Kerr space-time: K µ = (∂t )µ , Rµ = (∂φ )µ and K µ + ΩH Rµ , with ΩH been a constant 1 2. Axisymmetric: The line element ds does not depend explicitly of φ. As consequence there is the Killing vector Rµ . However, the Kerr spacetime is: 1. Not static: The line element is not an invariant under temporal inversion transformation, in other words, the transformation: t → −t 1 In general, every linear combination of K µ y Rµ is a Killing vector of the Kerr spacetime. 4 is not an isometry. 2. Invariant under simultaneous inversion of t and φ. The last statements are evident for spinning bodies; the temporal inversion produces a body spinning in the opposite direction. 1.1.1 Event Horizon Black holes have a region of the spacetime that characterizes them which is known as the event horizon. This region is the reason why black holes are given such a name, because once something crosses the event horizon, there is no way for it to be found or known about it outside of the black hole again. Identifying where an event horizon is given a metric, is a complex task because it is a global concept, i.e., it depends on the global structure of spacetime where the black hole is located. According to [5], the event horizon is defined as: An event horizon is a hypersurface separating those spacetime points that are connected to infinity by a timelike path from those that are not. In fact, the event horizon is the boundary of the black hole, which is generated by the light paths that move eternally on that edge [1]. In other words, the event horizon is a null hypersurface, never reaching the far future. In general, finding the event horizon for a given spacetime is not a easy task. However, there are cases, like the one of the Kerr space in (1.1), where an appropriate coordinate system reveals this region of space relatively clear. For instance, since the metric (1.1) is a stationary metric, it has an asymptotically timelike Killing vector ∂t , therefore ∂t gµν = 0. On hypersurfaces where t = constant, we can choose the coordinate system (r, θ, φ), for which (1.1) at infinity looks like Minkowski metric in spherical polar coordinates. For values of r = constant we have timelike cylinders with topology S 2 × R at r → ∞. If a coordinate system is correctly chosen, such that as we decrease r from infinity, the r = constant hypersurfaces remain timelike until a certain value 5 r = rH for which it is everywhere null, then rH will be the position in the radial coordinate of the event horizon. The one-form ∂µ r is normal to the hypersurfaces described above, therefore, the event horizon in Kerr spacetime will be the hypersurface where: g µν ∂µ r∂ν r = g rr (rH ) = 0, (1.11) For the Kerr metric in Boyer-Lindquist coordinates (1.1) the condition above reads: ∆ = r2 − 2M r + a2 = 0. (1.12) Solving the equation (1.12) we get: r± = M ± √ M 2 − a2 (1.13) The value r+ denotes the radial position of the outer event horizon of the Kerr black hole, while r− denotes the radial position of the inner event horizon of the Kerr black hole (see figure 1.1). 1.2 Stationary Limit Rotating black holes (Kerr or Kerr-Newman) have a structure that differentiates them from non-rotating black holes called Stationary Limit Surface. Because the Kerr metric is not static, the event horizon at r± is not a Killing Horizon for the Killing vector K = ∂t . A Killing horizon is a null hypersurface Σ where the norm of a Killing vector field χµ is zero [5]. For example, if you want to find the Killing horizon related to the Killing vector K of Kerr spacetime, we would have to solve the following equation: Kµ K µ = 0, 6 (1.14) where Kµ K µ = −
1 2 2 ∆ − a sin θ . ρ2 (1.15) If in the equation (1.15) r = r+ , then, ∆ = 0, ρ2 > 0 and a2 sin2 θ ≥ 0 getting: Kµ K µ = a2 sin2 θ ≥ 0. ρ2 (1.16) Equation (1.16) indicates that at the event horizon, the Killing vector K is spacelike (except when θ = 0 or θ = π). Another way to write equation (1.14) is as follows: (r − M )2 = M 2 − a2 cos2 θ. (1.17) All those Kerr spacetime events that satisfy the equation (1.17) are part of the Stationary Kerr black hole Outer event horizon Inner event horizon Stationary limit surface 10 y 5 0 5 10 20 15 10 5 0 x 5 10 15 20 Figure 1.1: Horizons and stationary surface for a Kerr black hole. The stationary boundary surface is in purple, the outer event horizon is black and the inner event horizon is red. The area between the black and purple colored surfaces is known as the ergosphere. Limit Surface. Note that if a → 0, then the value of r that would solve the equation (1.17) 7 would be rS = 2M (the Schwarzschild radius), besides the singularity r = 0 that in any we disregard. In the figure 1.1 all these features of the Kerr metric are shown. The name of stationary limit surface is because for a radius smaller than rS but greater than r+ an observer can not remain at rest [4], in fact, a massive body must move in the direction of rotation of the black hole. If a photon is emitted in the direction φ at some radius r in the equatorial plane (i.e., θ = π/2) near a Kerr black hole, then the line element satisfies: ds2 = 0 = gtt dt2 + gtφ (dtdφ + dφdt) + gφφ dφ2 , which can be rewritten as: gφφ  dφ dt 2 + 2gtφ dφ + gtt = 0. dt (1.18) Solving the equation (1.18) for dφ/dt we have [5]: gtφ dφ =− ± dt gφφ s gtφ gφφ 2 − gtt . gφφ (1.19) Evaluating the equation (1.19) at the stationary limit, two solutions are obtained: dφ dφ a . = 0, = 2 dt dt 2M + a2 (1.20) The first solution describes the instantaneous angular velocity of a directed photon against the rotation of a rotating black hole, while the second solution describes the instantaneous angular velocity of a photon in the same direction of rotation as the black hole. In conclusion, since the speed of a massive particle is strictly less than the speed of light, it is shown that on this boundary surface, no observer can remain at rest as mentioned before. This is an example of the phenomenon known as “dragging of inertial frames”. If the equation is evaluated for the value r = r+ , an expression is obtained that describes 8 the angular velocity ΩH of the event horizon: ΩH = dφ dt = r+ a r2 + a2 (1.21) Indeed, the event horizon angular velocity of a Kerr black hole is defined as the minimum velocity of a particle on the horizon. 1.3 The Penrose Process The first physical phenomenon that historically opened up the possibility of talking about thermodynamics in black holes was the so-called Penrose process [26]. Suppose we want to study the motion of a massive particle in Kerr spacetime, hence its four-momentum is defined by [5]: pµ = m dxµ , dτ (1.22) where τ and m are the proper time and the rest mass of the particle respectively. Then, the conserved quantities of the particle will be [5]:  2M r dt 2mM ar sin2 θ dφ + , E =m 1− 2 ρ dτ ρ2 dτ 2mM ar 2 dt m(r2 + a2 )2 − m∆a2 sin2 θ 2 dφ L=− sin θ + sin θ . ρ2 dτ ρ2 dτ  (1.23) Within the ergosphere the Killing vector K µ is spacelike, that is, k µ ≥ 0. Therefore, in the words of Roger Penrose: ...it is possible for the energy p0 to be negative in this region even the vector pa may be timelike (or null) and future pointing (as it must be for a real particle) [27]. 9 That is, regardless of whether pµ pµ ≥ 0 it is possible that within the ergosphere: E = −kµ pµ < 0. (1.24) The above inequality establishes a method for extracting energy from a rotating black hole, which can be described as follows: suppose we have a particle 1 thrown from infinity into a rotating black hole. Once inside the ergosphere, particle 1 decays into two particles (see figure 1.2), one of the particles escapes from the ergosphere and is thrown towards infinity, while the second one falls into the event horizon. Now suppose that the energy from the particle falling into the black hole is negative. Then, by energy conservation the energy of the outgoing particles is greater to the original one. If the energy of the incoming particle is less than the energy of the outgoing particle, Figure 1.2: Penrose process. this means that energy is being extracted from somewhere, the black hole. Kerr’s solution 10 is stationary, therefore there is a killing vector that is a linear combination of the Killing vectors K µ and Rµ defined by: χ µ = K µ + ΩH R µ . The above Killing vector becomes zero on the outer event horizon r+ since it is also a Killing horizon. The quantity ΩH is the angular velocity of the event horizon defined by the equation (1.21). The particle falling into the black hole “moves forward in time”, that is, it moves along a future-oriented geodesic (see section A.1). This, mathematically means that [5]: (p(2) )µ χµ < 0, (1.25) replacing the definition of χµ in the equation (1.25) the following inequality is obtained: L(2) < E (2) . ΩH (1.26) Since E (2) < 0 and ΩH > 0, then, L(2) < 0. In other words, in order to extract energy from a rotating black hole by the Penrose process, the particle that is thrown into the black hole must be thrown against the rotation of the black hole. The discussion made so far allows us to deduce that the change in the angular momentum and the mass of the black hole denoted by δJ and δM respectively is: δM = E (2) , (1.27) δJ = L(2) . Replacing this in the equation (1.26), we obtain: δJ < δM . ΩH (1.28) In conclusion, the angular momentum of a rotating black hole should be reduced each time 11 a Penrose process is used to extract a given amount of energy. 1.4 Irreducible Mass In 1970 Demetrios Christodoulou showed that in Penrose processes there is a quantity proportional to the area of the event horizon that cannot be decreased called irreducible mass [28]. To calculate an expression of the irreducible mass Mirr of a rotating black hole in terms of the area A of the event horizon, the quantity Mirr must be precisely defined. According to [29], the irreducible mass of a rotating black hole is defined as: The irreducible mass is the final mass of a charged or rotating black hole when its charge or angular momentum is removed by adding external particles to the black holes. It is the mass observed at infinity. In other words, it is the mass of the Schwarzschild black hole that results from extracting all rotational (and electromagnetic) energy from a Kerr (Kerr-Newman) black hole using Penrose process. On the other hand, the event horizon area of a Kerr black hole is defined as the integral of the induced volume element: Z p A= |γ|dθdφ, (1.29) where |γ| is the absolute value of the determinant of the induced metric γij . The metric γij is obtained by evaluating the metric of the equation (1.1) for values r = r+ (outer radius of the event horizon) and t = t0 constant (see subsection 1.1.1). As a result, the following line element is obtained: γij dxi dxj = ds2 (dt = 0, dr = 0, r = r+ ) = 2 (r+  2 (r+ + a2 )2 sin2 θ dφ2 . + a cos θ)dθ + 2 2 2 r+ + a cos θ 2 2 2  (1.30) 12 Therefore, the matrix representation of the metric γ will be:  2 2 2  r + a cos θ γ= 0 0 2 +a2 )2 sin2 θ (r+ 2 +a2 cos2 θ r+   , (1.31) whose determinant is: 2 |γ| = (r+ + a2 )2 sin2 θ. (1.32) By replacing (1.32) in (1.29) and solving the integral, the event horizon area of a rotating black hole is given by: 2 A = 4π(r+ + a2 ). (1.33) If the limit is used when a → 0 in the equation (1.33), the following expression of the irreducible mass is obtained [5]: 2 Mirr = A . 16π (1.34) Using the value for r+ we find: 2 Mirr =  1 2 √ 4 M + M − M 2 a2 , 2 or equivalently: 2 Mirr  1 2 √ 4 2 = M + M −J . 2 (1.35) Differentiating the equation (1.35) and using (1.21), an expression can be extracted that describes how Mirr changes as the angular momentum J of Kerr black hole changes through the Penrose process [5]: δMirr =
a −1 √ δM − δJ . ΩH 4Mirr M 2 − a2 13 (1.36) Figure 1.3: Gravitational waves. Taking into account the inequality of the equation (1.28), the following is then obtained: δMirr > 0. (1.37) In most physical processes, the irreducible mass cannot decrease. Therefore, from the equation (1.34), it is immediately deduced that the area A of the event horizon cannot decrease. However, the above can be misleading because there are cases in which δMirr ≤ 0 and the area of the event horizon satisfies δA ≥ 0. For instance, suppose we have a set of Schwarzschild black holes {B1 , B2 , B3 , B4 , B5 } which merge to form a Schwarzschild black hole B6 (see figure 1.3). By definition, the irreducible mass of a Schwarzschild black hole is its total mass, however, in the process of merging to form the black hole B6 energy is released in the form of gravitational waves. The above would mean that: MB1 + MB2 + MB3 + MB4 + MB5 > MB6 , 14 (1.38) or equivalently: δMirr < 0, (1.39) however, in this process δA ≥ 0. The previous counterexample shows that the proposition: δMirr ≥ 0 → δA ≥ 0, (1.40) is false. To prove that the event horizon area of a black hole cannot decrease, another approach is needed which will be developed in the following. 1.5 Hawking’s Theorem In 1971 Stephen Hawking proved in general way that the event horizon area of a black hole cannot decrease [30]. In this section we will use the techniques reviewed in appendix A in order to demonstrate Hawking’s theorem. 1.5.1 The Area Theorem The event horizon Area Theorem cannot be stated correctly without the mathematical development presented in the appendix A. Despite, there are intuitive ways to understand the theorem, its formal development is not intuitive. One of the strongest restrictions that is imposed is to assume that the spacetime is asymptotically flat and in the region outside the black hole, the theory of General Relativity is deterministic, that is, outside the event horizon, spacetime is hyperbolic. A spacetime is asymptotically flat if and only if its conformal diagram is similar to the one shown in figure 1.4. More precisely, the spatial infinity of the conformal diagram for an asymptotically flat spacetime should coincide to the one for Minkowski. Figure 1.4 shows schematically the conformal diagram for this kind of spacetime. According to [24] a spacetime is defined to be a asymptotically flat spacetime if the physical 15 Figure 1.4: Asymptotically flat spacetime.In the gray region of this spacetime infinity looks like the infinity of Minkowski spacetime. spacetime can be mapped into a new, “unphysical” spacetime via a conformal isometry with properties similar to that of the Minkowski case. However, for Hawking’s Area Theorem the spacetime that we are interested in defining is a special type of asymptotically flat spacetime known as asymptotically predictable spacetime which is defined as follows [24]: Definition 1.5.1. Let (M, gab ) be a spacetime with associated unphysical spacetime (M̃ , g̃ab ). We say that (M, gab ) is strongly asymptotically predictable if in the unphysical spacetime there is an open region Ṽ ⊂ M̃ with ψ[M ] ∩ J − (I + ) ⊂ Ṽ such that (Ṽ , g̃ab ) is globally hyperbolic 2 . An asymptotically predictable spacetime has a black hole, if and only if, ψ[M ] 6⊂ J − (I + ), where the map ψ : M → ψ[M ] ⊂ M̃ is a conformal isometry [24]. Then generally a black hole, or more accurately a black hole region, is defined as follows3 : Definition 1.5.2. The black hole region, B, of a space-time (strongly asymptotically predictable) is defined to be B = [M − J − (I + )]. This definition of B allows us to formalize the discussion about the existence of a region B in a spacetime: 2 3 If in a spacetime there is a Cauchy surface then the spacetime is called globally hyperbolic From here M is ψ[M ]. 16 Proposition 1.5.1. Let (M, gab ) be a strongly asymptotically predictable spacetime, then, ∃B ⊂ M̃ , such that, B = [M − J − (I + )], if and only if, M 6⊂ J − (I + ). Proof. If M ⊂ J − (I + ), then by definition4 :
∀x x ∈ M → x ∈ J − (I + ) , (1.41) however, for the region B by definition we have: x ∈ B ↔ x ∈ M ∧ x 6∈ J − (I + ), (1.42) If B 6= ∅, then we would have a contradiction because x ∈ J − (I + ) by (1.41). Hence, ∃B if and only if M 6⊂ J − (I + ). This means that for a region B to exist, the spacetime M must not be contained in J − (I + ), that is, the causal past of the infinite null does not must be the spacetime M itself, since there must be a region that is causally disconnected from I + . For example, in Minkowski spacetime B = ∅ because J − (I + ) = M . Recall that black holes have a region of no return called event horizon, which had the problem of not being clearly identified or defined only using the metric of a spacetime. However, the topological techniques developed so far allow us to avoid this problem and give a general definition of the event horizon. The simplest definition of an event horizon H would be: H = ∂B, (1.43) however, there are equivalent definitions that can be useful in proving some theorems (such as Hawking’s theorem), for example [24]: Definition 1.5.3. Let (M, gab ) be a spacetime with associated unphysical spacetime (M̃ , g̃ab ). 4 S. Here J − [S] is the causal past of a set S, I + is the infinity null future and ∂S is the Boundary of a set 17 If B 6= ∅ then ∃H ⊂ M̃ called event horizon defined by: H = ∂J − (I + ) ∩ M. (1.44) Figure 1.5: Schwarzschild spacetime. For example, if we use the definition of Horizon in Schwarzschild spacetime (see figure 1.5) we can notice that J − (I + ) 6= ∅ (blue color), therefore the boundary ∂J − (I + ) is the black line labeled as “Horizon ”, then H = ∂J − (I + )∩M = ∂J − (I + ) because ∂J − (I + ) ⊂ M . Hawking’s area theorem describes the behavior over time of the event horizon of a black hole by answering the question of what happens if two black holes B1 and B2 merge to form a third black hole B3 . For any physical process allowed, Hawking’s area theorem guarantees that: δA ≥ 0, (1.45) where A is the area of the event horizon of a black hole. Formally, Hawking’s theorem is stated in the following way [24]: Theorem 1.5.1. Let (M, gab ) be strongly asymptotically predictable spacetime for which Rab k a k b ≥ 0 for all null k a . Let Σ1 and Σ2 be spacelike Cauchy surfaces for globally hyperbolic region Ṽ with Σ2 ⊂ I + (Σ1 ) and let H1 = H ∩ Σ1 , H2 = H ∩ Σ2 , where H denotes the event 18 horizon, i.e, the boundary of the black hole of (M, gab ). Then the area H2 is greater than or equal to the area of H1 . For this theorem there are two important things to mention: 1. H is generated by a null congruence (see A.5): it can be demonstrated by assuming that the spacetime under study is “well behaved” (see [31] section 4 or [24] chapter 8). An intuitive way to demonstrate it is shown in [32], specifically, it is shown that ∂I + [S] for a S ⊂ M is made up of null geodesic segments (the same can be shown for ∂I − [S]). 2. All null k a obeys the null condition (Rab k a k b ≥ 0): this condition allows us to know the behavior of the null congruence that generates H, and therefore, the behavior of θ5 in H. The following is the proof of the Area Theorem: Proof. Suppose that at p ∈ H, θ < 0. Let Σ be a Cauchy spacelike surface for Ṽ and consider the two-dimensional surface H = H ∩ Σ. Because at p, θ < 0, by the focusing theorem there is a point q which is a conjugate point of p, therefore the null geodesic segment γ that passes through p and q can be smoothly deformed into a curve γ ′ that is timelike (see figure 1.6). This leads to a contradiction since H = ∂J − (I + ) ∩ M is achronal (this can be understood either with the boundary theorem of the future set F or by the simple fact that H is generated by null geodesics), therefore θ ≥ 0 anywhere in H. Suppose p ∈ H1 and q ∈ H2 . Since θ ≥ 0, then, there is a null geodesic segment that connects p with q, that is, there is a map φ : H1 → H2 . Since there are no pairs of conjugated points (the map is not onto), then the map can be such that φ(H1 ) ⊆ H2 . So 5 This quantity describes the change in volume of the hypersurface that forms the congruence. 19 Area(H2 ) ≥ Area(φ (H1 )). On the other hand, since θ ≥ 0, then Area(φ (H1 )) ≥ Area(H1 ): Area(H2 ) ≥ Area(H1 ). (1.46) As a consequence, if two black holes B1 and B2 merge to form a single black hole B3 , we will have the following inequality: Area (B1 ) + Area (B2 ) ≤ Area (B3 ) (1.47) In conclusion, for any physical process allowed, Hawking’s Area Theorem states that δA ≥ 0. Figure 1.6: Since θ < 0 at p, the null geodesic γ (blue color) can be deformed outward in a timelike geodesic γ ′ (black color). 20 Chapter 2 Entropy and Information Theory 2.1 2.1.1 Information Theory Definition of Information Suppose we have a problem P , which the only thing known about it is its Ω0 possible outcomes. In other words, we start from an initial situation where information is lacking to solve the problem P , i.e., I0 = 0. A situation of total ignorance can be though as one for which a priori the Ω0 outcomes are equally probable. If enough information I1 , is obtained at a later time to solve the problem P , then we will be able to indicate that only one of the Ω0 outcomes is actually realized. In summary: Initial situation: I0 = 0 with Ω0 equally probable outcomes; Final situation: I1 6= 0 with Ω1 = 1, i.e., one single outcome selected. To define the information, I, it is is necessary first to define a quantity that measures the amount of unknown information that we have about a system. Following [23] we define the unkown information as: IU ≡ K ln Ω0 , 21 (2.1) where K is a constant, ln is the natural logarithm, and Ω0 the number of possible (a priori) outcomes of an arbitrary problem. In texts such as [33] the definition of the equation (2.1) is often referred to as “information necessary to solve the problem”, however, for the present report it will be assumed both meanings as equivalent. The definition given to measure the quantity of unknown information about a problem uses the natural logarithm because we want that the information IU to have an additive property, furthermore a set of other properties that can be seen in [23]. For example, let P be a problem made up of two other problems p1 and p2 which are independent. Starting from an initial situation of total ignorance about the problem P , the only thing that is known is that the possible results of the problems p1 and p2 are Ω01 and Ω02 respectively. Therefore, the number of results the problem P will be: Ω0 = Ω01 · Ω02 . (2.2) Replacing the value of Ω0 of the equation (2.2) in the equation (2.1) we obtain: I1 = K ln (Ω01 · Ω02 ) = IU 1 + IU 2 , (2.3) where: IU 1 = K ln Ω01 , and IU 2 = K ln Ω02 . (2.4) In conclusion, the amount of unknown information (a priori) that we have about the problem P is the sum of the amount of unknown information that we have about the problems p1 and p2 . 2.1.2 Unit Systems In Information Theory, information (known or unknown) is considered to be a dimensionless quantity. Therefore, if a dimensional analysis is made of the equation (2.1), we obtain that 22 the constant K is a dimensionless quantity. The most convenient modern unit system for measuring information is called binary system. Such a system is made up of two possible results: 0 or 1, which are known as bits. Consider the problem of doing n independent selections, such that each corresponds to choosing a bit. The total number of possible outcomes of this problem will be: Ω = 2n . (2.5) Using the definition of unknown information in the equation (2.1), we have for this problem: IU = K ln Ω = Kn ln 2. (2.6) Then, if the information is taken in bit units we have Iu = n and: K = 1/ (ln 2) = log2 e, (2.7) using equation (2.7) and the base change property logarithms, the equation (2.1) can be rewritten to: IU = log2 Ω. (2.8) Another system of Units that can be used to measure information is obtained comparing the concept of information with that of thermodynamic entropy; in this case both quantities are measured in the same units. In fact, the Boltzmann entropy formula is very similar to the equation (2.1), but with coefficient the Boltzmann constant: k = 1.38 × 10−16 . (2.9) When K = k, the information is being measured in entropy units. It is also possible to make the definitions of entropy and information dimensionless; for this, the temperature must be measured in energy units [10] [33]. The ratio between the constants K and k for 23 the dimensionless definitions of entropy and information is of the order of: k/K = k ln 2 ≈ 10−16 . 2.1.3 (2.10) Unknown Information and Known Information Suppose that we start from a situation of total ignorance about a problem P . The only thing known about the system is that there are Ω0 possible outcomes, which a priori are equally likely. After a time, information about the system is obtained, such that the number of possible results that solve the problem P is now Ω1 6= 1, and Ω1 < Ω0 , therefore, the information obtained or known from the problem will be: I = K ln(Ω0 ) − K ln(Ω1 ) = K ln  Ω0 Ω1  , (2.11) connecting our ignorance with the actually information we extract from a problem. Let us suppose we have N ≥ 2 possible outcomes in G independent selections. Therefore, the number of possibilities is: Ω = N G, so using the equation (2.1) we will have: IU = KG ln N (2.12) IU = G log2 N (2.13) Or in bits units: So, if N = 2, the problem P will have 2G different possible outcomes, and furthermore, the amount of information that is unknown about the problem measured in bits will be: IU = G bits of information. 24 (2.14) 2.1.4 Shannon’s Formula Let P be a problem, where G selections of a single symbols can be made, each of which can be made for a single symbol from a set of symbols A = {1, 2, ..., j, ..., M } with a probability of occurrence of p1 , p2 , ..., pj , ..., pM , regardless of other conditions and restrictions about these symbols. Then we can define the average unknown information: I = G · H = −GK M X pj ln pj , (2.15) j=1 where H is the average unknown information per symbol, and: M X pj = 1. (2.16) j=1 The proof of the formula (2.15) is done as follows: Proof. Let N1 , N2 , ..., Nj , ..., NM be the number of times the symbols 1, 2, ..., j, ...M are selected respectively. Therefore, the following property is true: G= M X Nj . (2.17) j=1 The probability that a symbol j takes place in a given selection of the problem P is defined as follows: Nj . G (2.18) pj = 1. (2.19) pj = Thus: M X j=1 The total number Ω of possible outcomes to the problem P with G selections that can be 25 made randomly with a single symbol is: G! Ω = QM , N ! j j=1 using the equation (2.1): IU = K ln G! QM j=1 " = K ln G! − Nj ! M X ! (2.20) , # ln Nj ! . j=1 Regarding G and N large and the Stirling formula for the logarithm of the factorial as well the relation between G and the N : " ≈ K G ln G − =K =K " M X j=1 M X # Nj ln Nj , j=1 Nj ln G − Nj ln j=1 = −KG M X M X  M X j=1 # Nj ln Nj ,  G , Nj pj ln (pj ), j=1 where in the end we used the definition for the single probability. Equation (2.15) is know as the Shannon formula and it is the general version of the equation (2.1), allowing different probabilities in the outcomes. Indeed, it is reduced to it, when the results that can be obtained from an arbitrary problem are equally likely. For example, we want to write a letter consisting of 104 symbols, which are letters of the English alphabet (see table 2.1). If the probability of occurrence of each letter throughout the letter is exactly the same, then the amount of unknown information associated with a 26 Symbol Probability, p Word Space 0.2 A 0.63 B 0.0105 C 0.023 D 0.035 E 0.105 F 0.0225 G 0.011 H 0.047 I 0.055 J 0.001 K 0.003 L 0.029 Symbol M N O P Q R S T U X YW Z V Probability, p 0.021 0.059 0.0654 0.0175 0.001 0.054 0.052 0.072 0.0225 0.002 0.012 0.001 0.008 Table 2.1: The probability of occurrence p for letters of the English language [33]. letter of this size will be: IU = 104 (log2 27) = 4.76 × 104 bits. (2.21) Thus, the average amount of unknown information per letter will be: H= 47600 = 4.76 bits per letter. 10000 (2.22) If we consider the actual probability of occurrence of a letter in the English language (see table 2.1), then the unknown information associated with a letter of 10000 symbols will be: IU = −104 K 27 X pj ln pj j=1 ! = 40300 bits, (2.23) therefore, the average amount of unknown information per letter will be: H = −K 27 X pj ln pj j=1 ! = 4.03 bits. (2.24) Comparing the equation (2.22) with the equation (2.24) we realize that by making some 27 restriction on the probabilities of occurrence of a letter the amount of unknown information IU decreases. This is essentially due to the fact that the constraints indicate some degree of knowledge about a given problem. That is, there is information I 6= 0 of the problem. The amount H is usually used more than IU to measure the degree of uncertainty about an arbitrary problem. Due to the similarity of the equation (2.15) with Boltzmann entropy equation, H is also often called entropy [23]. In general, obtaining some information I 6= 0 about a problem leads to a decrease in entropy, that is [10] [33]: ∆I = −∆H 2.2 (2.25) Thermodynamics and Entropy It was the physicist Sadi Carnot (1796-1832) who described and formulated an upper limit to the efficiency of thermal machines. A Carnot engine is a heat engine that performs a cycle consisting of 4 reversible processes (see figure 2.1): 1. Isothermal expansion: from a volume V1 to a volume V2 , where V1 < V2 at temperature Th . 2. Adiabatic expansion: from volume V2 to volume V3 , where V2 < V3 and δQ = 0. 3. Isothermal contraction: from the volume V3 to a volume V4 , where V4 < V3 at temperature Tc . 4. Adiabatic contraction: from volume V4 to volume V1 , where V1 < V4 and δQ = 0. In this cycle called Carnot cycle, the variation of the internal energy is zero, that is: ∆UCarnot = 0 28 (2.26) Figure 2.1: Carnot cycle. However, the above is not surprising because: I dU = 0; (2.27) since dU is an exact differential. Now, if you evaluate the amount of total heat exchanged with the medium in a Carnot machine, you have: ∆Q = N KTh ln  V2 V1  + N KTc ln  3/2 = V4 . V1  V4 V3 V4 V3  . (2.28) Recalling the relations for adiabatic processes:  Tc Th 3/2 V2 = V3 and  Th Tc The above allows to establish that: ∆Q1 ∆Q3 + = N K ln Th Tc  V2 V1  + N K ln  = 0. (2.29) The expression (2.29) is valid for any cyclical reversible process [34]. The amount ∆Q/T is called reduced heat. If the Carnot process is decomposed into infinitesimal processes, the 29 equation (2.29) can be written as follows: I δQrev = 0. T (2.30) In general, according to [34]: As can be experimentally confirmed, δQrev /T is an exact differential not only for ideal gases, but for any other reversible thermodynamic process. This would indicate that there is a state function whose exact differential is δQrev /T . The state function S is called entropy and is defined as follows: dS = δQrev . T (2.31) By the equation (2.31) δQrev = T dS, therefore, rewriting the first law of thermodynamics for reversible processes we obtain: dU = T dS − P dV + · · · (2.32) As can be seen, each term on the right hand side must be a pair of quantities intensiveextensive, in the case of the pair P -V , we have that the pressure is an intensive quantity, while V is an extensive quantity, so since T is an extensive quantity, S must be an extensive quantity and also an amount analogous to V . The above is understood if we propose the following mental experiment. Suppose you have two air balloons connected through a tube that has a moving part, and whose state variables are P1 , V1 and P2 , V2 respectively, such that, P1 > P2 (see figure 2.2). Because the system is not in equilibrium, then it will evolve to a state of equilibrium. Such a state is achieved when P1 = P2 , however, the volume of both balloons will change, that is, to achieve balance it is necessary to pay a price: a change in volume. Now, suppose we have two systems in contact at different temperatures, T1 and T2 respec30 Figure 2.2: Two balloons connected through a tube with a plunger. There is a system of two balloons connected through a tube (gray) with a plunger (brown). Because the system is not in a state of mechanical equilibrium, because P1 > P2 , it will evolve to one where P1 = P2 thus changing the volume of both balloons. Figure 2.3: Systems in thermal contact. Red and blue colors have two systems with temperatures T1 and T2 respectively. In this case T1 > T2 , therefore, the system composed of the two systems described above is not in equilibrium. Then after some time, a state will be reached in which Tf 1 = Tf 2 (in magenta). tively. Since T1 6= T2 , the system will evolve to a state where T1 = T2 (see figure 2.3), in this case, the macroscopic quantity that will vary is the entropy S of the system, in an analogous way to V in the mechanical evolution of the system. 31 To describe the entropy change of a system, it is necessary to introduce The Second Law of Thermodynamics, which tells us the following [34]: Law 2.2.1. For isolated systems in equilibrium it holds that: dS = 0, S = Smax (2.33) and for irreversible processes it holds that: dS > 0. (2.34) In summary, for any allowed physical process: dS ≥ 0 (2.35) For irreversible processes, the system evolves until it reaches thermodynamic equilibrium. In this kind of processes, the entropy will grow until it reaches its maximum value. It is possible to decrease the entropy of a system, however in general the entropy of the universe will increase. Experiments to date demonstrate that the inequality (2.35) is maintained in a wide variety of settings, including black holes. 2.3 Boltzmann Entropy Suppose the following situation: you have a glass on a table at a height from the floor of approximately 70 cm. Also suppose that the glass is on the edge of the table and that by an act of carelessness we pass and hit the table; as a consequence the glass will fall and break. If we invert the direction of time, the situation that would be observed would be a glass recomposing itself, jumping from the floor to the table and finding itself in the initial situation from which we started earlier. Strangely, the last situation is compatible with the 32 laws of Classical and Quantum Mechanics, that is: that this is likely to occur in nature, the following proposition guarantees that this is true: Proposition 2.3.1. The laws of Quantum Mechanics are symmetrical under time inversion. However, there is no historical (or experimental) record that supports the possibility of observing a broken glass rearranging itself spontaneously. Processes in which its process associated with the temporal inversion are not observed or verified in reality is called irreversible [35]. As already seen in the previous section, in this type of process the entropy increases according to the second law of thermodynamics, that is, ∆S > 0. As the inverse process to that of the falling-breaking of the glass decreases the entropy of the universe, this process would violate the second law of thermodynamics, which must be maintained for any physically admissible process, this would indicate that of symmetrical laws under temporal inversion one cannot deduce the second law of thermodynamics [36]. 2.3.1 Statistical Interpretation If two paints, for example, of red and blue colors, are poured into a container and mixed until a uniform purple color is obtained (see figure 2.4), by the second law of thermodynamics, in the tin of paint system entropy increases until it reaches its maximum value. However, what connection does the entropy of a system have with the components of the system? Matter is currently known to be made of atoms, and also by associations of them called molecules. To simplify the reasoning, paint molecules can be idealized as small red and blue balls, just like in [36]. Suppose that each molecule (red or blue) can be found in a specific position within the volume of the container, which we will consider as a large box that is made up of N × N × N cells (see figure 2.4). To estimate the color in a place inside the container, the relative density of red balls with blue balls is averaged. To calculate the color associated with a zone, the container is divided into cubic zones of n × n × n, such that, N = kn, where k is a constant 33 Figure 2.4: Two-dimensional representation of mixing blue paint with red paint. The red and blue circles represent the molecules of the red and blue paints respectively. The mixture of paints can be represented as a system of N × N cells that can be occupied by either a molecule of blue paint or a molecule of red paint. An example of an intermediate box is given in yellow, which have 4 cells. that indicates the number of “intermediates” boxes that fill one side of the large box. Inside the intermediate boxes the paint molecules are restricted to being in a single cell. Each intermediate box has an associated color, which is measured by calculating the “average color” of the box. For example, let r be the number of molecules of red color, and b be the number of molecules of blue color inside the intermediate box, so that: n3 = r + b, (2.36) therefore, the ratio of r and b will be that which allows the color of a certain intermediate box to be established. If r/b is greater than 1, then the box has a more reddish hue, if r/b is less than 1, then the box has a more bluish tone. Therefore, if the state of our container is one in which the color is uniformly purple, it is because on average each intermediate box has a ratio r/b is close to 1. In reality, the number of molecules in a can of paint could be of the order of Avogadro number, that is, N = 1023 . If we suppose that in a container there is 34 a mixture of blue and red paints, such that N = 108 and k = 103 , such that, n = 105 , then there are about 102.357×10 25 configurations corresponding to the state in which half of the positions in each intermediate box are red paint molecules and the other half are blue paint molecules, i.e., the color of the mix is purple. On the other hand, there are only 104.657×10 13 settings associated with a situation where the system layout is the initial setting, where the blue paint is on top of the red paint [36]. From the above, it follows that by allowing the system described above to evolve over time, its most probable state is that of having a purple hue. Entropy is a quantity related to the count of allowed configurations associated with a system that is in a given macrostate. However, using such large numbers directly gives many problems, therefore, the natural logarithm of the number of configurations Ω, is calculated, that is: S ∼ ln(Ω). 2.3.2 (2.37) Boltzmann Entropy The theoretical reasons for using the natural logarithm of the number of allowed configurations associated with a system that is in a certain macrostate, is because we want to impose on the statistical measure of entropy two properties [37]: (i) The entropy S of the system is a monotonic increasing function of “degree of disorder” Ω, that is, S(Ω) ≤ S(Ω + 1). (ii) The entropy S is assumed to be an additive function for two statistically independent systems with degrees of disorder Ω1 and Ω2 , respectively. The entropy of the composite system is given by S(Ω1 · Ω2 ) = S(Ω1 ) + S(Ω2 ). This leads us to the following theorem: Theorem 2.3.1. If the entropy function S(Ω) satisfies the above two axioms (i) and (ii), 35 then S(Ω) is given by: S = k ln(Ω), (2.38) where k is a positive constant depending on the unit of measurement of entropy. In conclusion, the entropy of a system composed of a large number of elements is proportional to the natural logarithm of the number of configurations physically allowed for that system associated with a given macrostate. The constant k is known as Boltzmann constant, whose units of measurement in the international system are J/K. And whose approximate value is: k = 1.38 × 10−23 J/K. (2.39) However, the definition of entropy requires a subtle change in its definition. Suppose the following example, water and oil in a container, and we want to mix it, as a result of the mixture there is a combination of water and oil, however, after a long time the oil molecules begin to “sort”, finally obtaining a mixture where you can clearly distinguish the water from the oil (see figure 2.5). We already saw in the example of the mixture of paints that if something similar happened in the container where the mixture takes place (that is, the red paint is above the blue paint), the number of physically allowed configurations associated with this macrostate is less than that associated with a uniformly purple mixture, that is, in this case the final state of the system would not be that of maximum entropy. We conclude that there are systems that contradicts our definition, and subsequent interpretation, of entropy. This apparent paradox is resolved if we include information about the phase space of the system. To use the definition of entropy in general physical systems, where the motion of the components obey certain physical laws, a volume V of states of the space phase must be measured associated with the system, such that, at the macroscopic level, they are indistinguishable (see figure 2.6). This division of phase space into regions of volumes where states are macroscopically indistinguishable is called Coarse-Graining [36]. Therefore, if P is the 36 Figure 2.5: Mix of water and oil. Olive oil is added to a container of water. Subsequently, the water and oil are mixed so that an approximately homogeneous mixture is obtained, however, after a sufficient time the mixture will reach a state in which the olive oil can be differentiated (in the upper part of the container) of the water. Figure 2.6: Coarse-Graining. C is a subset of the phase space of physical states (or microstates) of an arbitrary system. Each separation within C contain a set of states that are macroscopically indistinguishable. For example, ǫ1 and ǫ2 are macroscopically indistinguishable states. 37 phase space associated with a system Σ and p is a state within a volume V of macroscopically indistinct states, that is, indistinguishable with respect at macroscopic parameters (such as temperature, pressure, density, direction, etc.), the entropy of the system is defined as follows: S = k ln(V ). 2.3.3 (2.40) Connection between Statistical Mechanics and Thermodynamics For a classical system (a collection of particles, for example) with a discrete set of microstates, with Ei the energy of the microstate i and pi the probability of it occurring during system fluctuations , then, the entropy of the system is: S = −kB X pi ln pi , (2.41) i then, with the energy: U= X pi E i , (2.42) 1 k 1 T Ei e B , pi (2.43) i and the partition function: Z= we have that the entropy can be written as: S= U + kB ln Z, T (2.44) where T is the absolute temperature of the system. From the First Law of Thermodynamics we can be deduced that: 1 ∂S = β, kB ∂U 38 (2.45) where β = 1/kB T and is known as Thermodynamic Beta. If we use the usual entropy’s definition, that is, S = kB ln(Ω), then the thermodynamic beta according to [35] can be defined as: β≡ ∂ ln(Ω) . ∂U (2.46) Recall that in thermodynamics entropy was manifested as a quantity analogous to volume, but related to the phenomenon of heat exchange. Therefore, it is logical to use in this same phenomenon our statistical definition of entropy; let Σ be a system which absorbs a small amount of heat Q, so the final and initial energies of the system will be U + Q and U , P respectively, where U = i pi Ei . Performing a Taylor Expansion in ln Ω(U ) we will have neglecting term of order Q2 : ln Ω(U + Q) − ln Ω(U ) ≈ βQ. (2.47) Now, if the heat transfer Q were infinitesimal, we would obtain the following expression: d(ln Ω) = βδQ, which is the microscopical equivalent to the thermodynamic relation (2.31). 39 (2.48) Chapter 3 Black Hole Thermodynamics 3.1 Bekenstein Entropy Expressing a formula that allows to measure the entropy of a black hole is not a trivial task, in fact, in the 60’s physicists were not sure how to approach this problem. The widespread belief that time was that black holes should be physical systems whose entropy was zero. However, if a black hole is a zero entropy object, then through it the Second Law of Thermodynamics is violated [26]. The root of the problem arises when taking into consideration the so-called no hair theorem of black holes, which can be stated as follows: A stationary black hole is characterized by only three quantities: mass, angular momentum and charge [7]. This indicates that in the formation of a black hole the phase space of a system that collapses into a black hole, from the point of view of an observer outside, is reduced. In other words, the formation of a black hole appears to violate the Second Law of Thermodynamics. According to the area theorem of black holes shown in chapter 1, for any allowed physical process, the area of the event horizon cannot decrease [24], that is, δA ≥ 0. However, Hawking’s theorem has an evident similarity with the Second Law of Thermodynamics. It was the physicist Jacob Bekenstein who saw the similarity between the two inequalities as 40 an indication of the existence of a relationship between the area of the event horizon of a black hole and the entropy associated with such a system. The theoretical basis for relating the entropy of a black hole to the area of its event horizon comes from the Information Theory discussed in the chapter 2. According to Shannon, entropy is a quantity that measures the uncertainty (or unknown information) of a source of information. In the case of an object that collapses into a black hole it is evident that there is a loss of information. For example, if two systems made of different objects collapse into two black holes of the same mass, angular momentum and charge, then according to the no-hair theorem there would be no way to distinguish them. Bekenstein’s idea was to relate the event horizon area of a black hole to this information lost, then, as a consequence, the function that measures the entropy associated with a black hole must be a monotonic function of the area: Sbh = f (α), (3.1) where α is the so called rationalized area defined as follows, α≡ A , 4π (3.2) where A is the area of the event horizon. The equation (3.1) can be used for any type of black hole because the area of a black hole is a well-defined quantity even when a black hole evolves and changes in time [10]. Equation (3.1) together with Hawking’s theorem allows to reestablish the Second Law of Thermodynamics for black holes: ∆Sbh ≥ 0. 41 (3.3) The simplest expression that can be chosen for (3.1) that obeys equation 3.3 and the black holes mechanics is the following [10]: Sbh = γα, (3.4) where γ is a constant of proportionality. In information theory, entropy is a dimensionless quantity (see chapter 2), therefore, a dimensional analysis of equation (3.4) implies that γ has dimensions of the inverse of the square of the length1 , which can be decomposed into two constants: a dimensionless one and a universal constant with dimensions of L−2 .In Classical Mechanics there is no universal constant with the dimensions of γ, however, in Quantum Mechanics there is such a constant; the inverse of the square of the Planck length p lp = G~/c3 ∼ 10−35 m. As the unit system being used takes c = G = 1, the Planck length √ takes the form lp = ~. Taking this into account, the equation 3.4 can be rewritten as: Sbh = η~−1 α, (3.5) where η is a dimensionless constant yet to be determined. 3.2 Semi-Classical Estimation of the Constant η Before stating the four laws of black hole mechanics, there are a couple of conclusions that should be mentioned because they are a direct consequence of the equation (3.5), but it is first necessary to have some insights on the value of η. A precise evaluation of the dimensionless constant η can only be done using Quantum Field Theory in curved space-times, which is not the subject of interest in this text. However, an estimate of its value can be made using a semi-classical approach to the particle absorption phenomenon in black holes [10]. 1 A consequence of expressing entropy as a dimensionless quantity is that Boltzmann constant will be dimensionless, thus, temperature must be measured in energy dimensions. 42 As mentioned before, the entropy of a black hole Sbh is associated with the information that is lost in the formation of a black hole and the area of the event horizon of a black hole allows to establish an adequate measure for this loss. Therefore, since a bit is the minimum amount of information that can be lost or obtained from a system, then the minimum amount of information that can be lost when a particle is thrown into a black hole is ln 2, which arises from answering the question: does the particle exist or not? [10]. Because there is a minimal change in the entropy of a black hole (∆S)min = ln 2 and the entropy is related to the area of the event horizon, then there is a minimal change in area of the horizon (∆α)min associated with this process. Following Bekeinstein the minimum increase in the area of the event horizon of a black hole when adding a spherical particle of rest mass µ and proper radius b is: (∆α)min = 2µb. (3.6) However, b is bounded from below since the proper radius of a particle cannot be smaller than its Compton Wavelength ~µ−1 or than its gravitational radius 2µ, while the Compton wavelength is greater than the gravitational radius when µ ≤ 2−1/2 ~1/2 . The gravitational radius is greater than the Compton wavelength when µ > 2−1/2 ~1/2 . If µ ≤ 2−1/2 ~1/2 , then bmin = ~µ−1 , hence (∆α)min = 2~. Instead if µ > 2−1/2 ~1/2 , then b > 2µ, hence, 2µb > 4µ2 . In conclusion, Quantum Mechanics states that: (∆α)min = 2~. Now, from S = f (α), then: (∆S)min = df (∆α)min . dα 43 (3.7) With (∆S)min = ln 2 and (∆α)min = 2~ we obtain an explicit expression for the entropy of black holes: Sbh = Finding η = ln 2 2 ln 2 −1 ~ α. 2 (3.8) in (3.5). Later, when discussing Hawking Radiation, we will realize that the actual value of η is different from that just found in this section, however, this estimate (which was made by Bekeinstein [10]) it is in the same order of magnitude to the correct one. In conventional units the entropy expression takes the form: Sbh =  ln 2 2  kc3 ~−1 G−1 = (1.46 × 10 48 A 4π −1 (3.9) −2 erg.K cm )A, where k is Boltzmann constant. The equation (3.9) shows that the entropy of a black hole is enormous. 3.3 The Four Laws of Black Hole Mechanics The work in [10] resulted in the formulation of a series of laws for black holes analogous to the four laws of thermodynamics, that we state in the following. The Second Law The Second Law has two formulations: one weak and one strong. The weak formulation established by Stephen Hawking in [12] tells us the following: The area A of the event horizon of each black hole does not decrease with time, i.e. δA ≥ 0. 44 The strong formulation is called Generalized Second Law which states that: The ordinary entropy in the black-hole exterior plus the black hole entropy never decrease. [10] That is: ∆Sbh + ∆Sc = ∆(Sbh + Sc ) ≥ 0, (3.10) where ∆Sc denotes the change in common entropy on the outside of the black hole. The Second Generalized Law is telling us that the entropy of a black hole is a real one that contributes to the entropy of the universe. This is the reason why Hawking’s statement of the second law is weak, since for him the relationship between the Second Law of Thermodynamics and his area theorem is nothing more than a mere analogy. For an observer who is outside a black hole, the Second Law of Thermodynamics is violated, however, the Generalized Second Law guarantees us that the increase in the event horizon of a black hole compensates the apparent decrease in entropy of the universe. The First Law ~ whose rationalized Consider a black hole with mass M , charge Q, and angular momentum L, area α is given by [10]: 2 α = r+ + a2 , (3.11) 2 = 2M r+ − Q , where: ~ ~a = L/M, (3.12) is the angular momentum per unit of mass and we define: r± = M ± (M 2 − Q2 − a2 )1/2 . 45 (3.13) Differentiating the rationalized area and solving for dM we obtain: ~ · dL ~ + φdQ, dM = Θdα + Ω (3.14) where: 1 Θ = (r+ − r− )/α, 4 ~ = ~a/α, Ω (3.15) φ = Qr+ /α. The equation (3.14) is the so-called First Law of the mechanics of black holes. Defining: p M 2 − a2 − Q2 r+ − r− p κ= , = 2α 2M (M + M 2 − a2 − Q2 ) − Q2 (3.16) know as surface gravity, equation (3.14) takes the following form: dM = κ ~ · dL ~ + φdQ, dA + Ω 8π (3.17) For black holes, the surface gravity is associated with the Killing horizons and is usually defined as follows: Let χµ be a Killing field and Σ the Killing horizon associated with χµ . For each Killing horizon there is a quantity κ defined by the following equation [24]: χµ ∇µ χν = −κχν , (3.18) evaluating the equation (3.18) an explicit formula is derived for κ [5] [24]: 1 κ2 = − (∇µ χν )(∇µ χν ). 2 46 (3.19) the name of “surface gravity” comes from the fact in static and asymptotically flat spacetimes, κ is the acceleration experienced by a static observer near the event horizon, measured by a static observer in the infinite [5]. If we compare the equation (3.14) with the First Law of Thermodynamics, we find that κ dA 8π ~ · dL ~ + φdQ are the terms analogous to T dS and −P dV respectively. Therefore, in and Ω this context κ 8π is a quantity analogous to the state variable T of thermodynamics [12]. On ~ refers to the rotational angular frequency and φ to the electric potential. the other hand, Ω ~ · dL ~ + φdQ could be interpreted analogous to the work done on a rotating and Hence, Ω charged body [10]. The Zeroth Law The event horizon κ obeys the following equation [24]: χ[d ∇c] k = 0, (3.20) where χµ = K µ + ΩH Rµ (see section 1.3). The foregoing can be stated as follows: The surface gravity, κ of a stationary black hole is constant over the event horizon [12]. The above statement is called the Zeroth Law of black hole mechanics given its similarity to the Zeroth Law of Thermodynamics. The Third Law Since κ tells us about the temperature of a black hole, then, following the comparisons between the mechanics of black holes and Thermodynamics, it is inevitable to formulate a law of black holes that is analogous to the Third Law of Thermodynamics. The Third Law states: 47 It is impossible by any procedure, no matter how idealized, to reduce κ to zero by a finite sequence of operations [12]. According to the equation (3.16), κ = 0 when: M 2 = a2 + Q2 . (3.21) A black hole that satisfies the equation (3.21) is called extreme black hole [24]. When the Third Law of black hole mechanics was first formulated in [12], there was no rigorous proof to establish the validity of that law. However, there were two arguments in favor of the Third Law: 1. If you try to reduce the value of κ by adding particles that increase the value of a (and also Q), then you find that the change of κ per particle is smaller and smaller, in fact, it would take an infinite amount of time for a non-extreme black hole to transform into an extreme black hole [12]. 2. If κ = 0, is obtained through a number of finite processes, then the process can presumably be taken even further by creating a naked singularity [12]. The existence of a naked singularity would result in the violation of many theorems of General Relativity, among them, Hawking’s area theorem. 48 Chapter 4 Holographic Principle 4.1 4.1.1 Motivations Hawking Temperature The non zero entropy for black hole motivated in the previous chapter seems to contradict one of the fundamental properties of black holes: if a black hole has a nonzero entropy, then it should also have a nonzero temperature. This would indicate that black holes should radiate; however, as we have seen, according to General Relativity, black holes are physical objects characterized by having a Event Horizon from which not even light can escape. Classically, it could be concluded that black holes should not radiate, and therefore, their effective temperature would be zero. Hawking showed that if Quantum Field Theory is used in curved spacetimes (where the metric obeys Einstein Field Equations) it is found that black holes can radiate [38]. Heuristically, the mechanism that generates radiation in black holes is due to vacuum quantum fluctuations near the event horizon of a black hole. Vacuum quantum fluctuations can be interpreted as the formation of virtual particle-antiparticle pairs. Close to the event horizon, one of the two elements that make up the particle-antiparticle pair can pass through the event horizon while the other escapes to infinity (see figure 4.1). A distant observer will 49 Figure 4.1: Hawking Radiation. According to Quantum Theory, near the event horizon the vacuum exhibits quantum fluctuations. These fluctuations can be interpreted as the formation of virtual particle-antiparticle pairs (colored white and black respectively). One of the elements of the pair can cross the event horizon by “abandoning” its partner. The latter ceases to be a virtual particle and becomes a real particle, which can go to infinity and be detected by an distant observer. The shower of particles that arises from the process described above is called Hawking radiation. detect a thermal spectrum of particles coming from the black hole, at a temperature given by [9] [7] [38]: κ . 2π T = (4.1) The discovery of this type of radiation made it possible to give physical reality to the entropy discovered by Bekeinstein. By the First Law of Thermodynamics we know that dM = T dSbh or equivalently, dM = T ηdA. Now, the first law of the Mechanics of Black Holes dictates that: dM = κ dA, 8π (4.2) κ , 8π (4.3) thus, Tη = 50 replacing the Hawking temperature in equation (4.3) we obtain: 1 η= . 4 (4.4) The radiation process causes black holes to lose their mass until they finally disappear. The D−1 evaporation of the black hole turns it into a radiation cloud in a time of the order of M D−3 , where D is the dimension of space -time where our black hole is. In summary, in the process of formation and death of a black hole, the common entropy (that which is outside the event horizon of a black hole) becomes horizon entropy and therefore, in the evaporation process, the horizon entropy becomes radiation entropy. The above finally reaffirms that the entropy associated with a black hole discovered through the theoretical framework of Information Theory is a genuine entropy contribution to the entropy of the universe. 4.1.2 Bekenstein Bound When a matter system (that is, containing matter) with entropy Smatter is thrown into a black hole and crosses the event horizon, its entropy disappears to a distant observer. The black hole that absorbs such a system grows because it gains mass. It is expected, therefore, that the Generalized Entropy S = Sbh + Smatter cannot decrease, that is, that the Second Generalized Law holds. That the Second Generalized Law is true for any physical phenomenon is not something that can be deduced directly from the Laws of General Relativity. In fact, the growth of the area of the event horizon depends on the mass-energy of the matter system that falls into the black hole, not on its entropy. If a system of matter is allowed to have an arbitrary amount of entropy, then situations can arise where the Generalized Second Law can be violated. For example, in the absorption process described above, a matter system of arbitrary entropy initial Smatter and mass-energy E, the growth of the area of the event horizon of a black hole is at least 8πER (if the system is weakly gravitating) [7] [21], where R is the radius of the smallest 51 sphere that contains the entire system of matter. Assuming a situation where absorption minimizes the growth of the event horizon area, we get: ∆Sbh = 2πER. While ∆Smatter = −Sinitial . Thus, Without an entropy limit for the matter system, it is initial possible that 2πER < Smatter , can be imposed, thus violating the Second Generalized Law of black holes because ∆Sbh + ∆Smatter < 0. Therefore, postulating the Second Generalized Law as a law of Nature we establish that for a weakly gravitating matter system captured by a process that minimizes the growth of the area of the event horizon of a black hole, the entropy limit of a matter system will be [21]: Smatter ≤ 2πER, (4.5) known as the Bekenstein bound. In the International System of Units the above inequality is written as follows: Smatter ≤ 2πkb ER/(~c), where the absence of the Newton constant implies its independency on gravity. Furthermore, the above calculation is independent of the scale of R, since the only consideration made for the absorption process is that the matter system is much smaller than the black hole [21]. Applying the entropy bound to strongly gravitating systems is complicated due to the difficulty of defining the radius R of a system in highly curved geometry. However, this problem does not arise for weakly gravitating systems nor for symmetrically spherical systems. For example, for a Schwarzschild black hole R = 2E (Schwarzschil radius), hence its entropy is Sbh = A 4 = 2πER (in D = 4). Thus, a Schwarzschild black hole is a system that exactly saturates the Bekenstein Bound, i.e., a system with the maximum possible entropy [21]. 52 Figure 4.2: Susskind Process. Suppose a stable matter system of mass-energy E found in asymptotically flat spacetime, and enclosed in a sphere of surface area A (colored gray). Near the matter system the metric is approximately spherically symmetric (at least, to be A a well-defined quantity). Around this system there is a shell of matter (brown in color) of mass-energy M − E, where M is the mass of a black hole of surface area A. As the shell collapses over the matter system, a black hole will form. 4.1.3 Susskind Process and Spherical Entropy Bound So far, the physical processes in which the Generalized Second Law has been tested have in common the use of already existing black holes, with which a universal bound for the entropy of a matter system found in the exterior of a black hole is found. The consequences of the GSL can be taken even further by proposing a process that is related to the formation of black holes. Suppose an isolated system of matter of mass-energy E and entropy Smatter which is in a spacetime M that is asymptotically flat and allows the formation of black holes. Let A be the area of a spherical surface circumscribing the system of matter, whose dependence with time is negligible1 . The gravitational stability of the matter system requires that the mass-energy E be less than the mass M of a black hole whose event horizon has an area A. If a mass-energy shell of 1 In other words, the system is assumed stable on a time scale greater than A1/2 . Consequently, the system does not expand or collapse rapidly [7]. 53 matter M − E is collapsed on the system, the matter system will form a black hole of surface area A. This physical process is known as Sussking process (see figure 4.2). According to the GSL: Smatter + Sshell ≤ A . 4 (4.6) Since Sshell is also an entropy contribution associated with a system composed of matter, then: Smatter ≤ A . 4 (4.7) The above inequality is known as Spherical Entropy Bound. From (4.7) it is concluded that given a spherical region of the area space A, the maximum entropy value that the matter content within the region can possess is the entropy of a black hole of surface area A. 4.1.4 Bekenstein Bound and Spherical Bound Although for different physical processes two upper bounds have been derived for the entropy associated with a system made of matter, equations (4.5) and (4.7), ultimately it can be shown that the Spherical Entropy Bound is a restricted case of the Bekenstein Bound. More precisely, with (4.5) and after the following are assumed: 1. The Bekenstein Bound holds for strongly gravitating systems and, 2. gravitational stability over the systems being studied (matter, radiation, or black holes); the maximum entropy that can be found in a spherical region of surface area A with matter content is A/4 [7]. The last of the restrictions mathematically indicates that: 2M ≤ R, (4.8) where R and M are the effective radius and mass-energy of the matter system. are the effective radius and mass-energy of the matter system. Multiplying by πR the members of 54 the inequality (4.8): 2πM R ≤ πR2 = A , 4 (4.9) and by the Bekenstein Bound (4.5): Smatter ≤ A . 4 In conclusion, the Spherical Entropy Bound is a weak version of the Bekenstein limit, despite this, it is the latter that is explicitly related to the Holographic Principle and later on can be stated in a general way in the so-called Covariant Entropy Bound [7]. 4.2 4.2.1 A Holographic Principle Degrees of Freedom Let (M, gab ) be an asymptotically flat spacetime and C a finite region of volume V and surface area A. The Spherical Entropy Bound forces C to be enclosed in a spherical region Σ of surface area A′ . In C gravity is weak enough to allow a clear definition of the quantities A and V . Also near C the metric is assumed not to be strongly time dependent. That is, ∂C will not be a trapped surface inside a black hole [39]. If C is a quantum-mechanical system, then at a fundamental level the number of degrees of freedom N will be the natural logarithm of the dimension of its Hilbert space (or equivalently, to the natural logarithm of the number of independent quantum states), denoted by N [7]: N ≡ ln N = ln dim(H). (4.10) The number of degrees of freedom is equivalent to the information necessary to characterize the state of a physical system. For example, the state of a system made up of 105 spins, has 105 ln 2 degrees of freedom, therefore, 105 bits of information are needed to describe the 55 state of such a system. 4.2.2 Fundamental System Suppose that the region C described above is a spherical region without any special restrictions on the content of matter. At a fundamental level, from C we want to find the maximum number of degrees of freedom that describes all the physics on it, or equivalently, its Hilbert space. The Hilbert space H that we want to find in C is not related to the Hilbert space of a particular system (for example, an atom), after all, in a fundamental theory all physical entities are reduced to their constituents. So, when we asked about the Hilbert space H of C we are referring to the Hilbert space of such constituents given the size of C. This system is called Fundamental System [39]. 4.2.3 Degrees of Freedom According to Local Field Theory Due to the absence of a Unified Theory of Gravity, to calculate the available states of a Fundamental System, an approximate theoretical framework is used. In [6] it is assumed that the Fundamental System is a Local Quantum Field Theory with a background spacetime that satisfies the Einstein equations (that is, with a classical metric). In a Quantum Field Theory (QFT), each point in space has one or more Harmonic Oscillators. However, it is well known from Quantum Mechanics that the Hilbert space related to a single harmonic oscillator is infinite in dimension (N = ∞). Furthermore, in a space region of volume V there are infinite points, no matter how small V is. Therefore, it follows that the number of degrees of freedom of the fundamental system seem to be N = ∞. However, if the gravitational effects that affect the Fundamental System are taken into account, a finite value of N can be found. Discretizing the space to a grid whose minimum distance is the Planck length (lp = 1.6 × 10−33 cm), the number of oscillators per Planck volume (lp3 ∼ 10−99 cm) is restricted to 56 1. On the other hand, according to [39] the Planck mass (Mp = 1.3 × 1019 GeV) is the maximum amount of energy that can be located in a Planck volume without producing a black hole. Therefore, we are finding an Ultraviolet Cut-off. This establishes a limit for the number of states of a harmonic oscillator bounded by a maximum energy value equal to the Planck Mass. The arguments above leads us to the following finite estimation: In a spherical region of volume V in Planck length units, it is expected that there will be V oscillators with finite n independent quantum states. Therefore, in the Fundamental System the dimension of the Hilbert space is: N ∼ nV , (4.11) and, the number of degrees of freedom according to the Quantum Field Theory is: N ∼ V ln n ≥ V (4.12) The expression (4.12) allows us to conclude that 1.) the number of degrees of freedom in the world is local, that is, that the degrees of freedom are well defined over a time of a point in space, and 2.) that complexity increases with the volume. In other words, that the information needed to describe the physics in a spherical region of space grows with its volume. 4.2.4 Degrees of Freedom According to Spherical Entropy Bound In chapter 2 we recall that for a thermodynamic system the entropy S is the natural logarithm of the number of microscopic states (that is, independent quantum states) that are compatible with a given macrostate, described by a set of macroscopic variables such as temperature, volume, energy, etc. Consequently, according to the previous definition: N = eS . 57 (4.13) Talking about entropy, therefore, is analogous to talking about the uncertainty or the ignorance faced with the microscopic details of a system. Given the practical impossibility of measuring a macroscopic parameter with zero uncertainty, it is allowed, for example, to let the energy of a system lie in a finite interval, thus increasing the number of microscopic states compatible with a certain macroscopic state. The more uncertainty, the more microscopic states can describe the state of a thermodynamic system, and in this way, the entropy becomes larger and larger. Every thermodynamic system should be described by the same Underlying Theory, including the Fundamental System. According to the Spherical Entropy Bound, in a spherical region of the surface area space A, the entropy of the content of that region must be less than or equal to A/4. This allows an estimate of the entropy enclosed in that region in terms of surface area, without the need to worry about microscopic details. A black hole that fits perfectly in this region has an entropy equal to: Sbh = A , 4 (4.14) saturating the Entropy Bound of the region. Then the number of independent quantum states of a spherical region of the surface area space A is: N = eA/4 , (4.15) and the number of degrees of freedom: N= A . 4 (4.16) Then, contrary to the QFT, estimation the entropy in this approach increase with the area and not with the volume, as stated by QFT. Thermodynamic systems are generally assumed to be greater than the Planck scale. So the volume will always be greater than the 58 surface area, in Planck units. In conclusion, when comparing the estimates of the number of degrees of freedom N obtained by the Local QFT, equation (4.12), and the Theory of General Relativity, equation (4.16) ,we realize that there is a disagreement between both theories. 4.2.5 Local Field Theory Estimation is Wrong? The disagreement that exists in the estimation of N made through the QFT and the Theory of General Relativity arises from not properly including gravitational effects in the first. Restricting the modes of a harmonic oscillator to a finite region of space provides an infrared cut-off, avoiding the input of entropy due to long wavelength states. Consequently, the greatest contribution to entropy is due exclusively to high-energy states. By General Relativity, in a spherical region of surface area A, the mass enclosed by such region cannot be greater than the mass of a black hole of surface area A. According to the Schwarzschild solution, the mass of a black hole is determined by its radius (Schwarzschild radius), so the mass M contained within the spherical region obeys the following inequality: M . R. (4.17) The ultraviolet cut-off that arises from imposing the Planck mass as the upper bound of energy obeying the inequality (4.17) for R = lp . However, at scales larger than the Planck scale, the ultraviolet cut-off proposed so far allows M ∼ R3 , violating restriction (4.17). For instance, then in a spherical region of size R = 1 cm or 1033 lp there are on the order of 1099 Planck cells. Assuming that each cell has a mass-energy on the order of Planck mass, then the mass-energy contained in the region is on the order of 1099 Planck mass units. However, according to General Relativity, in a spherical region with a radius of 1 cm a mass-energy of the order of 1033 Planck mass units is found at most. That is, there are on the order of 1066 quantum states that should not be taken into account because they are massive states 59 that violate (4.17). According to Bousso [7], before any of these quantum states can be excited a black hole will form, thus saturating the Spherical Entropy Limit. Which in principle, resolves the apparent contradiction in the estimates of N made by QFT and GR. 4.2.6 Unitarity and Holographic Principle By the Spherical Entropy Bound, it follows that the maximum number of degrees of freedom that completely describes a stable region of a spacetime asymptotically flat enclosed by a sphere of area A is N = A/4. From the point of view of a distant observer, the most entropic object that fits inside the region would be a black hole of surface area A. A restricted interpretation of the equation (4.7) suggests that the imposition of gravitational stability limits in a practical way the information that can be contained within a spatial region [7]. If we are willing to pay the price of a gravitational collapse, we can in principle have more degrees of freedom than A/4. However, we would have to fall into the black hole to corroborate the existence of these degrees of freedom. According to [7] there are two objections to the previous interpretation. Both from the theoretical framework of an asymptotically flat spacetime described by means of a Scattering Matrix. Since it contains information about the amplitudes between the initial and final states defined at infinity. For an observer at infinity this theoretical framework is satisfactory. The first objection to the restricted interpretation arises from what [39] calls “economics.” A fundamental theory should have no more information than is necessary. Consequently, if only A/4 degrees of freedom are needed to describe the physics of a spatial region, then, that is the amount of information needed in the theory. However, this argument is weak, since it is based on an aesthetic judgment. The second objection, which is stronger, is based on unitarity. In Quantum Mechanics, the evolution from one state to another is unitary, that is, pure states are transformed into pure states. Suppose that what is shown in the QFT is true, that is, that the number of 60 independent quantum states that describe the physics found within a spherical region is exactly eV , where V is the volume of that region. However, when the system evolves to the black hole state, the number of independent states decreases to eA/4 . In conclusion, to save unitarity, it is assumed, relying on the laws of quantum mechanics, that the dimension of Hilbert space that initially describes the system is as well eA/4 . Given the insistence of reconciling gravity collapse with the quantum mechanical postulates t’ Hooft [11] and Susskind [40] they formulated a strong holographic interpretation of equation (4.7) as follows: Holographic Principle (preliminary formulation). A region with boundary of area A is fully described by no more than A/4 degrees of freedom, or about 1 bit of information per Planck area. [7]. 4.2.7 Unitarity and Black Hole Complementarity Now that the complete history of the birth and death of a black hole is known, we wonder if the transformations that the physical system undergoes are unitary in nature. The ultimate goal is to reconcile the physics described by General Relativity with the physics described by Quantum Mechanics, where unitarity is required. Hawking [41] suggests that the slow transformation of a black hole into a radiation cloud is not a unitary process, in other words, there is a loss of information associated with such a transformation. So the S-matrix unitarity problem is not only a problem associated with the formation of black holes, but it is also related to the evaporation of a black hole. The Holographic Principle solves the Unitarity problem in the first scenario (the formation of a black hole), however, if the study of the evaporation of a black hole rules out the possibility that this latter process is unitary, there would not be sufficient reasons to assume this to be the case in the formation of black holes. According to Susskind, Thorlacius and Uglum [42], a distant observer and an observer who falls into a black hole see the same quantum information. The first observer obtains information about the formation 61 Figure 4.3: Complementarity. An observer outside a black hole (astronaut on the right) gets the same information about the formation of the black hole as one inside the black hole (astronaut on the left); the first one obtains it in the form of Hawking radiation (colored yellow and orange) and the second from the objects that originally collapsed. of the black hole from Hawking radiation and the second from the matter that originally collapsed (see figure 4.3). The fact that there are two copies with the same information seems to be problematic, since quantum mechanics prohibits these simultaneous photocopies of information with the so called no-cloning theorem [43]. This apparent paradox in fact can be resolved as it can be shown that no observer can obtain two copies of the same information, for the following two reasons: 1. An observer who falls into a black hole cannot escape it. 2. If a distant observer outside a black hole decides to store 1 bit and then crosses the event horizon to obtain the copy of the bit that it stored, then , you would have to face the following difficulties: (a) To obtain a single bit of information in the form of Hawking radiation, a distant observer would have to wait for a time on the order of the evaporation time of the black hole, that is, t ∼ M 3 , where M is the mass of the black hole. (b) If the distant observer gets 1 bit of information (which is associated with the 62 formation of the black hole) and proceeds to jump into the black hole, it would have to register the existence of the copy bit long before let that information hit the singularity. In order for a photon to avoid singularity in such a long time, it 2 would need an energy of the order eM . In short, in practice it would be impossible for an observer to have both copies of information. So, there are two complementary and self-consistent versions of black hole formation. Thus, the information paradox arises as a consequence of trying to describe the formation of a black hole from a global point of view, and in this way, quantum gravity assigns a new role to the observer, which he has to abandon a global description in space-times with event horizons [7]. In conclusion, holography and complementarity save unitarity at expenses of losing globality. 63 Chapter 5 Discussion and Conclusions In this monographic work we did a detailed review of the classical theoretical foundations that motivate the preliminary formulation of the Holographic Principle. In the first chapter, the Kerr solution to Einstein’s field equations is studied. In particular we studied the so called the Ergosphere, which in turn allows energy to be extracted from a black hole using a physical process known as the Penrose Process. A detailed analysis of this processes shows, however, that there is a remaining energy, known as reducible mass, proportional to the horizon area. Then, using some general arguments on these kind of processes, we demostrated what is known as the Hawking theorem stating that the area of a black hole cannot decrease in time by any classical process. In the second chapter a chronological review of the concept of entropy is made from its formulation in thermodynamics to its use in Information Theory. Boltzmann’s statistical interpretation establishes a bridge between the macroscopic world (governed by the laws of Thermodynamics) with the microscopic world, establishing that entropy is a count of microstates (quantum states) that are compatible with the macroscopic state of a themodynamic system. The word “compatibility” can be interpreted as possible microscopic states in which a system can be found given a macrostate, which indicates that there is a certain degree of ignorance or uncertainty about of the microscopic details of a system. This last 64 statement is the key idea behind the Shannon interpretation of entropy, and a starting point for forthcoming discussion on the Second Law of Thermodynamics in presence of gravity. In the third chapter, the no-hair theorem and the problem that arises from its apparent contradiction with the Second Law of Thermodynamics is stated. To resolve this incompatibility between General Relativity and Thermodynamics, it is postulated that the area of the event horizon is a measure of the entropy associated with black holes (given the obvious similarity between Hawking’s area theorem and the Second Law of Thermodynamics). The theoretical foundation that relates the horizon area to the entropy concept lies in the theoretical framework of Information Theory, whose main idea is summarized as follows: the area of the event horizon of a black hole measures the uncertainty of the internal state of this object. The entropy of a black hole is directly proportional to the area of its event horizon whose constant of proportionality is of the order of 1. In this way, the Second Law of Thermodynamics acquires a natural extension called the Generalized Second Law, which says that Generalized Entropy, i.e., the entropy associated with black holes plus common entropy (that associated with physical systems that are found outside black holes) cannot decrease. This theoretical framework allows finally to establish a formalism analogous to the four laws of Thermodynamics for the dynamics of black holes. Since the Generalized Second Law is expected to be obeyed for all allowed physical processes, it is then postulated as the law of the Universe. This leads to an universal upper bound for the entropy associated with a system of matter known as the Bekenstein Bound. For highly symmetric systems of matter, the Bekenstein Bound is reduced to the Spherical Entropy Bound, which establishes that for a spherical region of surface area A, the maximum entropy value that can be found enclosed in such region is the entropy of a black hole and equal to its horizon area, i.e., A/4. Estimates made through the theoretical framework of the Local Quantum Field Theory establish that the number of degrees of freedom of a spherical region of space in an asymptotically flat spacetime is of the order of the volume of that region, on the other hand, the 65 Spherical Entropy Bound tells that the entropy in a system increaces only with the surface area. This contradiction arises as a consequence of an inappropriate inclusion of the gravitational effects that affect the Fundamental System. Due to the absence of a Quantum Theory of Gravity, the Holographic Principle is imposed to reconcile General Relativity with the postulates of Quantum Mechanics. In conclusion, the information necessary to describe all the possible physics found in a spherical region of space in asymptotically flat spacetime is A/4, i.e., it is fully contained in the surface that encloses it, motivating the given name to the principle. The theoretical development that motivates the Holographic Principle, that is, the Spherical Entropy Bound and Unitarity are not applicable to all types of spacetimes. In fact, only certain special spacetimes can satisfy the Spherical Entropy Bound given the number of restrictions to impose and the high degree of symmetry required. On the other hand, there are space-times that allow describing the formation of black holes by means of a scattering matrix and that satisfy Unitarity, however, these descriptions lack generality. In the years after the formulation of these concepts, there were theoretical developments that supported the unitary descriptions in the formation and evaporation of black holes. More precisely examples with theoretical evidence are found in the String Theory. In certain asymptotic AdS spacetimes, the formation and evaporation of black holes are described by a Unitary Conformal Field Theory [7], showing that at less in certain spacetimes the evolution of a system that has collapsed into a black hole is unitary. Complementarity requires that the laws of physics describe the experience of any observer, even one falling into a black hole. However, the Spherical Entropy Bound need not be satisfied inside a black hole. Also, in [7] important scenarios are shown as in Cosmology where this entropy boundary is not maintained. In general, there is no way to do a clear a priori correspondence between areas and the number of fundamental degrees of freedom in a given region. To date the clearest manifestation of Holography is the so-called AdS/CFT correspondence conjecture [20]. Where gravity in a special background can be described by 66 a QFT in D − 1 dimensions, making explicit the final idea behind the Holographic Principle. Given the need to motivate the Holographic Principle through a general Entropy Boundary, developments subsequent to those shown in this report postulate two possible generalizations to the Spherical Entropy Boundary: the Spacelike Entropy Boundary and the Covariant Entropy Boundary. The Spacelike generalization although intuitive lacks generality, there are cosmological scenarios where it is violated. On the other hand, the Covariant Entropy Boundary, although it cannot be derived in a fundamental way (through the physics of black holes), has more generality than the Spacelike generalization. In fact, there are no known examples that exceed it (see [7] for a deeper discussion). Therefore, the Covariant Entropy Boundary should be the object of study for later works, since it allows a refinement of the ideas shown in this report. 67 Appendix A Foundations of General Relativity A.1 Introduction The theoretical foundation of General Relativity is found in the concept of manifold, however, what we are interested in studying is a special kind of manifold which is called: spacetime. A spacetime is defined as follows 1 : Definition A.1.1. A spacetime M is to be a real , four dimensional connected C ∞ Hausdorff manifold with a globally defined C ∞ (or C 2 would do) tensor field g of type (0, 2), which is nondegenerate and Lorentzian. By Lorentzian (or hyperbolic normal) is meant that for any x ∈ M there is a basis in Tx = Tx (M ) (the tangent space to M at x) relative to which gx has the matrix diag(−1, 1, 1, 1). We will call the mathematical object v ∈ Tx as vector. The reason why we define vectors this way is because: 1. Each point x of a spacetime (M, gab ) has associated a space Tx by definition. In this way, defining vectors does not depend on external structures. 2. The space Tx is a vector space. 1 The development of the tools shown in this appendix closely follows Penrose’s developments in [31]. 68 The following definition allows vectors to be classified into three types: Definition A.1.2. Let M be a spacetime, with x ∈ M . Then, any tangent vector X ∈ Tx is said to be: timelike, spacelike or null according as g(X, X) ≡ gab X a X b is negative, positive or zero. The null cone at x is the set of null vectors in Tx . The null cone disconects the timelike vectors into two separate components. The previous definition allows to recover the concept of light cones for curved spacetimes. The light cone is a subset of the tangent space Tx , that is, each point x ∈ M has an associated light cone Cx . In Minkowski spacetime, defining the past and the future can be trivially carried out as follows: Let M4 be Minkowski spacetime and x ∈ M4 . For each x ∈ M4 the future is one half of the cone of light Cx and the past is the other half [24]. However, there are spacetimes for which it is not possible to establish a continuous choice of past and future, these types of spacetimes are called non-time-orientable spacetimes. Throughout the monographic work we only concern ourselves with studying time-orientable spacetimes which can be defined as [31]: Definition A.1.3. A spacetime M is said to be time-orientable if it is possible to make a consistent continuous choice all over M , of one component of the set of timelike vectors at each point of M . To label the timelike vectors so chosen future-pointing and the remaining ones past-pointing is to make the spacetime M time-oriented. In this case, the non-zero null vectors are termed future-pointing or past-pointing according as they are limits of futurepointing or past-pointing vectors. That is, at each point x ∈ M , the timelike vectors of the tangent space Tx can be divided into two classes. An equivalence class that we can define is the following [44]: 69 Let M be a time-orientable spacetime and x ∈ M . Let X, Y ∈ Tx , we say that X and Y are equivalent (X ∼ Y ) if and only if g(X, Y ) < 0. Once the kind of spacetime we will work on is defined, we have to define two relevant concepts: path and endpoint. Definition A.1.4. A path is a continuous map µ : Σ → M , where Σ is a connected subset of R containing more than one point. This is a smooth path if µ is smooth with nonvanishing derivative dµ (the degree of smoothness being C ∞ unless otherwise stated). Thus, a path carries a parameter range Σ being the path domain. The concept of path is usually taken as a synonym of the concept of curve, however, we will call curve image of the map µ or the equivalent class of paths that transform smoothly under a change of coordinates, i.e., homeomorphisms and diffeomorphisms of the path domains (see figure A.1). The paths as the vectors can be generally classified into three types: 1. Timelike paths: these are paths whose tangent vectors are all timelike. 2. Causal paths: these are paths whose tangent vectors are all non-spacelike. 3. Spacelike paths: these are paths whose tangent vectors are all spacelike. In this way, curves can also be classified as timelike, causal, and spacelike, depending on the type of path that defines the curve. A curve λ(t) is said oriented if and only if ∀t0 , t1 , such that t0 < t1 then, λ(t0 ) < λ(t1 ). An oriented curve has two possible orientations: past and future. Consequently, a timelike curve is said to be future directed, if and only if, under the canonical orientation defined for the time-orientable spacetime, each temporal tangent vector is of the future equivalence class, that is, if all the tangent vectors of this curve are directed towards the future (see figure A.2). 70 Figure A.1: Path. On a spacetime M a map of the form µ : R → M is called a path. On M in red is a curve, which is the image of the path µ. Figure A.2: Future directed curve. Red is a curve on a spacetime M . All tangent vectors of the red curve are pointing towards the canonical future. 71 The curves in General and Special Relativity represent the movement of particles, people, planes, stars, etc. For example, if on a spacetime (M, gab ) there is a curve λ that is future directed and extends infinitely; it can be understood that said line corresponds to an object or person that will have no end in time. However, there are events (points) associated with a curve of a space-time that are special, since they are places where “the curve has an end”. Indeed, these points indicate an end in time and space. For this reason, these points are called endpoints. An endpoint is defined as follows [31]: Definition A.1.5. Let Σ be the domain of µ and let a = inf Σ, b = sup Σ (possibly a = −∞ or b = ∞). Then x ∈ M is an endpoint if for all sequences {ui } ∈ Σ, ui → a implies µ(ui ) → x or ui → b implies µ(ui ) → x. If µ is timelike (or causal) and futured oriented, then in the first case x is a past endpoint and in the second case a future endpoint. For convenience, every end point will be a point that belongs to its associated curve (see figure A.3). If a curve does not have a future endpoint it is said to be future endless, similarly, if a curve does not have a past endpoint, it is called past endless. A.2 Causal Structure The causal structure of a spacetime refers to the causal relationship that exists between events in that spacetime. Two events are said to be connected, if and only if, there is a causal curve connecting them. For example, in spacetime M4 an event p0 is causally connected to an event p1 if the latter is inside the cone of light of p0 . However, if an event p2 is outside the light cone of the event p0 , then p2 will be causally disconnected from p0 and p1 , since the only way to connect p0 or p1 with p2 is by means of a curve that describes a movement whose speed is higher than that of the speed of light (see figure A.4). 72 Figure A.3: Endpoint. Let (M, gab ) be an arbitrary spacetime and the family of curves F = {λi }5i=1 , such that λi ⊂ M . The point p0 ∈ M (colored orange) is said to be the future endpoint of the family of curves F. If p0 , p1 ∈ M4 are causally connected in the way described above p1 , it is said to be part of the causal future of p0 (as long as the curve that connects them is future directed). However, in arbitrary spacetimes, if the event p1 belongs to the causal future of p0 generally p1 6∈ Cp0 . For that reason, the definition of causal connection is independent of the light cone concept. In this way, we define an even more general concept to that of the curve to talk about the causal connection, future and past of an event called trip [31]. Definition A.2.1. A trip is a curve which is piecewise a future-oriented timelike geodesic. A trip from x to y is a trip with past endpoint x and future endpoint y. We write x ≪ y (read x chronologically precedes y) if and only if there exists a trip from x to y. Thus, the relation x ≪ y states the existence of points x0 , x1 , · · · , xn with n ≥ 1, a timelike geodesic called a segment having past endpoint xi−1 and future endpoint xi , for each i = 1, · · · , n, where we set x0 = x, xn = y. In the figure A.5, an example of trip is shown: in red there are events that belong to 73 Figure A.4: Connection between two points p0 and p1 . an arbitrary spacetime, which are causally connected by means of timelike geodesics. Of gray, green, blue, etc., we have what according to the definition of trip are called segments, which are timelike. The definition of trip clearly establishes an order relationship in which it would be possible to understand precisely how two events are related to each other and whether they are connected or not. If between two events x and y there is a connecting trip made up of causal geodetic segments, then we can generalize the concept of trip to causal trip, which is defined as follows [31]: Definition A.2.2. A causal trip is defined in the same way as a trip except that causal geodesics, possibly degenerate, replace the timelike geodesics of the trip definition. We write x ≺ y (read x causally precedes y) if and only if there is a causal trip from x to y. From the definitions of trip and causal trip the following proposition is derived: 74 Figure A.5: Trip from x to y in an arbitrary spacetime. Proposition A.2.1. Let M be a spacetime and a, b, c ∈ M , then: a ≪ b implies a ≺ b a ≪ b, b ≪ c implies a ≪ c (A.1) a ≺ b, b ≺ c implies a ≺ c Proof. 1. a ≪ b implies a ≺ b: a ≪ b implies the existence of a trip, that is, a curve composed of timelike geodesics. Because timelike geodesics are also causal curves (nonspacelike), then a ≺ b. 2. a ≪ b, b ≪ c implies a ≪ c: a ≪ b implies the existence of a trip that we will call γ1 that connects a with b, while b ≪ c implies the existence of a trip γ2 that joins the events b and c. Thus, if a is connected to b through γ1 and b is connected to c through γ2 , then a and c are connected through the trip γ3 = γ1 ∪ γ2 , which implies 75 by definition that a ≪ c. 3. a ≺ b, b ≺ c implies a ≺ c: the procedure is analogous to that shown in point 2, the only thing to do is to change the word trip for causal trip. Now that it is possible to formally establish a causal relationship between any two events in an arbitrary spacetime through the trip and causal trip curves, we will define the future and past of an event as follows [31]: Definition A.2.3. The set I + (x) = {y ∈ M | x ≪ y} is called the chronological (or open) future of x; I − (x) = {y ∈ M | y ≪ x} is the chronological past of x; J + (x) = {y ∈ M | x ≺ y} is the causal future of x; J − (x) = {y ∈ M | y ≺ x} is the causal past of x. The chronological S S or causal future of a set S ⊂ M is defined by I + [S] = x∈S I + (x), J + [S] = x∈S J + (x), repectively, and similarly for the pasts of S. Figure A.6 shows approximately the chronological future (gray color) of an event x (red color) in a spacetime (M, gab ). The causal future J + (x) is equivalent to the union of the sets I + (x) and ∂I + (x), where ∂I + (x) denotes the boundary of I + (x) (black color). For instance, in the case of spacetime M4 the boundary of the chronological future of an event x is known as future light cone. Therefore, in M4 , I + (x) represents the possible future events of a massive particle found in x, while ∂I + (x) represents the possible future events of a photon or a massless particle at x. In general the following proposition is true [31]: Proposition A.2.2. I + [S] is open, for any S ⊂ M . The above is also true for the set I − [S]. On the other hand, in general J + (x) is not a closed set. To demonstrate this, figure A.7 is used as a counterexample. 76 Figure A.6: Chronological future of an event. To continue advancing in the study of the causal structure of spacetimes, we must define the concept of exponential map, which is defined as follows [31]: Definition A.2.4. For any a ∈ M the exponential map is a smooth (C ∞ ) map, denoted expa from some open subset of the tangent space Ta , into M . If V ∈ Ta , we define expa (V ) to be the point p of M (if such exists) such that the affinely parameterized geodesic with tangent vector V at a and parameter value 0 at a acquires the parameter value 1 at p.. If expa maps all vector V ∈ Ta to M for all a ∈ M , then M is said to be a spacetime geodesically complete, since all affinely parameterized geodesics can be extended to arbitrarily large values of the parameter. There are spacetimes like the one shown in figure A.8, where two vectors V and V ′ are mapped at the same point a ∈ M (whether or not are geodesically complete). Finally, at the local level (in the neighborhood of an event) all spacetime recovers the causal properties of the spacetime M4 , for instance, I + (x) is generated by null geodesics. The following theorem guarantees the above [24]: 77 Figure A.7: Spacetime with a point removed at ∂I + (x) (in Minkowski spacetime). J + (x) is not closed because xis causally disconnected from events beyond the removed point. Hence, the complement of J + (x) is not open. Theorem A.2.1. Let (M, gab ) be an arbitrary spacetime and let p ∈ M . Then there exists a convex neighborhood of p, i.e., an open set U with p ∈ U such that for all q, r ∈ U there exists a unique geodesic γ connecting q and r and stay entirely within U . Furthermore, for any such U , I + (p)|U consists of all points reached by future directed timelike geodesics starting at p and completely within U , where I + (p)|U denotes the chronological future of p in the spacetime (U, gab ). In addition, ∂I + (p)|U is generated by future directed null geodesics in U emanating from p. The first part of the theorem is proved in [45], while the second part in [3]. The theorem states that if p, q ∈ M , such that q ∈ I + (p) and γ is a null geodesic whose past and future endpoints are p and q respectively, then there is a finite family of normal convex neighborhoods {Ui } that covers the curve γ. In general, if γ is a compact curve of M then 78 Figure A.8: The Riemannian 2-space M is the surface of a finite cylinder [31]. As can be seen in the figure, in this spacetime, at point a when we use the exponential map expa on two different vectors V, V ′ ∈ Ta the same image is obtained p ∈ M . there is such a finite family of neighborhoods (see figure A.9). Figure A.9: Normal convex neighborhoods covering a curve. 79 A.3 Future and Past Definition A.3.1. A set F ⊂ M is called a future set if F = I + [S] for some S ⊂ M . Clearly F is a future set if and only if F = I + [F ]. A future set F therefore has the property that: if x ∈ F and x ≪ y, then y ∈ F . The theorems and proofs shown below are applicable if the sense of time is reversed, that is, if instead of thinking about the chronological future of a set of events, we think about the chronological past. In fact, a set called past is defined, which is denoted by P as a subset of M , such that P = I − [S] and S ⊂ M . Once F and P are defined, the history of a set of events will be defined: Definition A.3.2. Let M be a spacetime and S ⊂ M . The set H = F̄ ∪ P̄ is called history. Where F̄ = ∂F ∪ F and P̄ = ∂P ∪ P . The future set has the following properties: Proposition A.3.1. Let F be a future set . Then: F̄ = ∼ I − [∼ F ],  ∂F = x | I + (x) ⊂ F and x ∈ F (A.2) − =(∼ F ) ∩ (∼ I [∼ F ]), F =I + [F̄ ]. The first property is demonstrated as follows: Proof. Let M be a spacetime and S ⊂ M . So, by definition F = I + [S]. Imposing x 6≪ x for all x ∈ M (non-existence of timelike closed curves). Then ∼ F = {x ∈ M | x 6∈ F }; that is, x ∈ ∂F or x ∈ P̄ or x ∈ M − H. I − [∼ F ] means that if x ∈ ∂F , then the events y ∈ M that precede chronologically x 80 Figure A.10: Future set of a set S in an arbitrary spacetime. (y ≪ x) are those y, such that y ∈ M − H or in y ∈ P̄ (see figure A.10). If x ∈ M − H, then again x is chronologically preceded by all y, such that y ∈ M − H or in y ∈ P̄ . If x ∈ P̄ , then x is causally preceded by all y ∈ P . Therefore, ∼ I − [∼ F ] will be all y, such that y 6∈ P̄ and y 6∈ M − H, that is, y ∈ ∂F or y ∈ F , in other words, y ∈ F̄ . In time-orientable spacetimes there is a special kind of subset called achronal. This set is relevant in General Relativity because it allows defining structures that imply causal conditions on a given spacetime. The definition of an achronal subset is as follows [31]: Definition A.3.3. A set S ⊂ M is called achronal if no two points of S are chronologically related (i.e., if x, y ∈ S, then x 6≪ y, for all x and y in S). Although in principle it seems that ∀S ∈ M , such that, S is spacelike is also an achronal subset; the truth is that this proposition is false, since there is a counterexample in the spacetime M4 (see figure A.11). One special kind of achronal subset is the known achronal boundary which is defined as follows: 81 Figure A.11: A spacelike surface Σ ∈ M4 that is not an achronal subset. In Minkowski space-time it is possible to draw a spacelike curve Σ, however, this does not mean that it is achronal because there are two points x, y ∈ M4 that are they can connect causally through a causal curve (blue color). Definition A.3.4. A set B ⊂ M is called an achronal boundary if it is the boundary of a future set, i.e., B = ∂I + [S]. Therefore, the following statement is true for any spacetime: Proposition A.3.2. Let M be a spacetime and S ⊂ M . If B = ∂F then B is an achronal boundary. However, to facilitate the proof of the previous proposition we will demonstrate the following lemma: Lemma A.3.1. Let A, B ⊂ M . I + [A ∪ B] = I + [A] ∪ I + [B]. Proof. By definition: x ∈ I + [A ∪ B] ↔ ∃y ∈ M (y ∈ A ∪ B ∧ y ≪ x) (A.3) y ∈A∪B ↔y ∈A∨y ∈B (A.4) On the other hand: 82 In other words, we have the following proposition: x ∈ I + [A ∪ B] ↔ ∃y ∈ M ((y ∈ A ∨ y ∈ B) ∧ y ≪ x) (A.5) x ∈ I + [A ∪ B] ↔ ∃y ∈ M ((y ∈ A ∧ y ≪ x) ∨ (y ∈ B ∧ y ≪ x)) (A.6) Then: For the extension axiom (of set theory) we will have: I + [A ∪ B] = I + [A] ∪ I + [B] (A.7) Proof. To prove Proposition 6.4 it is necessary to show that I + [∂F ] ∩ ∂F = ∅. Recall that F̄ = ∂F ∪ F , therefore: I + [F̄ ] = I + [∂F ] ∪ I + [F ] (A.8) I + [F̄ ] ∩ ∂F = (I + [∂F ] ∪ I + [F ]) ∩ ∂F (A.9) Then: Since I + [F̄ ] = F , then, since F is an open set, we will have: ∅ = (I + [∂F ] ∩ ∂F ) ∪ (I + [F ] ∩ ∂F ) (A.10) Which allows to conclude that: ∅ = I + [∂F ] ∩ ∂F (A.11) If (M, gab ) is a spacetime and Σ ⊂ M is achronal, then there is a subset D+ [Σ] ⊂ M called future domain of dependence defined as follows [24]: Definition A.3.5. Let (M, gab ) be a spacetime and Σ ⊂ M is closed (and possibly with 83 Figure A.12: An achronal subset with edges. In a neighborhood of p, there are two points q and r, such that q ∈ I + (p) and r ∈ I − (p), However, there is a curve γ (blue color) that causally connects q and r such that Σ ∩ γ = ∅. edges). The future domain of dependence of Σ, denoted by D+ [Σ], is defined by: D+ [Σ] =      p∈M   Every past inextensible causal curve    through p intersects Σ. (A.12) A closed achronal subset Σ has edges if and only if there is a set of points p ∈ Σ, such that, at a neiborghood O of p there are points q ∈ I + (p)|O and r ∈ O that are connected through a timelike curve λ that does not cross Σ (see figure A.12). There is also another set related to Σ which is the temporary opposite of D+ [Σ], which is called past domain of dependence; its definition is analogous to that used in D+ [Σ] except that the past is exchanged for the future and vice versa. Therefore, if D+ [Σ] and D− [Σ] exist for a Σ ⊂ M that is achronal and closed (possibly with edges), then, there is a D[Σ] ⊂ M called full domain of dependence defined by: D[Σ] = D+ [Σ] ∪ D− [Σ] (A.13) The reason why these sets are defined is because it allows us to study which events x ∈ M are entirely determined by the events y ∈ Σ. If D[Σ] = M , then Σ is a hypersurface called 84 Figure A.13: The conjugate point of the Earth’s North Pole is the South Pole. Due to the existence of conjugated pairs, it is always possible to make a longer trip to the country P through a geodesic. For example, to go around the back of the North Pole, to get to the South Pole, and to go from the South Pole to the country P . Cauchy surface. As by definition, a Cauchy Σ is always traversed by all timelike curves λ at M , then edge(Σ) = ∅. Surfaces Σ (Cauchy surfaces) are interpreted as instants of time throughout the universe [24]. Now, if in a spacetime there is a Cauchy surface Σ, then the spacetime is called globally hyperbolic [24]. In other words, a hyperbolic spacetime is one in which the past and the future can be determined from the conditions found on the surface of Cauchy Σ. A.4 Conjugate Points If we want to travel from the North Pole of the Earth to a country P that is in the Equator, the fastest way to make such a trip would be through a maximum circle, in other words, through a geodesic. However, in presence of conjugate points in the manifold the geodesic 85 Figure A.14: Conjugate points. is both the path of shortest and longest distance (see figure A.13) [31] [46]. Definition A.4.1. If V is a Jacobi field defined on γ and V vanishes at two distinct points p, q ∈ γ, while not vanish at all points of γ, then p and q are called a pair of conjugate points on γ. For example, the North Pole and the South Pole are conjugate points. Figure A.14 shows a pair of conjugate points p and r on a path γ that is causal. A pair of conjugate points can be thought of roughly as intersection points between a geodesic γ and its neighbor γ ′ . Analogously, the following statement is true for any spacetime [31]: Proposition A.4.1. Let (M, gab ) a spacetime and N ⊂ M . Let N be a simple region and let p, q ∈ N with pq future causal. Then if η is the causal trip pq and η ′ is another causal trip in N from p to q, we have l(η) > l(η ′ ). The previous proposition establishes that locally a causal geodesic maximizes the functional length l (or proper time). What has been discussed so far allows us to conclude that although locally a geodesic can be identified as a spacetime curve that maximizes proper time, globally this may not be the case due to the existence of conjugate points. 86 A.5 Congruence The modern method for predicting conjugate points in a spacetime is given by studying a type of set of curves called geodesic congruence. A congruence is defined as follows [5]: Definition A.5.1. A congruence is a set of curves in an open region of spacetime such that every point lies on precisely one curve (geodesic). Suppose we have a congruence and its tangent vector field is U µ = dxµ . dτ A restriction that can be imposed on the vector field without losing generality is the following: Uµ U µ = −1, (A.14) that is, U µ is a normalized vector field and timelike. Recall that U µ by definition satisfies the geodesic equation: U λ ∇ν U µ = 0. (A.15) On the congruence, other vector field V µ is defined, which satisfies the deviation equation: DV µ ≡ U ν ∇ν V µ = B µν V ν , dτ (A.16) B µν = ∇ν U µ . (A.17) where The vector field V µ is called separation vector [5] or Jacobi field [31]. The tensor B µν measures the failure of parallel-transporting the vector V µ , that is, it measures the effect that the curvature has on two curves that at first were parallel (see figure A.15). In order to describe the congruence, we define a set of three vectors that are normal to the timelike geodesics and follow their evolution along to the congruence. The failure to parallel transport these three vectors along a reference geodesic will give information on 87 Figure A.15: Vector field V µ between two neighbor geodesics. how neighboring geodesics evolve. However, to describe the evolution of the congruence, it is only necessary to study the behavior of the tensor Bµν [5]. Proposition A.5.1. Bµν is normal to the space Tp . Proof. The proposition is demonstrated by making use of an element U µ ∈ Tp for a p ∈ M . For example, let U µ be the vector that satisfies the equations (A.14) and (A.15), then: U µ Bµν = U µ ∇µ Uν = 0, ν (A.18) ν U Bµν = U ∇µ Uν = 0. The first of the equations is true because U µ satisfies the equation (A.15), while the second expression is true because: U ν ∇µ Uν = ∇µ (U ν Uν ) = ∇µ (−1) = 0. (A.19) The above is equivalent to that two vectors ~v and ~u are orthogonal if: ~v · ~u = 0. (A.20) Which leads us to conclude that Bµν is normal to the space Tp . Proposition 6.6 can be taken as a valid argument to justify the use of Bµν to describe the behavior of a timelike congruence. From the tensor calculus is known that every tensor 88 (0, 2) can be decomposed into a symmetric part and an antisymmetric part, and also the symmetrical part can be decomposed into a trace and traceless parts. The outcome of this decomposition results in the following identity: 1 Bµν = θPµν + σµν + ωµν , 3 (A.21) where Pµν is called projection tensor, θ is called expansion, σµν is called shear, ωµν is called rotation and: P µν = δνµ + U µ Uν , θ = P µν Bµν , 1 σµν = B(µν) − θPµν , 3 (A.22) ωµν = B[µν] . Expansion describes the change in volume of the hypersurface that forms the congruence, shear describes the distortion of the form of the hypersurface that forms the congruence and finally rotation describes the rotational movement of the hypersurface that forms the congruence. Therefore, the evolution of a congruence is described through the covariant derivative of Bµν along of D/dτ ≡ U σ ∇σ : DBµν = U σ ∇σ ∇ν Uµ , dτ (A.23) however, for any vector V ρ associated with each x ∈ M (with M being a torsion-free spacetime) the following relationship holds: [∇µ , ∇ρ ] V ρ = Rρσµν V σ , 89 (A.24) then, the equation (A.23) can be rewritten as follows: DBµν = U σ ∇ν ∇σ Uµ + U σ Rλµνσ Uλ , dτ = ∇ν (U σ ∇σ Uµ ) − ∇ν U σ ∇σ Uµ − Rλµνσ U σ U λ , = −∇ν U σ ∇σ Uµ − Rλµνσ U σ U λ , (A.25) = −∇ν U σ ∇σ Uµ − Rλµνσ U σ U λ , = −B σν Bµσ − Rλµνσ U σ U λ . Taking the trace of DBµν /dτ , the Raychaudhuri equation is derived [5]: dθ 1 = − θ2 − σµν σ µν + ωµν ω µν − Rµν U µ U ν dτ 3 (A.26) The equation (A.26) describes the evolution of the size of a congruence formed by timelike geodesics and its null version, shown below, it is used to prove the area theorem. Suppose that for all U µ , ωµν = 0 and satisfies the Strong Energy Condition [5, 24, 3]: Tµν U µ U ν ≥ 0, (A.27) with Tµν the Stress-Energy Tensor. Then: Rµν U µ U ν ≥ 0, (A.28) with Rµν the Ricci Tensor.In addition to ωµν ω µν ≥ 0 and σµν σ µν ≥ 0 2 , then: dθ 1 ≤ − θ2 . dτ 3 (A.29) To prove that ωµν ω µν ≥ 0 and σµν σ µν ≥ 0 only it is required to demonstrate that the inner product of these two tensors with the vector U µ is null. 2 90 Solving differential inequality (A.29): 1 − θ−1 + θ0−1 ≤ − [τ − τ0 ] . 3 (A.30) If τ0 = 0, then, we will finally get: 1 θ−1 (τ ) ≥ θ0−1 + τ 3 (A.31) The equation (A.31) states that at a point p ∈ M (M being a spacetime) θ0 < 0, then, in a finite proper time τ ≤ −3θ0−1 congruence will hit a caustic (a place where geodesics intersect). The previous statement is known as the focusing theorem and is formally stated in the following way [24]: Theorem A.5.1. Let (M, gab ) be a spacetime satisfying Rab U a U b ≥ 0 for all timelike U a . Let Σ be a spacelike hypersurface with K = θ < 0 at a point q ∈ Σ. Then, within proper time τ ≤ 3/ |K| there exist a point p conjugate to Σ along the geodesic γ orthogonal to Σ and passing through q, assuming that γ can be extended that far. In summary, the focusing theorem allows us to know “when” a conjugate point will be present and under what conditions. Furthermore, remembering the meaning of the existence of a pair of conjugate points in a spacetime, the following theorem can be stated as a consequence of focusing theorem [24]: Theorem A.5.2. Let γ be a smooth timelike curve connecting a point p ∈ M to a point q on a smooth spacelike hypersurface Σ. Then, the necessary and sufficient condition that γ locally maximize the proper time between p and Σ over smooth one-parameter variations is that γ be a geodesic orthogonal to Σ with no conjugate point to Σ between Σ and p. The null version of the equation (A.26) is the following [5]: 1 dθ = − θ2 − σ̂µν σ̂ µν + ω̂µν ω̂ µν − Rµν k µ k nu , dλ 2 91 (A.32) where k µ is a null vector. Using the equation (A.32) and the null condition (i.e., Tµν k µ k ν ≥ 0) we can derive a null version of the focusing theorem, which establishes [24]: Theorem A.5.3. Let (M, gab ) be a spacetime satisfying Rab k a k b ≥ 0 for all null k a . Let S be a smooth two-dimensional spacelike submanifold such that the expansion, θ, of, say, the “outgoing” null geodesics has the negative value θ0 at q ∈ S. Then within affine parameter λ ≤ 2/ |θ0 |, there exists a point p conjugate to S along the outgoing null geodesic µ passing through q. Whose consequence theorem is the following [24]: Theorem A.5.4. Let S be a smooth two-dimensional spacelike submanifold and let µ be a smooth causal curve from S to p. 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