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Energy, entropy, and spacetime : lessons from semiclassical black holes Costa, Bruno Arderucio 2020

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ENERGY, ENTROPY, AND SPACETIME: LESSONS FROMSEMICLASSICAL BLACK HOLESbyBruno Arderucio CostaA thesis submitted in partial fulfillment of the requirements for the degree ofDoctor of PhilosophyinThe Faculty of Graduate and Postdoctoral Studies(Physics)The University of British Columbia(Vancouver)August 2020© Bruno Arderucio Costa, 2020The following individuals certify that they have read, and recommend to the Faculty ofGraduate and Postdoctoral Studies for acceptance, the dissertation entitled:Energy, Entropy, and Spacetime: Lessons from Semiclassical Black Holessubmitted by Bruno Arderucio Costa in partial fulfillment of the require-ments forthe degree of Doctor of Philosophyin Physics.Examining Committee:William G. Unruh, Physics and AstronomySupervisorGordon W. Semenoff , Physics and AstronomySupervisory Committee MemberRobert Raussendorf , Physics and AstronomySupervisory Committee MemberKristin Schleich, Physics and AstronomyUniversity ExaminerSven Bachmann, MathematicsUniversity ExamineriiAbstractThis doctoral thesis explores semiclassical effects on black hole physics. Semiclassicaltheory refers to the application of quantum field theory in curved, classical backgroundgeometries, which respond to the expectation value of the regularised stress-energy tensorof the quantum matter.Among the original findings, I develop a few useful techniques to help regularise thestress-energy tensor in two dimensions. I apply them to a model of stellar collapse toanalyse the importance of quantum mechanical effects in the collapse itself. I find anexplicit example showing that the behaviour of the late-time Hawking radiation doesnot depend on the details of the collapse and argue that any quantum mechanical effectis negligible for the collapse of an astrophysical object (whose mass is comparable to thesolar mass).In the realm of black hole thermodynamics, I prove the first law for stationary blackholes and propose a definition for the entropy in piecewise stationary black holes whichI show to obey the generalised second law of thermodynamics. After also discussingthe zeroth law, it becomes clear that this set of laws is rooted in semiclassical physicsand give the hypotheses which are necessary for it to hold. My derivation of the lawsof black hole thermodynamics also contributes towards the answer to the long-standingquestion of interpreting the Bekenstein-Hawking entropy. My work suggests that it isunderstood from the information perspective as accounting for the information hiddenbehind the horizon.iiiLay SummaryBlack holes (that is, regions of spacetime from which no signal can escape to infinity)emit thermal radiation as a consequence of the laws of quantum mechanics. Since thediscovery of this effect, questions of how this radiation itself can affect the black holehave constantly been asked. This thesis provides simplified models for addressing relatedquestions.I also clarify a related forty-year-old question of applying the laws of thermodynamicsto systems containing a black hole. Most importantly, this thesis proves the law ofincrease of entropy, a measure of the amount of information fundamentally missingabout a system, under a controlled set of assumptions.ivPrefaceThis thesis begins presenting a literature review in chapter one that supports laterchapters.Chapter two contains unpublished original work by the author in its sections 2.1.1and 2.1.2. Section 2.2 is original work coauthored by his research supervisor publishedin Arderucio-Costa, B.; Unruh, W.G. Phys Rev. D 97 024005 (2018) [1]. The researchsupervisor was responsible for the design the model therein. The author of this dis-sertation was responsible for the details of its implementation, all the calculations, fulltext, and graphs of the published paper. The material here has more details than in itspublished version. Figures 2.1 and 2.2, as well as part of the text of that section is usedwith permission of the American Physical Society (APS).Chapter three contains unpublished original work by the author in section 3.1.2, andoriginal work by the author submitted to a single-authored publication in section 3.2.The pre-print is available in the arXiv gr-qc: 1905.10823 [2]. Again the section containsmore details than in the version submitted for publication.vContentsAbstract. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vContents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiNotation and Conventions. . . . . . . . . . . . . . . . . . . . . . . . ixAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . xiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Evolution of Concepts of Ordinary Thermodynamics . . . . . . . . . . . 41.2 Survey of Black Hole Thermodynamics . . . . . . . . . . . . . . . . 121.2.1 Review of Quantum Fields in Curved Spacetimes . . . . . . . . . . 121.2.2 Geometry of Killing Horizons and the Zeroth Law of Black Hole Mechanics 201.2.3 From the Laws of Black Hole Mechanics to Black Hole Thermodynamics . 262 Role of Energy . . . . . . . . . . . . . . . . . . . . . . . . . . 372.1 Regularisation of Stress-Energy Tensor in 2D . . . . . . . . . . . . . . 392.1.1 Method for Finding the Conformal Factor in 2D Spacetimes . . . . . . 452.1.2 Relating Spherically Symmetric 4-Dimensional Solutions and 2-DimensionalSolutions of Field Equations. . . . . . . . . . . . . . . . . . . 512.2 Example: Modelling Quantum Mechanical Effects in Stellar Collapse. . . . . 532.2.1 4-Dimensional Model . . . . . . . . . . . . . . . . . . . . . 54vi2.2.2 Expectation Value of the Regularised Stress-Energy Tensor . . . . . . 582.3 Difficulties with the Semiclassical Theory . . . . . . . . . . . . . . . 633 Role of Entropy. . . . . . . . . . . . . . . . . . . . . . . . . . 663.1 Nature of Entropy. . . . . . . . . . . . . . . . . . . . . . . . . 663.1.1 Geroch’s Box. . . . . . . . . . . . . . . . . . . . . . . . . 673.1.2 Criticism to Proposed Entropy Bounds . . . . . . . . . . . . . . 693.2 Laws of Black Hole Thermodynamics in the Semiclassical Context. . . . . . 753.2.1 Zeroth Law . . . . . . . . . . . . . . . . . . . . . . . . . 763.2.2 First Law . . . . . . . . . . . . . . . . . . . . . . . . . . 773.2.3 Second Law . . . . . . . . . . . . . . . . . . . . . . . . . 843.2.4 Third Law. . . . . . . . . . . . . . . . . . . . . . . . . . 933.2.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 944 Conclusions and Prospects . . . . . . . . . . . . . . . . . . . . . 98Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102viiList of Figures1.1 Geometry of a Killing horizon, depicting the orbits of the Killing fieldχa, which is null over the bifurcated horizon h+ ∪ h− and vanishes at thebifurcation surface B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.1 Carter-Penrose diagram adapted from [48]. Regions III and V repre-sent Minkowski patches; region IV is Schwarzschild; region I is an ingo-ing Vaidya patch, and region II an outgoing one. Our simplified two-dimensional version is concerned with regions I and II only. . . . . . . . 482.2 m2〈0|Tuu|0〉 as a function of r coincides with 〈0|Tvv|0〉 on the onset ofthe collapse, here represented by large values of R and with ref. [6] forvery late times, for R → 2m. The horizontal plane on top of the graphrepresents the constant κ2/48pi. . . . . . . . . . . . . . . . . . . . . . . . 612.3 All appropriate observables as measured by someone following the col-lapse obtained from the regularised energy-momentum tensor are finiteeverywhere except at the singularity. The graph shows exclusively theregion around r = 2m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.1 Geometry that leads to eq. 3.25. The worldline labeled as “wall of box”denotes the timeline along which the boundary conditions on the fields areimposed. The shaded areas represent the regions of M that are isometricto stationary spacetimes. The contour of integration is in blue. . . . . . 88viiiNotation and ConventionsI use the metric signature (−+ ++) and natural units such } = c = G = 1.Ricci tensor is Rab = Rcacb, where Rabcd denotes the Riemann curvature tensor. Weyltensor is denoted by Cabcd, and R denotes the scalar curvature, except in section 2.2,where it is denoted by R.1 denotes the identity operator in an arbitrary vector space.Penrose’s abstract index notation is used throughout the text. Indices belonging tospacetime are represented by lower case Latin alphabet letters; indices belonging to phasespace by upper case Latin letters. Greek letters denote components in a specific basis. Imake use of the following abbreviations: A[ab] = 12!(Aab−Aba), and A(ab) = 12!(Aab+Aba)and similar expressions for more than two indices. I also use lower case Latin lettersfor tensors in⊗nH , the symmetrised or antissymetrised tensor products of a Hilbertspace by itself n times, but no confusion arises since context will make it clear when thisset of indices has this less common meaning throughout the text.When no confusion regarding their rank is present, differential forms are representedby bold symbols, making the notation more concise. This is particularly useful when anobject is a differential form both in space-time and in phase space.The boundary in a topological sense is given by A˙ = A¯ \ int(A), where the bar overa letter denotes the closure and int the interior of a set. The boundary in a manifoldsense is denoted by ∂M , which is the subset of M whose image under a local chart’smap lies on the boundary of Hn(= [0,∞)× Rn−1) in Rn.Chronological and causal future (past) are denoted respectively by I± and J±. Fu-ixture (past) null infinities (I ±) and related objects are defined according to Hawking &Ellis [3]. Their notation is preserved.Complex conjugation is denoted by a bar over the symbol.The tangent space at a point P of a manifold M is symbolized by TPM , the tangentbundle TM . The cotangent space at P , as any dual space, by T ∗PM , and cotangentbundle T ∗M . The set of vector fields on M is denoted by X (M).The Heaviside step function is denoted by θ : R → R, θ(x) = 1 for x > 0 and 0otherwise. The sign function sgn : R→ R is defined by sgn(x) = θ(x)− θ(−x).The notation x→ x0 + 0 means the lateral limit limx→x+0 .xAcknowledgmentsThe author wishes to thank numerous people who played a myriad of different rolesduring the course of the programme. First and foremost, I thank my research supervisor,Bill Unruh, for accepting me as his student, for several discussions about this projectand many unrelated topics in physics, and for all the time he devoted to me over thecourse of the last few years.I was lucky to have had the chance of discussing research-related topics with manyother people, such as Marek Radzikowski, Ricardo Correa da Silva, Colby DeLisle, Jor-dan Wilson, Gabriel Cozzella, and Daniel Sheinbaum. Marek and Ricardo played animportant role in teaching me some topics related to the algebraic approach to quantumfield theory.I was lucky to be a teaching assistant for courses instructed by Kristin Schleich,which whom I had a fruitful relation throughout my time at UBC. I thank her for herkindness and for caring for my own research and for me as a person. The atmosphereshe created made it much more pleasant to develop my own teaching skills.My friends who gave continued support in areas not related specifically to my re-search, but helped in other crucial ways that allowed me to pursue my degree withoutmajor struggles, Erich, Taís, Mateus, Carolina, Acza, and Rogerio.I thank Green College UBC for the lovely opportunity of being a resident member,and the experience it provided me. Countless valuable conversations happened over thedinner table, and good laughs at the 7:30 am breakfast. The fellow members (or formermembers) Emily, Yasmin, Bronwyn, Shannon, Yotam, and Weiyu were particularlyxihelpful.Even at distance, my mother Adriane offered continued support and care. I amextremely grateful for her always believing in what I can achieve.This project was partially supported by Conselho Nacional de Desenvolvimento Cien-tífico e Tecnológico under the process 200339/2014-1. It is impossible to overstate theimportance of public investment in basic science, as in projects like this one.xiiDedicationTo the memory of Stephen W. Hawking, a long-time source of inspiration who passedaway while the work that led to this thesis was in progress.xiii1. IntroductionIn this introductory chapter, I present my take on existing literature that is relevantfor the rest of this thesis. The adaptation of the original derivation of the Unruh effectby Unruh in [4] to other stationary spacetimes with a bifurcate horizon that led to eq.(1.28) is believed to be original, but the result is not. Kay and Wald’s [5] derivation ismore rigorous and more general than mine.The next chapters are organised as follows: I start chapter two by reviewing thecovariant point-splitting method for regularisation and the axiomatic approach to definethe regularised expectation value of the stress-energy tensor in quantum field theoryin curved spacetimes. The chapter as a whole is dedicated to the energy-momentumtensor in semiclassical gravity. With the exception of the explanation of Israel’s junctionconditions in subsection 2.2.1, all the material from subsection 2.1.1 on is original to thiswork. Among this material, I present technical advances in regularising the expectationvalue of this object in two dimensions (series expansion for finding the conformal factorand its derivatives) and discuss what can be extrapolated to a spherically symmetric four-dimensional problem. These advances have potential to be useful in future problems.The most scientifically relevant content is a model for a stellar collapse. The model heredeveloped is different from one previously proposed by Davies, Fulling, and Unruh [6],yet it produces the same value of the regularised stress-energy tensor at late times atinfinity. This late-time concurrence provides a concrete example in support of the beliefthat the quantum radiation emitted by the black hole in this regime does not dependon the details of the collapse. Furthermore, this model shows that the values of all1observables are finite and small except near the singularity. This is important since itis in contradiction of some claims in the literature (see e.g. Mersini-Houghton [7]).Chapter three is dedicated to the notion of entropy. The derivation of Tolman’srelation in subsection 3.1.1 is my adaptation of an argument by Landau [8] to modernterms. Subsection 3.1.2 is original from this work, with the exception of the estimate ofthe collapse time, which is due to Wald [9]. Section 3.2 is entirely original. It containsthe separation of the g and ψ degrees of freedom in the classical theory and the propertiesof the forms (the symplectic current and the pre-symplectic form) after this separation,the semiclassical extension of the Hamiltonian theory, and its applications to the laws ofthermodynamics. As a whole, the chapter starts by examining some proposed boundsthat establish a maximum value of entropy that a system can have for a fixed amountof energy it carries. I give a concrete example that can be used to violate some of theseproposals. Then I move on to deriving the laws of black hole thermodynamics froma semiclassical theory. In the formulation of these laws in this chapter presented, theentropy is the von Neumann entropy of the quantum fields. The laws themselves andtheir derivation are interpreted from the information perspective.***2Our universe obeys the rules of quantum mechanics, which have succeeded in everypossible experiment ever designed, and have been applied to a myriad of different phys-ical systems. Although the relation between classical and quantum physics is generallycomplicated, there are many instances when the quantum description is approximatedby classical physics. Some problems can be understood purely by the laws of classi-cal physics, some require a fully quantum-mechanical picture; and some are adequateto a semiclassical approach, that is, when some of its degrees of freedom are treatedclassically, and some quantum mechanically.Probably the most traditional example of this semiclassical approach is in atomicphysics, say when an atom decays from an excited state emitting electromagnetic radia-tion. This problem is properly treated using quantum electrodynamics, the full quantummechanical description of both the atom and the electromagnetic field. However, if oneis interested, for example, in describing the effect of an external magnetic field in thefrequency of the emitted radiation, to a very good approximation it suffices to treat elec-trons quantum mechanically, and the electromagnetic field classically (see, for example,the discussion about the Zeeman effect in Dirac’s book [10]).Gravity has proven to create conceptually interesting phenomena in quantum matter,and since, despite great efforts, we currently do not have a full consistent quantummechanical theory of gravity, the semiclassical approach is as far as this dissertation willgo.As the title suggests, the goal of this doctoral thesis is to explore semiclassical gravitywith particular interest to the laws of thermodynamics, which have two actors verydifferent in nature: energy and entropy. An equation relating these actors has thereforeconceptually interesting interpretations. This chapter exposes how thermodynamicsmade its way into semiclassical gravity through black hole physics.31.1 Evolution of Concepts of Ordinary ThermodynamicsThe aim of this section is not to give a historically accurate survey of the developmentof thermodynamics, but to briefly discuss what level of the understanding of the generalprinciples of thermodynamics is possible to be gained within the limitations of a giventheoretical framework. I refer to a logical, rather than a historical sense of the word“evolution”. Later we will see that the way concepts evolve in black hole thermodynamicsin many ways mirrors the evolution of concepts in ordinary thermodynamics.Our understanding of thermodynamics has changed since its birth in the age of theIndustrial Revolution. It is possible to obtain experimental input from observationsthat measure macroscopic quantities alone, laws for the behaviour of matter. Directlyfrom these observations, one could establish some “laws of nature”, like the zeroth law,stating that thermal equilibrium is transitive; the first law, stating that there is a formof energy called heat that needs to be added to the energy balance so that the variationof the total energy1 of a closed system is zero; and the second law, stating that there isa preferred direction for the spontaneous heat flow. In possession of these laws, one can,for example, determine an upper bound for the efficiency of heat engines or refrigerators.Once one accepts that matter is made out of atoms, with a few additional reasonableassumptions, one can in fact “derive” these “laws” from more basic laws of physics, likethe laws of classical mechanics2. Two approaches deserve being mentioned for thistask, a more traditional one, following, for example, Landau and Lifshitz [8]; and oneinspired by black hole physics [11]. Starting from the former, which is conceptually moreilluminating, one chooses a set of macroscopic observables denoted by Λ, like energy, andnumber of particles, and define the Boltzmann entropy of the system in a given state asS = logW, (1.1)where W is the volume of the region of the phase space accessible given the set of1This notion becomes clearer after the advent of statistical mechanics.2For this reason, it is misleading the use of the word “law” for these results, but since this terminologyis widespread, I will keep adopting it.4constraints, typically the intersection of surfaces of constant Λ. S is thus a function ofthese constants. Under the assumption that every point in the phase space is a prioriequally probably, if we take a collection of system with the same constraints, we expectto be more likely to select one in a given state that has a higher value for the entropy,because there are more states with that entropy. The situation where the entropy is atits maximum under some constraints is called thermodynamic equilibrium. Statisticalindependence between two parts implies that S is extensive [8], i.e., the total value ofthe entropy is the sum of the entropies of each part. Roughly speaking, an extensivesystem can be thought of possessing the property that, if it is divided into subpartssatisfying the thermodynamic limit, these subparts do not interact with one another.We are in a position to derive the zeroth law as follows. In a system in thermodynamicequilibrium consisted by two parts for which can trade energy freely but respectingE = E1 + E2 for a constant E, we impose that a small amount of energy transferredfrom part 1 is related to the receiver 2 by dE1 = −dE2 and that dS = dS1 + dS2 = 03to obtaindE1dS1=dE2dS2≡ T,where the common value T is defined as the temperature of the system. If besidesthe energy we had let other values of the macroscopic variables Λ to be traded freelybetween the parts, similar equalities would have been found, each imposing the equalityof a different potential.To derive the second law (in Kelvin’s statement, see, e.g. [12]), we suppose wehave two systems, 1 and 2, in internal equilibrium, but not in mutual equilibrium.Assume T1 > T2. If we now allow these systems to trade energy freely, but respectingdE1 = −dE2, we have0 ≤ dS1 + dS2 =(1T1− 1T2)dE1 ⇒ dE1 ≤ 0,meaning that energy has to flow out of the part with higher temperature.3This is a necessary condition for thermodynamical equilibrium, requiring that the entropy is amaximum.5It is clear from eq. (1.1) that only differences in entropy can be physically mean-ingful. More specifically, since W is dimensionful, and a change in units of phase spacecoordinates produce an additive constant in the entropy, instead of a simple rescaling.When quantum mechanics comes into play, we are given a way of counting thediscrete number of accessible microstates that generate the same set of Λs (at least inthe quasi-classical case, when the Sommerfeld quantisation rule is applicable [13]). Wecan then redefine Boltzmann’s entropy as the logarithm of this number. This meansthat, for a system of distinguishable particles, we replace eq. (1.1) by eq. (1.2) below:S = logW(2pi~)2n, (1.2)where 2n is the dimension of the submanifold of accessible states. In case of indistin-guishable particles, to correctly account for the number of microstates, one has furtherto divide W by the number of permutations of particles in their individual states. Ikept the constant ~ in eq. (1.2) to make it clear the quantum-mechanical origin of thefactor. Now the argument of the logarithm is dimensionless and it is conceivable toassign physical meaning to the absolute value of the entropy.The least possible value of S, which is zero, occurs when only one microstate is possi-ble. The observation that this value happens for extensive systems at zero temperatureis referred to as the Nernst theorem or the third law of thermodynamics. The prooffollows by imposing second derivative conditions to maximise the entropy, leading tothe positivity of the specific heat for extensive systems ∂S∂T ≥ 04. There is also a weakformulation of the third law of thermodynamics: that it is impossible with a finite num-ber of physical processes to reduce the temperature of a system to zero starting fromany positive value. As we shall see, self-gravitating systems are not extensive. There ishope for the weak formulation to hold for non-extensive systems, unlike the former. Iwill discuss this possibility on chapter three.There is another, originally independent, notion of entropy, the Shannon entropy,4The equality can hold for phase transitions, in which case higher order derivatives have to beanalysed in order to impose that the entropy is a maximum.6that we shall discuss. Let ρ(X) be a the probability density function of a set of randomvariables X. The Shannon entropy is defined as the functionalS[ρ] = −∫ρ(X) log ρ(X) dX, (1.3)This functional can, as argued by Caticha [14], be interpreted as a measure of the amountof ignorance a given probability density function has for a system. Roughly speaking,the more “assumptions” the probability distribution contains5, the lower the value of itsShannon entropy. In the particular case where we have W equally probable states, wehave a uniform (discrete) distribution whose probabilities are all equal to 1/W . In thiscase (1.3) reduces to (1.1).We can apply this notion of entropy to quantum mechanical systems for which welack complete information about its state, in which case we can use the density matrixto characterise what we know about it. The idea is to associate a projection operatorPi = |i〉〈i|/〈i|i〉 to each vector |i〉 of a Hilbert space. For any observable O, its expec-tation value obeys 〈i|O|i〉 = Tr(PiO), so one can think of the projector Pi representingstates. Naturally all vectors are identified with an operator, the projector operator,but we can generalise the notion of states to include operators that do not come fromvectors. Namely, we promote all Hermitian, positive semi-definite, trace-one operators,henceforth referred as density matrices to states. Some of these density matrices willnot be idempotent; in such a case the states they represent are said to be mixed. Expec-tation values of operators O in a state ρ are calculated the same way, using the trace,〈O〉ρ = Tr(ρO).If the density matrix is written as ρ =∑i pi|i〉〈i|, where pi ≥ 0,∑i pi = 1 and |i〉denotes state vectors, then S[ρ] = −∑i pi log pi can be written in an explicitly basis-invariant formS[ρ] = −Tr ρ log ρ, (1.4)5For example, a distribution containing correlations has a lower value for the entropy. And, if thereare constraints expressed as integrals over part of the system, the distribution maximising the entropywill not contain correlations with complementary part.7commonly referred as von Neumann entropy. The entropy of any vector state — hence-forth referred as pure state as opposed to a mixed state — is zero.When we have a system that is composed of two parts for which the total Hilbertspace can be decomposed as a tensor product, H = H1⊗H2, there is a process to obtaina density matrix ρred operating on H2 alone such that the expectation value of any localoperator O2 on H2 is given by Tr(ρredO2). The process is the partial trace, consistingof taking the trace of the operator over H1, leaving an operator on O2. The details ofthis process and examples in simple systems can be found, for example, in ref. [15]. Inprobability language, taking partial traces is merely the marginalisation over some of therandom variables. Taking a partial trace over a pure entangled state generates a mixedstate with non-vanishing entropy6, this is the reason why sometimes the definition (1.4)is referred as “entanglement entropy”, but this terminology can be misleading and willbe avoided here.Another particular application of Shannon’s notion of entropy is to recover statisticalmechanics. I briefly describe it below following Caticha [14]. In Boltzmann’s approach,the macroscopic quantities Λ were imposed by restricting the set in which microstatesare counted. In Shannon’s approach, this is done by implementing a constraint in theproblem of extremising the entropy functional. We seek a probability density function forthe occupation in the many-body phase space subjected to the macroscopic constraintsΛ, plus the ubiquitous normalisation constraint∫ρ(X)dX = 1. We can then use thisformula for the entropy to derive once again a number of properties of macroscopicsystems, in particular the laws of thermodynamics. Boltzmann’s formula for entropy(1.2) is already inviting us to interpret the entropy as an epistemological concept, but inShannon’s formula (1.3) this interpretation is mandatory; one does not have to count thenumber of microstates that give rise to a given macrostate, instead we simply state thatthe certain number of macroscopic observations allowed us to determine the probabilitydensity function on phase space with a remaining amount of ignorance encoded in thevalue of S. If we are honest in the process of inferring the probability function, we6As long as, of course, the partial trace is carried over entangled parts of the system.8have to choose the one that maximises S given certain constraints Λ, which can beimplemented with the method of the Lagrange multipliers. In case Λ is the total energyE, extremisation of S[ρ] − λ1(∫ρ(X)u(x)dX) − λ2 ∫ ρdX, where u is the energyfunction of the point X in phase space, and λ1,2 are Lagrange multipliers, givesρ(X) =1Ze−u(X)/T , Z ≡∫e−u/TdX. (1.5)One hidden assumption is that the partition function Z converges. This is not true forall functions u(X). It is divergent for functions that decay as slow as or slower than1/|X| for large |X|. In typical physical systems in which u is only a function of thecoordinates of configuration space, not of the generalised momenta, this convergencecriterion is seen as a condition of short-range interactions. It is clear that this conditionis not met by the Newtonian gravitational potential, meaning that the self-gravitatingsystems are not expected to obey extensivity or any of its consequences.An ensemble leading to eq (1.5) is often named the canonical ensemble. Whenthis ensemble exists, the heat capacity C is non-negative. An explicit evaluation ofC = T ∂/∂〈E〉 S[ρ(〈E〉)] revealsC =1T 2(〈E2〉 − 〈E〉2) ≥ 0.This inequality is violated by black holes, as we will see below, but this should beno surprise, given the dominance of a long range interaction, gravity. This derivationassumes extensivity simply by assuming that the canonical ensemble exists.One can show extensive systems have C > 0 directly from requiring the entropy tobe a maximum (adapting for example from ref. [8]). Imagine a system composed by twoparts, labelled as 1 and 2. The condition for the entropy S(E1, E2) to be a maximumareδS =∂S∂E1δE1 +∂S∂E2δE2 = 0 (1.6)andδ(2)S =∂2S∂E21(δE1)2 +∂2S∂E22(δE2)2 + 2∂2S∂E1∂E2δE1δE2 ≤ 0. (1.7)9The statements that the combined system is both isolated and extensive can be expressedmathematically as δE2 = −δE1 and S(E1, E2) = S1(E1) + S2(E2) respectively, whereS1,2 are some single variable functions. Under these assumptions, eq. (1.6) reduces asbefore to T1 = T2. The extensivity property also kills the cross term in eq. (1.7), leavingus with ∂2S1,2/∂E21,2 ≤ 0. Using∂2S∂E2=∂∂E(1T)= −∂T/∂ET 2= − 1T 2C,we see that the second condition reduces to C1,2 ≥ 0. This derivation makes it clearlyevident the role played by extensivity.We can see that any system N with negative heat capacity can be in equilibriumwith a reservoir P with positive heat capacity, since the equilibrium condition is ob-tained by simply imposing the first-derivative relations, in particular, equality of theirtemperatures. But this equilibrium cannot be stable, since if P loses (gains) a smallamount of energy to N , the temperature of P remains constant, but the temperatureof N decreases (increases), meaning that more and more energy will flow in the samedirection. In order to the equilibrium to be stable, there must be an upper bound onthe energy of P so that under that loss (gain) in energy, its temperature drops (raises)by the same amount as the temperature in N . This is an intuitive way of seeing that,even when it is possible to define a state — called the Hartle-Hawking state — of aquantum field on a background spacetime containing a black hole and representing athermal equilibrium between the field and the black hole, it cannot be stable in most ofthe physically relevant cases.After Shannon’s proposal, another useful notion was introduced by Shore and John-son [16] and has since been used in probability theory, the “MaxEnt” [14], which givesone the freedom of introducing a prior µ(X) other than the uniform distribution. It isdefined asS[ρ|µ] = −∫ρ logρµdX. (1.8)This notion can also be used in quantum mechanics. In the language of density matrices,10we defineS[ρ|µ] = Tr ρ log ρ− Tr ρ logµ, (1.9)as the relative entropy between any two density matrices ρ and µ. Note that I alsoreversed the sign in the definition above so that our definition agrees with the conventionsfound in quantum mechanics literature. It turns out that the relative entropy (1.9), whencompared to the von Neumann entropy, can be more precisely defined for general statesin quantum field theory in the algebraic approach [17], when states are viewed as linearfunctionals over the algebra of operators, which make them more general entities thanour density matrices7. The notion of relative entropy will be very useful for us in chapterthree. It measures “how distant” two states (or probability density functions) are apart,and it is zero when they coincide. We shall see that the relative entropy is always non-negative and can only decrease after a state reduction (i.e., if the ρ and µ are bothsubjected to partial traces over a same subspace) [17, 18].Before ending this subsection, let us briefly discuss the work by Sasa and Yokokura[11], because we can obtain an interesting interpretation for their results. Inspired byblack hole physics, especially the Noether charge approach for black hole entropy, aswe shall see below, they were motivated to investigate mechanical systems derived fromthe action I =∫L(q, q˙, t; ν) dt, where ν is an extensive control variable. The functionν of t is not dynamically determined and is used to introduce constraints coming frominteractions not accounted in the system. Sasa and Yokokura considered the case whenthere is symmetry with respect to inhomogeneous time translations, more precisely thetransformation t → t′ = t + η(q, q˙; ν), and simultaneous redefinition q → q′(t′) = q(t),keeping ν unchanged. If the Lagrangian after this transformation changes, withoutimposing the equations of motion, by an amount δL = dψ/dt, i.e., a total derivative,7In this approach, one can still define a pure state as one that cannot be written as a combination oftwo other states with positive coefficients strictly smaller than unity. When applied to finite-dimensionaldensity matrices, this definition is equivalent to the one I gave above.11then after imposing the equations one has the conservation ruledSdt≡ ddt(ψ + Eη) = 0,where E ≡ (∂L/∂q˙)q˙ − L. This constant of motion cannot be independent of the totalenergy, viewed also as a function of ν. The authors of ref. [11] compare the functionaldependence of the conserved energy and of S on ν and then proceed to argue that thequantity S — if supposed to be extensive — has to be a multiple of the thermodynamicentropy except for an additive constant, which is immaterial in classical physics (as Iargued above). This works as a proof of concept for the idea of understanding the entropyin the formalism of Noether conserved quantities, but two remarks have to be made.Firstly, I do not know of a physical interpretation for this calculation, and secondly, thisapproach has an important difference with respect to the Noether charge approach forblack hole entropy below, because the former depends heavily on the quantities ν in orderto its results to be non-trivial and the latter assumes the theory to be diffeomorphismcovariant, and any structure which could play the role of ν would spoil this property.1.2 Survey of Black Hole Thermodynamics1.2.1 Review of Quantum Fields in Curved SpacetimesIn this section, I review the basic theory of quantum fields in curved spaces, introducingtools that are going to be useful for us later on. To particularly emphasise the symplecticstructure of phase space, I start by describing classical mechanics in this formalism. Fora more thorough treatment, I refer the reader to the books by Abraham and Marsden[19] and Arnold [20].Most standard approaches to classical mechanics start by choosing a configurationspace G representing all possible “positions” of the system. For a particle in one dimen-sion, G = R, and each x ∈ G is thought of a position of that particle; for a rigid body inthree dimensions, G = R3×SO(3). In possession of G as a vector space, one defines thephase space P as its cotangent bundle P = T ∗G, and each point in P is referred as a12state of the system. One chooses a set of local coordinates q for a point x in G, viewedas a set of 0-forms in P. Denoting by p the 0-forms on T ∗xG in the coordinate dual basisassociated with q, we can define the two-form Ω byΩ = dp ∧ dq, (1.10)which is closed and non-degenerate, so it admits an inverse. When there is a functionH : P → R so that the evolution of the system through time t obeysdydt= Ω−1 · ∂H∂y, (1.11)where ∂/∂y denotes the gradient in P, the system is said to be Hamiltonian and thefunction H is called the Hamiltonian of the system.It turns out that in some applications it is not fruitful to distinguish between q andp, normally referred to as coordinates and momenta, all the structure one needs is thatof a symplectic manifold (P,Ω) where P is the cotangent bundle of a vector space andΩ is any closed non-degenerate two-form over P. We turn our attention to this moregeneral theory. An important example is the theory of fields, for which the distinctionbetween coordinates and momenta is unclear and the phase space is infinite dimensional.It is useful to use the abstract index notation for tensors on P, for which we use uppercase Roman letters, e.g., y is denoted by yA, and Ω by ΩAB, and its inverse by ΩAB.We denote yA = ΩAByB, representing an isomorphism between vectors and one-forms.For example, in this notation, eq. (1.11) readsdyAdt= ΩAB∂H∂yB.From the symplectic manifold, we define the Poisson brackets between two functionsf, g : P → R by[f, g]PB ≡ ΩAB ∂f∂yA∂g∂yB, (1.12)where ∂/∂yA denotes the gradient on phase space. The Poisson brackets obey[yA, zB]PB = ΩCD ∂yA∂ηC∂zB∂ηD= ΩCDΩAEΩBF∂yE∂ηC∂zF∂ηD= ΩCDΩAEΩBF δEC δFD= δDAΩBD = −ΩAB.(1.13)13The relation (1.13) corresponds to the usual relations between the Poisson brackets ofcanonical coordinates and momenta, written in a basis-independent manner.An approach to find a quantum version of the theory is to replace the phase space bya (usually infinite-dimensional) Hilbert spaceH and a subset of the classical observablesby a set of self-adjoint operators yˆ : H → H satisfying a Lie algebra. Formally,8 thisalgebra mimics eq. (1.13),[yˆA, zˆB] = −iΩAB1. (1.14)Of particular interest is the quantum theory of fields defined on a fixed globallyhyperbolic spacetime (M, g), whose classical version is derived from a Lagrangian. Forexample, for a real scalar field φ the action isI =∫ML√−g d4x, L = 12gab∇aφ∇bφ−m2φ2.To obtain a natural symplectic structure, we use the Arnowitt-Deser-Misner (ADM)[22] 3+1 decomposition for which I =∫dtL, withL = 12∫Σt[(na∇aφ)2 − hab∇aφ∇bφ−m2φ2]N√h d3x,where the parameter t identifies the particular Cauchy surface Σt in the family {Σt}t.All globally hyperbolic manifoldsM admit a foliation of this type,M ∼ R×Σt (see, e.g.,Wald’s book [23] for a proof). The normal to Σt is na, and the vector T a defined throughT a∇at = 1 is decomposed in its part normal and tangential to Σt : T a = Na + Nna.The induced metric on Σt is hab = gab + nanb.The “coordinate” in configuration space can be taken to be the field value φ, and itscanonically conjugated momentum isp =δIδ(∂φ/∂t)=√h na∇aφ, (1.15)8I use the word “formally” to acknowledge that for a state |s〉 in the domain of zˆB , zˆB |s〉 does notneed to belong to the domain of yˆA, so the relation below is ill defined. As shown by Weyl [21], it ispossible to reformulate these relations in terms of bounded operators exp(iyˆA) and exp(izˆB), but theformal relation presented here suffices for our particular applications.14so the symplectic form acting on a pair of vectors y1,2 = (φ1,2, p1,2) is obtained byΩ(y1, y2) =∫Σt(p1φ2 − p2φ1) d3x.For different Lagrangians, e.g., for different spins, one can use the same definition ofthe momentum from the varied action and the same definition for the symplectic form.An “inner product” between a pair of solutions y1 and y2 of the equations of motionderived from the Lagrangian9 by(y1, y2) = −iΩAB y¯A1 yB2 , (1.16)which does not depend on the choice of the Cauchy surface Σt if appropriate asymptoticconditions are imposed over y1,2.From the set of complex solutions of the field equations, we wish to obtain a Hilbertspace with the product as in eq. (1.16). However, because this product is not positive-definite, it is not a proper inner product when applied to the full set of complex solutions.The quantum theory is obtained on a subspace H of the space of complex solutions forwhich (1.16) is an inner product. There are, in general, many choices of such subspacesand different choices can lead to non-unitarily equivalent theories, and hence to differentphysics. There is, however, a procedure for stationary spacetimes in which the choice ofH is natural as follows. Let ξa denote the timelike Killing field, and let φα be complexsolutions that are eigenfunctions of the Lie derivative with respect to ξa,£ξφα = iωαφα, (1.17)then the subset of φα whose eigenvalues ωα are positive, often named “positive normed”or “positive frequency” modes, can be taken as basis elements of the Hilbert space H ,since ωα > 0⇒ (φα, φα) > 0.Here I emphasise that the choice of H , or equivalently, the choice of the set ofpositive normed modes is not unique. The prescription for stationary spacetimes as9For the Lagrangian used as example here, this product if often referred to as Klein-Gordon product,whereas for spinor fields, Dirac product, but I will avoid this nomenclature here.15above is merely one possible choice, “natural” in the sense of exploiting the symmetryof spacetime, but by no means mandatory.The quantum version of the field theory has as its Hilbert space the Fock spaceF (H ) constructed out of H , which is called the “single particle space” in this context.The Fock space is defined asF (H ) ≡ C⊕H ⊕ (H ⊗H )⊕ (H ⊗H ⊗H )⊕ . . . ,where all the tensor products are understood to be either symmetrised or antissymetrisedaccordingly to the spin of the field.A “vacuum” state |0〉 ∈ F (H ) is defined as the state whose only non-vanishingcomponent is the one coming from the first term in this direct sum. This state obviouslydepends on the choice of the Hilbert space and it is in this respect arbitrary. Thisfact limits the physical relevance of the notion of vacuum, and therefore has importantconsequences in the quantum theory of fields.We can use the abstract index notation for vectors in products of H as follows. Wedenote a state in the Fock space by Ψ = (ψ,ψa1 , . . . , ψa1...an , . . .) ∈ F (H )10. UsingRiesz’ lemma, there is a dual vector ϕ¯a ∈ H ∗ so that (ϕ,ψ) = ϕ¯aψa. Although we areusing lower case Latin alphabet indices, these are not to be confused with spacetimeindices.The inner product in F (H ) is defined as (Ψ, H) ≡ ψη · φ¯aηa · ψ¯abηab · . . ., whereΨ = (ψ,ψa, ψab, . . .) and H = (η, ηa, ηab, . . .) are vectors in F (H ).To find the algebra of operators satisfying eq. (1.14), we start by defining (as Waldin [24] and [25]) the creation and annihilation operators associated with the mode ξ bytheir action on a generic state asa†ξΨ = (0, ψξa1 ,√2ψ{a1ξa2}, . . .) and aξΨ = (ξ¯a¯ψa,√2ξ¯aψaa1 ,√3ξ¯aψaa1a2 , . . .),(1.18)respectively. Here, each ψ is positive normed, and the curly brackets denote eithersymmetrisation or antissymetrisation in the products. For instance, ξ{aψb} in the above10Here we assume that H is a Hilbert space, so we suppose each of the ψ is positive normed.16has to be understood as ξ(aψb) for bosons and ξ[aψb] for fermions. The factors√2,√3were added for normalisation purposes. For example, if ψ and ξ are normalised, so is√2× 12(ψaξb ± ξaψb).Henceforth, for sake of simplicity, we suppose we are dealing with bosons.Applying eq. (1.18) to |0〉 we see that aξ|0〉 = 0 ∀ξ ∈H , giving a way of character-ising the vacuum |0〉.These operators obey the commutation relation[aξ, a†η] = (ξ, η)1. (1.19)In possession of these operators, if we can find a basis formed by positive normedsolutions φα as above and their complex conjugate, then any given complex solution φscan be written as a sum of a positive-normed and a negative-normed term. The formeris the projection of φs onto the subspace generated by the positive-normed basis vectorsonly, and the latter is the projection onto the complement subspace11. We denote thefirst of these projections by Kφs. Then the operatorsφˆ(ψ)A = iaKψ(Kψ)A − ia†Kψ(Kψ)A, (1.20)together with the operators obtained by replacing the field configuration in the expres-sion for the momentum (1.15) obey the desired relations (1.14). Conversely, if one hasthe set of operators satisfying (1.14), then the operators for positive normed modes ξaξ = (ξ, φˆ) and a†ξ = (ξ¯, φˆ) (1.21)obey the algebra of (1.19).As our notation suggests, the creation and annihilation operators are indeed theadjoints of each other, as it can be verified directly from eq. (1.18) or eq. (1.21).We now turn our attention to the important analysis of our quantisation procedurewith respect to the choice of the single particle Hilbert space. Different choices of these11It is clear from this argument that the Hilbert space here constructed H and its complex conjugateH are orthogonal subspaces of the space of all complex solutions of the field equations.17spaces will lead in general to different physics. Suppose the field operators φˆ1 : F (H1)→F (H1) and φˆ2 : F (H2) → F (H2) coming from the two quantisation processes arerelatable by a unitary transformation U , that is φˆ2 = Uφˆ1U †. These two quantisationprocesses generate the projectors K1 and K2 as above, each one projecting a complexsolution to its respective Hilbert space H1 and H2.To obtain the operator U above, we start from the norm-preserving mappings C :H1 →H2 and D :H1 →H2 between the two independently constructed single-particleHilbert spaces defined respectively as the restriction of K2 and K2 to H1. These mapsare often referred to as Bogoliubov transformation.They obey two useful identities, derived from the fact that these operators preservethe inner product and from the identity K + K¯ = 1. To derive them, take ξ, η ∈ H1,then(ξ, η) = −iΩ(ξ¯, η) = −iΩ((K + K¯)ξ, (K + K¯)η)= (Cξ,Cη)− (Dξ,Dη) == (ξ, (C†C −D†D)η)⇒ C†C −D†D = 1,where the third equality equates the inner product in H1 on the left hand side and theinner product in H2 on the right hand side. A similar reasoning supposing η ∈ H2instead leads toC†D¯ −D†C¯ = 0.The matrix elements of the operators C and D are usually denoted respectively by αand β¯ and referred to as Bogoliubov coefficients.Using the expression (1.20) for both set of operators, the creation and annihilationoperators must satisfyUa1,ξ¯U† = a2,Cξ − a†2,Dξ ∀ξ ∈H1.Applying this to the vacuum state |01〉, represented generically as |01〉 = (c, cψ, cψa1 , cψa1a2 , . . .),we geta2,η¯ − a†2,D¯C¯−1η¯ = 0,18where η ≡ C−1ξ. As an equality for operators in the Fock space, it is translated to anequality for each of the direct sum addends representing n-particle amplitudes,√n η¯aψaa1...an =√n− 1 D¯C¯−1η¯(a1ψa2...an). (1.22)For n = 1, eq. (1.22), i.e., η¯aψa = 0, implies ψa = 0. Substituting this into theequation for n = 3 we get ψa1a2a3 = 0, and so on. Hence ψa1...an vanishes whenever nis odd. For n = 2, we can compare the symmetry properties of both sides of eq. (1.22)to argue that C¯D¯−1 acting on η¯a can be uniquely identified with a vector of H ⊗S Hdenoted by εab =√2/c ψab. Hence,|02〉 = U |01〉 =(c, 0,c√2εa1a2 , 0, . . . ,√2n!2nn!ε(a1a2εa3a4 . . . εan−1an), . . .). (1.23)There is one technical detail that I avoided here for sake of fluidity of the text, thatwe turn now our attention to. In order to make more precise sense of commutationrelations like (1.14) for quantum fields, where the identity operator 1 is a delta functionand the field operators have to be viewed as operator-valued distributions [26]. Thetest functions are taken to be compactly supported elements of C∞(M). Therefore, wereplace the definition of the quantum operator in eq. (1.20) by a “smeared version”Φˆ(f) = iaK(Ef)− ia†K(Ef), (1.24)where f is a test function and Ef is defined as [25] Ef = Af −Rf , where Af and Rfare the advanced and retarded solutions of the field equations with inhomogeneity f . Inother words, Af is zero outside the chronological past of the support of f , J−(suppf),and Rf is zero outside of J+(suppf). Because the field equations are linear, Ef solvesthe homogeneous field equation.This replacement of eq. (1.20) by eq. (1.24) is satisfactory because there is a corre-spondence between a solution ϕ of the field equations and a function Ef for some testfunction f preserving the symplectic structure.The proof for this is (found, for example, in refs. [25] and [26]) constructive. Thecorrespondence has the form f = −E(χϕ), where E is the operator so that the equation19of motion for φ reads Eφ = 0 and χ is any function that makes Af and Rf have correctsupports. Functions χ always exist, they can be taken to be χ = 0 in the past of aCauchy surface Σt1 and χ = 1 in the future of another Σt2 ⊂ I+(Σt1). Furthermore,using Green’s identity for the integration in a spacetime region bounded by two Cauchyslices Σt1 and Σt2 outside of which f = 0 and writing the volume element as√−gd4x =√hdtd3x, we have∫ √−gd4x ϕf = ∫ √−gd4x ϕE(Af) = ∫Σt1√hd3x(ϕ∂Af∂t−Af ∂ϕ∂t)= Ω(Ef, ϕ),where in the second step we chose the spatial boundary of the region outside the supportofAf , and used the fact thatAf = 0 in Σt2 , and on the last step, we subtracted 0 = Rf |t1from each occurrence of Af .1.2.2 Geometry of Killing Horizons and the Zeroth Law of Black HoleMechanicsLet (M, g) be a spacetime with a globally defined Killing field χa that vanishes in a 2-surface B, and whose derivatives across B are nonzero. Then there is a pair of 3-surfacesh+ and h− where χa is null, with h+ ∩ h− = B, as in figure (1.1). In this situation B isreferred to as the bifurcation surface and h± the bifurcate horizon. See chapter 5 of ref.[25] for a proof and more details.Since g(χ, χ) is a constant (zero) along the bifurcate Killing horizon, ∇ag(χ, χ) mustbe normal to these null surfaces; hence we have∇a(χbχb) = −2κχa (1.25)for a function κ at a point in the bifurcate horizon. κ is referred to as surface grav-ity. Taking the Lie derivative along χa of both sides of eq. (1.25), using the identity∇a∇bχc = −Rbcadχd valid for any Killing field, and the skew-symmetry properties ofthe curvature tensor, it follows that £χκ = 0, meaning that the value of κ is constantalong the orbits of χa.20It also follows that χb∇bχa = κχa, which can be either contracted at each point witha null vector `a defined such that χa`a = −1 to obtainκ = −χa`b∇bχa; (1.26)or multiplied by `c and antisymmetrised to obtain `[bχc]∇cχa = κab, where  is thevolume form on the two-dimensional section of h± orthogonal to `a (B is an exampleof such 2-surface) [27]. The antisymmetry between b and c on the left hand side isguaranteed by the Killing equation, so the brackets can be removed. If one completesthe (`, χ) basis to obtain a Newman-Penrose basis and expresses the inverse metric interms of direct products of this vector basis, using the definition of κ,−∇bχa = κab. (1.27)Figure 1.1: Geometry of a Killing horizon, depicting the orbits of the Killing field χa,which is null over the bifurcated horizon h+∪h− and vanishes at the bifurcation surfaceB.When discussing quantum field theory in spacetimes with a bifurcate Killing horizon,it will be useful to have a coordinate chart near the horizon. We introduce the coordi-21nates (u, %, xα) where χ = ∂/∂u, ` = ∂/∂% with % = 0 at h± and xα are the remainingcoordinates.In these coordinates, with the abbreviations F = −χaχa and guα = χa(∂/∂xα)a themetric can be written asds2 = −Fdu2 − 2dud%+ 2guαdudxα + gαβdxαdxβ,where none of the metric coefficients depends on u, since it is the Killing parameter.Thus, only keeping terms up to the first order in % and usingdFd%= `b∇b(−χaχa) = −2nbχa∇bχa = 2κ,near the horizon where F = 0,ds2 = −2κ%du2 − 2dud%+ 2guαdudxα + gαβdxαdxβ.The change of coordinates from (u, %) to (t, r) with t ≡ u+∫ F−1d% and r = √2%/κbrings the line element near the bifurcate horizon to the formds2 = −κ2r2dt2 + dr2 + 2guαdudxα + gαβdxαdxβ, (1.28)whose (t, r) section is identical to the Minkowski spacetime’s in Rindler coordinates.We are now in position to prove (following Wald in [25]) a version of the zeroth lawof black hole mechanics, stating that the surface gravity is constant over h+ and h−.This differs from the “usual” zeroth law of black hole mechanics as in first developed byBardeen, Carter, and Hawking [27] because use a different set of assumptions, namely weare supposing a bifurcate horizon, whereas most approaches only require a Killing hori-zon, but they also assume the dominant energy condition and that Einstein’s equationsare satisfied, both of which are undesired assumptions for us in chapter three.From eq. (1.27), (∇bχa)(∇bχa) = κabκab = −2κ2,taking a derivative of this equation along a direction ta tangent to h± and orthogonalto χa,2κta∇aκ = −tc (∇c∇aχb) (∇aχb) = tcR dabc χd∇aχb = 0,22where the Killing identity and the symmetry properties of the curvature tensor wereused. For a bifurcate horizon (for which κ 6= 0), this means ta∇aκ = 0, or that κ isconstant throughout the whole h±, the zeroth law.If besides χa, (M, g) admits a globally defined timelike Killing field ξa, which in thechart of eq. (1.28) can be taken as ∂/∂t, we wish to obtain the result of the restrictionof the vacuum with respect to ξa to the region I+(h−) ∩ I−(h+), where h− ∪ h+ is thebifurcate Killing horizon of χa.We choose the one-particle Hilbert space of solutions of the field equations to factoras H = HL ⊕HR, where the first term represents solutions that are positive normedwith respect to the Killing field −χa and whose initial data have support contained ona surface contained I+(h+) ∩ I−(h−), and the second are positive normed with respectto +χa and initial data with support contained the a surface of I+(h−) ∩ I−(h+). TheFock space F (H ) constructed over H factors as F (H ) ∼= F (HL)⊗F (HR)12, so therestriction of a state to the Cauchy development of an observer following the orbits ofthe Killing field can be obtained by tracing out degrees of freedom in F (HL).We work in the coordinates of the line element (1.28) and reproduce Unruh’s stepsof ref. [4] in the derivation of the Unruh effect. Let U and V denote the retardedand advanced null coordinates constructed from the timelike coordinate t. A solutionis positive normed with respect to ξa if and only if it is positive normed with respectto its retarded null coordinate U or the advanced null coordinate V . This fact canbe verified by choosing the integration surface to be parallel to the spacelike inertialcoordinates. Since the value of the Klein-Gordon product is invariant under the choiceof the causal integration surface, this integration can be chosen to be either over h− orover h+, and the Bogoliubov coefficients relating positive-normed solutions with respect12An isomorphism can be easily constructed as follows. A Schauder basis of F (A) ⊗F (B) can betaken of the form {ξa1...anηb1...bm}, for n,m ∈ N ∪ {0}, ξ running over a basis of A and η over a basisof B. The isomorphism identifies this basis with the Schauder basis {(ξ ⊕ 0)a1...an + (0 ⊕ η)b1...bm}of F (A ⊕ B). An entirely analogous construction works for symmetrised or antisymmetrised tensorproducts.23to ξa and to χa can be evaluated on these surfaces. From eq. (1.28) we identify t as theKilling parameter corresponding to ξa. On h+, the relationship between the advancedcoordinate V built from the timelike coordinate t and the Killing parameter v definedas χa∇av = 1 isv =log |V |κ, (1.29)and similarly, on h−, with similar definitions for the retarded coordinates u and U ,u = − log |U |κ. (1.30)We want to find HR,L, that is, we want to find positively-normed modes that vanishoutside the regions of interest. A possible choice for modes generating HR is pure-frequency modes in the coordinate v, say Rφω(v) = f(xα)e−iωvθ(V ). Viewed as afunction of V , these modes have the Fourier transformR˜φω(σ) ∝∫ ∞0eiσV−iωv(V )dV ≡ I(σ).It is possible to relate I(σ) and I(−σ) using eq. (1.29) interpreting them as analyticfunctions of a complex-valued frequency σ by using the analytic extension of the loga-rithm function with branch cut in R−. To evaluate the transforms, fix σ > 0. For I(σ),we make the substitution V = iy, so that log V = log i + log y andI(σ) = iepiω2κ∫ ∞0e−σy−iωκlog y dy.And for I(−σ), we substitute V = −iy:I(−σ) = −ie−piω2κ∫ ∞0e−σy−iωκlog y dy,leading to the following relation between the Fourier transformed modesR˜φω(−σ) = −epiωκ R˜φω(σ). (1.31)From eq. (1.28) we see that in the appropriate neighbourhood of the bifurcate hori-zon, the transformation that brings the pair (U, V ), or equivalently (t, %), to (−U,−V )24or (−t,−%) is an isometry. So the Fourier transforms of the modes L˜φω on HL obeyL˜φω(σ) = R˜φω(−σ), meaning that the combinationsΦω = Rφω + e−piωκ Lφω and Ψω = Lφω + e−piωκ Rφωgenerate HL ⊕HR, since (Φ˜ω(σ), Φ˜ω(σ)), (Ψ˜ω(σ), Ψ˜ω(σ)) > 0 ∀σ.We identify the operator C by its action on this basis:CΦω = Rφω and CΨω = Lφω,and the operator D acting on the same vectors byDΦω = e−piωκ Lφω and DΨω = e−piωκ Rφω,leading toDC−1Rφω = e−piωκ Lφω and DC−1Lφω = e−piωκ Rφω. (1.32)In order to construct the factorisation of the “vacuum” state |0〉 ofF (HL)⊗F (HR),it is useful to write eq. (1.23) using a Schmidt decomposition for the operator εab =∑i ciξ(ai ηb)i asU |0〉 =∏i∞∑n=0cni |ni〉 ⊗ |ni〉, (1.33)where |ni〉 denotes the state in Fock space we would denote as (0, . . . , ψa1...an , 0, . . .),with ψ denoting either ξ or η depending on the position of the factor.The form (1.33) makes it easy to take partial traces. In our case of interest, thedecomposition factors out states inF (HL) andF (HR). If |0〉 is the vacuum annihilatedby modes Φω and Ψω, we can read the operator εab from eq. (1.32) as its Schmidtdecomposition and obtain the reduced density matrix by tracing out degrees of freedomin F (HL) from ρ = |0〉〈0| to obtain ρred : F (HR)→ F (HR)ρred =∏i∞∑n=0e−2pinωiκ |ni〉〈ni|.If instead of χa, we take the normalised χ′a = χa/√−g(χ, χ) that could represent thefour-velocity of an observer, and adapt the coordinate v so it still satisfies χ′a∇av = 1,25the effect of this replacement is the same as replacing κ by κ/√−g(χ, χ) in eqs. (1.29,1.30) and in all following equations. This transformed vector field leads to the reduceddensity matrixρred =∏i∞∑n=0e−2pinωi√−g(χ,χ)κ |ni〉〈ni|. (1.34)The trick used to obtain eq. (1.34) from (1.31) was originally developed by Unruh[4].When operating on number eigenstates |ni〉 associated with the modes of the form(1.17), the Hamiltonian Hξ, i.e. the generator of infinitesimal time translations alongthe orbits of ξa, has the eigenvalue of niωi. Indeed, the state evolves in t by the operatorexp(−iHξt), whose infinitesimal version 1−iHξt has to be identified with 1+t£ξ. HenceHξ = i£χ, and for a one-particle state, H(φω, φˆ)|0〉 = ω(φω, φˆ)|0〉; for the n-particle stateHξ|ni〉 = niωi|ni〉.Comparing the spectrum of the Hamiltonian and eq. (1.34), we see that ρred is aGibbs state with temperatureT =κ2pi√−g(χ, χ) .1.2.3 From the Laws of Black Hole Mechanics to Black Hole Thermo-dynamicsIn the 1960s and 1970s, much about the nature of black holes was discovered thanks tothe advent of global techniques originally introduced by Penrose [28]. For example, anumber of singularity theorems were proven meaning that the existence of a singularityin a black hole was not a consequence solely of the symmetry of models for collapsingmatter, like Oppenheimer-Snyder’s (see, e.g. [29]), but a mandatory feature in anyglobally hyperbolic spacetime with a non-compact Cauchy surface, with the presence ofa closed trapped surface (or similar condition), and an energy condition [3].Other results, summarised by Bardeen, Carter, and Hawking, [27] became famous asthe four laws of black hole mechanics, named this way due to their remarkable resem-blance to the laws of thermodynamics. I review them below.26The zeroth law, the constancy of the surface gravity for a stationary black hole, wasdiscussed in the last section. The second law is discussed in detail in [30] and in [3]. Thetheorem can be stated as follows:Let Σ1 and Σ2 ⊂ I+(Σ1) be Cauchy surfaces for the asymptotically flatspacetime for which the null energy condition holds. Then the sum of theareas of the intersection of Σ1 and all the connected components of J˙−(I +)is no larger than the sum of the areas of the intersection between Σ2 and allconnected components of J˙−(I +).There is a conjecture in black hole physics which is often referred to as the “thirdlaw of black hole mechanics”, which can be stated as followsIt is impossible for a physical process to bring a black hole starting from apositive surface gravity to one with zero surface gravity.In the Kerr-Newman solution to Einstein-Maxwell equations, κ ≤ 0 implies [3, 23]the existence of a naked singularity, i.e., a singularity lying in a region outside blackholes, so that it is possible to connect the singularity to the future null infinity via acausal curve. In fact, Kerr-Newman solution with κ ≤ 0 does not possess a bifurcateKilling horizon. [3] For this reason, the third law is related to the weak cosmic censor-ship conjecture, roughly speaking, the conjecture that spacetime singularities must besituated inside black holes, rather than being visible from infinity. A thorough discussionof this conjecture can be found in ref. [23].I will now briefly present the first law of black hole mechanics in a way that differsfrom the original derivation as in ref. [27], but that will be useful for us later in chapterthree. This derivation follows Wald [31], based on a construction originally publishedby Lee and Wald [32]. Although the construction of the symplectic current below is notexplicitly covariant, Iyer and Wald later published a paper [33] showing that it is alwayspossible to choose one that is.Starting from the variational principle for a classical diffeomorphism-covariant theoryover an n-dimensional manifoldM , the action I0 =∫M L is the integral of the Lagrangian27n-form L, which is a function of the matter fields and spacetime metric, collectivelydenoted by φ, and their derivatives. The variation of the Lagrangian can be generallyexpressed asδL = E · δφ+ dΘ(φ, δφ), (1.35)where Θ is a (n−1)-form in spacetime, which is linear in δφ and its derivatives, referredto as sympectic potential. The second term is produced after “integration by parts” andcan be converted, via Stoke’s theorem, into a boundary term. Thus, the equations ofmotion are recognised as E = 0.If one views the field φ as an element of the configuration space G and its variationsδφ as vectors in the tangent space TφG, the potential Θ can naturally be seen as aone-form in the configuration space. Then we can produce a two-form ω in this spaceby taking its exterior derivative. I will reserve the symbol d for exterior derivatives inspacetime, and simply write insteadω(φ, δ1φ, δ2φ) = δ2Θ(φ, δ1φ)− δ1Θ(φ, δ2φ), (1.36)for two independent variations (“directions” in G) δ1φ and δ2φ13. Eq. (1.36) defines aclosed and antisymmetric (n− 1)-form in spacetime. The particular caseωχ ≡ ω(φ, δφ,£χφ). (1.37)will be extremely useful. This (n− 1)-form in spacetime is ready to be integrated overa (n− 1)-surface C, which we choose to be a partial Cauchy surface.Ω(φ, δ1φ, δ2φ) =∫Cω(φ, δ1φ, δ2φ), (1.38)We wish to make this 2-form in G a symplectic form. This requires it to be closedand non-degenerate. By construction, Ω is exact and consequently closed, but it isdegenerate. For example, ΩAB (δφ)B is zero for all non-zero (δφ)A whose restrictionover C happens to vanish14.13A more precise definition of these variations comes from forming a two-parameter family of pointsof G, φ(λ1, λ2). Then, for example, δ1Θ ≡ dΘ/dλ1.14Here I use the abstract index notation with capital letters to represent tensor arguments of X (P),i.e. vector fields in phase space, and with lower case symbols for arguments in X (M).28However, if one follows the prescription originally introduced by Marsden and Wein-stein [34] called the Marsden-Weinstein quotient or symplectic quotient, it is possible todefine a proper phase space P of the theory by setting an appropriate quotient of G sothat P is automatically equipped with a symplectic 2-form ΩAB, induced by the 2-formin phase space ωχ, which admits an inverse. The prescription, adapted to this contextby Lee and Wald [32], goes roughly as follows.Let ηA and ζA denote degenerate fields, i.e., ΩABηB = ΩABζB = 0. Then the (phasespace) commutator ψA = [η, ζ]A is also degenerate. Indeed,ΩABψB = ΩAB£ηζB = £η(ΩABζB)− ζB£ηΩAB = 0,where the first term is zero by hypothesis, and the second is zero as a consequence ofCartan’s formula £ηΩAB = ηC(dΩ)CAB+(d(Ω·η))AB = 0. Consequently, by Frobenius’theorem [35] the subbundle of TG consisting of the degeneracy subspaces of ΩAB admitsan integral submanifold. Then one defines an equivalence relation ψ ∼ φ of G if ψ andφ lie on the same integral submanifold. As a set, the phase space P is the quotient ofG by this relation. To endow it with a symplectic structure, one uses the projection$ : G → P to identify points on the manifolds (i.e. $ maps every element of G to itsequivalence class in P) and its pullback to identify vectors on the tangent space. Thesymplectic form on P defined so thatΩAB($∗ψA) ($∗φB)= ΩABψAφB, ∀ ψ, φ ∈ G,where ΩAB on the left-hand side denotes, with a small abuse of notation the symplecticform on P.In possession of a symplectic manifold (P,ΩAB), the missing ingredient for makingclassical mechanics is a Hamiltonian function H : P → R generating a desired flow. If ϕand ϕ+ δϕ satisfy the equations of motion E = 0, then dωχ = 0 and Ωχ is independentof the choice of the Cauchy surface C, so £τΩχ = 0, where τA = (£χϕ)A representsthe time evolution in the solution subspace of P. If this subspace is simply connected(if the equations of motion are linear, this subspace is a topological vector space, and29hence automatically simply connected [35]), there exists a Hχ such that(dPHχ)A = (Ωχ)AB τB, (1.39)where dP represents the exterior derivative in that space (which coincides with thedifferential since Hχ is a function). Or, in our customary notation, a Hamiltonianconjugate to the parameter along the orbits of χa must satisfyδHχ = Ω(φ, δφ,£χφ) ≡ Ωχ. (1.40)Eq. (1.40) is equivalent to(£χϕ)A = (Ω−1)AB(∇Hχ)B,the usual canonical equations in the dynamically accessible phase space [20]. Indeed,when we can apply the inverse symplectic form to eq. (1.39), we obtain the familiarcanonical equations τA = ΩABχ (dPH)B.The Lagrangian is covariant under diffeomorphisms, its Lie derivative is determinedby the Lie derivative of the fields φ,£χL =∂L∂φ£χφ,then the (n− 1)-formjχ = Θ(φ,£χφ)− χ · Lis closed whenever the equations of motion (both on the matter fields and on the metric)are satisfied, more explicitly,djχ = £χL−E ·£χφ− d(χ · L) = £χL−E£χφ− χ · dL−£χL == −E ·£χφ = 0, (1.41)where the identity χ · L = £χL − d(χ · L), valid for any form L [35] was used in thesecond step, and that dL = 0, since L is a top form.For a simply connected domain of jχ, as it will be the case for a spacetime wedgelike I+(h−)∩ I−(h+), bounded by a bifurcated Killing horizon of an asymptotically flat30spacetime [23], this means that jχ is also exact, jχ = dQχ. Taking a variation of jχ andusing the covariant property of Θ, we relate jχ and ωχ:δjχ = δΘ(φ,£χφ)− χ · δL = δΘ(φ,£χφ)− χ ·E · δφ− χ · dΘ(φ, δφ) =δΘ(φ,£χφ)−£χΘ(φ, δφ) + d(χ ·Θ(φ, δφ))− χ ·E · δφ⇒ ωχ = δjχ − d(χ ·Θ) + χ ·Eδφ. (1.42)We can transform δHχ for solutions into a pure boundary term using eqs. (1.42) and(1.40) and applying Stokes’ theorem.In order Hχ to be defined, we assume the necessary and sufficient condition thatthere exists a (n− 1)-form B such that15δ∫∂Σχ ·B =∫∂Σχ ·Θ. (1.43)Using eqs. (1.40-1.42) whenever the equations of motion are obeyed,Hχ =∫∂ΣQ[χ]− χ ·B. (1.44)This Hamiltonian (1.44) is only defined up to a constant, since the equations ofmotion only impose requirements over its variation, not on its absolute value.If such a form B exists, then instead of the original action I0, we adopt the modifiedactionI =∫ML−∫∂MB, (1.45)which generates the same equations of motion as I0, and has the advantage that I, andnot I0, is extremised when we keep the boundary of spacetime fixed, which is of greatimportance in approaches to quantum gravity, and crucial for the so-called Euclideanmethods to compute gravitational entropy. In general relativity, i.e., when L is theEinstein-Hilbert Lagrangian, B on the boundary is proportional to the trace of the15If this form does not exist, we cannot define a Hamiltonian for the field χa. B is known [33] toexist in general relativity when χa is an assymptotic time translation under reasonable behaviours ofthe matter fields at infinity.31extrinsic curvature of ∂M times the induced volume form, up to a constant (see, forexample, the appendix E of Wald’s book [23] for a proof). The common choice, referredto as the Hawking-Gibbons-York term in the action, is choosing this constant to beminus the value of the trace of the extrinsic curvature if the boundary is embedded inflat space, a procedure that is well defined for asymptotically flat spacetimes.We are now in position to derive the first law of black hole mechanics, but before wedo so, I show an example of a theory for the reader to gain familiarity with the formalismI described so far.Consider a real massless scalar field ψ in a fixed, Minkowski background(R4, η). The action is I0 = 12∫d4x ∂µψ∂µψ, whose variation isδI0 =∫d4x ηµνδ(∂µψ∂νψ) =∫d4x {∂µ (∂µψδψ)− (∂µ∂µψ) δψ} ,from which we identify (−∂µ∂µψ)d4x as E and ∂µ(∂µψδψ) as dΘ(ψ, δψ).Thus, in coordinates so that ds2 = −dt2 +d~x2, we write the symplectic formas an integral over a constant t surface:Ω(ψ, δ1ψ, δ2ψ) =12∫d3x {∂t(δ1ψ)δ2ψ − (∂tδ2ψ)δ1ψ} ,which is proportional to the Klein-Gordon product between δ1ψ and δ2ψ aswritten in standard texts.Before integration, when the second variation is taken to be the Lie derivativewith respect to a Killing field ξa,−2µνρλωξ(ψ, δψ)νρλ = £ξ(∂µψδψ)− δ(∂µ£ξψ),leading to the familiar Hamiltonian associated with ξ = ∂/∂t given byδHξ =∫d3x{∂t(∂tψδψ)− δ(∂2t ψ)}=∫d3x{δψ∇2ψ − ∂tψ∂tδψ}=∫d3x {div (∇ψδψ)−∇ψ · ∇δψ − ∂tψ∂tδψ} =12δ∫d3x{(∂tψ)2 + (∇ψ)2}+ boundary term,32where the equation of motion was used in the second step, and the symbol∇ denotes the spatial gradient. This is recognised as a variation of theusual Hamiltonian for the real massless scalar field. It is also clear that theintegrand above is identified with the Noether current j, and the boundaryterm with ∂∂t ·Θ, by direct substitution. For the real scalar field, the initial value problem is well posed in the sensethat producing a δψ that vanishes over a constant t surface makes the fieldzero everywhere and the form ΩAB is non-degenerate. To illustrate the casewhere it is degenerate, consider the electromagnetic field in the same back-ground, whose action is I0 = −14∫d4x FµνFµν , which leads toΩ(Aµ, δ1Aµ, δ2Aµ) =12∫d3x ξν(δ1Aµδ2Fνµ − δ2Aµδ1Fνµ),which is degenerate. Whenever δ2Aµ is the four-gradient ∂µf of a scalarfunction f , Ω(Aµ, δ1Aµ, δ2Aµ = ∂µf) = 0 ∀ δ1Aµ. The symplectic quotientthen identifies two variations δ2Aµ that differ by this four-gradient. In par-ticular, ∂µf itself is identified with the null vector in TAµG, removing thedegeneracy in this direction. From this example we see that the originaldegeneracy of ΩAB is related to the gauge freedom of the field. In the following I consider only bosonic matter fields ψ to avoid introducing addi-tional structures like spinors. Following Wald in ref. [31], with the definition of Hχabove, an integration of the identity (1.42) over a slice bounded by a surface at far dis-tances of the black hole denoted by “∞” and the bifurcation surface B leads to a versionof the first law of black hole mechanics,0 =∫BδQ[χ] + δ∫∞(Q[χ] + χ ·B),where we already made use of the fact that χ|B = 0 and converted the integral of jχ intoboundary terms. Defining Eχ as the integral under the variation on the second term wehave33δ∫BQ[χ] = δEχ.The left hand side is the variation of a locally defined quantity over B times itssurface gravity since all the dependence on χa can be eliminated by setting its valueto zero over B, its first derivatives can be eliminated in terms of κ and the volumeform, and higher derivatives can be eliminated by repeated use of the Killing identity,∇a∇bχc = −Rbcadχd, which is linear in the field. So we can put∫BQ[χ] =κ2piSNC, (1.46)defining geometrically the entropy of the stationary black hole, i.e. the geometricalquantity that is necessary to make the first law of black hole mechanics analogous tothe first law of thermodynamics16. This notion coincides with the Bekenstein-Hawkingfor general relativity in the vacuum to be one quarter of the area of the event horizonin Planck units. To see this, for L = R16pi,Θabc =116pidabcgdegfh(∇f£χgeh −∇e£χgfh),jgabc =18pidabc∇e∇[eχd],Qab = − 116piabcd∇cχd,(1.47)so thatSNC = − 18κ∫abcdκcd =14∫ab =A4.To recognise this form with the more traditional ways of writing, as in Bardeen,Carter and Hawking [27], which I reviewed in ref. [30], we just need to interpret thequantities Eχ. If we take the familiar example of one of the asymptotically flat regions ofthe extended Kerr solution and the Killing field tangent to the horizon χa = ξa + Ωψa,16The fact that SNC above is a locally defined, geometrical quantity that obeys the first law of blackhole mechanics was enough for Wald to name it the black hole entropy in [31]. Much later, Ford andRoman [36] managed to show that this notion obeys also a version of the generalised second law ofthermodynamics in classical physics. Hence, this nomenclature is now common.34where ξa is the asymptotically timelike Killing field and ψa the axial Killing field,Eχ =∫∞(Q[ξ + Ωψ]− ξ ·B− Ωψ ·B).The surface of integration is everywhere tangent to ψa, so that the last term vanishes.Because Q[V ] is linear in its argument [33], we getδEχ = δM − ΩδJ ,whereM =∫∞(Q[ξ]− ξ ·B) and J = −∫∞Q[ψ] (1.48)are the canonical ADM energy and angular momentum [31, 33]. They deserve the nameof conserved quantities, since they do not depend on the choice of integration surface, ifit is taken to be a full Cauchy surface.In summary, there is a parallel between the four laws of thermodynamics and thelaws of black hole mechanics. The first hint that there could be something more aboutthis parallel than a mere mathematical analogy comes from the fact that Eχ is indeedto interpreted in the same way as the energy (and possibly other potentials used todistinguish macrostates) in thermodynamics. But from the purely classical point ofview, the temperature of a black hole is exactly zero since the impossibility of emittingenergy, only absorbing it (say in the spherically symmetric case) forces it to be colderthan any positive temperature if the second law of thermodynamics is to be respected.In this case, the entropy of a black hole would be necessarily infinite.This view changed dramatically after Hawking’s discovery [89] that a black hole emitsthermal radiation as in eq. (1.34), and is consequently a black body with temperatureκ/2pi. It is because of considerations such as the above that led to eq. (1.34), that wechose the proportionality constant on eq. (1.46). Once one accepts this consequence ofquantum mechanics, the aforementioned laws have to be understood as having physicalsignificance, as applications of the laws of thermodynamics to systems containing blackholes. Then two questions naturally arise: whether we can prove these laws, and what35the physical interpretation of black hole entropy could be. I dedicate a good part ofchapter three on these issues.362. Role of EnergyIn this chapter, I explore a different, yet conceptually related notions of “energy” incomparison to the chapter above. That notion, is of the total energy of an asymptoticallyflat spacetime, as in the ADM formulation in eq. (1.48)1. It is possible to physicallyinterpret this global notion based on measurements of the total energy at infinity byobserving the orbits of test masses, just like we can determine the total electric chargewithin a region by measuring the electric field far from the charge distribution andintegrating Gauss’ law.The notion I discuss here, which is not properly the notion of energy alone, which hasno geometrical meaning, but of the energy-momentum tensor Tab. It is clearly related toenergy. Not only do global notions like the ADM energy depend on this tensor, but alsoTabξaξb is the local energy density as measured by an observer following the orbits of ξaat a given point. But the importance of this object goes well beyond this interpretation.In general relativity, it is this object that is the source of gravity, so it has an undeniableinherent physical meaning.Nevertheless, it must be stressed that, unlike in electrodynamics, where the totalcharge is merely the integral of the charge density, this is not so for gravity, since thereare non-vanishing terms in eq. (1.48) even when Tab = 0. A possible interpretation comes1There are other similar definitions (see, e.g., Misner, Thorne, and Wheeler [22]) which do notnecessarily coincide with the one I gave. Examples are the Komar definition, dropping the B term ineq. (1.48), and the Bondi definition, replacing the spacelike surface for a null surface. They are useful indifferent contexts (for example, depending on whether or not one wishes to include energy carried awayby gravitational waves in this total), but they are inspired by the same general ideas here discussed.37when studying the linearised version of Einstein’s equations around certain backgroundmetric g(0)ab (assumed to satisfy the vacuum equations for simplicity), which means ex-panding the Einstein’s tensor up to order one G(1)ab in the metric perturbation hab andsolving the equation G(1)ab [hab] = 0. Then second-order corrections to the metric h(2)ab aredetermined by G(2)ab [hab] 6= 0, which can is sometimes interpreted as a constant timesan energy-momentum pseudo-tensor of the gravitational field, but this procedure is, ofcourse, background-dependent.The scope of this doctoral thesis is, however, the semiclassical context, for whichobtaining a version of the energy-momentum tensor is far from being obvious or unique.The semiclassical approach replaces the classical energy-momentum tensor by the ex-pectation value 〈Tab〉 of the stress-energy operator in a given state, after a regularisationsince otherwise this value would be infinite. Symbolically, for general relativity we haveGab = 8pi〈Tab〉reg. (2.1)This semiclassical coupling occurs in other areas of physics as well. Most notably inatomic/molecular physics, when instead of solving problems using the existing full quan-tum electrodynamics, one treats the electrons orbiting an atom quantum mechanically,but the electromagnetic field classically. It is also used in nuclear physics, when describ-ing direct nuclear reactions [38]. However, one important difference, again a consequenceof non-linearity of gravity, manifests itself here if we believe in a future quantum theoryof gravity for which the Einstein tensor is promoted to an operator that satisfies anoperator equation. Because Einstein’s tensor is not linear in the metric, even if one hasan operator-valued metric, the expectation value of the Einstein tensor evaluated on thisoperator will not be the same as the Einstein tensor evaluated at the expectation valueof the metric operator. And any attempt to use perturbation theory to write a metricaround certain background will generate equations depending on the pseudo-tensor aswell as the matter stress-energy tensor, making us believe that gravitons are as impor-tant as matter fields. In the classical theory, there are pure gravity solutions (Tab = 0everywhere) with a positive total energy as seen at infinity.38I devote this chapter to the task of understanding this object and using it to studythe effects of the back reaction of the phenomena of quantum mechanical origin in curvedspacetimes.2.1 Regularisation of Stress-Energy Tensor in 2DLet us start with the metric in the form ds2 = C(u, v)dudv (as in eq. (2.14) below), forwhich the non-vanishing Christoffel symbols areΓuuu =1C∂C∂u, Γvvv =1C∂C∂v. (2.2)One of the most common methods for dealing with the divergences that occur inquadratic functions in quantum field theory in Minkowski spacetime is the point-splittingmethod, which consists in evaluating vacuum expectation values in two different pointsin a bi-distribution and then taking the limit when the points are close together andremoving the divergent terms. There is no guarantee that this general prescription isunique, but we now follow Davies, Fulling, and Unruh [6] and references therein to finda covariant formulation of point-splitting, which can be applied in curved spacetimesand we will deal with the remaining ambiguities later.We start at a point x and pick a timelike or spacelike geodesic through it and definethe points x+ (and x−) by propagating forward (backward) a proper distance  > 0along it. In the end we are going to make  → 0, so we can always make this processwell defined in a sufficiently small normal convex neighbourhood of x.The idea is to use a Taylor approximation for small proper distance along a geodesicconnecting two points for both the geodesic equation for its tangent vector, and forthe equation requiring parallel transport of the vector we are evaluating at the relevantpoints. Symbolically, for the tangent vector ta, in a coordinate basisdtµd+ Γµνρtνtρ = 0, tµ() = tµ1 + tµ2 +22!tµ3 + . . . (2.3)anddaµd(x±) + Γµνρ(x±)aνtρ = 0, aµ() = aµ1 + aµ2 +22!aµ3 + . . . (2.4)39for the a vector field aa. For the position x± we havex±µ = xµ ± tµ1 ±22tµ2 ±36tµ3 ,which can be used as the argument for the connection coefficients. Expanding thesecoefficients and collecting all terms in the same power in ,aµ1 = aµ(x)au2 = −∂uCCtu1au1 , av2 = −∂vCCtv1av1au3 = C[3(∂uC)2 − C∂2uCC3tu1 −R4tv1]tu1au1 , av3 = C[3(∂vC)2 − C∂2vCC3tv1 −R4tu1]tv1av1,(2.5)whereR =4C3[∂uC∂vC − C∂u∂vC]is the curvature scalar.The energy-momentum tensor for a massless scalar field is obtained fromTµν(x+, x−) =12〈{∇αΦ(x+),∇βΦ(x−)}〉×[eα(µ()eβν)(−)−12gµνeασ()eσβ(−)], (2.6)where eαµ are the parallel transport bivectors, that is, the function from TxM to Tx+Mdefined as the result of the parallel transport by solving eq. (2.4), and the curly bracketsdenote the anticommutator2.From eqs. (2.5),euv () = evu() = 0,euu() = 1−tu1∂uCC+Ctu12[3(∂uC)2 − C∂2uCC3tu1 −R4tv1]2 +O(3),evv() = 1−tv1∂vCC+Ctv12[3(∂vC)2 − C∂2vCC3tv1 −R4tu1]2 +O(3).(2.7)To evaluate eq. (2.6) we expand the quantum field Φ in normal modes for whichφω =e−iωu√4piωor φω =e−iωv√4piω,2This is the regular symmetrisation process coming from promoting quadratic forms on the fields tooperators in quantum mechanics. See for example [40].40which are normalised so that (φω, φω′) = δ(ω − ω′) and substitute this expansion intoeq. (2.6), leading to a double integral over ω and ω′.The delta functions make the integration over ω′ straightforward, and the integrationover ω is dealt with as follows.Starting from ∫ ∞0eiωsω dω,we complete a contour moving from the origin of the complex ω-plane to the negativeradius of a quarter circle, that extends over the third quadrant and a straight line fromthe negative imaginary radius back to the origin. The integral over the circumferencearc vanishes as we make the radius go to infinity, so Cauchy’s theorem transforms theoriginal integral into ∫ ∞0ze−zs dz = − ∂∂s∫ ∞0e−sz dz = +1s2.From this evaluation and from eq. (2.7), one finally hasTµν(x+, x−) = −(14pitρ1t1ρ2+R24pi)(t1µt1νtρ1t1ρ− 12gµν)+ θµν +O(), (2.8)whereθuv = 0, θuu =√C12pi∂2∂u21√C, θvv =√C12pi∂2∂v21√C. (2.9)From the result (2.8), we identify a term that grows with −2, which diverges when → 0. This term is discarded for regularisation purposes. We also identify direction-dependent terms, depending on the direction of the first-order tangent vector ta1. Theseterms cannot be physically meaningful, and common approach is to “average them out”3This prescription is still ambiguous. In conformally symmetric spacetimes, the Weyl3The idea is to integrate these direction-dependent terms over all possible directions of vectors tan-gent to straight lines in a Riemann normal coordinate system. However, the volume of a hypershpereof squared radius tµtµ is divergent. To overcome this issue, one has to analytically continue the com-ponents tµ to a metric with signature (++++), average over a hypershpere, and then continue backto the Lorentzian metric. See Adler, Liberman & Ng [41] and Christensen & Fulling [42] for a detaileddiscussion.41tensor vanishes, meaning that there are other geometrical scalars creating degeneracy.Following Davies, Fulling, and Unruh [6], we impose the condition that the regularisedtensor ought to be divergence-free (at expense of having a non-zero trace). We willdiscuss the axioms defining the regularisation process in more details below. This choiceeliminates the first term proportional to t1at1b, leading us to the regularised form〈Tµν〉reg = R48pigµν + θµν . (2.10)One striking consequence of eq. (2.10) is that the trace of the regularised stress-energy tensor is non-zero. This result is known as trace anomaly, conformal anomaly,or Weyl anomaly, since the classical tensor has to be trace-free for a massless field intwo-dimensions. An easy way to see this is directly from the behaviour of the classicalaction S under a conformal transformation gab → g¯ab = Ω2gab:S[g¯ab] = S[gab] +∫δSδg¯cdδg¯cd d4x.Using δg¯ab = −2g¯abδΩ/Ω [29], from the definition of the energy-momentum tensor onehasS[g¯ab] = S[gab]−∫ √−g¯ T cc [g¯de]δΩΩ d4x.In particular the trace of the above expression isT cc = −Ω√−gδS[g¯ab]δΩ∣∣∣∣Ω=1,which is zero if the action is invariant under conformal transformations.This anomaly does not come entirely as a surprise. Since the conformal symmetry isinterpreted as a rescaling symmetry, anything that breaks the scale invariance, such asa mass in the classical action, a cut-off or a separation distance  in the renormalisationprocess, is expected to break consequences of this symmetry.For higher spins, the reader can find expressions for the trace of the regularisedstress-energy tensor in the paper by Christensen and Duff [43]. It is noticeable that, likethe spin-0 case above, part of the trace can be written in terms of scalars obtained from42the curvature tensor. For massive fields, I refer the reader to chapter 6 of the book byBirrell and Davies.[39]Expression (2.10) picks a particular state for the fields. In most applications however,one is not necessarily interested in the vacuum state annihilated by the normal modesproportional to eiωu or eiωv. But if one calculates the regularised stress-energy tensorfor the modes as above, it is possible to find the expectation value with respect to adifferent state as follows: [44, 45]〈0|Tuu(u˜)|0〉 =(dudu˜)2〈0˜|Tuu(u(u˜))|0˜〉+ 124pi[u′′′(u˜)u′(u˜)− 32(u′′(u˜)u′(u˜))2], (2.11)whose the first term is identical to the usual tensor transformation law, and the termsin square brackets are known as the Schwarzian derivative [46].The most straightforward proof of eq. (2.11) is obtained by writing the tensor trans-formation rule for the conformal factor in the new coordinates (u˜, v˜), so that the modesare written as imaginary exponentials eiωu˜ and eiωv˜, g˜µν = (∂xρ/∂x˜µ)(∂xλ/∂x˜ν)gρλ andreapplying eq. (2.10) in these coordinates.The covariant point-splitting procedure, or the regular point splitting process inMinkowski spacetime for that matter, may seem ad-hoc and unjustified. The methoduses intuitive generalisations to make sense of infinite results, but these prescriptionscannot be derived from first principles and ultimately require experimental validation.Luckily the results so obtained are stronger than they seem at first sight. Since reg-ularisation is necessary even in Minkowski spacetime, where it has survived a multitudeof testing, it seems reasonable to expect the method to work at least for calculating thedifferences between the expectation values for a given state and a state of reference4.Symbolically, we can consider the functionF (x, x′) = 〈{Φ(x),Φ(x′)}〉ρ − 〈{Φ(x),Φ(x′)}〉ref4This is because in Minkowski, there is the natural vacuum state for which one sets 〈Tab〉 = 0, andthere is no preferred vacuum state in curved spacetimes.43which is the difference in the two point functions5 of the field Φ between a state ρ anda reference state, and we can write instead of eq. (2.6)〈Tab〉 = limx′→x(∇a∇b′ − gab2∇c∇c′)F (x, x′). (2.12)As it stands, there is no guarantee that the expression (2.12) converges. As an extrahypothesis, we say that only states satisfying the Hadarmard condition are physicallyacceptable. For these states, F (x, x′) is a smooth function of its variables. [25]Once the point-splitting method is accepted, it is possible to generically define theregularised expectation value of the tensor in a given state by taking an axiomaticapproach first introduced by Wald in ref. [47] and later revised by him and reformulatedin ref. [25]. The axioms can be taken as follows6.1. For all states, ∇a〈Tab〉 = 0,2. In Minkowski spacetime, for its naturally-defined vacuum state 〈Tab〉 = 0,3. For any two given states ρ and σ, the difference 〈Tab〉ρ−〈Tab〉σ is obtained from thedifference in the corresponding two-point functions by the point-splitting processsuch as in eq. (2.12),4. For each state, 〈Tab〉(x) is local as a function of the point x ∈M . More precisely,for any two globally hyperbolic spacetimes (M1, g1) and (M2, g2) that are isometric(with isometry φ : M1 →M2) around an open globally hyperbolic neighbourhoodO of x ∈M1 for which the intersection between a Cauchy surface of M1 and O isitself a Cauchy surface for O. If for all test functions f whose support is containedin O, the expectation value of an operator Ψ(φ∗f) in φ(O) coincides with theexpectation value of Ψ(f) in O and, then on 〈Tab〉2(φ(x)) = 〈Tab〉1(x).5Here we take the anticommutator between them, but we could adapt for the commutator in thefermionic case or even different Green functions depending on the application. For further discussionsabout the choice of two-point functions I refer the reader to [39].6Axiom (4) is here phrased in our approach for quantum field theory, albeit it was originally formu-lated in the algebraic approach as in ref. [25].44From axiom (3), the difference between two different definitions, 〈Tab〉 and 〈T˜ab〉 thatsatisfy these axioms in any given pair of states ρ and ω have to obey〈T˜ab〉ρ − 〈T˜ab〉ω = 〈Tab〉ρ − 〈Tab〉ω,or equivalently,〈T˜ab〉ρ − 〈Tab〉ρ = 〈T˜ab〉ω − 〈Tab〉ω, (2.13)meaning that the difference between two nonequivalent regularisation prescriptions doesnot depend on the state. Furthermore, from axiom (4), the common value (2.13) is alocal geometric tensor, which is sometimes expressed in terms of the curvature tensorin some proposed approaches in the literature. For example, Christensen & Fulling [42]and Adler, Liberman & Ng [41].2.1.1 Method for Finding the Conformal Factor in 2D SpacetimesAll two-dimensional Lorentzian manifolds are locally conformally flat. The proof followsdirectly from the locally Euclidean property of differentiable manifoldsM : for any givenpoint p ∈M there is an open neighbourhood O ⊂M,p ∈ O which is homeomorphic toR2. Let φ : O → R2 be such homeomorphism and `a andma be orthogonal (with respectto the metric in R2) null vectors in Tpφ(O), then the metric on M will be manifestlyconformally flat, that is in the formds2 = C(u, v)dudv, (2.14)in coordinates along the integral lines of the vector fields φ−1∗ `a and φ−1∗ ma. The atlascovering M can be taken to be the union of all φ−1(O), completing the proof. This is,of course, no guarantee that the manifold is globally conformally flat.This proof is not constructive, though. I now show a method that can be used toexplicitly find the conformal factor in a not necessarily unique coordinate chart around agiven point and an arbitrary number of its derivatives. From the form of the line elementin generic coordinates one can always “complete squares” to write it in the form ds2 =−gttdt2 + grrdr2. Let t˜(r, v) denote a solution for the null geodesic dr/dt = −√grr/gtt45with integration parameter v satisfying the initial condition t˜(r0, v) = v for some conve-nient choice of r0. From this definition (∂t˜/∂v)dv = dt˜+√grr/gtt dr and, consequentlyds2 = −gtt(∂t˜/∂v)dv(dt−√grr/gtt dr) = −gtt(∂t˜/∂vdv2 − 2 ∂t˜/∂v√grr/gtt)dvdr. Weare then left with the line element in the formds2 = gvvdv2 + 2gvrdvdr. (2.15)Similarly, now define a new coordinate u as the integration constant which selects thenull geodesic r˜(v) satisfying the equation dr/dv = −gvv/(2gvr) and the initial conditionr˜(v0, u) = u, leading to the final form ds2 = 2gvr(∂r˜/∂u)dvdu leading to the final formds2 = 2gvr∂r˜∂udvdu. (2.16)Henceforth I assume, for simplicity, that we already have the metric in the form ofeq. (2.15). We seek solution for the orthogonal null geodesic obtained from (2.15)dr˜dv= − gvv2gvr≡ 12F (v, r˜), (2.17)in the power seriesr˜(v, u) =∞∑n=0an(v)un, (2.18)satisfying the initial condition r˜(v0, u) = u meansa1(v0) = 1 and an≥2(v0) = 0. (2.19)The arbitrary coefficient a0(v) represents our choice of the point around which wewant to find the conformal coefficient.One can then substitute a truncated version of eq. (2.18) into eq. (2.17) expandingthe composition F (v, r˜(v, u)) up to the desired power of u. Collecting the same powersof u leads to a set of first order ordinary differential equations for the coefficients an(v).A concrete example of this method is shown below.If one wishes to find the expectation value of the regularised stress-energy tensor ofa massless scalar field at a given point, one does not need to know the entire conformal46factor, but only its derivatives up to fourth order [6], so one can use the method describedabove to find its exact value at that point.Let us exemplify this process by calculating the regularised expectation value ofthe energy-momentum tensor in a spacetime that is of interest in semiclassical gravity;namely the two-dimensional version of a simplification of a model proposed by Ringin [48] for a slowly evaporating black hole. The original model approximates the backreaction of quantum effects into the metric to describe an evaporating black hole byconsidering a spacetime composed by five charts: two Minkowski patches to represent thedistant past and future, an ingoing Vaidya patch to represent the behaviour of quantumradiation near the black hole, one outgoing Vaidya patch to represent its behaviourfarther from the hole, and a Schwarzschild patch for near the spatial infinity (see figure2.1 for a reproduction of a Carter-Penrose diagram representing the geometry).Since the energy-momentum tensor is a local quantity, and that the details of thecollapse do not seem to influence the quantum state long after the black hole is formed[1], we simplify the model and concentrate on the region where the Hawking radiationis visible, leaving a metric in the formds2 =−(1− 2(m0−µv)r)dv2 + 2dvdr r < 3(m0 − µv)−(1− 2(m0−µu)r)du2 − 2dvdr. r ≥ 3(m0 − µv),(2.20)where m0 and µ are positive constants. The charts are limited so that the coordinatesv and u are, of course bounded to (m0 − µv), (m0 − µu) > 0. The matching conditionscan be met exactly as in ref. [48].For later use, let us localise the apparent horizon for this metric. Ingoing null rays inthe inner region are described as dv = 0 (tangent covector na = −(dv)a), and outgoingby gvvdv+2dr = 0, whose tangent covector is then `a = gvv2 (dv)a+(dr)a. The parameterv is not an affine parameter, which means one has to be careful when calculating theexpansion of this congruence.The expansion θ of a geodesic congruence is defined with respect to the affine param-eter λ and its tangent vector ka = d/dλ as θ = ∇aka (see for example sec. 4.2.1 of ref.47Figure 2.1: Carter-Penrose diagram adapted from [48]. Regions III and V representMinkowski patches; region IV is Schwarzschild; region I is an ingoing Vaidya patch, andregion II an outgoing one. Our simplified two-dimensional version is concerned withregions I and II only.[30] for the its interpretation). In a different parametrisation so that V b∇bV a = κ(v)V a,then the parameter v is related to λ by dλdv = exp∫ vκ(s)ds, hence V a = exp∫ vκ(s)ds kaand∇aV a = ∇a(e∫ v κ(s)ds) ka + θ,usingka∇a = Ddλ=dvdλDdv,one hasθ = ∇aV a − κ.The expansion of this congruence of outgoing null rays is θ = −gvv/r, meaning thatthe apparent horizon is described by rah = 2(m0 − µv). Similarly, the apparent horizon48in the outer region is described by rah = 2(m0 − µu).In contrast, the event horizon is identified as the outgoing null ray containing theevaporation endpoint. This gives in the outer regionreh =4(m0 − µv)1 +√1 + 16µ,which is the same as the apparent horizon except for terms of order µ2, symbolicallyreh(v) = rah(v) + (64m0 + 8v)µ2 +O(µ3).And similar expressions for the outer chart.We now use the series approach for finding the conformal factor. Because eq. (2.16)includes a first order derivative, we need to carry out the series (2.18) up to fifth orderin u:25∑n=0a′n(v)un =12− (m0 − µv)[1a0+a1a22u+a0a2 − a21a30u2+a31 − 2a0a1a2 + a20a3a40u3 +a30a4 − 2a20a1a3 − a20a22 + 3a0a21a2a50u4+a51 − 4a0a31a2 + 3a20a1a22 + 3a20a21a3 − 2a20a2a3 − 2a30a1a4 + a40a5a60u5]. (2.21)We also choose a0(v) = 2(m0 − µv), meaning making the expansion around theapparent horizon. With this substitution and imposing the equality (2.21) to holdidentically, we geta1(v) =(m0 − µv0m0 − µv) 18µ,a2(v) =(m0 − µv0)18µ−1 [(µv0 −m0)(m0 − µv0)18µ + (m0 − µv)(m0 − µv)18µ](2 + 16µ)(m0 − µv)1+14µ,and more complicated expressions for the other coefficients. Because we model theevaporation to be slow, we can largely simplify the equations by keeping the expressionsonly up to the first order in µ.If one choose the state that plays the role of the Unruh state, |0〉 that is, annihilatedby modes proportional to eiωv¯, at I −, where v¯ denotes the advanced null coordinate at49infinity, one gets v¯ = v, so the expectation value of the vv component of the stress-energytensor is calculated directly by taking the derivatives of the conformal factor above togive〈0|Tvv|0〉 = 1768pim20+v − 8m0384pim30+ev−v08m0 u24pim30µ, (2.22)where only terms up to the first order in u and µ were kept. The first term of eq. (2.22)represents a constant ingoing flux of negative energy with the same magnitude as thezero-th order approximation of the constant flux of positive energy radiation at I +, asone would expect.For the uu-component, we first calculate the expectation value for the nonphysicalstate |0˜〉 annihilated by modes proportional to eiωu and then convert the result to amore meaningful state |0〉, annihilated by modes proportional to eiωu˜ using the formula(2.11).The result up to the first order in u and µ is〈0˜|Tuu|0˜〉 =3(1− ev−v04m0)24pim20− 48m0(v0 − 4m0) + 3ev−v04m0 (v20 − v2 + 64m20 − 16m0v)12pim30u+24m0v0 + 6(v20 − v2 − 16m0v + 64m20) + 3uev−v04m0 (v2 − v20 − 128m20 + 16m0v − 8m0v0)96pim50µ.(2.23)For the off-diagonal components, we only need to know the scalar curvatureR =4(∂uC∂vC − C∂2uvC)C3,directly calculated from eq. (2.14). Or, from our results, keeping only first powers of µand u,R =14m0− 3ev−v08m0 u8m30+µv2m30− 3(v2 − v20 + 48m0v)ev−v08m0 u128m50µ.Directly from eq. (2.20) we can, of course, find the classical energy-momentum tensor50from Einstein’s equations, which isTab =− µ4pir2(dv)a(dv)b r < 3(m0 − µv)+ µ4pir2(du)a(du)b r ≥ 3(m0 − µu).(2.24)In the next subsection, I discuss some general arguments in order to be able to saysomething about a four-dimensional case from the results computed in two dimensions,at least when spherical symmetry is present. In particular, we expect that the four-dimensional regularised expectation value of the stress-energy tensor to be proportionalto r−2 times the expression we obtained for the two-dimensional metric. In this way, theresults for the classical energy-momentum tensor for metric (2.20) can be compared tothe quantum ones. We can see that the off-diagonal term Tvr produced by the quantumradiation cannot be approximated by the Vaidya metric. However, the results for theleading order terms in the tangent directions Tuu and Tvv obtained around the apparenthorizon can be approximated by the Vaidya geometry. If one is interested in study-ing, for example, the evaporation effects on the apparent horizon, this model is usefulsince only tangent contractions of the energy-momentum tensor appear in Raychaud-huri’s equation, which governs the local effects of the curvature in the flow of geodesiccongruences.Consequently, our results above show that a convenient choice of µ makes the metric(2.20) useful as an approximation of the evaporation of a black hole around the apparenthorizon when µ ∼ m−20 is small, or conversely for black holes with large masses.2.1.2 Relating Spherically Symmetric 4-Dimensional Solutions and 2-Dimensional Solutions of Field EquationsOne of the main reasons why the full four-dimensional problem is much more complicatedthan a two-dimensional version is because of the presence of an angular momentumbarrier that produces scattering of incoming radiation and back-scattering of outgoingradiation. This difficulty is naturally ameliorated if, for some reason, we restrict ourselvesto the spherically symmetric solutions for the four-dimensional problem, so we now turn51our attention to this problem.The two-dimensional Lagrangian for a massless scalar field φ2 with a potential U isgiven byL2 =√−g2 (∇aφ2∇aφ2 + U(φ2)) , (2.25)and for the four-dimensional isL = √−g∇aφ∇aφ. (2.26)We know that the equations of motion derived from varying the Lagrangian areunchanged if one adds a total divergence to it, since it would only contribute to aboundary term after applying Gauss’ theorem. In particular, this will be the case if weadd a term that is the derivative with respect to a coordinate λ, since∂λ[√−gf(λ)φ(x)] = √−g∇a[`af(λ)φ(x)],where ` = ∂/∂λ.For example, in Schwarzschild metric and coordinates, the two-dimensional La-grangian (2.25) after the substitution φ2 = rϕ is given byU(rϕ)−r2(∂tϕ)2(1− 2mr)−1+(1− 2mr)r2(∂rϕ)2+(1− 2mr)ϕ2+(1− 2mr)2r∂rϕ,where the last term can be rewritten as(r − 2m)∂r(ϕ2) = ∂r(rϕ− 2mϕ2)− ϕ2,and only the second term above contributes for the equations of motion. Therefore ifone choosesU(φ2) =2mr2φ22 (2.27)the Lagrangian isr2[−(1− 2mr)−1(∂tϕ)2 +(1− 2mr)(∂rϕ)2],which is proportional to the four-dimensional Lagrangian (2.26) in the same coordinatesif one supposes φ to be spherically symmetric.52This shows that spherically symmetric solutions of the field equations in the four-dimensional Schwarzschild metric can be built from the solutions in two dimensions witha potential given by eq. (2.27) by φ = ϕ/r.But for Vaidya, a different approach is required since there is no such potential inthe chart (u, r) for whichds2 = −(1− 2m(u)r)du2 − 2dudr + r2dΩ2,because the equations of motion do not separate in a simple form in these coordinates,not even if m(u) is a constant, i.e., Schwarzschild metric in Eddington-Finkelstein coor-dinates. To relate solutions we need to introduce a new set of coordinates as followsdη = −G(u, r)du− dr,where G(u, r) =(1− 2m(u)r). In these coordinates,ds2 = G˜(η, r)(−dη2 + dr2) + r2dΩ2 G˜(η, r) = 1/G(u(η, r), r).Because the section of the manifold dΩ = 0 is consequently written in a explicitlyconformally flat manner, it is straightforward to write eq. (2.25) asU(rϕ)− r2(∂ηϕ)2 + (∂rφ2)2,whose last term is ϕ2 + r∂r(ϕ2) + r2(∂rϕ)2 = ∂r(rϕ2) + r2(∂rϕ)2 and, disregardingthe total derivative, we see that the Lagrangian takes the desired form with vanishingpotential.2.2 Example: Modelling Quantum Mechanical Effects inStellar CollapseI now apply the techniques above described to a concrete problem as an example ofapplication, motivated by the fact that over the course of the last few years, there hasbeen proposed a number of models to prevent information loss in a black hole. Most53popular versions propose small modifications to either General Relativity or QuantumMechanics. But it has been argued [49, 7, 50, 51, 52] that black holes could simply notform as a result of the back reaction of its Hawking emission. While it is well knownthat all outgoing null rays that carry low energy at infinity pile up at the horizon, Ishow that this fact does not mean that one can find an infinitely large amount energyon the horizon itself as a result of the non conservation of the expectation value of theoutgoing energy flux along these lines, as a consequence of the conformal anomaly. Thisexample, which agrees with Davies, Fulling, and Unruh’s [6] collapsing null shell, waspublished by Arderucio Costa and Unruh in ref. [1].2.2.1 4-Dimensional ModelWe model our 4-dimensional collapsing star as dust (zero pressure) as a FRW solution7in the interior and Schwarzschild in the exterior of the star.ds2 =a2(η)(−dη2 + dχ2 + sin2 χ dΩ2) interior− (1− 2mr ) dt2 + (1− 2mr )−1 dr2 + r2dΩ2 exterior, (2.28)where a(η) = a02 (1 + cos η), 0 ≤ η ≤ pi is a standard solution of Friedmann’s equationfor positive-energy density dust [23].The surface of the star is described in the inner region χ = χ0 and in the exterior isdescribed by a radial timelike geodesic R(t) passing through the point r = a0 sinχ0 ≡ R0at zero speed. Exterior region’s geodesic equations for stellar surface leads todtdR=√1− 2m/R02m/R− 2m/R0(1− 2mR)−1. (2.29)In order to eq. (2.28) properly represent the physics we are modelling, we need toimpose that the metric is continuous across the junction interface and we have to imposeEinstein’s equations over it; more specifically, that the energy-momentum tensor is not7The form of FRW solution below, referred as k = 1 is necessary so Einstein’s equations are com-patible with initial condition of the dust being at rest in the comoving frame.54singular on the junction8. These are the Israel junction conditions [53, 54], which Ioutline below.The exterior metric restricted to the surface described by eq. (2.29) isds2ext =r/2mr/R0 − 1dr2 + r2dΩ2,and with the change of coordinates r = R02 (1 + cosλ),ds2ext = −R308m(1 + cosλ)2dλ2 +R204(1 + cosλ)2dΩ2, (2.30)which can be compared to the restriction of the interior metric to the surface χ = χ0ds2int =a204(1 + cos η)2(−dη2 + sin2 χ0dΩ2)to give λ = η andm =R02sin2 χ0 and R0 = a0 sinχ0. (2.31)if the induced metrics are imposed to coincide over the interface. As stated above, westill have to impose that the metric across the interface is continuous. Since eqs. (2.31)imply that the metric is continuous along the interface, this missing requirement meansthe whole 4-metric is continuous. Continuity of the metric guarantees that its firstderivatives are regular (even though the first derivative of the metric in the directionacross the interface is discontinuous). We can then proceed to impose that the energy-momentum tensor does not include delta-like singularities, which means that the Einsteintensor does not contain any delta functions.Let us write an expression for the metric derivatives and curvature tensor in terms ofa chart straddling the interface with respect to a coordinate ` defined so that n = ∂/∂`is the normal to the interface and ` = 0 on it, and tangent coordinates xα. Denotingthe difference in a quantity A when one jumps across the surface by ∆A ≡ A+ − A− –A+ and A− using the charts valid for ` > 0 and ` < 0 respectively –, we have [53]8Here I stress that this is the classical, background energy-momentum tensor, nothing is imposed onthe one coming from quantum mechanical origin at this point.55Γαγδ = θ(`)∂δΓ+αβγ + θ(−`)Γ−αβγ + δ(`)∆Γαβγnδ,Rαβγδ = θ(`)R+αβγδ + θ(−`)R−αβγδ + 2δ(`)∆Γαβ[δnγ].(2.32)Because, as I argued above, the discontinuity of the first derivative of the metric isorthogonal to surface, we write ∆∂cgab = Kabnc for some symmetric tensor Kab. Hence,∆Γαγδ =12(2Kα(βnγ) −Kβγnα),Rαβγδ = θ(`)R+αβγδ + θ(−`)R−αβγδ + δ(`)(Kβ[γnδ]nα +Kα[δnγ]nβ)︸ ︷︷ ︸≡Aαβγδ,Gβδ = G+βδθ(`) +G−βδθ(−`) + δ(`)(Aαβαδ −12gβδAαγαgγ)== G+βδθ(`) +G−βδθ(−`) + δ(`)12(2nαKα(βnδ) −Kβδ +Kαα(gβδ − nβnδ)− gβδKαγnαnγ)︸ ︷︷ ︸≡Sβδ.(2.33)The condition we seek is then Sab = 0. But one can go further to simplify it by identifyingthis object geometrically, first by explicitly verifying from the definition in eq. (2.33)that Sabnb = 0, then writing its components in the basis of tangents to xα, denoted by{eα)a}:Sab = −2Kβδeβaeδb + habhcdKαβeαc eβd , (2.34)where hab = gab − nanb is the induced metric.We can also make use of the fact that our field na is geodesic to write the extrinsiccurvature of the surface to which it is orthogonal as Kab = ∇bna, and using the first ofeqs. (2.33),∆Kab =12Kabncnc −Kc(anb)nc. (2.35)The extrinsic curvature also obeys Kabnb = 0 and can be expanded in the basis eα)agiving, after comparing to eq. (2.34),∆Kab =12Kαβeαaeβb ,meaning that requiring Sab = 0 implies requiring continuity of the extrinsic curvature.56We can now compute the extrinsic curvature for each “side” of eq. (2.28). In the innercoordinates, na = a(η) (∂/∂χ)a and the basis {eα)a} can be taken to be the coordinatebasis with ∂/∂χ subtracted. Direct evaluation of ∇anb leads toK−θθ =a2sin 2χ0, K−φφ =a2sin 2χ0 sin2 θ. (2.36)as the non-vanishing components of the tensor in this basis.For the outer region, it is convenient to start with the chart in eq. (2.30) and to usecoordinates in which again we can use a subset of the coordinate vectors to form {eα)a}.This is the proper time τ of the surface of the collapsing star; and the coordinate `which is orthogonal to the surface (τ, `, θ, φ) so that {eα} = {(∂/∂τ, ∂/∂θ, ∂/∂φ)}. Therelationship between τ and the other coordinates is obtained by solving the geodesicequation leading toτ(λ) = −R3/20√8m(λ+ sinλ), r(τ) =R02(1 + cosλ).From the orthogonality properties of the coordinate basis, we can find the extrinsiccurvature by K+ab = 1/2 £n(gab + nanb) = 1/2 ∂/∂` gab, where n = ∂/∂` is normalised.Writing this vector in the {∂/∂τ, ∂/∂r} basis we getK+θθ =√1− 2mR0R02(1 + cosλ) K+φφ = K+θθ sin2 θ. (2.37)Comparing eqs. (2.36) and (2.37) one sees that no supplementary condition is necessaryto impose continuity of the extrinsic curvature after eqs. (2.31) are accounted for.The surface of the star is described by χ = χ0 in the inner region and byη = arccos(2R(t)a0 sinχ0− 1)(2.38)in terms of the coordinates common to both charts.The event horizon is located by finding the outgoing null geodesic that coincides withr = 2m in the exterior region to giveη = χ− χ0 + arccos(4mR0− 1). (2.39)572.2.2 Expectation Value of the Regularised Stress-Energy TensorNow we consider the section dΩ = 0 of spacetime manifold so we can cast the innermetric in a conformally flat formds2 = −a2(U, V )dUdV, a(U, V ) = a02(1 + cosU + V2).For later convenience we choose the origin of our advanced and retarded coordinates sothat U = η − χ + χ0 and V = η + χ − χ0 so that the surface of the collapsing body issimply U = V . The curvature scalar can be readily computed from the two dimensionalresulting metric to giveR =2a4(aa′′ − a′2) = 8a20(1+cos η)3 interior4mr3= 2a0 sin3 χ0r3, exterior(2.40)where prime denotes differentiation with respect to η.One can employ the prescription of ref. [6] to compute the expectation value ofthe energy momentum tensor after covariant point splitting regularisation with respectto the vacuum state defined by coordinates u¯ and v¯, using the same notation of thatreference.Our modes u¯ and v¯ are defined with respect to I − to mimic the so-called Unruhvacuum so v¯ = t+r+2m log(r−2m) is the usual Schwarzschild advanced coordinate. Aningoing null geodesic coming from I − passes through the surface of the collapsing objectand a mode proportional to eiωv¯ at I − will be reflected at χ = 0⇐⇒V = U − 2χ0 =U − 2 arcsin√2mR0(using eqs. (2.31) and (2.38) in the last step) and come out to I +like eiωu¯ withu¯ = v(U(u)− 2 arcsin√2mR0)(2.41)at the interface, where the composition U(V ) is evaluated at the surface of the collapsingobject using eq. (2.38) to relate these coordinates on the interface:U(u) = arccos(2RR0− 1)⇒ U ′(u) = R− 2m2R√R(R0 −R)(2.42)58andv(V ) = t(R02(1 + cosV ))+R∗(R02(1 + cosV ))−R∗0 ⇒v′(V ) =R20 sinV cos2 V22[R0(1 + cosV )− 4m]√2(R0 − 2m) cot2 V2m− 2 , (2.43)where eq. (2.29) was used for taking derivative of the first term in eq. (2.43) andR∗(r) ≡ r + 2m log(r − 2m) and R∗0 is an abbreviation for R∗(R0).There are no contributions to the expectation value of the energy-momentum tensorcoming from the matching conditions of the field modes across the collapsing surface.Because the surface where the matching conditions are applied is a geodesic and satisfiesthe Israel junction conditions with no δ terms in its energy-momentum tensor, it isrequired that the field modes and its derivatives across the surface are continuous [55].These conditions are satisfied, since the solutions to the field equations are merely thecomposition between the solution in one chart and the matching conditions (2.42) and(2.43), both C 1 functions in their variables.As before, after covariant point-splitting regularisation, the components of the ex-pectation value of the stress-energy tensor for the 2-dimensional space-time is given byeq. (2.10),〈Tµν〉 = R48pigµν + θµν . (2.44)Introducing the abbreviationF(f(x);x) ≡√f12pid2dx21√f=124pif(3(df/dx)22f− d2fdx2),F(fg;x) = gF(f ;x) + fF(g;x)− f′g′24pifg,the components of expectation value of the renormalised stress-energy tensor are givenby [6]〈0|Tvv|0〉 = m(3m− 2r)48pir4〈0|Tuv|0〉 = m(2m− r)24pir4〈0|Tuu|0〉 =(du¯du)2F((dudu¯)(1− 2mr); u¯).(2.45)59outside the collapsing body and [45, 39]〈0|TUU |0〉 = F(v′(U − 2χ0);U)−F(a(U);U)〈0|TV V |0〉 = F(v′(V );V )−F(a(V );V )(2.46)inside it.Inside the matter, explicit evaluation givesF(a(U);U) = csc2 U2 (3 + cosU)96pi,F(v′(V );V ) = 1192pim(R0 +R0 cosV − 4m)2(ξ − 2)2×{m3(256(1− 4 cos 2V )− ξ(1792− 512 cos 2V ) + 256 cosV csc2 V (9ξ − 10)) +m2R0(1088 + ξ(224 + 96 cos 3V )− 4 cos 2V (48 + 56ξ)− 32 cosV csc2 V (3ξ − 34)) +mR20(56 cos 3V − 412 + 2 cos 4V (2− ξ)− 6ξ − 16 cos 2V (1− ξ)− 472 cosV csc2 V ) +R30(45− cos 4V − 20 cos 2V + 64 cosV csc2 V )}, (2.47)where ξ ≡√2(R0−2m) cot2 V2m . The missing term F(v′(U − 2χ0);U) is obtained fromabove by replacing V by U − 2χ0 in F(v′(V );V ) and F(a(V );V ) by replacing U by Vin F(a(U);U).The outgoing energy flux as measured by a stationary observer within the star〈0|TUU |0〉 on the event horizon U = arccos(4m/R0 − 1) is finite for all χ0 6= 0, in-cluding in its formation, and all the way until the horizon intersects the surface of thestar, in explicit disagreement with Mersini-Houghton in ref. [7].Outside the matter, the formulas for 〈0|Tvv|0〉 and 〈0|Tuv|0〉 are the same as inref. [6], whilst the explicit formula for 〈0|Tuu|0〉 is several pages long and it is moreelucidative to show a plot (figure 2.2) than writing it. There are some properties thatare worthwhile elaborating. First, we note that at the onset of the collapse from eq.(2.41) u¯ = u, so that 〈0|Tuu|0〉 = 〈0|Tvv|0〉, while at very late times (R → 2m + 0) itrestores Hawking’s result κ2/48pi = 1/(768pim2) as r →∞ for any initial radius, as onewould expect. More generally, for very late times, 〈0|Tuu|0〉 as a function of r turns out60to be the same as in ref. [6], suggesting that the nature of the collapse does not affectthe final energy-momentum tensor of the radiation emitted by the black hole. The onlydivergence present in this expression is for the true singularity at the end of the stellarcollapse at r = 0.Figure 2.2: m2〈0|Tuu|0〉 as a function of r coincides with 〈0|Tvv|0〉 on the onset of thecollapse, here represented by large values of R and with ref. [6] for very late times, forR→ 2m. The horizontal plane on top of the graph represents the constant κ2/48pi.To study possible effects of back-reaction, it is interesting to analyze observablesmeasured by an observer following the surface of the collapsing star extracted fromthe results above, namely 〈Tab〉`a`b, 〈Tab〉`amb and 〈Tab〉mamb where (`a,ma) is a diadadapted to this observer, so `a is the tangent vector to the ingoing timelike geodesicdescribing the stellar surface, and ma is a spacelike unit vector orthogonal to `a. Usingthe Eddington-Finkelstein coordinate system9 to write these vectors,9These coordinates are more adapted to study the neighbourhood of the future horizon.61Figure 2.3: All appropriate observables as measured by someone following the collapseobtained from the regularised energy-momentum tensor are finite everywhere except atthe singularity. The graph shows exclusively the region around r = 2m.`a =(1− 2mr)−1(√1− 2mR0−√2mr− 2mR0)(∂∂v)a−√2mr− 2mR0(∂∂r)a, (2.48)and−ma =√√√√r (R0r + 2mR0 − 4mr − 2rR0√(2m/r − 2m/R0)(1− 2m/R0))R0(r − 2m)2(∂∂v)a+R0√(2m/r − 2m/R0)(1− 2m/R0) +R0 − 2mR0mv(∂∂r)a. (2.49)All three contractions above are regular everywhere except at the singularity, asshown in figure 2.2.For an observer in the interior of the matter, all these observables are also finite, asone can see from the results from sec. (2.47) and by realizing that the vector componentsof the vector tangent to timelike geodesics in coordinates (2.28) are regular, and so arethe components of the spacelike vector orthogonal to it, since the metric components inthese coordinates are regular and invertible.62For observers outside the the star, e.g. for a geodesic observer starting from infinity,observables can be found by the same method. If the observer starts at rest at infinity,the tangent vector and the spacelike vector orthogonal to it can be found from eq. (2.48)and eq. (2.49) respectively by making R0 →∞.From eqs. (2.44), (2.40) and (2.45-2.46) we see that, in contrast to ref. [7], theregularised energy-momentum tensor is perfectly regular at the classical event horizonand small for astrophysical black holes, (∼ 1/768pim2 for r = 2m). The only divergenceoccurs at the singularity when the star collapse entirely at η = pi, where the classicalstress-energy tensor is also divergent as the scale parameter goes to zero. The same istrue for observables like the expectation values of energy density 〈Tab〉`a`b or energy flux−〈Tab〉`amb which are only divergent near the true singularity.Also, from the coincidence of the behaviour from 〈0|Tuu|0〉 for very late times for ourstar and for a collapsing null shell of ref. [6], our results strengthen the notion that thefinal regularised expectation value of the energy-momentum tensor of Hawking radiationignores all the details of the collapse.One could argue that, despite the dynamical similarities between 2D and 4D modelswith respect to the piling of outgoing rays on the horizon, the situation could be verydifferent in four dimensions, but it must be noted that near I +, the usual conservationlaw ∂νTµν = 0 implies [48] that the expectation value for the 4D stress-energy tensoris proportional to r−2 times the one in 2D for spherical waves. Assuming this to betrue, conservation of energy arguments imply that the 4D values cannot explode in anynon-zero measure set contained in a domain of integration, such as the event horizon.2.3 Difficulties with the Semiclassical TheoryPerhaps the most immediate question one may have about the semiclassical Einstein’sequationsGab = 8pi〈Tab〉reg63is on which grounds we expect it to be true. First, as discussed in section 2.1, there areambiguities in the definition of this regularised object that appears in the right-handside. Second, as we also discussed in the beginning of this chapter, the non-linearityof Einstein’s equations poses a serious problem in any attempt to describe quantumgravity as a linearised theory. Any terms containing gravitons are not included on theright-hand side, but they should be, since a pseudo-tensor of the gravitational fieldover a background is of the same order of magnitude as the contribution of a speciesof matter. Consequently, we may also expect that if there are many different speciesof matter, then the contribution of gravitons would be diluted and the semiclassicalequation would become a better approximation, and this view is endorsed by someapproaches to quantum gravity [25].The general task of evaluating the regularised stress-energy tensor in general four-dimensional spacetimes is extremely difficult10. But progress has been made in partic-ular cases. For example in ref. [56], Candelas and Howard managed to numericallycompute the regularised expectation value 〈φ2〉reg of a scalar field using the covariantpoint splitting method in Schwarzschild spacetime using a WKB approximation for thesolutions of the field equations. From this, Anderson [57] managed to find the regularisedstress-energy tensor under the same hypotheses. In spacetimes nearly conformally flat,Horowitz and Wald [58] managed to analytically obtain formulas directly from the ax-ioms that define the tensor. And much more recently, Levi and Ori [59] numericallyimplemented the point-splitting regularisation for spacetimes in two different circum-stances: for spacetimes possessing azimuthal symmetry or for stationary spacetimes.Another difficulty in the semiclassical Einstein’s equations is that, unlike their clas-sical counterparts, contain fourth derivatives of the metric in its right-hand side (seeeq. (2.10) for example). It has been argued [60] that even Minkowski spacetime in itsusual vacuum is unstable under small perturbations when these semiclassical corrections10This is the reason why we based our previous discussions in models in two dimensions. If one isinterested in general properties and qualitative results, this difficulties can be bypassed using thosetechniques. Generally, four-dimensional results are nevertheless desirable.64are accounted for. This problem is analogous to the existence of runaway solutions inclassical electrodynamics when the radiation field is included, leading to a third orderequation of motion for a point charge (this problem is discussed by Abraham & Becker[61] and a relativistic generalisation was accomplised by Dirac [62]. See also Unruh’snotes [63] for a review and excellent discussion). Since not even in the present day thereis consensus about a proper elimination of runaway solutions in electrodynamics, it isnot surprising that this issue is still open in our current understanding.653. Role of Entropy3.1 Nature of EntropyEntropy and energy are conceptually remote notions. On the one hand, the energy-momentum tensor has direct physical importance as a source of gravity; on the otherhand, as discussed in chapter one, entropy is not as easily localised: it may measurethe extent to which a subsystem of interest is entangled with a larger system. A set ofmacroscopic constraints is necessary to define the total entropy of a system of interest,but not for its total energy. Finally, entropy does not play a role in as many physicaltheories as energy does.Nevertheless, entropy is arguably a very useful notion when it comes to determiningthe macroscopic behaviour of matter, as it encodes all the degrees of freedom whosedetails are immaterial for the specification of the chosen set of macroscopic variables.And in many applications, there are natural choices of this set of variables, for examplein case of black hole physics — where all observers that do not cross the horizon agreethat no information can be retrieved from within the black hole — a natural set ofmacroscopic variables is restricted to observables in its exterior region.Particularly in the case of black hole physics, it is clear from our discussion in chapterone that the relation between energy and entropy needs to be explored globally, bycomparing total energies and total entropies. There are two types of these relations onenormally finds in the literature, the first type is in the form of entropy bounds, that Idiscuss in this section, and the other in the form of the laws of thermodynamics, which66I discuss in the next section.3.1.1 Geroch’s BoxAt first glance, it seems to be possible to challenge the Generalised Second Law (GSL),which states that no process can decrease the entropy outside the black hole summedwith the Bekenstein-Hawking entropy of a black hole. The idea, proposed in 1971 by R.Geroch in a seminar at Princeton University, is to adiabatically lower a box with energyE∞ as measured at infinity and entropy S then dropping its content into a black holeof a stationary spacetime with Killing field ξa. For the box to be lowered adiabatically,some device, say a rope, must be attached to it. The energy of the box seen by theend of the rope attached to it as it is lowered can be approximated by the energy asmeasured by a family of stationary observers situated at each point of the trajectory,that is, at each point it is given by E = E∞/√−g(ξ, ξ). When the content of this boxis released, the energy seen from infinity transfered to the black hole is the differencebetween E∞ and the work performed by the force at infinityW =∫ horizon∞dd`(E∞√−g(ξ, ξ))d` = E∞(1−√−g(ξ, ξ)|horizon).But, at the horizon, ξa is null; consequently the energy transferred to the black holeapproaches zero, the area of the black hole is unchanged, and the entropy S originallyin the box disappeared as viewed from the outside the black hole.The generalised second law is safe, though. The reconciliation was found by Unruhand Wald in [64]. The argument above is erroneous because it fails to account for thequantum nature of the generalised second law, more specifically, for it to be appliedconsistently, one has to take Hawking, and most importantly Unruh radiation1 intoconsideration. As discussed in section 1.2, the temperature measured by an observerfollowing the orbits of the Killing field χa depends on its position p asTp =κ2pi√−g(χ, χ)|p .1More precisely the analog of Unruh effect in curved spacetime.67A physical interpretation for the formula above is possible in terms of Tolman’s relationwhich describes how the temperature observed of a system in thermal equilibrium varieswith the position.Tolman’s original derivation of the formula carrying his name [65] made use of equa-tions of state of a particular material and consequently hides the truly relevant hypothe-ses for the result to hold. Thus, I here reproduce the derivation presented by Landauand Lifshitz [29, 8], but I rewrite it in modern, geometrical terminology. Imagine amaterial in thermal equilibrium in a stationary spacetime with Killing field ξa. If twoparts of the system trade 4-momentum pa geodesically from point 1 to point 2, theng(p, ξ) is a constant along its worldline denoted by −E0, so that the energy as measuredby a stationary observer in 1 or 2 is given in terms of E0 asE1,2 = −g(p, u1,2) = −gp, ξ√−g(ξ, ξ)∣∣∣∣∣1,2 = E0√−g(ξ, ξ)∣∣∣1,2.Supposing the system is extensive, the entropy of these parts are additive and, be-cause the system is in equilibrium, dS1 + dS2 = 0. UsingddE1= −√−g(ξ, ξ)|1√−g(ξ, ξ)|2 ddE2 ,where the minus sign comes from the fact that the energy that leaves 1 arrives at 2, wehaveT1√− g(ξ, ξ)|1 = T2√− g(ξ, ξ)|2, (3.1)where T1,2 ≡ dS1,2/dE1,2. Equation (3.1) is called Tolman relation, and means thatthe temperature is not uniform within a material in equilibrium in the presence of astationary gravitational field.The gradient of temperature generates, via Gibbs-Duhem relation, a gradient ofpressure that has to be taken into account when computing the work performed bythe tension of the rope at infinity. If s is the entropy density of the thermal bath,68Gibbs-Duhem relation gives dP = sdT , so the work becomesW =∫ final∞dd`√−g(ξ, ξ) (E∞ + V P ) d` == E∞(1−√−g(ξ, ξ)∣∣∣final)−√−g(ξ, ξ)∣∣∣finalV P,where “final” is taken as the limit of integration because the most threatening menacethis box can pose to the second law is no longer when the final point is the horizon.Rather, if the box’s content is released where the work above is a maximum, the valueof the energy transferred to the black hole will be the minimum, therefore leading to theminimum change in area. The condition for the maximum work dW/d` = 0 is simplythe condition that the force at infinity is zero, so the optimal point for releasing thecontent of the box is at the “floating point”. More explicitly, also using Gibbs-Duhemand Tolman relations we have for the floating point0 = (E∞ + PV )d√−g(ξ, ξ)d`+√−g(ξ, ξ)V dPd`== (E∞ + PV )d√−g(ξ, ξ)d`+√−g(ξ, ξ)V sdTd`= (E∞ + PV − V sT )d√−g(ξ, ξ)d`,leading toE∞ + PV − TsV = 0.The Euler relation for homogeneous functions applied to the entropy of the thermal bathas a function of energy and volume reads uV + PV − V sT = 0, where u is the energydensity of the thermal bath; a comparison with the above shows E∞ = uV , which is theregular condition for floating in hydrodynamics, justifying the choice of nomenclature.With this result the energy delivered to the black hole is at most TsV , leading to achange in area of 4sV , exactly the amount needed to preserve the (saturated) generalisedsecond law.3.1.2 Criticism to Proposed Entropy BoundsA number of entropy bounds ordinary matter is supposed to obey has been proposedin the literature. The first was due to Bekenstein in ref. [66]. To avoid violation of69the second law in the Gedankenexperiment above, he proposed that the entropy S in aspherical box of radius R had to obey S ≤ 2piRE, where E is the total energy in the box.Later, in 1982, Unruh and Wald [64] (see also ref. [67] for a correction that does not alterthe original conclusion) found the resolution above, showing that the GSL is preservedeven for matter violating Bekenstein’s bound. In 1994, motivated by unrelated ideas,Susskind [68] proposed a bound limiting the amount of entropy matter could carry bythe Bekenstein-Hawking entropy of a black hole occupying the same space.Although not central to that reference, Susskind suggested that the bound wouldcome as a consequence of the GSL, since if the matter contained in that region were tocollapse to form a black hole, the final total entropy would be smaller than the initial.This reasoning is, however, a misapplication of the second law of thermodynamics, thecorrect conclusion would be that the collapse of the highly entropic matter to a blackhole is, in this case, not thermodynamically favourable. Indeed, in ref. [9], Wald presentsan elegant and compelling argument to show that such collapse is indeed expected tobe disfavourable. He considers a gas of n different species of ultrarelativistic particlesconfined in a spherical box of radius R. These particles have their Helmholtz free energyF proportional to nV T 4, where V = 4piR3/3 and T is the temperature. The entropy isS = −∂F∂T∝ nV T 3 ∝ n1/4(ER)3/4, (3.2)where in the last step we eliminated the temperature in favour of the total energy E,which is proportional to the free energy [8]. The initial configuration is such that R > 2Eand S > piR2, since the configuration is not a black hole, and it has more entropy thana black hole occupying the same volume. This means n > R2. The problem comes whenone considers an estimate of the evaporation time for a black hole in this model withthat many different species. We expect that for a Schwarzschild hole with mass M andradius R, the rate of energy loss to obeydM/dt = −nσ(4piR2)(1/8piM)4 ∝ −n/M2,where σ is the Stefan-Boltzmann constant. Integrating this relation in our case, theestimate of the evaporation time is less than R, which means that it is shorter than70any causal collapse time, indicating that the matter indeed simply will not undergo toa collapse.In 1999, Bousso [69] conjectured a modified version of Susskind’s bound, stating thatthe entropy flux through a null 3-surface is bounded by the change in area of a spacelike2-surface from which the null surface departs. This bound has been proved by Flana-gan, Marolf & Wald [70] in classical physics and an adaptation to the quantum versionformulated by Bousso et al [71]. Although this conjecture has had great repercussions(see, e.g. [72]), it has the serious limitation of requiring “localising” the entropy so onecan define an entropy flux. And, as discussed in chapter one, this is not the case inseveral physically relevant problems.To illustrate how energy and entropy are, in general, independent quantities, considera quantum system composed by N independent spins for which the Hamiltonian H isgiven by the number of excited spins in a given state. Now we consider the mixedstate obtained by non-coherently superposing all eigenstates of the Hamiltonian withprobabilities given by pn, where n is the number of excited spins of that particulareigenstate. The expectation value of the Hamiltonian is then given by〈H〉 =N∑i=1(Ni)pi =N∑i=1(Ni)pi1N−i = (1 + p)N − 1. (3.3)The von Neumann entropy of this state is given byS = −N∑i=1(Ni)pi log(pi) = − log pN∑i=1(Ni)ipi = −p log pN∑i=1(Ni)ipi−1 == −p log p ∂∂pN∑i=1(Ni)pi = −Np log p (1 + p)N−1. (3.4)From eqs. (3.3) and (3.4) we see that the ratio S : 〈H〉 can be made arbitrarily large bychoosing p small enough and N large.A different proposal by Marolf, Minic, and Ross [73] acknowledges that the entropyat least for inertial frames does not need to obey any bounds, but claims it does foraccelerated frames. They defined the entropy of a system as the difference in von Neu-mann entropies associated with the Minkowski vacuum restricted to the right Rindler71wedge and a slightly perturbed state. This difference is bounded by ∆E/T , where ∆Eis the difference between the expectation values of the energy corresponding to these twostates, and T is the Unruh temperature. I give a dramatically different counterexampleto this proposed bound for quantum fields in a state that cannot be described as a smallperturbation around Minkowski’s vacuum. Although this example may seem artificialand complicated, it might be useful for different purposes.The physical system, suggested by Unruh, is chosen to be an accelerated box in a2-dimensional Minkowski space-time. The walls of the box are located at ξ = a andξ = b, b > a in coordinates such ds2 = −g2ξ2dτ2 + dξ2. The walls are such thatthey impose Neumann boundary conditions on a free massless scalar φ field, that is,∂φ∂ξ∣∣∣ξ=a= ∂φ∂ξ∣∣∣ξ=b= 0 up to a frequency cut-off, which is needed to ensure modes arenormalisable and that certain quadratic functions of field operators converge.The equations of motion∂µ(√−det(gµν)gµν∂νφ)= − 1gξ∂2τφ+ ∂ξ (gξ ∂ξφ) = 0inside the box admit oscillating solutions,φn = Aeiωnτ cosωn logξag, ωn =npiglog ba,where n ∈ Z goes from 1 up to a maximum representing the frequency cut-off. Besidesthose, they also admit solutions like φ0α = α (1 + iα˜τ), where α is a constant and α˜ isa constant dependent on α, which we refer as “zeroth modes”. These can be normalisedsuch that (φ0α, φ0α) = 1 by an appropriate choice of α˜ satisfying Re(α˜) = g2|α|2 log(b/a) .Furthermore, they are orthogonal to both φn and φ¯n and if α˜ ∈ R, then we have(φ0α, φ0α′) = 0 for α′ 6= α.The oscillating modes are not orthogonal, but we can always choose A so that(φn, φn) = 1. Because of the cut-off, we always have A <∞.The following identities for inner product and creation/annihilation operators are72easily verified for all α, α′, β, β′, n,m:(φ0α, φn) = (φ0α, φn) = 0,(φn, φm) ∈ iR,aαala†ja†α′∼= (φ0α, φ0α′)(fl, fj),aαa†l aja†α′∼= 0,aα′aβ′a†βa†α∼= (φ0α′ , φ0β′)(φ0β, φ0α),aα′aβ′a†βa†α∼= (φ0α′ , φ0α)(φ0β′ , φ0β)− (φ0α′ , φ0β)(φ0β′ , φ0α),(3.5)where the symbol ∼= denotes equality after the vacuum (i.e., the state annihilated by allthe a0α and ai) expectation value.We denote by Φ the quantised field. We consider the system in the mixed statedescribed by the density matrixρ =∑c,wpc,w|gc,w〉|O〉〈O|〈gc,w|, (3.6)where |O〉 lives in the subspace spanned by oscillating modes only and is annihilatedby annihilation operators associated with all the non-zero modes,∑c,w pc,w = 1 for0 ≤ pc,w < 1, and|gc,w〉 =∫dα gc,w(α)(Φ, φ0α)|0〉is a state representing a wave packet of zeroth modes in the formatgc,w(α) =N exp[− w2w2−(α−c)2]−w + c < α < w + c0 otherwise,where N is a constant chosen such that ∫ dα|gc,w(α)|2 = 1 and our state is normalised.Note that these packets are infinitely differentiable functions whose support’s centresand width are controlled respectively by c and w so we can choose a density matrix sothat no two packets’ support intersect, i.e., by non-coherently superimposing orthogonalpackets. Henceforth I will be assuming that the density matrix (3.6) is constructed insuch a manner.73The von Neumann entropy associated with this mixed state, by virtue of the orthog-onality, is obviously S = −∑c,w pc,w log pc,w, which can be made arbitrarily large bysuperposing as many different packets as needed.The energy-momentum tensor is given by Tab = ∇(aΦ∇b)Φ∗ − 12gab∇cΦ∇cΦ∗, andits expectation values are readily found by Tr(ρTab). To evaluate this, we first considersome general results.To find the expectation value of an operator constructed from a bilinear function ofthe classical fields FII(Φ,Φ) (such as the energy-momentum tensor) in a vacuum state|0〉, we can expand the field operator as in eq. (1.20) in terms of the creation and anni-hilation operators for that vacuum. From the bilinearity of FII , and the commutationrelations (1.19), which led to eq. (3.5), it follows that for this state the expectation valueis〈0|FII(Φ,Φ)|0〉 =∑i,kFII(φi, φ¯k)(φi, φk).Or for the zeroth-modes excited state packet (3.6) we haveTr(ρTab) =∑cwpcw∫dα′∫dβ′∫dβ∫dα |gcw(α)|2{(φl, φj)(φ0α′ , φ0α)FII(φl, φj) ++ FII(φ0β′ , φ0β)[(φ0α′ , φ0α)(φ0β′ , φ0β)− (φ0α′ , φ0β)(φ0β′ , φ0α)]++ FII(φ0β′ , φ0β)(φ0α′ , φ0β′)(φ0β, φ0α)}, (3.7)where φi represent, as before, the ith oscillating mode.The need of introducing a ultraviolet cut-off is evident here, when FII is replaced bysecond derivatives coming from the classical energy-momentum tensor. Without it, wecannot guarantee the convergence of the energy density near the walls of the box or theits total energy. This is not particularly surprising, since we know that the same wouldhappen in Casimir effect, replacing Neumann by Dirichlet boundary conditions, wheneven the regularised energy density of near the plates diverges. For Casimir effect, thisdifficulty can be bypassed because one is essentially interested in the difference between74the energies for different separation distance between the plates, which give rise to theCasimir force [40].It turns out that FII(φ0α, φ0β) ∝ 1/αβ, so one needs to restrict the packets to makesure that the support of gcw(α) is always to the right of a positive constant so that theintegrals over α, β, α′ and β′ converge on eq. (3.7).The right-hand-side of eq. (3.7) can be written concisely in the form∑cw pcwqcw, forsome finite coefficients qcw. Hence, to violate the proposed entropy bound, it is sufficientto superpose a large enough number of packets so that for each (c, w),pcw < exp[−b− aTqcw]≤ e(a−b)/T esupcw{qcw}.The Marolf-Minic-Ross bound is formally identical to an inequality derived from thenon-negativity property of the relative entropy [74], which can be derived from (1.9) asfollows. When σ = exp(−βH)/Z is a Gibbs state, the relative entropy S(ρ|σ) can bewritten as0 ≤ S(ρ|σ) = βTr(ρH) + logZ − S[ρ] = 1T(〈H〉ρ − 〈H〉σ)− S[ρ] + S[σ], (3.8)where the subscript of expectation value 〈H〉 specifies the state the expectation valueis calculated at. But this inequality is to be interpreted differently from the one Idiscussed above. The relation (3.8) imposes constraints only on the differences in energyand entropy when comparing two states, whereas Marolf, Minic, and Ross explicitly state[73] that one should not consider contributions from the thermal acceleration radiation,and the differences 〈H〉ρ − 〈H〉σ and S[ρ] − S[σ] necessarily include contributions thatare not localised in the box in any manner. The counterexample above cannot be usedto disprove eq. (3.8) since the presence or absence of the box is not described as a changein the state of a single system.3.2 Laws of Black Hole Thermodynamics in the Semiclas-sical ContextWe turn now our attention to the second type of relation between energy and entropy.75All the discussion above, quite noticeably in the Geroch’s box, clearly shows thatif one aims to obtain a general description of the laws of black hole thermodynamics,one has to consider both classical gravitational theory and quantum effects on matter.Sadly, as seen in chapter two, it is extremely challenging to incorporate back reactioneffects of these phenomena of quantum mechanical origin. To tackle these difficulties wenow explore general properties of what we expect from semiclassical theories instead ofworking on approximate examples. This approach may not be the most useful to applyin toy models since the semiclassical Einstein’s equations are just too complicated tobe solved even approximately; but this approach teaches us about the fundamentals ofblack hole thermodynamics and role of each hypothesis behind the conclusions.3.2.1 Zeroth LawIn general relativity, the dominant energy condition guarantees the validity of the zerothlaw of black hole mechanics at any Killing horizon [3]; but the dominant energy conditionis not expected to hold when semiclassical effects are taken into account. Even the nullenergy condition is known to be violated as we saw in chapter two. Another example ofnull energy condition violation is for fermion fields around a Kerr black hole [30]. Theseenergy conditions are only expected to hold at far distances from the black hole, butcertainly not on the horizon.This version of the zeroth law is then lost in a semiclassical context, but luckily theseare only sufficient conditions, not necessary conditions for the zeroth law. As we saw inchapter one, a kinematic (hence independently of any dynamics of the theory) propertyof a bifurcated horizon ensures the validity of the zeroth law in our context as long asκ 6= 0. This means that as long as the metric is treated classically, there is no change inthis result.763.2.2 First LawThe set-up is a semiclassical theory (M, g, ψ, ρ) for (bosonic) matter fields ψ that arequantised in a state ρ, in the stationary spacetime (M, g), which is treated classically.Varying the classical action I0 =∫L(g, ψ,∇g,∇ψ, . . .) of the theory, we constructthe symplectic current j and potential Θ associated to a vector field χa in the standardway [31, 33] as described in chapter one and write them as a sum j = jg + jψ, wherethe first term represents the vacuum contribution only, independent of the matter fields.And a similar sum for the symplectic potential.Remarkably, both contributions jψχ and jgχ to the the symplectic current remain closedif we require the field equations to hold and £χg = 0. I now prove this.Writing the Lagrangian L as the vacuum part Lg and the matter part Lψ, a variationof the Lagrangian can be written as in (1.35)δL = Eψ · δψ + Eg · δg + R · δg + dΘg + dΘψ, (3.9)for some form R. Here Eg = 0 would represent the equations of gravity in absence ofmatter and Eg + R = 0 the full equations of gravity. Requiring the first variation ofthe Lagragian is zero for any δψ means Eψ = 0, the equations of motion for the matter.Repeating the calculations that led to eqs. (1.41) and (1.42) for the variation (3.9), wenow havedjψχ = −Eψ£χψ −R£χg and djgχ = −Eg£χg, (3.10)which vanish after imposing Eψ = 0 and £χg = 0.We can manipulate the expression for djgχ similarly to Iyer and Wald’s constructionfor the symplectic current in a fixed background in [33]:djgχ = −(Eg)ab(£χg)ab = −2(Eg)ab∇(aχb) = −2{∇a[(Eg)abχb] + χb∇a(Eg)ab} == −2d[(χb(Eg)ab) · ] + 2χb∇a(Eg)ab,where we have written Eg£χg in terms of the volume form as (Eg)ab(£χg)ab and madeuse of the fact that owing to the symmetry of gab, (Eg)ab can be taken to be symmetric77without loss of generality. The left-hand side of the first line is an exact form, and so isthe first term of the last line. This means that the second term of the last line has alsoto be exact. Moreover, this identity has to hold for any field χa at this stage, which isonly possible when ∇b(Eg)ab = 0.This means that, imposing the equations of motion only for the fields, one canintegrate the g-part of the symplectic current asjgχ = −2 · (χ · Eg) + dQgχ, (3.11)where  · (χ ·Eg) is the shorthand for (χb(Eg)ab) · , and the integration constant Qgχ isthe g-part of the Noether charge. This can be seen by noticing that if one could imposeEg = 0, eq. (3.11) would reduce to the definition of gravitational part of the Noethercharge accordingly to the above prescription.The exact same manipulation for the ψ-part of the symplectic current using the firstof eq. (3.10) revealsjψχ = −2 · (χ ·R) + dQψχ , (3.12)where R = R was defined.Instead of varying the exterior derivative of the current, we can vary the currentitself directly from its definition. Following the same steps as in eq. (1.42),ωψχ = δjψχ − d(χ ·Θψ) + χ ·Rδg and ωgχ = δjgχ − d(χ ·Θg) + χ ·Egδg, (3.13)with the forms ωψ,gχ constructed from the symplectic potentials Θψ,g as in (1.36) and(1.37). For the last equation, we suppose both ψ and ψ + δψ obey Eψ = 0.As expected when the equations of motion for gravity Eg + R = 0 are satisfied,ωχ = ωψχ + ωgχ and dωχ = 0.Can we exploit this formulation in a semiclassical context? To address this question,I start with a toy model of as a warm-up. Consider the LagrangianL(q,Q, q˙, Q˙) ={12(q˙2 + Q˙2)− 12(Q2 + 2c qQ)}dt,78of which only the first term represents the analogue of Lg, the other three terms composeLψ. In this expression c is a coupling constant. From a variation of the aboveδL = −[(Q¨+Q+ cq)︸ ︷︷ ︸EψδQ+ q¨︸︷︷︸Egδq + cQ︸︷︷︸Rδq]dt+ d( q˙δq︸︷︷︸Θg+ Q˙δQ︸︷︷︸Θψ)we obtain the symplectic current of the field ξ = ∂/∂tjψξ =12[Q˙2 + q˙2 +Q2 + c qQ].And using the first of eqs. (3.13),δHψξ = QδQ+ PδP + cqδQ,where p = q˙ and P = Q˙.We can now promote Q and P to operators. In our example (P,Q) is a symplecticcoordinate system. Hence, if the (classical) Hamiltonian function is evaluated at thecorresponding operators P and Q to obtain the quantum Hamiltonian operator, thesame Hamilton’s equations, now interpreted as operator-valued, are satisfied2.Satisfying the operator-valued Hamilton’s equations is not a peculiarity of our toy-model Lagrangian. In general, the formal relations i[f(pi), qk] = −∂f/∂pi δik andi[f(qi), pk] = ∂f/∂qi δik3 obeyed by the operators {(pi, qi)}i associated with a sym-plectic coordinate system ensure that Heisenberg’s equations of motion are equivalentto the operator-valued Hamilton’s equations at any given time. This result is a versionof Ehrenfest’s theorem, see, e.g., ref. [75].2Explicitly, the quantum equations of motion in the Heisenberg picture for the Hamiltonian operatorHψ = 12(Q2 + P 2)+ cqQ areQ˙ = −i[Q,Hψ] = P = δHψ/δP and P˙ = −i[P,Hψ] = −Q− cq1 = −δHψ/δQ,as claimed.3These relations can be verified when f is a power of its argument by induction, and subsequentlygeneralised for other functions by making use of the density property of polynomials over the set ofcontinuous functions.79It is justifiable to write a semiclassical Hamiltonian of the g-part of this interactingtheory formally asδHgξ = pδp+ c〈Q〉δq (3.14)since it generates the desired Hamilton’s equations for the classical degrees of freedomδHξ/δp = p and δHξ/δq = c〈Q〉. And overall, we can interpret ωξ = ωgξ + 〈ωψξ 〉 as thepre-symplectic form generating the full semiclassical theory. Variations δ of functions ofoperators are understood as “derivatives” as in their classical counterparts.Returning to a more general theory, I consider, for simplicity, only quadratic, freefields, to avoid dealing with counterterms in the action and with altered equations ofmotion. The Hamiltonian operator δHψχ whose flow is generated by χa for the quantumfields is obtained from the classical function by preserving the format of jψχ , Θψ, andR as functions of the fields to be evaluated on the field operators and then take theirregularised expectation values 〈jψχ〉, 〈Θψ〉, and 〈R〉 respectively and integrating the firstof eq. (3.13) over a Cauchy surface C. As in the toy-model, the classical degrees offreedom respond according toωgχ = δjgχ − d(χ ·Θg)− χ · 〈R〉δg, (3.15)which is requiring that the semiclassical equations of motion Eg + 〈R〉 = 0 are satisfiedin eq. (3.13).This construction requires that the semiclassical couples gravity to expectation valuesof the fields as if they were classical objects, and that the matter field operators satisfytheir Hamilton equations under their expectation values for a state ρ. When there existssuch (M, g, ψ, ρ) for this semiclassical theory, imposing Eψ = Eg + 〈R〉 = 0 produces,unlike usual quantum field theory in a fixed background, a self-consistent theory4. Sokinematic properties of the Killing horizon like the constancy of the surface gravitythroughout h± (see chapter one) are automatically true5.4In other words, back reaction of phenomena originated in the quantum mechanical nature of matterare automatically accounted for.5Another identity for Killing horizons is Rabχaχb = 0, which for semiclassical general relativity thisimplies 〈Tab〉χaχb = 0, which exemplifies how constrained the set of theories (M, g, ψ, ρ) are.80Before moving on, I give a concrete example of a minimally coupled masslessscalar field in General Relativity, for whichL =R16pi+12gab∇aψ∇bψ , (3.16)whereR is the scalar curvature and  is the volume element on the spacetime.For this Lagrangian we obtain〈Θψabc〉 = abcd∇d〈ψδψ〉,Θgabc =116pidabcgdegfh(∇fδgeh −∇eδgfh),(jψχ )abc = dabc〈∇eψχe∇dψ〉 −12eabc〈∇dψ∇dψχe〉,(jgχ)abc =18pidabc∇e∇[eχd].(3.17)Integrating eq. (1.40) over a Cauchy surface C and using the second of(3.17), we can write the g-part of the varied Hamiltonian in the ADM form[22, 23]− 132pi∫Cδhab£χpab −£χhabδpab,where hab is the induced metric over C, and pab =√deth(Kab − habKcc ) isits conjugate momentum with Kab the extrinsic curvature of C.The ψ-part is easily identified — using the first and third of (3.17) — withδ∫C〈Tdf 〉χf dabc, Tab = ∇aψ∇bψ −gab2∇cψ∇cψ,which is in the familiar form for the Hamiltonian derived from the fieldequations.The same Hamiltonian is generated by the dynamics of Gab = 8pi〈Tab〉, theusual semiclassical Einstein’s equations. The first law of black hole thermodynamics is obtained as follows. For asymptoticallyflat stationary spacetimes and stationary metric perturbations, ωgχ = £χΘg(g, δg) −δΘg(g,£χg) = 0. Then, from eq. (3.15),0 = dδQgχ + 2δ[ · (χ · 〈R〉)]− d(χ ·Θg)− χ · 〈R〉δg, (3.18)81where we have used eq. (3.11) and Eg = −〈R〉.We now integrate eq. (3.18) over a spacelike surface C bounded by the bifurcationsurface B and a (n − 2)-sphere at spatial infinity, denoted by ∞. We can eliminatethe terms containing 〈R〉 in favour of the regularised expectation value of the Hamilto-nian 〈δHψχ 〉 =∫C〈ωψχ〉 of the field theory defined over the domain of dependence of C.Explicitly, from eqs. (3.13) and (3.12),∫C{2δ[ · (χ · 〈R〉)]− χ · 〈R〉δg} = −〈δHψχ 〉+∫C{dδ〈Qψχ〉 − d(χ · 〈Θψ〉)}.Plugging into eq. (3.18),− 〈δHψχ 〉+∫∂C{δQχ − (χ ·Θ)} = 0, (3.19)where Qχ = Qgχ + 〈Qψχ〉, Θ = Θg + 〈Θψ〉, and Stokes’ theorem has been used.The right-hand side of eq. (3.19) is evaluated as an integral over B plus an integralover ∞. Over B, the second term vanished and, for a black hole with constant surfacegravity, the first is simply the variation of Noether charge entropy SNC multiplied by−κ/2pi [31, 33]. And the integral over ∞ is (as defined in chapter one, and assumingthe existence of a form B such that Θ|∞ = δB) the variation of the canonical quantityEχ conjugated to the Killing parameter of χ, i.e., the corresponding ADM conservedquantity [76] (“conserved” in the sense of not depending on the choice of the integrationthree-surface, see chapter one for more details). Thus,κ2piδSNC + 〈δHψχ 〉 = δEχ. (3.20)Equation (3.20) is already a form of the first law of black hole thermodynamics. Butwe can also write it in a different way.In some applications it is useful to understand expectation value of variations ofoperators, say δHψχ , in the Schödinger picture instead of Heisenberg’s. This is alwayspossible since the transformation that maps an operator O on a Hilbert space to anotheroperator O + δO ought to be unitary, meaning there is a unitary operator U so that82O → O+ δO = U OU †. In Schrödinger picture, we then keep the operators unchangedand produce the same expectation value 〈δO〉 by transforming the state ρ by ρ→ U †ρU .Adopting this picture, we investigate the case where the unperturbed state definedover the right wedge (i.e., the domain of dependence of C) is the Hartle-Hawking stategiven byρ0 =1Zexp(−2piκHψχ), (3.21)where Hψχ is the Hamiltonian operator with respect to the “time translation” defined byχa as above, and 1/Z is a normalisation factor. Strictly speaking, this density matrixis only defined when the spectrum of Hˆχ is discrete, so that Z = Tr e−2piHχ/κ is afinite quantity, and this problem is normally bypassed by confining the field in a boxwith certain boundary conditions. This is not a problem of fundamental physical signif-icance, and thermal states can be defined precisely through the Kubo-Martin-Schwingercondition [5].The difference between von Neumann entropies associated with the states ρ0 andρ0 + δρ is given byδSvN = Tr(ρ0 log ρ0)− Tr[(ρ0 + δρ) log(ρ0 + δρ)] =− Tr(ρ0ρ−10 δρ)− Tr(δρ log ρ0) +O(δρ2) =2piκ〈δHψχ 〉+O(δρ2), (3.22)where we used the fact that normalisation of both ρ0 and ρ0 + δρ implies Tr δρ = 0and used (3.21) in the last step. Eq. (3.22) is sometimes referred to “the first law ofentanglement entropy” (as in Fareghbal, Hakami and Shalamzari [77]).Combining eqs. (3.22) and (3.20) we get a statement of the first law in terms of theentropy of the matter:κ2piδ(SvN + SNC) = δEχ. (3.23)833.2.3 Second LawDifferently from the last section, results concerning the generalised second law of ther-modynamics (GSL) have to be established in non-stationary spacetimes. In the classicaltheory, substantial progress has been made by Ashtekar and Krishnan [79] in extending alocal version of the first law for dynamical black holes, generalising the version obtainedby Hawking and Hartle. [80] Although it seems possible to rewrite their results (espe-cially sec. V) in the semiclassical regime, this would not replace the version obtainedin section 3.2.2, as the classical version does not replace the version of the first law inref. [27]. Since the classical area theorem deals with the event horizon (as opposed toAshtekar and Krishnan’s “dynamical horizons”), we now seek a version of the second lawbased on globally defined quantities as above.The notion of black hole entropy as Noether charge is not so solidly establishedfor the dynamical case. For stationary black holes, classical fields and stationary per-turbations, it has been shown by Jacobson, Kang, and Myers [81] that the integral ofthe Noether charge does not depend on the integration surface, in particular does notneed to be evaluated at the bifurcation surface. It is a local geometrical object andall the dependence on the Killing vector tangent to the horizon can be eliminated onthe bifurcation surface as follows: its value restricted on the surface is χa = 0, its firstderivative is written in terms of the volume form (1.27), and any higher derivatives canbe eliminated by applying successively the Killing identity ∇c∇aχb = −Rabcdχd. Fornon-stationary black holes on the other hand, or even for stationary black holes butnon-stationary perturbations, these properties are lost.Iyer and Wald [33] listed the following desirable conditions the entropy of dynamicalblack holes ought to satisfy: (i) coincides with the Noether charge entropy in any crosssection for stationary black holes, (ii) its first variation evaluated on the bifurcationsurface obeys a version of the first law of black hole mechanics, (iii) is invariant byaltering the Lagrangian by adding an exact form, and (iv) obeys a version of the secondlaw of the black hole mechanics.84They proposed a definition satisfying the first three conditions, but not the fourth.The proposal here obeys (i), (ii) and (iv) corrected for effects of semiclassical origin, butfails to obey (iii). Fortunately, relaxing condition (iii) is not problematic since, althoughtheories with identical dynamical contents can generate different entropies, argumentscoming from the path integral approach to quantum gravity suggest that the boundaryterms on the Lagrangian can be physically meaningful for a future theory. For example,Hawking [82, 83] considers that the transition amplitude from a state |ψ, hab〉 to a state|ψ′, h′ab〉 should be given by a Feynman integral over fields ψ and induced metrics hab.If this amplitude is to factor out 〈ψ, hab|ψ′, h′ab〉 =∑ψ′′h′′〈ψ, hab|ψ′′h′′ab〉〈ψ′′h′′ab|ψ′, h′ab〉when an intermediate state |ψ′′h′′ab〉 is placed, the classical actions must add. For theEinstein-Hilbert action, this only happens when a boundary term proportional to thetrace of the extrinsic curvature is included in the action (see, for example, appendixE of [23]). This term is, apart from an additive constant, referred to as the Hawking-Gibbons-York term in the action.We consider the semiclassical theory (M, g, ψ, ρ) consisting of a future-predictable,asymptotically flat, “piecewise stationary” (definition below) spacetime (M, g) and con-sistent matter fields obeying the semiclassical equations E = 06. Intuitively speaking,“piecewise stationarity” means that the spacetime behaves like a stationary one up un-til a surface S1, evolves dynamically up until a second surface S2, when it becomesstationary again.More rigorously, a spacetime is called piecewise stationary if there are two Cauchysurfaces S1 and S2 ⊂ I+(S1) such that J−(S1) is isometric (with isometry ϕ1) toJ−(S′1) ⊂ M1 and J+(S2) to J+(S′2) ⊂ M2 (ϕ2 denotes the isometry), where (M1, g1)and (M2, g2) are (unphysical) stationary spacetimes and S′1 and S′2 are Cauchy surfacesin their respective manifolds.The case of interest is when there is a black hole on (M, g). As argued by Hawking[78], being a system with negative heat capacity, black hole can only be in a “stablethermal equilibrium” with an extensive system when the latter has its energy bounded,6In our notation from previous section, this denotes satisfying both Eψ = 0 and (Eg + 〈R〉) = 085so that if the extensive system gains (loses) energy to the black hole, its temperatureincreases (decreases) faster than the black hole’s. An alternative way of interpreting theimpossibility of a stable thermal equilibrium in a system with a black hole in an infinitereservoir is picturing a localised perturbation which requires an infinite amount of timeto get to infinity, so an originally stationary spacetime cannot become stationary againin a finite amount of time.Thus, in order to make the hypothesis of piecewise stationarity attainable, we confinethe fields ψ in a “box” on whose walls we impose as a boundary condition that anycomponent of 〈?jψχ〉 orthogonal to them must vanish (i.e., no energy can flow into or outof the box). The walls are outside the black hole in the region between S1 and S2 (as infigure 3.1). We further assume that all the disturbance responsible for the breakdownof stationarity comes from localised changes in the states of the fields within the box.Let χa1 denote a generator of the event horizon on (M1, g1). This field can seenas a field in the physical spacetime J−(S1) through the isometry, χa = ϕ−11∗ χa1, as agenerator of the event horizon in (M, g). As any generator of the event horizon, it isfuture-inextendible and entirely contained in the event horizon ofM [3]. In particular, itgenerates the event horizon in J+(S2), where it can be used to define the field χa2 = ϕ2∗χaon (M2, g2) which generates the event horizon of (M2, g2). According to the rigiditytheorem (see Proposition 9.3.6 of the book by Hawking and Ellis [3]), χa2 generates aKilling horizon in (M2, g2).In contrast, although there is no guarantee that χa1 will be a Killing field, this isthe case in the limit where the horizon approaches i−. In this limit, the event horizonand the apparent horizon coincide. The former has zero shear and vorticity, and thelatter zero expansion, so in that limit χa1 obeys the Killing equation and is null. In otherwords, the event horizon approaches a Killing horizon generated by χa1. This fact can beillustrated using an ingoing Vaidya metric7 ds2 = −(1− 2m(v)/r)dv2 + 2dvdr+ r2dΩ2,with m(v) = m1 > 0 in the past of a surface of constant v = v0, and m(v) = m2 > m17This spacetime is not strictly speaking piecewise stationary, but it serves to the purpose of illus-trating the coincidence of the event and apparent horizon.86on its future. In the past of the surface v = v0, the event horizon is the surface ofconstant u = v − 2r∗, r∗ = r + 2m1 log(r − 2m1) which coincides with r = 2m2 onv = v0. This surface approaches the apparent horizon r = 2m1 exponentially fast as onemoves to the past. For a solar-mass object, the characteristic Schwarzschild time 2m1is of order 10−5 seconds.The initial data of the field modes φ in S′1 is identified with the pullback of modesϕ∗1φ on S1. In other words, we choose the modes φ that solve the field equations on Mand whose restriction on J−(S1) coincides with the ϕ∗1φ and we carry the quantisationprocess using these modes. Similarly, we can use the isometry ϕ2 to identify modes onJ+(S2) with modes on J+(S′2) and quantise the fields on M2.If the semiclassical equations of motion hold on the entirety of the spacetime, ωχ =δjgχ + 〈δjψχ〉+ d[χ · (Θg + 〈Θψ〉)] is closed:dωχ = £χEδφ− δE£χφ = 0, (3.24)meaning that the integral of ωχ over the a closed three-surface is zero.We choose this surface to be C1 ∪C2 ∪T ∪h, where C1,2 are portions of two Cauchysurfaces Σ1,2 of (M, g) outside the black hole, with Σ2 ⊂ J+(S2), Σ1 ⊂ J−(S1). Later Σ1will be pushed down to the past so that it intercepts the horizon near i−. The segmenth is the intervening portion of the event horizon between Σ1 and Σ2, and T is a timelikesurface outside the causal future of the box in which the fields are confined, so that anypoint of T is spacelike separated to any point within the box, wherein the disturbancein stationarity happens. With this choice, T does not contribute to the integral of exactforms for causal theories8. See figure (3.1) for a scheme of this construction.The integral becomes (−∫C1+∫C2−∫h)ωχ = 0.The integrals over C1 and C2 can be evaluated from the isometries, and ωgχ = 0 due8Using Stoke’s theorem, integration of any exact form over T can be converted into integration overT ∩ C1 and T ∩ C2, where the integrands must share a common value.87Figure 3.1: Geometry that leads to eq. 3.25. The worldline labeled as “wall of box”denotes the timeline along which the boundary conditions on the fields are imposed.The shaded areas represent the regions ofM that are isometric to stationary spacetimes.The contour of integration is in blue.to stationarity, as in the previous section, leaving∫hωχ = ∆〈δHψχ 〉, (3.25)where ∆ denotes the difference in a quantity caused by the change in the integrationsurface from a domain in Σ1 to its corresponding domain in Σ2, i.e., a change “in time”.A direct calculation of a variation in the full symplectic current, or the addition ofωψχ and ωgχ in eq. (3.13) together with Eg + 〈R〉 = 0 revealsωχ = δjχ − d(χ ·Θ).Furthermore, if both (g, ψ) and (g + δg, ψ + δψ) satisfy the equations of motion E = 0,then adding eqs. (3.11) and (3.12),δjχ = δdQχand, under these hypotheses, both δjχ and ωχ are closed and the integral over h in eq.88(3.25) can be converted into a boundary integral,∫hωχ =(∫C2∩h−∫C1∩h){δQχ − χ ·Θ} ,If both (M1,2, g1,2) posses a bifurcation surface, the second term does not contributeto the integral. To see this, write(∫ϕ1,2(C1,2∩h)−∫B1,2)χ ·Θ =∫h′χ ·Θ, where h′ isthe intervening portion of the Killing horizon between ϕ1,2(C1,2∩h) and the bifurcationsurfaces B1,2. But χ is tangent to h′, so the right-hand side is zero, showing that theoriginal integrals can have their domains of integration replaced by B1,2, where χa = 0.Putting these elements together, eq. (3.25) can be written as∆〈δHψχ 〉 = −∆∫δQ[χ]. (3.26)It is well known (see Wald [31], Iyer & Wald [33], and Jacobson, Kang & Myers[81]) that the decomposition in the Noether charge is not unique, but rather carriesambiguities coming from three different sources, namely, the addition of an exact formin the Lagrangian, which although does not alter the classical dynamical content ofthe theory, is believed to be important in the quantum theory (as argued above); theaddition of an exact form in the Noether charge itself, which since it does not contributefor its integral, is immaterial both classically and quantum mechanically; and finallythe addition of an exact form in the current j′ = j + dY, whose presence is entirelyexpected, since no requirement to the absolute value of the Hamiltonian was imposed,only to its variations. We can physically interpret this last ambiguity as the addition ofa “second conserved current” that obeys a separate continuity equation, and thereforeis completely independent of the conservation of Q. This ambiguity is also present onthe definition of the expectation value of the Hamiltonian of the quantum fields andcomes as no surprise since no particular renormalisation procedure has been specified. Ifa renormalisation process uniquely defines 〈Hψχ 〉, a prescription for determining Y willfollow from it9. For this reason, after fixing Y and the boundary term in the action, we9As discussed in chapter two below eq. (2.13), using an axiomatic approach for defining 〈Tab〉reg89can defineSNC ≡ 2piκ∫Q[χ]uniquely in the semiclassical theory. Nevertheless, as it will be clear below, a “generalisedsecond law” holds for each choice of Y. In terms of the entropy, eq. (3.26) reads∆〈δHψχ 〉 = −∆[ κ2piδSNC.](3.27)In the density matrix approach, each expectation value of Hχ can be written asTr(ρ1,2Hχ), where ρ1,2 is obtained by taking the partial trace over degrees of free-dom lying in S1,2 − C1,2. The partial trace is a linear positive map between op-erators defined on their respective spaces, so the theorem by Lindblad in ref. [18]can be applied, which states that the relative entropy between two states ρ and σS(ρ|σ) ≡ Tr(ρ log ρ) − Tr(ρ log σ) cannot increase after the operation is used to obtainρred and σred, symbolicallyS(ρred|σred) ≤ S(ρ|σ), (3.28)where the suffix “red” denotes the reduced density matrix obtained after the partialtrace.Consistently with the hypothesis that (M, g) is stationary in J−(S1), we preparethe fields in a state σ0, annihilated by the annihilation operators corresponding to theeigenmodes of £ξ, with ϕ1∗ξa is an asymptotically timelike Killing field of (M1, g1). Letσ1 denote the reduced density matrix obtained from σ0 by tracing out degrees of freedomlying in S1−C1. All the observables locally constructed on J−(S1) can be calculated viaJ−(S′1), where the eigenmodes above diagonalise the Hamiltonian. Hence σ1 is describedas a Gibbs state associated with Hψχ and temperature κ1/2pi, exp(−2piHψχ /κ1)/Z1, i.e.,the corresponding Hartle-Hawking state.The evolution of σ0 in M is supposed to be Hamiltonian, then modes evolve alongthe orbits of ξa according to σ0 → U †σ0U , with U = exp(iHψξ t), with ξ = ∂/∂t. To bedefines this object only up to a local geometric function. This means that the classical ambiguity ofthe presence of the form Y manifests itself in the same way as the quantum ambiguity one has for therenormalisation process.90consistent with the piecewise stationarity of the unperturbed solution, we conclude thatin I+(S2), the Hamiltonian has to be a constant, so U can be written as U = exp(iω˜t)for some constant ω˜10. The density matrix σ2 is obtained after tracing out the degreesof freedom lying on S2−C2 of σ0. Again, σ2 viewed fromM2 is a Gibbs state associatedwith the Hamiltonian Hχ and temperature κ2/2pi.From the definition of a black hole, all causal curves intercepting S1−C1 necessarilyintercept S2 − C2, but since the black hole is dynamical, there are curves interceptingS2−C2 that do not intercept S1−C1, meaning that ρ2 can be obtained by reducing ρ1,not the reverse. This is how an arrow of time is introduced in the GSL, it comes fromthe time-asymmetric nature of black holes.If ρ1,2 is the physical (mixed) state on the algebras defined over C1,2, a simpleevaluation showsS(ρ1,2|σ1,2) = 2piκ1,2(〈Hχ〉ρ1,2 + logZ1,2)− SvN, (3.29)and as we know from standard statistical mechanics (e.g., ref. [8]),− κ1,22pilogZ1,2 = F [σ1,2] = 〈H〉σ1,2 −κ1,22piSvN[σ1,2]. (3.30)Combining eqs. (3.29) and (3.30),S(ρ1,2|σ1,2) = 2piκ1,2(〈Hχ〉ρ1,2 − 〈Hχ〉σ1,2)− (SvN[ρ1,2]− SvN[σ1,2]) ,together with (3.28) and (3.27),∆δ(SvN + SNC) ≥ 0. (3.31)I emphasise the definition SNC 1,2 = 2piκ1,2∫Q[χ] is only evaluated in the regionswhere the spacetime is stationary, i.e., where the all the previous discussions apply. Anydefinition of black hole’s entropy which agrees with the Noether charge’s in stationaryspacetimes ought to obey eq. (3.31).10This can be seen from the non-degeneracy property of the symplectic form. The evolution of ascalar quantity F is governed by dF/dt = Ω(∇F,∇H), which is zero for a generic F when δH = 0.91If one can show that the GSL is true for the state σ0 then it follows from eq. (3.31)that it is true for any state described as a linear perturbation around it. Two ingredientsare needed to complete this task.First, from the closure of full symplectic current when the full semiclassical equationsof motion are satisfied everywhere, an integral of jχ over the same circuit of figure (3.1)reveals∆〈Hψχ 〉σ0 + ∆∫h∩C1,2Qχ + ∆∫C1,2jgχ = 0, (3.32)where we put 〈Hψχ 〉 =∫C1,2[〈jψχ〉 − d(χ · 〈Bψ〉)] and made use of the fact that∫h∩C1,2 χ ·〈Bψ〉 = 0 = ∆ ∫C1,2∩T χ · 〈Bψ〉, the first equality obtained from arguments similar to theones above eq. (3.26).Second, for the first time, arguments coming from thermodynamics are invoked.More specifically, we suppose that variations of the von Neumann entropy of the opensystem behave as variations of its thermodynamic entropy. For this system, 〈Hψχ 〉 cannotbe interpreted as a conserved energy. For even if the integration of 〈jψχ〉 − d(χ · 〈Bψ〉)is taken over entire Cauchy surfaces, its value will still depend on the choice of theintegration surface because d〈jψχ〉 6= 0, meaning that the matter degrees of freedomover a Cauchy surface constitute an open system. Hence, the first law of ordinarythermodynamics applied to the exterior of the black hole isTdSvN = d〈Hψχ 〉+ d∫Cjgχ, (3.33)the last term is added to represent the amount of “conserved energy” leaves the matterdegrees of freedom over C1,2 to the gravitational ones over the same surface, d(〈jψχ〉+jgχ) =0 used. And since the variation ∆SvN of the entropy of the matter field is a functionof state, it is the same as in any process that share the same initial and final statesfor the matter, including one passing through infinitely many equilibrium states withtemperature T varying between κ1/2pi and κ2/2pi in an ensemble where macrostatesare distinguishable by the the expectation value of the Hamiltonian 〈Hχ〉. A directintegration of eq. (3.33) gives∆SvN = −∆SNC, (3.34)92after using eq. (3.32) and the definition of the Noether charge entropy.Putting this equality and eq. (3.31) together we have the GSL∆(SvN + SNC) ≥ 0, (3.35)where the equality holds for the “equilibrium state” σ0.Variations ∆ above are finite even though the quantities being varied themselvesmay diverge. This seems reasonable since one does not expect the ultraviolet behaviour(which is the one responsible for such divergences) to play an important role in theseeffects of semiclassical origin.The use of the monotonicity of the relative entropy as in (3.28) has been used be-fore by Sorkin [85] and Wall [84] to argue in favour of the GSL, albeit not as rigor-ously applied to non-stationary spacetimes. Our semiclassical Noether charge approachhas the following advantages. First, it is possible to generalise it to other geometricaldiffeomorphism-invariant theories of gravity. Second and most importantly, it avoidsthe assumption that all the change in the horizon’s area of a dynamical black hole asone moves to the future is caused by the expansion of the existing geodesic congruencesgenerating the horizon, when it is not clear whether or not the emergence of new horizongenerators dominates this change in area. This makes this proof distinct of any existingattempts.3.2.4 Third LawUnlike the zeroth, first, and second laws, I do not currently have a proof for any versionof the third law. However, the absence of counterexamples for producing a degeneratehorizon from a horizon with κ 6= 0 in black hole physics (commonly referred to as “over-spinning” or “overcharging” a Kerr-Newman black hole to produce a naked singularity)suggests that it may be possible to find a version that holds in semiclassical physics.For example, superradiance prevents the use of incoming bosonic waves to increasea Kerr black hole’s angular momentum faster than its energy [30]. In order to reduce κ,the black hole would have to absorb waves ei(ωt−mφ), t and φ the Kerr-Schild coordinates,93obeying ω < mΩ, where Ω is the angular velocity of the black hole (∂/∂t+ Ω ∂/∂φ gen-erating the horizon), but this is precisely the regime for which the reflection coefficientis higher than unity, meaning that if one tries to feed the hole with these bosonic waves,one is actually driving it farther from the extreme case. Fermions do not experiencesuperradiance [30] and, at first sight, this seems to be exploitable to overspin a Kerrblack hole. Interestingly, the reason why this is not the case is quantum mechanical [30]!Pauli’s exclusion principle forbids one from creating coherent waves ei(ωt−mφ) with arbi-trarily large intensity, and Hawking radiation for Kerr black holes favours the emissionof these waves with ω < mΩ just enough to compensate the absorption of such incomingwaves. See Hawking [89] for the calculation of the quantum emission and my MSc thesis[30] to show it compensates for the absorption. This suggest that if there is a version ofthe third law of thermodynamics, it must also be rooted in the semiclassical regime.Any version of the third law ought to be mathematically independent of the weakcosmic censorship conjecture, since it normally applies to spacetimes satisfying the dom-inant energy condition [23], which is not expected to hold true in semiclassical physics,as extensively discussed in this thesis.It should also be pointed out that much of the interest on the third law comes solelyfrom the sake of analogy. It is not needed for most of the processes that one wantsto apply the principles of thermodynamics, be it in ordinary context or in black holephysics. It is therefore not particularly concerning that we lack of a derivation for thislaw.3.2.5 DiscussionThis section shows that explicit forms of the zeroth, first, and second laws of black holethermodynamics emerge from a semiclassical theory of gravity. The meaning of eachof their ingredients — at least in the version of the laws here presented — was madeclearer than ever before since we kept a controlled set of assumptions. In particular, thematter entropy that enters these laws is, per construction, the von Neumann entropyassociated with the state restricted to the outside of black holes, as assumed in the94literature since the 1970’s (see Hawking [78]). A possible criticism to this framework isthe assumption of the existence of consistent semiclassical theories (M, g, ψ, ρ) and theconsequent enormous difficulties in applying our results to particular cases since evenapproximate back reaction effects are already challenging to calculate.It is noteworthy that the approach for verifying both first and second laws in semi-classical theory restores the use of “global techniques” of the classical theory, whichprevious attempts had ignored (see, e.g. [85] for the second law). While it is true thatthere are two distinct theorems referred to indistinctively by “the first law of black holemechanics”; prior to this work, there was only a semiclassical version of the so-called“physical process” version of the first law [25], which makes use of the semiclassical Ein-stein’s equations and Raychauhuri’s equation. As in the classical theory, the semiclassicalversion of these two results complement each other. For the second law, non-trivial clas-sical results necessarily appealed to global techniques, and so did our derivation for theGSL. I believe the takeaway is that entropy, generally a non-local quantity, needs globaltechniques to be fully accounted in gravitational physics.The construction of the semiclassical Hamiltonian as in section 3.2.2 can be appliedin broader contexts. For example, Jacobson and Visser [86] recently postulated a semi-classical correction in the first law in causal diamonds11. I believe it is possible to adaptthe constructions here to justify their assumption. They also write a relation betweenthe von Neumann entropy as in eq. (3.22), whose right-hand-side represents a sourceof gravity. The meaning of their variations in expectation values can be assigned witha more precise significance using the definition of the semiclassical Hamiltonian usedin this section. Finally, it would be interesting to investigate if there is an analog ofthe second law (a stronger result than their stationarity condition for the generalisedentropy) in causal diamonds. This is an excellent direction to look further into usingthe methods here developed.The form of expressing the first and second laws of thermodynamics in the presence of11Strictly speaking, a generalisation of the classical first law using conformal Killing fields, which wasobtained using the Noether charge methods I reviewed in chapter one of this thesis.95a semiclassical black hole allows the existence of processes that convert ordinary entropyfrom the exterior of the black hole to its Noether Charge. A prominent example of sucha process is the evaporation of black holes, as it emits thermal radiation and loses mass,its Noether-charge entropy decreases while the von Neumann entropy of the radiatingfields increases. A crucial hypothesis in deriving eq. (3.22) was that ρ0 is thermal withrespect to the Killing parameter (Hartle-Hawking state), but a similar situation alreadyhappens in the ordinary first law of thermodynamics, when one expresses heat exchangedto or off the system in terms of its temperature, which is only defined in very specialcases. In this sense, equation (3.20) can be thought of as the first law of thermodynamicswritten in terms of the heat exchanged rather than in terms of variations in entropy,and remains valid for all physically meaningful states.The assumption of “piecewise stationarity” plays the role of the quasi-steady as-sumption in the ordinary second law of thermodynamics. In ordinary thermodynamicstransitions between different thermodynamic states are thought to be approximated bya succession of equilibrium states. The thermodynamic notion of entropy written interms of the amount of heat exchanged is only defined under these conditions. Simi-larly, the notion of the black hole entropy as Noether charge only works under certainapproximations, when one can think of the geometry evolving from one stationary stateto another, remaining in each stationary stage long enough so that the event horizoncan be approximated by a Killing horizon, as discussed above. This approximation isbelieved to get better the slower the evolution of the geometry.The derivation of the laws of black hole thermodynamics is consistent with the inter-pretation of the entropy of a black hole (apart from an additive constant) as accountancyof the amount of information inaccessible its exterior, a view with a strong intuitive ap-peal that has been discussed since the 1980s by Bombelli et al [87]. This interpretationalso explains the origins of the ordinary laws of thermodynamics (see Jaynes’ seminalwork [88]) without appeal to Boltzmann’s microcanonical ensemble, providing unifiedgrounds to both ordinary and black hole thermodynamics. Consequently, while the ex-istence of these laws is the only reason to assign a black hole with an entropy, no further96microscopic description of it is needed.All discussion here was based on changes in entropy. It is possible to speculate thatthe specification of the absolute value of entropy in terms of the Noether charge willonly be possible after something further about quantum gravity is learnt. That wouldmirror the ordinary thermodynamics since Boltzmann entropy was defined only up toan additive constant before the introduction of the Sommerfeld quantisation rule (andconsequently the appearance of the quantum mechanical constant ~ in his formula).Until this happens, we can apply and interpret the macroscopic laws of black holethermodynamics in the semiclassical level as I developed here.974. Conclusions and ProspectsThis thesis dealt with semiclassical black hole physics. At the first sight, it may seemthat the sole purpose of studying this approach is to model the backreaction of effectscoming from the quantum nature of matter, such as Hawking radiation and the conse-quent prediction that black holes evaporate. But, as much interest of its own it has,this approach has led to several other investigations, particularly considerations aboutthermodynamics.This work explored a number of techniques useful for regularising the stress-energytensor particularly in two dimensions applying the results of ref. [6] and extrapolatingthem to four dimensions in particular cases. As an application, a model for a collapsingstar was proposed and the regularised stress-energy tensor for a scalar field in thisgeometry was found using some of the aforementioned techniques. The most importantconclusions of this model were two. First, the explicit agreement of the quantum emissionat very late times with a different collapse mechanism (namely Davies, Fulling, andUnruh’s collapse of a null shell in ref. [6]), illustrating in a concrete example (for whichthe formulae for the emission as a function of time were found) that this emission doesnot depend on the details of the collapse, a result that has been argued using differentconsiderations since the 1970’s (see Hawking [89] and Unruh [4]). Second, the regularisedexpectation value of the stress-energy tensor is small everywhere except in a vicinity ofthe singularity, meaning that quantum mechanics cannot have an important effect inthe collapse of an astrophysical object.I also explored the notion of entropy in quantum mechanics, which historically has98led to a number of debates in black hole physics. I addressed the debate about entropybounds and concluded that albeit there is evidence that entropy cannot assume arbitrar-ily large values for a given amount of energy in the matter, there are no comprehensiveproofs of entropy bounds in laws of quantum mechanics or gravity. In particular, Ishowed a counterexample to a recent proposal by Marolf, Minic, and Ross [73]. If thereexist such bounds which prevent the spontaneous creation of highly entropic objects, Ispeculate that they are a consequence of the particular equations of state of the matter.The origin of all the discussion involving the notion of entropy in black hole physicsis, of course, the laws of black hole thermodynamics, whose roots are both in classicalgeneral relativity and in the quantum theory of free fields. They were conjectured in the1970’s [78] and no counterexample has been found since. This thesis demonstrates themunder a controlled set of assumptions1. Besides increasing confidence in their validity,having proofs also brings other considerable advantages. First, the meaning of each oftheir ingredients is elucidated, in particular, the entropy in the GSL is indeed the vonNeumann entropy of the quantum fields after tracing out the degrees of freedom hiddenunderneath the event horizon; and the Bekenstein-Hawking entropy is well describedin the Noether charge approach. Second, the derivation being carried out using themonotonicity of the relative entropy and the causal structure of spacetime is consistentwith the interpretation of black hole’s entropy accounting for the amount of inaccessibleinformation to observers outside the black hole. The laws of black hole thermodynamicare the reason why the notion of entropy was brought into gravitational physics. Thepossession of a derivation for them compatible with an interpretation is then enough tomake this interpretation fully satisfactory until an independent reason for consideringabsolute value of entropy (rather than merely their variations) is known.Limitations of the semiclassical approximation are hard to predict in lack of a fullquantum interactive theory of gravity, but there are some general expectations one has.1I believe the zeroth and first laws as presented here are as general as possible within the limitationsof the semiclassical theory, but I suspect it must be possible to generalise the version of the second lawthis thesis established.99For example, interesting dynamical results were recently found byWang, Zhu, and Unruhin ref. [90] for a cosmological solution to a stochastic gravity theory, that is, the energy-momentum tensor for the matter scalar field was constructed such that it fluctuates overspacetime in a way that it matches the expectation value of the quantum field and itsvariance when the average is carried over regions of spacetime. Since this theory stillobeys the regular dynamical equations, everything we discussed in chapter three aboutthe zeroth, first, and second laws of black hole thermodynamics remains valid in thiscase without any modification. This comes as no surprise since for the zeroth and firstlaws (at least in the “heat exchange” formulation in eq. (3.20)) we only used the generalform of a classical geometric theory of gravity with classical sources to represent theeffects of quantum matter; and for the second law we used the von Neumann entropy forthe quantum fields, which is a measurement of the entanglement between the interiorand the exterior of the black hole and no stochastic modelling of the energy-momentumtensor can reproduce the correlations caused by this entanglement. This is an exampleof the different nature of the two objects that give the title to this thesis; energy, deeplyconcerned with the dynamics of a theory on the one hand; and entropy, which only caresabout the information on the other.Black hole thermodynamics (and its more speculative repercussions, like the holo-graphic principle for example, as in Bousso [91]) is currently an active topic in theoreticalphysics. Investigations of versions of the energy conditions satisfied quantum fields incurved spacetimes (like the averaged null energy condition, for example) and how toadapt the classical global structure theorems (e.g. Penrose’s singularity theorem) is atopic that receives attention (see the review by Fewster [92]). Obeying the null energycondition (plus other reasonable assumptions) implies that the Bekenstein-Hawking en-tropy is non-decreasing; when we allow this condition to be violated, a more generalresult remains valid, namely the GSL as stated in eq. (3.35). If one believes there isan inequality that plays the role of a classical energy condition and that remains truein quantum field theory in curved spacetimes, this result suggests it may have in someway to involve a different quantity, the entropy. It is far from clear how, but this same100suggestion has been proposed by Wall in ref. [93] using completely different methods! 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