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Advanced General Relativity and Quantum Field Theory in Curved Spacetimes Lecture Notes Stefano Liberati SISSA, IFPU, INFN January 2020 Preface The following notes are made by students of the course of “Advanced General Relativity and Quantum Field Theory in Curved Spacetimes”, which was held at the International School of Advanced Studies (SISSA) of Trieste (Italy) in the year 2017 by professor Stefano Liberati. Being the course directed to PhD students, this work and the notes therein are aimed to inter- ested readers that already have basic knowledge of Special Relativity, General Relativity, Quantum Mechanics and Quantum Field Theory; however, where possible the authors have included all the definitions and concept necessary to understand most of the topics presented. The course is based on di↵erent textbooks and papers; in particular, the first part, about Advanced General Relativity, is based on: “General Relativity” by R. Wald [1] • “Spacetime and Geometry” by S. Carroll [2] • “A Relativist Toolkit” by E. Poisson [3] • “Gravitation” by T. Padmanabhan [4] • while the second part, regarding Quantum Field Theory in Curved Spacetime, is based on “Quantum Fields in Curved Space” by N. C. Birrell and P. C. W. Davies [5] • “Vacuum E↵ects in strong fields” by A. A. Grib, S. G. Mamayev and V. M. Mostepanenko [6] • “Introduction to Quantum E↵ects in Gravity”, by V. F Mukhanov and S. Winitzki [7] • Where necessary, some other details could have been taken from paper and reviews in the standard literature, that will be listed where needed and in the Bibliography. Every possible mistake present in these notes are due to misunderstandings and imprecisions of which only the authors of the following works must be considered responsible. Credits Andrea Oddo (Chapters 1, 2, 3, 6, 7, Appendix A; revision) • Giovanni Tambalo (Chapters 8, 9, 10, 11) • Lumen Boco (Chapters 4, 5) • Also thanks to Giovanni Tricella and Paolo Campeti for useful scientific discussions. 1 Contents I Advanced General Relativity 5 1 Foundations of Relativity 6 1.1 Definitions and postulates . 6 1.2 Axiomatic derivation of Special Relativity . 8 1.3 EquivalencePrinciples .................................... 11 2 Di↵erential Geometry 13 2.1 Manifolds,Vectors,Tensors ................................. 13 2.1.1 Vectors ........................................ 13 2.1.2 Dualvectors ..................................... 14 2.2 TensorDensities ....................................... 15 2.3 Di↵erentialforms....................................... 16 2.4 TheVolumeelement ..................................... 17 2.5 Curvature........................................... 18 2.6 IntrinsicandExtrinsiccurvature . 21 2.6.1 Example: Intrinsic-Extrinsic (Gaussian) curvature of two surfaces . 21 2.6.2 Hypersurfaces and their Extrinsic curvature . 21 2.7 LieDerivative......................................... 23 2.8 KillingVectors ........................................ 25 2.9 Conservedquantities ..................................... 27 3 Kinematics 29 3.1 Geodesic Deviation Equation . 29 3.2 RaychaudhuriEquations................................... 31 3.2.1 Time-like geodesics congruence . 31 3.2.2 Null geodesics congruence . 33 3.3 EnergyConditions ...................................... 35 4 Variational Principle 37 4.1 Lagrangian Formulation of General Relativity . 37 4.1.1 Gibbons-Hawking-York counterterm . 40 4.1.2 Schr¨odinger action . 42 4.1.3 Palatini variation . 42 2 5 Alternative Theories of Gravity 44 5.1 Holding SEP: GR in D =4 ................................. 44 6 5.1.1 D<4......................................... 44 5.1.2 D>4......................................... 45 5.2 Holding SEP: Lanczos-Lovelock theories . 46 5.2.1 More about the Euler characteristic . 48 5.3 Relaxing SEP: F (R) Theories as a first example of higher curvature gravity . 49 5.3.1 Palatini variation . 50 5.4 Relaxing SEP: Generalized Brans–Dicke theories . 51 5.4.1 F (R) theories as generalized Brans–Dicke theories . 52 5.5 Relaxing SEP: Scalar-Tensor Theories . 52 5.5.1 Einsteinframe .................................... 53 5.5.2 Jordan frame . 53 5.5.3 Horndeski theory . 54 5.5.4 DHOSTTheories................................... 54 5.6 Goingfurther......................................... 56 6 Global Methods 57 6.1 Carter–Penrose Diagrams . 57 6.1.1 Minkowski spacetime Carter–Penrose diagram . 58 6.1.2 Schwarzschild spacetime Carter–Penrose diagram . 60 6.2 Black Holes: Singularities and Event Horizon . 63 6.2.1 Singularities...................................... 63 6.2.2 Horizons . 64 6.3 Killing Horizons and Surface Gravity . 66 6.3.1 Surface gravity: alternative definitions . 68 6.4 Cauchy Horizon . 69 6.5 Cosmological Horizon . 72 6.6 Local black hole horizons . 73 6.6.1 Trapped Surfaces and Trapped Regions . 73 6.6.2 Apparent Horizon . 75 6.6.3 Trapping Horizon, Dynamical Horizon, Isolated Horizon . 75 6.7 SingularityTheorems..................................... 77 7 From Black Hole Mechanics to Black Hole Thermodynamics 81 7.1 Rotating Black Holes . 82 7.1.1 AringSingularity .................................. 83 7.1.2 Horizons . 84 7.1.3 Killingvectors .................................... 85 7.2 Ergosphere and Penrose Process . 86 7.2.1 Superradiance..................................... 88 7.3 Black Hole Thermodynamics . 90 7.3.1 Null hypersurfaces, Surface integrals and other tools . 90 7.3.2 Surfaceelement.................................... 91 7.3.3 Komar Mass and the Smarr formula . 92 7.3.4 On the necessity of Hawking radiation . 97 7.3.5 Hawking radiation and black hole entropy . 98 3 II Quantum Field Theory in Curved Spacetimes 101 8 Preliminaries 102 8.1 Second quantization in curved spacetime . 102 8.2 Particle production by a time-varying potential . 105 8.3 CoherentandSqueezedstates................................ 107 9 E↵ects of Quantum Field Theory in Curved Spacetime 111 9.1 The Casimir e↵ect ...................................... 111 9.1.1 1+1 dimensional case: . 113 9.2 The Unruh E↵ect....................................... 115 9.2.1 TheRindlerWedge.................................. 115 9.2.2 Quantum field theory on the Rindler Wedge . 117 10 Evaporating Black Holes 121 10.1MovingMirrors........................................ 121 10.1.1 Field theory with moving boundaries . 121 10.1.2 Quantum field theory with moving boundaries . 122 10.1.3 Special case: the exponentially receding mirror . 123 10.2 Hawking Radiation . 125 10.3 Open issues with Hawking radiation . 129 10.3.1 Trans-planckian problem . 129 10.3.2 The information loss problem . 132 10.4 Black Hole Entropy Interpretations . 137 11 Semiclassical Gravity 141 11.1 Semiclassical Einstein Equations . 141 11.1.1 Approximation Techniques . 142 11.2 Renormalization of the Stress-Energy Tensor . 143 11.2.1 Point-splitting Renormalization . 144 11.3 Semiclassical Collapse . 149 12 Time Machines and Chronology Protection 156 12.1Timetravelparadoxes .................................... 156 12.2TimeMachines ........................................ 157 12.2.1 Time machines from globally rotating solutions . 158 12.2.2 Wormholes . 161 12.2.3 Alcubierre Warp Drives . 165 12.3 Possible solutions . 166 12.3.1 Preemptive chronology protection? . 167 12.3.2 Superluminal warp drive instability . 167 Bibliography 172 4 Part I Advanced General Relativity 5 1 Foundations of Relativity 1.1 Definitions and postulates We often refer to spacetime as a metaphysical entity, but in physics this would never be the case; therefore, our first task is to define both space and time in an operative fashion. We can replace the concepts of time with duration measured by a clock and space with distance measured with a rod. This is also because duration and distance are experimental quantities, that are directly meaningful, and that don’t need any interpretation whatsoever. With this, the properties of space and time are now directly related to the respective properties of rods and clocks. This gives us an operative definition of spacetime, rather than a metaphysical definition. We can then define an event. We will assume this to be a primitive notion in physics, in the same way a point is a primitive notion in geometry. In particular, an event is something that happens for a very small period time ∆t in a very small region of space ∆x; in this way, we can define our spacetime as the collection of all possible events: given a generic event , a spacetime is E M . (1.1) M⌘ E We can postulate that every measurement is either[ local or reducible to local, and in this way we can introduce the notion of an observer. We can imagine to place an observer in each point of space, and hence define a set of observers in our spacetime , that measure in di↵erent locations Oi M of spacetime. All these observers have trajectories in spacetime, and therefore we can define the set of these worldlines as a congruence of observers, one for each point in , in the sense that the M worldlines of all the observers never cross. The set of these worldines is . U⇢M O1 O2 O3 O4 O5 Figure 1.1: Pictorial representation of a congruence of worldlines, each one representing one observer (or a point in ). M One can go from an observer to another by performing a boost from the frame of the first observer to the frame of the second observer through a coordinate transformation. We want now to relate 6 observations in di↵erent frames, and therefore we want to agree on a system of rulers and clocks. Once we agree on a recipe to build a clock (e.g. quartz oscilation, period of radiation from atomic transitions,
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