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Energy Conditions in Semiclassical and Quantum Gravity

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Energy Conditions in Semiclassical and Quantum Gravity Energy conditions in semiclassical and quantum gravity Sergi Sirera Lahoz Supervised by Professor Toby Wiseman Imperial College London Department of Physics, Theory Group Submitted in partial fulfilment of the requirements for the degree of Master of Science of Imperial College London MSc Quantum Fields and Fundamental Forces 2nd October 2020 Abstract We explore the state of energy conditions in semiclassical and quantum gravity. Our review introduces energy conditions with a philosophical discussion about their nature. Energy conditions in classical gravity are reviewed together with their main applications. It is shown, among other results, how one can obtain singularity the- orems from such conditions. Classical and quantum physical systems violating the conditions are studied. We find that quantum fields entail negative energy densities that violate the classical conditions. The character of these violations motivates the quantum interest conjecture, which in turn guides the design of quantum energy conditions. The two best candidates, the AANEC and the QNEC, are examined in detail and the main applications revisited from their new perspectives. Several proofs for interacting field theories and/or curved spacetimes are demonstrated for both conditions using techniques from microcausality, holography, quantum infor- mation, and other modern ideas. Finally, the AANEC and the QNEC will be shown to be equivalent in M4. i Acknowledgements First of all, I would like to express my gratitude to Prof. Toby Wiseman for supervising this dissertation. Despite the challenges of not being able to meet in person, your guidance and expertise have helped me greatly to stay on track. The simple and clear way in which you communicate ideas has stimulated my interest in the topic and deepened my comprehension. Besides, thanks to your kindness and empathy, I left every meeting with renewed energy and motivation, which has made writing this dissertation a much easier and enjoyable activity. Second, I feel extremely lucky for the friends I have made at Imperial this year. I appreciate the support that we all have given each other to not only endure this MSc, but also have some fun along the way. I would also like to thank the rest of the theory group at Imperial for creating a nice and supportive learning environment. Third, to all the amazing friends I have made in London since I first came here. I am thankful for all the unforgettable experiences that we have lived together and the ones still to come. Fourth, to my friends and family back in Barcelona. Thank you for all the emotional support and encouragement. I am very lucky to have you in my life. A special thanks goes to everyone who has volunteered to proofread the dissertation through its different stages and to anyone who will read it in the future. Above all, I would like to thank my mom, my dad and my sister. Thank you for letting me follow my dreams. Without your unconditional love and support, this would not have been possible. For that I will always be deeply thankful. ii But, it is similar to a building, one wing of which is made of fine marble, but the other wing of which is built of low grade wood. Albert Einstein, 1936 about the equation Gab = 8πTab iii Conventions and Abbreviations EFEs Einstein Field Equations ECs Energy Conditions M4 Minkowski spacetime in 4 dimensions (A)dS (Anti) de Sitter spacetime NEC Null Energy Condition WEC Weak Energy Condition DEC Dominant Energy Condition SEC Strong Energy Condition ANEC Averaged Null Energy Condition AANEC Achronal Averaged Null Energy Condition AWEC Averaged Weak Energy Condition ASEC Averaged Strong Energy Condition ADEC Averaged Dominant Energy Condition QNEC Quantum Null Energy Condition QWEC Quantum Weak Energy Condition QSEC Quantum Strong Energy Condition GSL Generalized Second Law BCC Boundary Causality Condition QFC Quantum Focussing Conjecture Natural units G = c = ~ = 1 Metric signature convention (−; +; +; +) Z Fourier transform f^(k) = dnxeik·xf(x) a D’Alembert operator := r ra iv Contents Abstract i Acknowledgements ii Abbreviations and Conventions iv Introduction 1 1 The Nature of Energy Conditions5 1.1 What is an Energy Condition? . .5 1.2 Marble and wood: the GR asymmetry Gab vs Tab .................6 1.3 A twofold job . .9 1.4 A threefold formulation . 10 1.5 Fundamentality of Energy Conditions . 11 2 Energy conditions in classical gravity 13 2.1 Survey of pointwise Energy Conditions . 13 2.2 Implications . 17 2.3 Applications . 19 2.3.1 Singularity Theorems . 20 2.3.2 Cosmic Censorship Conjecture . 29 2.3.3 Other results . 32 2.4 Violations . 33 2.4.1 Traversable wormholes & time machines . 33 v 2.4.2 Negative cosmological constant Λ ...................... 36 2.4.3 Classical scalar field φ ............................ 37 3 Road to Quantum Energy Conditions: the Quantum Interest Conjecture 40 3.1 The Casimir Effect . 40 3.1.1 Casimir wormholes . 42 3.2 Quantum Coherence Effects . 45 3.3 The Quantum Interest Conjecture . 49 3.4 Foreword to chapters4 and5............................ 49 4 Averaged Energy Conditions 52 4.1 The Achronal Averaged Null Energy Condition and its siblings . 52 4.2 Applications revisited for the AANEC . 54 4.3 Proving the AANEC . 56 4.3.1 AANEC from the generalized 2nd Law ................... 56 4.3.2 AANEC from causality in QFT . 60 4.3.3 AANEC from holography . 74 4.3.4 AANEC from quantum information . 78 5 Quantum Energy Inequalities 83 5.1 The Quantum Null Energy Condition and its siblings . 83 5.2 Applications revisited for QEIs . 87 5.3 Proving the QNEC . 88 5.3.1 QNEC from causality + quantum information + quantum chaos . 89 5.3.2 QNEC for Rényi entropy . 92 5.4 QNEC & ANEC . 96 Conclusion 99 Bibliography 102 vi List of Figures 2.1 Logical relations between classical ECs. For instance, the line from the WEC to NEC means that if the WEC is true, then the NEC is also true. Roughly, the strongest conditions are displayed at the top and the weakest at the bottom. This diagram will be extended in the following chapters. 19 2.2 Conical singularity at r = 0.............................. 22 2.3 Trapped Surface (diagram idea taken from [71]) . 27 2.4 Carter-Penrose diagrams for Schwarzschild with M > 0 and M < 0. J + and J − are the future and past null infinity respectively. i0 is the spatial infinity. H+ is the future event horizon. 31 2.5 Carter-Penrose diagram for the gravitational collapse of Schwarzschild M < 0 . 31 π 2.6 Wormhole spacetime. Slice with θ = 2 and t = const. 34 3.1 Casimir setting . 41 3.2 Long traversable wormhole: 3 regions and magnetic field lines. 44 4.1 Logical relations of classical and averaged ECs. 54 4.2 Penrose diagram for the unperturbed classical spacetime. ∆T quantifies the region between the two slices. (taken from [127]) . 58 4.3 Spacetime diagram of a 4-point correlating function causality test with two back- ground operators (B,C) and two probe operators (A,D). 61 4.4 Euclidean (green) and Lightcone (blue) OPE limit. 62 4.5 Lightcone OPE double limit. First v ! 0, then u ! 1............... 66 4.6 Branch cut in analytic continuation of τ....................... 68 4.7 Operators of a 4-point function. x2 is evolved across the two lightcones created by x1 and x3. Causality entails the second branch cut will not occur before the second lightcone. 68 4.8 Rindler reflection in Lorentzian. 70 vii 4.9 ANEC setting. Operators O’s create a background that a signal from the probe operators has to go through. The ANE E is integrated along the null green line. 71 4.10 Sigma contour. The radius of the disk is (mention radius of the disk is η 1.. 73 4.11 Cylinder representing the boundary of AdS. A signal is sent from a point on the boundary to another point on the boundary. Causality in the CFT entails ∆t ≥ 0. 76 4.12 Poincaré patch showing trajectory running parallel and close to boundary . 77 4.13 Carter-Penrose diagram of the spacetime near the horizon (exterior region of extremal black hole) and its maximal extension. (taken from [106]) . 81 4.14 (Left) Causal domain D(A0). (Right) Deformed causal domain D(A). (taken from [106])....................................... 82 5.1 Cauchy-splitting spatial surface Σ. The entropy is evaluated in one of the sides of the Cauchy surface (purple). dλ represents a small variation in the null direction (diagram idea taken from [12]). 86 5.2 (Left) Inclusion property of causal domains D(B) ⊂ D(A). (Right) Causal domains of the two causally disconnected regions, D(B) and D(A¯). δxu is the null separation between the two. (diagram idea taken from [4]) . 89 5.3 Analiticity strip of f(s). The contour in the image, which is analogous to the contour we chose when proving the ANEC, is used in [4] to prove the QNEC. (taken from [4]).................................... 91 5.4 The entangling surface V (y) divides the cauchy surface Σ into regions R and R¯. The deformation of width A points towards region R and thus makes it smaller. 93 5.5 (Left) Carter-Penrose diagram showing the spatial region R where ρ is defined (green). R is unitarily equivalent to the two null regions on its future (green). (Right) Pencil decomposition, or null quantization. (taken from [91]) . 94 5.6 Logical relations between the relevant energy conditions. 102 viii Introduction Throughout the history of science, there is a very small number of concepts that have endured all the different ages until today. Most of them have either died or been replaced by stronger ideas. Energy is probably the utmost example of such everlasting concepts.
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