K-Causal Structure of Space-Time in General Relativity
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Introduction to Dynamical Triangulations
Introduction to Dynamical Triangulations Andrzej G¨orlich Niels Bohr Institute, University of Copenhagen Naxos, September 12th, 2011 Andrzej G¨orlich Causal Dynamical Triangulation Outline 1 Path integral for quantum gravity 2 Causal Dynamical Triangulations 3 Numerical setup 4 Phase diagram 5 Background geometry 6 Quantum fluctuations Andrzej G¨orlich Causal Dynamical Triangulation Path integral formulation of quantum mechanics A classical particle follows a unique trajectory. Quantum mechanics can be described by Path Integrals: All possible trajectories contribute to the transition amplitude. To define the functional integral, we discretize the time coordinate and approximate each path by linear pieces. space classical trajectory t1 time t2 Andrzej G¨orlich Causal Dynamical Triangulation Path integral formulation of quantum mechanics A classical particle follows a unique trajectory. Quantum mechanics can be described by Path Integrals: All possible trajectories contribute to the transition amplitude. To define the functional integral, we discretize the time coordinate and approximate each path by linear pieces. quantum trajectory space classical trajectory t1 time t2 Andrzej G¨orlich Causal Dynamical Triangulation Path integral formulation of quantum mechanics A classical particle follows a unique trajectory. Quantum mechanics can be described by Path Integrals: All possible trajectories contribute to the transition amplitude. To define the functional integral, we discretize the time coordinate and approximate each path by linear pieces. quantum trajectory space classical trajectory t1 time t2 Andrzej G¨orlich Causal Dynamical Triangulation Path integral formulation of quantum gravity General Relativity: gravity is encoded in space-time geometry. The role of a trajectory plays now the geometry of four-dimensional space-time. All space-time histories contribute to the transition amplitude. -
Causality and Determinism: Tension, Or Outright Conflict?
1 Causality and Determinism: Tension, or Outright Conflict? Carl Hoefer ICREA and Universidad Autònoma de Barcelona Draft October 2004 Abstract: In the philosophical tradition, the notions of determinism and causality are strongly linked: it is assumed that in a world of deterministic laws, causality may be said to reign supreme; and in any world where the causality is strong enough, determinism must hold. I will show that these alleged linkages are based on mistakes, and in fact get things almost completely wrong. In a deterministic world that is anything like ours, there is no room for genuine causation. Though there may be stable enough macro-level regularities to serve the purposes of human agents, the sense of “causality” that can be maintained is one that will at best satisfy Humeans and pragmatists, not causal fundamentalists. Introduction. There has been a strong tendency in the philosophical literature to conflate determinism and causality, or at the very least, to see the former as a particularly strong form of the latter. The tendency persists even today. When the editors of the Stanford Encyclopedia of Philosophy asked me to write the entry on determinism, I found that the title was to be “Causal determinism”.1 I therefore felt obliged to point out in the opening paragraph that determinism actually has little or nothing to do with causation; for the philosophical tradition has it all wrong. What I hope to show in this paper is that, in fact, in a complex world such as the one we inhabit, determinism and genuine causality are probably incompatible with each other. -
Causality in Quantum Field Theory with Classical Sources
Causality in Quantum Field Theory with Classical Sources Bo-Sture K. Skagerstam1;a), Karl-Erik Eriksson2;b), Per K. Rekdal3;c) a)Department of Physics, NTNU, Norwegian University of Science and Technology, N-7491 Trondheim, Norway b) Department of Space, Earth and Environment, Chalmers University of Technology, SE-412 96 G¨oteborg, Sweden c)Molde University College, P.O. Box 2110, N-6402 Molde, Norway Abstract In an exact quantum-mechanical framework we show that space-time expectation values of the second-quantized electromagnetic fields in the Coulomb gauge, in the presence of a classical source, automatically lead to causal and properly retarded elec- tromagnetic field strengths. The classical ~-independent and gauge invariant Maxwell's equations then naturally emerge and are therefore also consistent with the classical spe- cial theory of relativity. The fundamental difference between interference phenomena due to the linear nature of the classical Maxwell theory as considered in, e.g., classical optics, and interference effects of quantum states is clarified. In addition to these is- sues, the framework outlined also provides for a simple approach to invariance under time-reversal, some spontaneous photon emission and/or absorption processes as well as an approach to Vavilov-Cherenkovˇ radiation. The inherent and necessary quan- tum uncertainty, limiting a precise space-time knowledge of expectation values of the quantum fields considered, is, finally, recalled. arXiv:1801.09947v2 [quant-ph] 30 Mar 2019 1Corresponding author. Email address: [email protected] 2Email address: [email protected] 3Email address: [email protected] 1. Introduction The roles of causality and retardation in classical and quantum-mechanical versions of electrodynamics are issues that one encounters in various contexts (for recent discussions see, e.g., Refs.[1]-[14]). -
The Penrose and Hawking Singularity Theorems Revisited
Classical singularity thms. Interlude: Low regularity in GR Low regularity singularity thms. Proofs Outlook The Penrose and Hawking Singularity Theorems revisited Roland Steinbauer Faculty of Mathematics, University of Vienna Prague, October 2016 The Penrose and Hawking Singularity Theorems revisited 1 / 27 Classical singularity thms. Interlude: Low regularity in GR Low regularity singularity thms. Proofs Outlook Overview Long-term project on Lorentzian geometry and general relativity with metrics of low regularity jointly with `theoretical branch' (Vienna & U.K.): Melanie Graf, James Grant, G¨untherH¨ormann,Mike Kunzinger, Clemens S¨amann,James Vickers `exact solutions branch' (Vienna & Prague): Jiˇr´ıPodolsk´y,Clemens S¨amann,Robert Svarcˇ The Penrose and Hawking Singularity Theorems revisited 2 / 27 Classical singularity thms. Interlude: Low regularity in GR Low regularity singularity thms. Proofs Outlook Contents 1 The classical singularity theorems 2 Interlude: Low regularity in GR 3 The low regularity singularity theorems 4 Key issues of the proofs 5 Outlook The Penrose and Hawking Singularity Theorems revisited 3 / 27 Classical singularity thms. Interlude: Low regularity in GR Low regularity singularity thms. Proofs Outlook Table of Contents 1 The classical singularity theorems 2 Interlude: Low regularity in GR 3 The low regularity singularity theorems 4 Key issues of the proofs 5 Outlook The Penrose and Hawking Singularity Theorems revisited 4 / 27 Theorem (Pattern singularity theorem [Senovilla, 98]) In a spacetime the following are incompatible (i) Energy condition (iii) Initial or boundary condition (ii) Causality condition (iv) Causal geodesic completeness (iii) initial condition ; causal geodesics start focussing (i) energy condition ; focussing goes on ; focal point (ii) causality condition ; no focal points way out: one causal geodesic has to be incomplete, i.e., : (iv) Classical singularity thms. -
Time and Causality in General Relativity
Time and Causality in General Relativity Ettore Minguzzi Universit`aDegli Studi Di Firenze FQXi International Conference. Ponta Delgada, July 10, 2009 FQXi Conference 2009, Ponta Delgada Time and Causality in General Relativity 1/8 Light cone A tangent vector v 2 TM is timelike, lightlike, causal or spacelike if g(v; v) <; =; ≤; > 0 respectively. Time orientation and spacetime At every point there are two cones of timelike vectors. The Lorentzian manifold is time orientable if a continuous choice of one of the cones, termed future, can be made. If such a choice has been made the Lorentzian manifold is time oriented and is also called spacetime. Lorentzian manifolds and light cones Lorentzian manifolds A Lorentzian manifold is a Hausdorff manifold M, of dimension n ≥ 2, endowed with a Lorentzian metric, that is a section g of T ∗M ⊗ T ∗M with signature (−; +;:::; +). FQXi Conference 2009, Ponta Delgada Time and Causality in General Relativity 2/8 Time orientation and spacetime At every point there are two cones of timelike vectors. The Lorentzian manifold is time orientable if a continuous choice of one of the cones, termed future, can be made. If such a choice has been made the Lorentzian manifold is time oriented and is also called spacetime. Lorentzian manifolds and light cones Lorentzian manifolds A Lorentzian manifold is a Hausdorff manifold M, of dimension n ≥ 2, endowed with a Lorentzian metric, that is a section g of T ∗M ⊗ T ∗M with signature (−; +;:::; +). Light cone A tangent vector v 2 TM is timelike, lightlike, causal or spacelike if g(v; v) <; =; ≤; > 0 respectively. -
Causality in the Quantum World
VIEWPOINT Causality in the Quantum World A new model extends the definition of causality to quantum-mechanical systems. by Jacques Pienaar∗ athematical models for deducing cause-effect relationships from statistical data have been successful in diverse areas of science (see Ref. [1] and references therein). Such models can Mbe applied, for instance, to establish causal relationships between smoking and cancer or to analyze risks in con- struction projects. Can similar models be extended to the microscopic world governed by the laws of quantum me- chanics? Answering this question could lead to advances in quantum information and to a better understanding of the foundations of quantum mechanics. Developing quantum Figure 1: In statistics, causal models can be used to extract extensions of causal models, however, has proven challeng- cause-effect relationships from empirical data on a complex ing because of the peculiar features of quantum mechanics. system. Existing models, however, do not apply if at least one For instance, if two or more quantum systems are entangled, component of the system (Y) is quantum. Allen et al. have now it is hard to deduce whether statistical correlations between proposed a quantum extension of causal models. (APS/Alan them imply a cause-effect relationship. John-Mark Allen at Stonebraker) the University of Oxford, UK, and colleagues have now pro- posed a quantum causal model based on a generalization of an old principle known as Reichenbach’s common cause is a third variable that is a common cause of both. In the principle [2]. latter case, the correlation will disappear if probabilities are Historically, statisticians thought that all information conditioned to the common cause. -
Closed Timelike Curves, Singularities and Causality: a Survey from Gödel to Chronological Protection
Closed Timelike Curves, Singularities and Causality: A Survey from Gödel to Chronological Protection Jean-Pierre Luminet Aix-Marseille Université, CNRS, Laboratoire d’Astrophysique de Marseille , France; Centre de Physique Théorique de Marseille (France) Observatoire de Paris, LUTH (France) [email protected] Abstract: I give a historical survey of the discussions about the existence of closed timelike curves in general relativistic models of the universe, opening the physical possibility of time travel in the past, as first recognized by K. Gödel in his rotating universe model of 1949. I emphasize that journeying into the past is intimately linked to spacetime models devoid of timelike singularities. Since such singularities arise as an inevitable consequence of the equations of general relativity given physically reasonable assumptions, time travel in the past becomes possible only when one or another of these assumptions is violated. It is the case with wormhole-type solutions. S. Hawking and other authors have tried to save the paradoxical consequences of time travel in the past by advocating physical mechanisms of chronological protection; however, such mechanisms remain presently unknown, even when quantum fluctuations near horizons are taken into account. I close the survey by a brief and pedestrian discussion of Causal Dynamical Triangulations, an approach to quantum gravity in which causality plays a seminal role. Keywords: time travel; closed timelike curves; singularities; wormholes; Gödel’s universe; chronological protection; causal dynamical triangulations 1. Introduction In 1949, the mathematician and logician Kurt Gödel, who had previously demonstrated the incompleteness theorems that broke ground in logic, mathematics, and philosophy, became interested in the theory of general relativity of Albert Einstein, of which he became a close colleague at the Institute for Advanced Study at Princeton. -
Light Rays, Singularities, and All That
Light Rays, Singularities, and All That Edward Witten School of Natural Sciences, Institute for Advanced Study Einstein Drive, Princeton, NJ 08540 USA Abstract This article is an introduction to causal properties of General Relativity. Topics include the Raychaudhuri equation, singularity theorems of Penrose and Hawking, the black hole area theorem, topological censorship, and the Gao-Wald theorem. The article is based on lectures at the 2018 summer program Prospects in Theoretical Physics at the Institute for Advanced Study, and also at the 2020 New Zealand Mathematical Research Institute summer school in Nelson, New Zealand. Contents 1 Introduction 3 2 Causal Paths 4 3 Globally Hyperbolic Spacetimes 11 3.1 Definition . 11 3.2 Some Properties of Globally Hyperbolic Spacetimes . 15 3.3 More On Compactness . 18 3.4 Cauchy Horizons . 21 3.5 Causality Conditions . 23 3.6 Maximal Extensions . 24 4 Geodesics and Focal Points 25 4.1 The Riemannian Case . 25 4.2 Lorentz Signature Analog . 28 4.3 Raychaudhuri’s Equation . 31 4.4 Hawking’s Big Bang Singularity Theorem . 35 5 Null Geodesics and Penrose’s Theorem 37 5.1 Promptness . 37 5.2 Promptness And Focal Points . 40 5.3 More On The Boundary Of The Future . 46 1 5.4 The Null Raychaudhuri Equation . 47 5.5 Trapped Surfaces . 52 5.6 Penrose’s Theorem . 54 6 Black Holes 58 6.1 Cosmic Censorship . 58 6.2 The Black Hole Region . 60 6.3 The Horizon And Its Generators . 63 7 Some Additional Topics 66 7.1 Topological Censorship . 67 7.2 The Averaged Null Energy Condition . -
Problem Set 4 (Causal Structure)
\Singularities, Black Holes, Thermodynamics in Relativistic Spacetimes": Problem Set 4 (causal structure) 1. Wald (1984): ch. 8, problems 1{2 2. Prove or give a counter-example: there is a spacetime with two points simultaneously con- nectable by a timelike geodesic, a null geodesic and a spacelike geodesic. 3. Prove or give a counter-example: if two points are causally related, there is a spacelike curve connecting them. 4. Prove or give a counter-example: any two points of a spacetime can be joined by a spacelike curve. (Hint: there are three cases to consider, first that the temporal futures of the two points intesect, second that their temporal pasts do, and third that neither their temporal futures nor their temporal pasts intersect; recall that the manifold must be connected.) 5. Prove or give a counter-example: every flat Lorentzian metric is temporally orientable. 6. Prove or give a counter-example: if the temporal futures of two points do not intersect, then neither do their temporal pasts. 7. Prove or give a counter-example: for a point p in a spacetime, the temporal past of the temporal future of the temporal past of the temporal future. of p is the entire spacetime. (Hint: if two points can be joined by a spacelike curve, then they can be joined by a curve consisting of some finite number of null geodesic segments glued together; recall that the manifold is connected.) 8. Prove or give a counter-example: if the temporal past of p contains the entire temporal future of q, then there is a closed timelike curve through p. -
Theoretical Computer Science Causal Set Topology
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Theoretical Computer Science 405 (2008) 188–197 Contents lists available at ScienceDirect Theoretical Computer Science journal homepage: www.elsevier.com/locate/tcs Causal set topology Sumati Surya Raman Research Institute, Bangalore, India article info a b s t r a c t Keywords: The Causal Set Theory (CST) approach to quantum gravity is motivated by the observation Space–time that, associated with any causal spacetime (M, g) is a poset (M, ≺), with the order relation Causality ≺ corresponding to the spacetime causal relation. Spacetime in CST is assumed to have a Topology Posets fundamental atomicity or discreteness, and is replaced by a locally finite poset, the causal Quantum gravity set. In order to obtain a well defined continuum approximation, the causal set must possess the requisite intrinsic topological and geometric properties that characterise a continuum spacetime in the large. The study of causal set topology is thus dictated by the nature of the continuum approximation. We review the status of causal set topology and present some new results relating poset and spacetime topologies. The hope is that in the process, some of the ideas and questions arising from CST will be made accessible to the larger community of computer scientists and mathematicians working on posets. © 2008 Published by Elsevier B.V. 1. The spacetime poset and causal sets In Einstein’s theory of gravity, space and time are merged into a single entity represented by a pseudo-Riemannian or Lorentzian geometry g of signature − + ++ on a differentiable four dimensional manifold M. -
A WORLD WITHOUT CAUSE and EFFECT Logic-Defying Experiments Into Quantum Causality Scramble the Notion of Time Itself
A WORLD WITHOUT CAUSE AND EFFECT Logic-defying experiments into quantum causality scramble the notion of time itself. BY PHILIP BALL lbert Einstein is heading out for his This finding1 in 2015 made the quantum the constraints of a predefined causal structure daily stroll and has to pass through world seem even stranger than scientists had might solve some problems faster than con- Atwo doorways. First he walks through thought. Walther’s experiments mash up cau- ventional quantum computers,” says quantum the green door, and then through the red one. sality: the idea that one thing leads to another. theorist Giulio Chiribella of the University of Or wait — did he go through the red first and It is as if the physicists have scrambled the con- Hong Kong. then the green? It must have been one or the cept of time itself, so that it seems to run in two What’s more, thinking about the ‘causal struc- other. The events had have to happened in a directions at once. ture’ of quantum mechanics — which events EDGAR BĄK BY ILLUSTRATION sequence, right? In everyday language, that sounds nonsen- precede or succeed others — might prove to be Not if Einstein were riding on one of the sical. But within the mathematical formalism more productive, and ultimately more intuitive, photons ricocheting through Philip Walther’s of quantum theory, ambiguity about causation than couching it in the typical mind-bending lab at the University of Vienna. Walther’s group emerges in a perfectly logical and consistent language that describes photons as being both has shown that it is impossible to say in which way. -
A Brief Introduction to Temporality and Causality
A Brief Introduction to Temporality and Causality Kamran Karimi D-Wave Systems Inc. Burnaby, BC, Canada [email protected] Abstract Causality is a non-obvious concept that is often considered to be related to temporality. In this paper we present a number of past and present approaches to the definition of temporality and causality from philosophical, physical, and computational points of view. We note that time is an important ingredient in many relationships and phenomena. The topic is then divided into the two main areas of temporal discovery, which is concerned with finding relations that are stretched over time, and causal discovery, where a claim is made as to the causal influence of certain events on others. We present a number of computational tools used for attempting to automatically discover temporal and causal relations in data. 1. Introduction Time and causality are sources of mystery and sometimes treated as philosophical curiosities. However, there is much benefit in being able to automatically extract temporal and causal relations in practical fields such as Artificial Intelligence or Data Mining. Doing so requires a rigorous treatment of temporality and causality. In this paper we present causality from very different view points and list a number of methods for automatic extraction of temporal and causal relations. The arrow of time is a unidirectional part of the space-time continuum, as verified subjectively by the observation that we can remember the past, but not the future. In this regard time is very different from the spatial dimensions, because one can obviously move back and forth spatially.