Closed Timelike Curves, Singularities and Causality: a Survey from Gödel to Chronological Protection
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Arxiv:1911.08602V1 [Gr-Qc] 19 Nov 2019
Causality violation without time-travel: closed lightlike paths in G¨odel'suniverse Brien C. Nolan Centre for Astrophysics and Relativity, School of Mathematical Sciences, Dublin City University, Glasnevin, Dublin 9, Ireland.∗ We revisit the issue of causality violations in G¨odel'suniverse, restricting to geodesic motions. It is well-known that while there are closed timelike curves in this spacetime, there are no closed causal geodesics. We show further that no observer can communicate directly (i.e. using a single causal geodesic) with their own past. However, we show that this type of causality violation can be achieved by a system of relays: we prove that from any event P in G¨odel'suniverse, there is a future-directed lightlike path - a sequence of future-directed null geodesic segments, laid end to end - which has P as its past and future endpoints. By analysing the envelope of the family of future directed null geodesics emanating from a point of the spacetime, we show that this lightlike path must contain a minimum of eight geodesic segments, and show further that this bound is attained. We prove a related general result, that events of a time orientable spacetime are connected by a (closed) timelike curve if and only if they are connected by a (closed) lightlike path. This suggests a means of violating causality in G¨odel'suniverse without the need for unfeasibly large accelerations, using instead a sequence of light signals reflected by a suitably located system of mirrors. I. INTRODUCTION: GODEL'S¨ UNIVERSE In 1949, Kurt G¨odel[1]published a solution of Einstein's equations which provides what appears to be the first example of a spacetime containing closed timelike curves (CTCs). -
5 VII July 2017
5 VII July 2017 International Journal for Research in Applied Science & Engineering Technology (IJRASET) ISSN: 2321-9653; IC Value: 45.98; SJ Impact Factor:6.887 Volume 5 Issue VII, July 2017- Available at www.ijraset.com An Introduction to Time Travel Rudradeep Goutam Mechanical Engg. Sawai Madhopur College of Engineering & Technology, Sawai Madhopur ,.Rajasthan Abstract: Time Travel is an interesting topic for a Human Being and also for physics in Ancient time, The Scientists Thought that the time travel is not possible but after the theory of relativity given by the Albert Einstein, changes the opinion towards the time travel. After this wonderful Research the Scientists started to work on making a Time Machine for Time Travel. Stephon Hawkins, Albert Einstein, Frank j. Tipler, Kurt godel are some well-known names who worked on the time machine. This Research paper introduce about the different ways of Time Travel and about problems to make them practically possible. Keywords: Time travel, Time machine, Holes, Relativity, Space, light, Universe I. INTRODUCTION After the discovery of three dimensions it was a challenge for physics to search fourth dimension. Quantum physics assumes The Time as a fourth special dimension with the three dimensions. Simply, Time is irreversible Succession of past to Future through the present. Science assumes that the time is unidirectional which is directed past to future but the question arise that: Is it possible to travel in Fourth dimension (Time) like the other three dimensions? The meaning of time Travel is to travel in past or future from present. The present is nothing but it is the link between past and future. -
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. -
BLACK HOLES: the OTHER SIDE of INFINITY General Information
BLACK HOLES: THE OTHER SIDE OF INFINITY General Information Deep in the middle of our Milky Way galaxy lies an object made famous by science fiction—a supermassive black hole. Scientists have long speculated about the existence of black holes. German astronomer Karl Schwarzschild theorized that black holes form when massive stars collapse. The resulting gravity from this collapse would be so strong that the matter would become more and more dense. The gravity would eventually become so strong that nothing, not even radiation moving at the speed of light, could escape. Schwarzschild’s theories were predicted by Einstein and then borne out mathematically in 1939 by American astrophysicists Robert Oppenheimer and Hartland Snyder. WHAT EXACTLY IS A BLACK HOLE? First, it’s not really a hole! A black hole is an extremely massive concentration of matter, created when the largest stars collapse at the end of their lives. Astronomers theorize that a point with infinite density—called a singularity—lies at the center of black holes. SO WHY IS IT CALLED A HOLE? Albert Einstein’s 1915 General Theory of Relativity deals largely with the effects of gravity, and in essence predicts the existence of black holes and singularities. Einstein hypothesized that gravity is a direct result of mass distorting space. He argued that space behaves like an invisible fabric with an elastic quality. Celestial bodies interact with this “fabric” of space-time, appearing to create depressions termed “gravity wells” and drawing nearby objects into orbit around them. Based on this principle, the more massive a body is in space, the deeper the gravity well it will create. -
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 . -
Minkowski Space-Time and Einstein's Now Conundrum
NeuroQuantology | May 2018 | Volume 16 | Issue 5 | Page 23-30 | doi: 10.14704/nq.2018.16.5.1345 Sorli A., Minkowski Space-time and Einstein’s Now Conundrum Minkowski Space-time and Einstein’s Now Conundrum Amrit Sorli 1*, Steve Kaufman 1, Davide Fiscaletti 2 ABSTRACT The Minkowski formula X4=ict indicates that time t does not express the 4th dimension of space-time, i.e., X4 is not t. Therefore, Minkowski space-time is not a 3D+T manifold, it is 4D. The 4D manifold has 4 spatial dimensions and is timeless, in the sense that it does not contain some physical time in wh ich material changes run. In physics we experience the timelessness of 4D space as “Einstein’s Now.” From this perspective, the time that we measure with clocks is just the sequential numerical order of changes, i.e. motions that run in the timeless space of Now. On the other hand, past and future are nothing more than psychological or conceptual realities that derive from the neuronal activity of the brain. Therefore, “time flow” and “arrow of time” have only a mathematical existence. Key Words: Now, Time, Space, Space-Time, GPS, Closed Timeline Curve, Arrow of Time DOI Number: 10.14704/nq.201 8.16.5.1345 NeuroQuantology 2018; 16(5):23-30 Introduction system S, if complex enough, can be separated into 23 Today, 130 years after the pioneering work of a subsystem S2 whose dynamics is described, and Ernst Mach regarding the interpretation of time as another cyclic subsystem S1 which behaves as a a measure of the changes of things, the related clock and that, as a consequence of the gauge ideas that time cannot be considered as a primary invariance, this separation may be made in many physical reality that flows on its own in the ways. -
Tippett 2017 Class. Quantum Grav
Citation for published version: Tippett, BK & Tsang, D 2017, 'Traversable acausal retrograde domains in spacetime', Classical Quantum Gravity, vol. 34, no. 9, 095006. https://doi.org/10.1088/1361-6382/aa6549 DOI: 10.1088/1361-6382/aa6549 Publication date: 2017 Document Version Peer reviewed version Link to publication Publisher Rights CC BY-NC-ND This is an author-created, un-copyedited version of an article published in Classical and Quantum Gravity. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at: https://doi.org/10.1088/1361-6382/aa6549 University of Bath Alternative formats If you require this document in an alternative format, please contact: [email protected] General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. Take down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Download date: 05. Oct. 2021 Classical and Quantum Gravity PAPER Related content - A time machine for free fall into the past Traversable acausal retrograde domains in Davide Fermi and Livio Pizzocchero - The generalized second law implies a spacetime quantum singularity theorem Aron C Wall To cite this article: Benjamin K Tippett and David Tsang 2017 Class. -
Stable Wormholes in the Background of an Exponential F (R) Gravity
universe Article Stable Wormholes in the Background of an Exponential f (R) Gravity Ghulam Mustafa 1, Ibrar Hussain 2,* and M. Farasat Shamir 3 1 Department of Mathematics, Shanghai University, Shanghai 200444, China; [email protected] 2 School of Electrical Engineering and Computer Science, National University of Sciences and Technology, H-12, Islamabad 44000, Pakistan 3 National University of Computer and Emerging Sciences, Lahore Campus, Punjab 54000, Pakistan; [email protected] * Correspondence: [email protected] Received: 21 January 2020; Accepted: 3 March 2020; Published: 26 March 2020 Abstract: The current paper is devoted to investigating wormhole solutions with an exponential gravity model in the background of f (R) theory. Spherically symmetric static spacetime geometry is chosen to explore wormhole solutions with anisotropic fluid source. The behavior of the traceless matter is studied by employing a particular equation of state to describe the important properties of the shape-function of the wormhole geometry. Furthermore, the energy conditions and stability analysis are done for two specific shape-functions. It is seen that the energy condition are to be violated for both of the shape-functions chosen here. It is concluded that our results are stable and realistic. Keywords: wormholes; stability; f (R) gravity; energy conditions 1. Introduction The discussion on wormhole geometry is a very hot subject among the investigators of the different modified theories of gravity. The concept of wormhole was first expressed by Flamm [1] in 1916. After 20 years, Einstein and Rosen [2] calculated the wormhole geometry in a specific background. In fact, it was the second attempt to realize the basic structure of wormholes. -
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. -
Weyl Metrics and Wormholes
Prepared for submission to JCAP Weyl metrics and wormholes Gary W. Gibbons,a;b Mikhail S. Volkovb;c aDAMTP, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK bLaboratoire de Math´ematiqueset Physique Th´eorique,LMPT CNRS { UMR 7350, Universit´ede Tours, Parc de Grandmont, 37200 Tours, France cDepartment of General Relativity and Gravitation, Institute of Physics, Kazan Federal University, Kremlevskaya street 18, 420008 Kazan, Russia E-mail: [email protected], [email protected] Abstract. We study solutions obtained via applying dualities and complexifications to the vacuum Weyl metrics generated by massive rods and by point masses. Rescal- ing them and extending to complex parameter values yields axially symmetric vacuum solutions containing singularities along circles that can be viewed as singular mat- ter sources. These solutions have wormhole topology with several asymptotic regions interconnected by throats and their sources can be viewed as thin rings of negative tension encircling the throats. For a particular value of the ring tension the geometry becomes exactly flat although the topology remains non-trivial, so that the rings liter- ally produce holes in flat space. To create a single ring wormhole of one metre radius one needs a negative energy equivalent to the mass of Jupiter. Further duality trans- formations dress the rings with the scalar field, either conventional or phantom. This gives rise to large classes of static, axially symmetric solutions, presumably including all previously known solutions for a gravity-coupled massless scalar field, as for exam- ple the spherically symmetric Bronnikov-Ellis wormholes with phantom scalar. The multi-wormholes contain infinite struts everywhere at the symmetry axes, apart from arXiv:1701.05533v3 [hep-th] 25 May 2017 solutions with locally flat geometry. -
Quantum Time Machines Are Expectation Values of the Scalar field Squared and Stress- Quantum-Mechanically Stable
Quantum time machine Pedro F. Gonz´alez-D´ıaz Centro de F´ısica “Miguel Catal´an”, Instituto de Matem´aticas y F´ısica Fundamental, Consejo Superior de Investigaciones Cient´ıficas, Serrano 121, 28006 Madrid (SPAIN) (December 6, 1997) The continuation of Misner space into the Euclidean region is seen to imply the topological re- striction that the period of the closed spatial direction becomes time-dependent. This restriction results in a modified Lorentzian Misner space in which the renormalized stress-energy tensor for quantized complex massless scalar fields becomes regular everywhere, even on the chronology hori- zon. A quantum-mechanically stable time machine with just the sub-microscopic size may then be constructed out of the modified Misner space, for which the semiclassical Hawking’s chronology protection conjecture is no longer an obstruction. PACS number(s): 04.20.Gz, 04.62.+v I. INTRODUCTION some attempts intended to violate it [14,15], this con- jecture has survived rather forcefully [16]. However, the After the seminal papers by Morris, Thorne and Yurt- realm where chronology protection holds is semiclassical sever [1], the notion of a time machine has jumped from physics, as it is for all hitherto proposed time machines. the pages of science fiction books to those of scientific Actually, because spacetime foam [17] must entail strong journals, giving rise to a recent influx of papers [2-7] violations of causal locality everywhere, one would expect and books [8,9] on the subject. Several potentially use- the Hawking’s conjecture to be inapplicable in the frame- ful models for time machines have since been proposed, work of quantum gravity proper, and that the divergences including the wormhole of Thorne et al.