DOCSLIB.ORG

Black Holes Are Quantum Complete

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

Black holes are quantum complete Marc Michael Schneider Munchen¨ 2018 Black holes are quantum complete Marc Michael Schneider Dissertation an der Fakult¨at fur¨ Physik der Ludwig{Maximilians{Universit¨at Munchen¨ vorgelegt von Marc Michael Schneider aus Trier Munchen,¨ den 6. August 2018 Erstgutachter: Prof. Dr. Stefan Hofmann Zweitgutachter: Prof. Dr. Peter Mayr Tag der mundlichen¨ Prufung:¨ 14. September 2018 Only those who will risk going too far can possibly find out how far one can go. { Thomas Stearns Eliot { (Preface to Transit of Venus: Poems) Contents Zusammenfassung xi Abstract xiii 1. Concerning singularities 1 2. The fellowship of completeness 7 2.1. Classical completeness . .7 2.2. Quantum mechanical completeness . .9 2.3. Quantum-mechanical versus classical completeness . 14 2.4. Geodesic completeness . 17 2.5. Quantum-mechanical probes of space-time singularities . 21 3. The two singularities 27 3.1. The gravitational singularity . 28 3.2. The generalised Kasner singularity . 36 4. The return of regularity 45 4.1. The Schr¨odinger representation of quantum field theory . 46 4.1.1. Functional calculus . 47 4.1.2. Flat space-time formulation . 50 4.1.3. Curved space-time formulation . 59 4.2. The quantum completeness criterion . 66 viii Contents 4.3. Quantum probing of Schwarzschild . 72 4.3.1. Ground state analysis . 73 4.3.2. Gaussian deviations: Excited states . 79 4.3.3. Influence of polynomial self-interactions . 84 4.3.4. Stress-energy tensor of quantum probes . 91 4.4. Charge conservation inside the black hole . 96 4.5. Quantum probing of the Kasner space-time . 103 4.5.1. Ground state analysis . 104 4.5.2. Mode functions . 106 5. Conclusion 109 A. Fourier transformation 115 B. Riccati differential equation 117 C. Publication: Classical versus quantum completeness 119 D. Publication: Non-Gaussian ground-state deformations near a black-hole singu- larity 127 E. Publication: Information carriers closing in on the black-hole singularity 135 F. Kasner analysis and connexion to the Schwarzschild case 141 G. Heisenberg analysis of charge conservation inside a black hole 143 Acknowledgements 159 List of figures 2.1. Potential V(x) with steps. 15 2.2. Potential V(x) with spikes. 17 2.3. Penrose diagram of the negative mass Schwarzschild solution . 25 3.1. Penrose diagram of the Schwarzschild solution . 29 3.2. Penrose diagram of the fully extended Schwarzschild solution. 32 4.1. Relation between functions f(x) and functionals F[f]............. 47 4.2. Schematical illustration of the scattering operator from two to n particles. 60 4.3. Plot of the normalisation N(0)(t) for t (0; 0:25M) ............. 78 2 4.4. Plot of the stress-energy tensor T00 for N(Λ) f1; 5g and t [0; 0:25] .. 93 h i 2 2 4.5. Plot of the stress-energy tensor T00 for t [0; 0:5].............. 95 h i 2 4.6. Plot of the probabilistic current. 102 Zusammenfassung Vollst¨andigkeit ist ein ¨außerst wichtiges Konzept in der theoretischen Physik. Die Haupt- idee besagt, dass eine Bewegung oder ein Freiheitsgrad eindeutig und fur¨ alle Zeiten de- finiert ist. Die Kriterien, die Vollst¨andigkeit in klassischen und quantenmechanischen Theo- rien beschreiben, sind je nach Theorie unterschiedlich. Sie sind eng mit dem spezifischen Messprozess der jeweiligen Theorie verknupft.¨ Unvollst¨andigkeit geht oft mit der Pr¨asenz einer Singularit¨at einher. Die Existenz einer Singularit¨at ist untrennbar gegeben durch die Freiheitsgrade der Theorie. In Allgemeiner Relativit¨atstheorie sind Raumzeiten charakterisiert durch die L¨ange geo- d¨atischer Kurven. Dieses Kriterium fußt auf der klassischen Punktteilchenbeschreibung mittels Differentialgeometrie und fuhrte¨ zur Entwicklung der Singularit¨atentheoreme nach Hawking und Penrose. Ein Quantendetektor in einer dynamischen Raumzeit kann nicht durch Quantenmecha- nik im engeren Sinne beschrieben werden, denn es ist nicht m¨oglich eine konsistente re- lativistische Einteilcheninterpretation einer Quantentheorie zu formulieren. Infolgedessen ubertragen¨ wir den Terminus der Vollst¨andigkeit auf Situationen deren einzig ad¨aquate Be- schreibung durch Quantenfeldtheorie auf gekrummten¨ Raumzeiten erfolgen kann. Um Uni- tarit¨atsverletzungen, welche sich in endlicher Zeit ereignen, aufl¨osen zu k¨onnen, nutzen wir die Schr¨odingerdarstellung der Quantenfeldtheorie, da diese eine Zeitaufl¨osung erm¨oglicht. Sinnhaftigkeit der Persistenzamplitude des Wellenfunktionales, d.h. Wahrscheinlichkeits- verlust oder Stabilit¨at werden mit Vollst¨andigkeit in Verbindung gebracht. Anhand eines schwarzen Loches der Schwarzschildgattung wenden wir unser Kriteri- um an und testen Vereinbarkeit mit freien, massiven Skalarfeldern. Abweichungen vom xii Zusammenfassung Gauß’schen Ansatz fur¨ das Wellenfunktional, angeregte Zust¨ande und Selbstwechselwir- kung der Testfelder, werden betrachtet und deren gutartige Entwicklung wird gezeigt. Die Analyse wird auf eine weitere Klasse von Raumzeiten, den Kasner-Raumzeiten an- gewandt, die hohe Relevanz durch die Vermutung von Belinskii, Khalatnikov und Lifshitz haben. Abstract Completeness is a very important concept in theoretical physics. The main idea is that the motion or the degree of freedom is uniquely defined for all times. The criterion for completeness is different for classical and quantum theories. This corresponds to a specific measurement process in the corresponding theories. Incompleteness is often related to the occurrence of a singularity. The notion of a singularity is closely related to the corresponding degree of freedom. In general relativity space-times are characterised by the extendibility of geodesic curves. This criterion founded on the point particle description through differential geometry has given rise to the singularity theorems of Hawking and Penrose. A quantum probing of dynamical space-times can not be described with quantum me- chanics because there is no consistent relativistic one particle interpretation of a quantum theory. Hence, we extend the notion for completeness to situations where the only adequate description is in terms of quantum field theory on curved space-times. In order to analyse unitarity violations occurring during a finite time, we use the Schr¨odinger representation of quantum field theory which allow for time resolution. Consistency of the wave-functional's persistency amplitude, i.e. probability loss or stability, will be connected to completeness. For a Schwarzschild type back hole we apply the criterion and probe with free massive scalar fields for consistency. Furthermore, deviations from Gaussianity, i.e. excited states and self-interaction of the probing fields are derived and consistency is showed for those deformations. The analysis is furthermore applied to Kasner space-times, which have high relevance due to the conjecture of Belinskii, Khalatnikov, and Lifshitz. 1 Concerning singularities Considering the case that an random person on the street asks you about black holes and the big bang, most people which are educated in science would inevitably come to the point where they have to talk about singularities, but this will trigger the next question: What is a singularity?\. The concept is known to every physicist, although the notion is " not very familiar to non-physicists or non-mathematicians because it has no equivalence in their daily life, at least, however, it seems to be important in order to understand what a black hole or the big bang is. During a day without thoughts about physics or mathematics the concept of singularities is not needed. From a heuristic point of view one would deny that a system in nature becomes singular, e.g. reaches an infinite value of energy. Amongst physicists, it is clear that the occurrence of a singularity is equivalent of having a severe problem our theory. The black hole and the big bang singularities are a direct outcome of Einstein's equations, hence, they are predicted to occur by general relativity. It is, however, not clear that these solutions can be reached dynamically. Hence, they could also be a mathematical artefact. Usually the term singularity\ is connected to the situation where at least one observable " could be measured to grow unbounded. Experimentalists have never reported an infinite value of a measurable quantity. This fact motivates to question the physical justification for the concept. Let us recall the definition of singularities in mathematics. The definition of a singular point is [Simon, 2015a]: Definition 1. If Ω is a region, f A(Ω) (all analytic functions on Ω) and z0 ∂Ω, we 2 2 say that f is regular at z0 if and only if there is δ > 0, g analytic in Dδ(z0) (disk of radius δ about z0) so that f(z) = g(z) for all z Ω Dδ(z0). If f is not regular at z0, we say that 2 \ 2 1. Concerning singularities f is singular at z0. The above definition says that there exists a point z0 for which the function acquires an infinite value: f(z0) > M; M R. 8 2 The language of physics is mathematics, therefore, it is very natural that mathematical concepts are also present in physics. Singularities appear in different theories; in some they signal a breakdown of the description. Before we specify the notion in physical examples we want to elucidate what is generally meant by breakdown of the description\. Physical " theories are devoted to specific energy (or length) scales, below this energy scales the theory is effectively describing the system while beyond the scale the theory loses its predictability. In general, the pure presence of a singularity starts to become a problem only in conjunc- tion with the possibility to dynamically reach the singularity in a finite amount of time. Considering the mathematical definition, the first singularity, ever deduced in physics, oc- curred in the theory of gravitation. Newton's law of the attraction between two massive mM bodies admits a singular value at the origin. The formula F = -GN r2 suggests that the force at r = 0 will be infinitely strong. This is clearly a pathology of the theory signalling its breakdown close to this point.
Recommended publications
  • 5-Dimensional Space-Periodic Solutions of the Static Vacuum Einstein Equations

    5-Dimensional Space-Periodic Solutions of the Static Vacuum Einstein Equations

    5-DIMENSIONAL SPACE-PERIODIC SOLUTIONS OF THE STATIC VACUUM EINSTEIN EQUATIONS MARCUS KHURI, GILBERT WEINSTEIN, AND SUMIO YAMADA Abstract. An affirmative answer is given to a conjecture of Myers concerning the existence of 5-dimensional regular static vacuum solutions that balance an infinite number of black holes, which have Kasner asymp- totics. A variety of examples are constructed, having different combinations of ring S1 × S2 and sphere S3 cross-sectional horizon topologies. Furthermore, we show the existence of 5-dimensional vacuum solitons with Kasner asymptotics. These are regular static space-periodic vacuum spacetimes devoid of black holes. Consequently, we also obtain new examples of complete Riemannian manifolds of nonnegative Ricci cur- vature in dimension 4, and zero Ricci curvature in dimension 5, having arbitrarily large as well as infinite second Betti number. 1. Introduction The Majumdar-Papapetrou solutions [14, 16] of the 4D Einstein-Maxwell equations show that gravi- tational attraction and electro-magnetic repulsion may be balanced in order to support multiple black holes in static equilibrium. In [15], Myers generalized these solutions to higher dimensions, and also showed that such balancing is also possible for vacuum black holes in 4 dimensions if an infinite number are aligned properly in a periodic fashion along an axis of symmetry; these 4-dimensional solutions were later rediscovered by Korotkin and Nicolai in [11]. Although the Myers-Korotkin-Nicolai solutions are not asymptotically flat, but rather asymptotically Kasner, they play an integral part in an extended version of static black hole uniqueness given by Peraza and Reiris [17]. It was conjectured in [15] that these vac- uum solutions could be generalized to higher dimensions, possibly with black holes of nontrivial topology.
  • Supergravity Pietro Giuseppe Frè Dipartimento Di Fisica Teorica University of Torino Torino, Italy

    Supergravity Pietro Giuseppe Frè Dipartimento Di Fisica Teorica University of Torino Torino, Italy

    Gravity, a Geometrical Course Pietro Giuseppe Frè Gravity, a Geometrical Course Volume 2: Black Holes, Cosmology and Introduction to Supergravity Pietro Giuseppe Frè Dipartimento di Fisica Teorica University of Torino Torino, Italy Additional material to this book can be downloaded from http://extras.springer.com. ISBN 978-94-007-5442-3 ISBN 978-94-007-5443-0 (eBook) DOI 10.1007/978-94-007-5443-0 Springer Dordrecht Heidelberg New York London Library of Congress Control Number: 2012950601 © Springer Science+Business Media Dordrecht 2013 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc.
  • The Physics of Star Trek the Physics of Star Trek

    The Physics of Star Trek the Physics of Star Trek

    The Physics of Star Trek The Physics of Star Trek The Physics of Star Trek The Physics of Star Trek FOREWORD Stephen Hawking I was very pleased that Data decided to call Newton, Einstein, and me for a game of poker aboard the Enterprise. Here was my chance to turn the tables on the two great men of gravity, particularly Einstein, who didn't believe in chance or in God playing dice. Unfortunately, I never collected my winnings because the game had to be abandoned on account of a red alert. I contacted Paramount studios afterward to cash in my chips, but they didn't know the exchange rate. Science fiction like Star Trek is not only good fun but it also serves a serious purpose, that of expanding the human imagination. We may not yet be able to boldly go where no man (or woman) has gone before, but at least we can do it in the mind. We can explore how the human spirit might respond to future developments in science and we can speculate on what those developments might be. There is a two-way trade between science fiction and science. Science fiction suggests ideas that scientists incorporate into their theories, but sometimes science turns up notions that are stranger than any science fiction. Black holes are an example, greatly assisted by the inspired name that the physicist John Archibald Wheeler gave them. Had they continued with their original names of “frozen stars” or “gravitationally completely collapsed objects,” there wouldn't have been half so much written about them.
  • Spacetime Geometry from Graviton Condensation: a New Perspective on Black Holes

    Spacetime Geometry from Graviton Condensation: a New Perspective on Black Holes

    Spacetime Geometry from Graviton Condensation: A new Perspective on Black Holes Sophia Zielinski née Müller München 2015 Spacetime Geometry from Graviton Condensation: A new Perspective on Black Holes Sophia Zielinski née Müller Dissertation an der Fakultät für Physik der Ludwig–Maximilians–Universität München vorgelegt von Sophia Zielinski geb. Müller aus Stuttgart München, den 18. Dezember 2015 Erstgutachter: Prof. Dr. Stefan Hofmann Zweitgutachter: Prof. Dr. Georgi Dvali Tag der mündlichen Prüfung: 13. April 2016 Contents Zusammenfassung ix Abstract xi Introduction 1 Naturalness problems . .1 The hierarchy problem . .1 The strong CP problem . .2 The cosmological constant problem . .3 Problems of gravity ... .3 ... in the UV . .4 ... in the IR and in general . .5 Outline . .7 I The classical description of spacetime geometry 9 1 The problem of singularities 11 1.1 Singularities in GR vs. other gauge theories . 11 1.2 Defining spacetime singularities . 12 1.3 On the singularity theorems . 13 1.3.1 Energy conditions and the Raychaudhuri equation . 13 1.3.2 Causality conditions . 15 1.3.3 Initial and boundary conditions . 16 1.3.4 Outlining the proof of the Hawking-Penrose theorem . 16 1.3.5 Discussion on the Hawking-Penrose theorem . 17 1.4 Limitations of singularity forecasts . 17 2 Towards a quantum theoretical probing of classical black holes 19 2.1 Defining quantum mechanical singularities . 19 2.1.1 Checking for quantum mechanical singularities in an example spacetime . 21 2.2 Extending the singularity analysis to quantum field theory . 22 2.2.1 Schrödinger representation of quantum field theory . 23 2.2.2 Quantum field probes of black hole singularities .
  • A Domain Wall Solution by Perturbation of the Kasner Spacetime

    A Domain Wall Solution by Perturbation of the Kasner Spacetime

    A Domain Wall Solution by Perturbation of the Kasner Spacetime George Kotsopoulos 1906-373 Front St. West, Toronto, ON M5V 3R7 [email protected] and Charles C. Dyer Dept. of Physical and Environmental Sciences, University of Toronto Scarborough, and Dept. of Astronomy and Astrophysics, University of Toronto, Toronto, Canada [email protected] January 24, 2012 Received ; accepted arXiv:1201.5362v1 [gr-qc] 25 Jan 2012 –2– Abstract Plane symmetric perturbations are applied to an axially symmetric Kasner spacetime which leads to no momentum flow orthogonal to the planes of symme- try. This flow appears laminar and the structure can be interpreted as a domain wall. We further extend consideration to the class of Bianchi Type I spacetimes and obtain corresponding results. Subject headings: general relativity, Kasner metric, Bianchi Type I, domain wall, perturbation –3– 1. Finding New Solutions There are three principal exact solutions to the Einstein Field Equations (EFE) that are most relevant for the description of astrophysical phenomena. They are the Schwarzschild (internal and external), Kerr and Friedman-LeMaître-Robertson-Walker (FLRW) models. These solutions are all highly idealized and involve the introduction of simplifying assumptions such as symmetry. The technique we use which has led to a solution of the EFE involves the perturbation of an already symmetric solution (Wilson and Dyer 2007). Whereas standard perturbation techniques involve specifying the forms of the energy-momentum tensor to determine the form of the metric, we begin by defining a new metric with an applied perturbation: gab =g ˜ab + hab (1) where g˜ab is a known symmetric metric and hab is the applied perturbation.
  • Analog Geometry in an Expanding Fluid from Ads/CFT Perspective

    Analog Geometry in an Expanding Fluid from Ads/CFT Perspective

    Physics Letters B 743 (2015) 340–346 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb Analog geometry in an expanding fluid from AdS/CFT perspective ∗ Neven Bilic´ , Silvije Domazet, Dijana Tolic´ Division of Theoretical Physics, Rudjer Boškovi´c Institute, P.O. Box 180, 10001 Zagreb, Croatia a r t i c l e i n f o a b s t r a c t Article history: The dynamics of an expanding hadron fluid at temperatures below the chiral transition is studied in Received 16 February 2015 the framework of AdS/CFT correspondence. We establish a correspondence between the asymptotic AdS Accepted 5 March 2015 geometry in the 4 + 1dimensional bulk with the analog spacetime geometry on its 3 + 1dimensional Available online 9 March 2015 boundary with the background fluid undergoing a spherical Bjorken type expansion. The analog metric Editor: L. Alvarez-Gaumé tensor on the boundary depends locally on the soft pion dispersion relation and the four-velocity of Keywords: the fluid. The AdS/CFT correspondence provides a relation between the pion velocity and the critical Analogue gravity temperature of the chiral phase transition. AdS/CFT correspondence © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license 3 Chiral phase transition (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP . Linear sigma model Bjorken expansion 1. Introduction found in the QCD vacuum and its non-vanishing value is a man- ifestation of chiral symmetry breaking [8]. The chiral symmetry The original formulation of gauge-gravity duality in the form of is restored at finite temperature through a chiral phase transition the so called AdS/CFT correspondence establishes an equivalence of which is believed to be of first or second order depending on the a four dimensional N = 4 supersymmetric Yang–Mills theory and underlying global symmetry [9].
  • Particle Production in the Central Rapidity Region

    Particle Production in the Central Rapidity Region

    LA-UR-92-3856 November, 1992 Particle production in the central rapidity region F. Cooper,1 J. M. Eisenberg,2 Y. Kluger,1;2 E. Mottola,1 and B. Svetitsky2 1Theoretical Division and Center for Nonlinear Studies Los Alamos National Laboratory Los Alamos, New Mexico 87545 USA 2School of Physics and Astronomy Raymond and Beverly Sackler Faculty of Exact Sciences Tel Aviv University, 69978 Tel Aviv, Israel ABSTRACT We study pair production from a strong electric ¯eld in boost- invariant coordinates as a simple model for the central rapidity region of a heavy-ion collision. We derive and solve the renormal- ized equations for the time evolution of the mean electric ¯eld and current of the produced particles, when the ¯eld is taken to be a function only of the fluid proper time ¿ = pt2 z2. We ¯nd that a relativistic transport theory with a Schwinger¡source term mod- i¯ed to take Pauli blocking (or Bose enhancement) into account gives a good description of the numerical solution to the ¯eld equations. We also compute the renormalized energy-momentum tensor of the produced particles and compare the e®ective pres- sure, energy and entropy density to that expected from hydrody- namic models of energy and momentum flow of the plasma. 1 Introduction A popular theoretical picture of high-energy heavy-ion collisions begins with the creation of a flux tube containing a strong color electric ¯eld [1]. The ¯eld energy is converted into particles as qq¹ pairs and gluons are created by the Schwinger tunneling mechanism [2, 3, 4]. The transition from this quantum tunneling stage to a later hydrodynamic stage has previously been described phenomenologically using a kinetic theory model in which a rel- ativistic Boltzmann equation is coupled to a simple Schwinger source term [5, 6, 7, 8].
  • On Limiting Curvature in Mimetic Gravity

    On Limiting Curvature in Mimetic Gravity

    On limiting curvature in mimetic gravity Tobias Benjamin Russ München 2021 On limiting curvature in mimetic gravity Tobias Benjamin Russ Dissertation an der Fakultät für Physik der Ludwig–Maximilians–Universität München vorgelegt von Tobias Benjamin Russ aus Güssing, Österreich München, den 11. Mai 2021 Erstgutachter: Prof. Dr. Viatcheslav Mukhanov, LMU München Zweitgutachter: Prof. Dr. Paul Steinhardt, Princeton University Tag der mündlichen Prüfung: 8. Juli 2021 Contents Zusammenfassung . vii Abstract . ix Acknowledgements . xi Publications . xiii Summary 1 0.1 Introduction . .1 0.2 The theory . .7 0.3 Results & Conclusions . 12 0.3.1 Big Bang replaced by initial dS . 12 0.3.2 Anisotropic singularity resolution requires asymptotic freedom . 17 0.3.3 Modified black hole with stable remnant . 20 0.3.4 Higher spatial derivatives avert gradient instability . 26 0.4 Outlook . 30 1 Asymptotically Free Mimetic Gravity 31 1.1 Introduction . 32 1.2 Action and equations of motion . 33 1.3 The synchronous coordinate system . 33 1.4 Asymptotic freedom and the fate of a collapsing universe . 35 1.5 Quantum fluctuations . 39 1.6 Conclusions . 40 2 Mimetic Hořava Gravity 43 3 Black Hole Remnants 49 3.1 Introduction . 50 3.2 The Lemaître coordinates . 50 3.3 Modified Mimetic Gravity . 51 3.4 Asymptotic Freedom at Limiting Curvature . 52 3.5 Exact Solution . 53 3.6 Black hole thermodynamics . 55 3.7 Conclusions . 56 vi Contents 4 Non-flat Universes and Black Holes in A.F.M.G. 57 4.1 The Theory . 58 4.1.1 Introduction . 58 4.1.2 Action and equations of motion .
  • Symbols 1-Form, 10 4-Acceleration, 184 4-Force, 115 4-Momentum, 116

    Symbols 1-Form, 10 4-Acceleration, 184 4-Force, 115 4-Momentum, 116

    Index Symbols B 1-form, 10 Betti number, 589 4-acceleration, 184 Bianchi type-I models, 448 4-force, 115 Bianchi’s differential identities, 60 4-momentum, 116 complex valued, 532–533 total, 122 consequences of in Newman-Penrose 4-velocity, 114 formalism, 549–551 first contracted, 63 second contracted, 63 A bicharacteristic curves, 620 acceleration big crunch, 441 4-acceleration, 184 big-bang cosmological model, 438, 449 Newtonian, 74 Birkhoff’s theorem, 271 action function or functional bivector space, 486 (see also Lagrangian), 594, 598 black hole, 364–433 ADM action, 606 Bondi-Metzner-Sachs group, 240 affine parameter, 77, 79 boost, 110 alternating operation, antisymmetriza- Born-Infeld (or tachyonic) scalar field, tion, 27 467–471 angle field, 44 Boyer-Lindquist coordinate chart, 334, anisotropic fluid, 218–220, 276 399 collapse, 424–431 Brinkman-Robinson-Trautman met- ric, 514 anti-de Sitter space-time, 195, 644, Buchdahl inequality, 262 663–664 bugle, 69 anti-self-dual, 671 antisymmetric oriented tensor, 49 C antisymmetric tensor, 28 canonical energy-momentum-stress ten- antisymmetrization, 27 sor, 119 arc length parameter, 81 canonical or normal forms, 510 arc separation function, 85 Cartesian chart, 44, 68 arc separation parameter, 77 Casimir effect, 638 Arnowitt-Deser-Misner action integral, Cauchy horizon, 403, 416 606 Cauchy problem, 207 atlas, 3 Cauchy-Kowalewski theorem, 207 complete, 3 causal cone, 108 maximal, 3 causal space-time, 663 698 Index 699 causality violation, 663 contravariant index, 51 characteristic hypersurface, 623 contravariant
  • Arxiv:2008.13699V3 [Hep-Th] 18 Mar 2021 1 Rsn Drs:Dprmn Fpyis Ea Ae Olg,S College, Patel Vesaj Physics, of Department Address: Present Xadn Udi H Aetm Regime

    Arxiv:2008.13699V3 [Hep-Th] 18 Mar 2021 1 Rsn Drs:Dprmn Fpyis Ea Ae Olg,S College, Patel Vesaj Physics, of Department Address: Present Xadn Udi H Aetm Regime

    Anisotropic expansion, second order hydrodynamics and holographic dual Priyanka Priyadarshini Pruseth 1 and Swapna Mahapatra Department of Physics, Utkal University, Bhubaneswar 751004, India. [email protected] , [email protected] Abstract We consider Kasner space-time describing anisotropic three dimensional expan- sion of RHIC and LHC fireball and study the generalization of Bjorken’s one dimensional expansion by taking into account second order relativistic viscous hydrodynamics. Using time dependent AdS/CFT correspondence, we study the late time behaviour of the Bjorken flow. From the conditions of conformal invariance and energy-momentum conservation, we obtain the explicit expres- sion for the energy density as a function of proper time in terms of Kasner parameters. The proper time dependence of the temperature and entropy have also been obtained in terms of Kasner parameters. We consider Eddington- Finkelstein type coordinates and discuss the gravity dual of the anisotropically expanding fluid in the late time regime. arXiv:2008.13699v3 [hep-th] 18 Mar 2021 1Present address: Department of Physics, Vesaj Patel College, Sundargarh, India 1 Introduction AdS/CFT correspondence has provided a very important tool for studying the strongly coupled dynamics in a class of superconformal field theories, in particular, = 4 super N Yang-Mills theory and the corresponding gravity dual description in AdS space-time [1, 2]. One of the fundamental question in the field of high energy physics is to understand the properties of matter at extreme density and temperature in the first few microseconds after the big bang. Such a state of matter is known as Quark-Gluon-Plasma (QGP) state where the quarks and the gluons are in deconfined state.
  • "Behaviour of the Gravitational Field Around the Initial Singularity"

    "Behaviour of the Gravitational Field Around the Initial Singularity"

    "BEHAVIOUR OF THE GRAVITATIONAL FIELD AROUND THE INITIAL SINGULARITY" Gerson Francisco. t ? õ INSTIT0T0 01 flSICA TEÓRICA "BEHAVIOUR OF THE GRAVITATIONAL FIELD AROUND THE INITIAL SINGULARITY" Gerson Francisco Instituto de Física Teórica Rua Pamplona, 145 01405 - São Paulo-SP BRASIL. Abstract The ultralocal representation of the canonical^ ly quantized gravitational field is used to obtain the evolution of coherent states in the immediate neighborhood of the singulari- ty. It is shown that smearing functions play the role of classical fields since they correspond to cosrnological solutions around the singularity. A special class of ultralocal coherent states is shown to contain the essential aspects of the dynamics of the system when we choose a simple representation for the field operators of the theory. When the ultralocality condition is broken a conjec- ture will be made about the quantum evolution of coherent states in the classical limit. i-í . Intiüd^': I ion Cosmologica.1 solutions to Einstein equations a-o'jnd the Initial singularity have been thouroughly investigated _y b^Lin^kii, KhaiaLnikov and Lifshitz |1—3| (or BKL t\u short). [n fiLs connection ue also mention the u»ork of Eardley et al. |4| inn thu Hami ltoninr' tiaatment of Liang |5|. The main point of theso st'ciRs is ttip far:t thac a large class of solutions to Einstein rtn,iaf,ions \n r.ne asymptotic region close to the singularity exhi- ;•>: ' :. kifnj cr ?\.>antaneously decoupled dynamics in the sense that field ewoJution at a given initial point on a Cauchy hypersur- 5^ is totally independent from what occurs in its neighbor- hood.
  • STARFLEET COMMAND's SEVENTH FLEET Advanced Technical

    STARFLEET COMMAND's SEVENTH FLEET Advanced Technical

    STARFLEET COMMAND’S SEVENTH FLEET U.S.S. White Buffalo - NCC 80116 Advanced Technical Specifications for the Luna-Class Production Vehicle U.S.S. White STARFLEET COMMAND’S SEVENTH FLEET Buffalo NCC 80116 Quantum Singularity Core + Slipstream Drive Information Packet Revised: 07 - 2018 | U.S.S. White Buffalo - N.C.C. 80116 (Revised: 07-2018) STARFLEET COMMAND’S SEVENTH FLEET U.S.S. White Buffalo - NCC 80116 Quantum Singularity Core + Slipstream Drive Information Packet Quantum Singularity Core Intro: Since I started working on the White Buffalo’s QSC, I discovered that everything I knew about Quantum Singularities and how they could be manipulated to create enough power to not only sustain a warp reaction, but that they could also power a vessel as large as, or larger than, 1,041.65 Meters Long, 772.43 Meters Width, 285.47 Meters Tall, and with a Mass of 4,320,000 Metric Tons. With this kind of power a Federation Starship would be able to maintain a more stable warp reaction, power all of its systems at one time, (Provided your power network could also maintain the power flow) and even maintain power for systems outside of the ship, like shields, tractor beam, the deflector dish, or much more. I also discovered that few onboard understood the propulsion mechanics involved with the QSC, Including myself. Having to adapt to a power core, and warp system like this because the Orion Syndicate ripped it from a Romulan Warbird and placed it into a Federation Aerie Class starship was interesting enough, but then to figure out how to use it without destroying not only the ship, but possibly the fabric of space-time itself was almost daunting to say the least.