Einstein Field Equations for Diagonal, Vacuum, Separable Metrics
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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
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. -
Schwarzschild's Solution And
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server RECONSIDERING SCHWARZSCHILD'S ORIGINAL SOLUTION SALVATORE ANTOCI AND DIERCK EKKEHARD LIEBSCHER Abstract. We analyse the Schwarzschild solution in the context of the historical development of its present use, and explain the invariant definition of the Schwarzschild’s radius as a singular surface, that can be applied to the Kerr-Newman solution too. 1. Introduction: Schwarzschild's solution and the \Schwarzschild" solution Nowadays simply talking about Schwarzschild’s solution requires a pre- liminary reassessment of the historical record as conditio sine qua non for avoiding any misunderstanding. In fact, the present-day reader must be firstly made aware of this seemingly peculiar circumstance: Schwarzschild’s spherically symmetric, static solution [1] to the field equations of the ver- sion of the theory proposed by Einstein [2] at the beginning of November 1915 is different from the “Schwarzschild” solution that is quoted in all the textbooks and in all the research papers. The latter, that will be here al- ways mentioned with quotation marks, was found by Droste, Hilbert and Weyl, who worked instead [3], [4], [5] by starting from the last version [6] of Einstein’s theory.1 As far as the vacuum is concerned, the two versions have identical field equations; they differ only because of the supplementary condition (1.1) det (g )= 1 ik − that, in the theory of November 11th, limited the covariance to the group of unimodular coordinate transformations. Due to this fortuitous circum- stance, Schwarzschild could not simplify his calculations by the choice of the radial coordinate made e.g. -
General Relativity As Taught in 1979, 1983 By
General Relativity As taught in 1979, 1983 by Joel A. Shapiro November 19, 2015 2. Last Latexed: November 19, 2015 at 11:03 Joel A. Shapiro c Joel A. Shapiro, 1979, 2012 Contents 0.1 Introduction............................ 4 0.2 SpecialRelativity ......................... 7 0.3 Electromagnetism. 13 0.4 Stress-EnergyTensor . 18 0.5 Equivalence Principle . 23 0.6 Manifolds ............................. 28 0.7 IntegrationofForms . 41 0.8 Vierbeins,Connections . 44 0.9 ParallelTransport. 50 0.10 ElectromagnetisminFlatSpace . 56 0.11 GeodesicDeviation . 61 0.12 EquationsDeterminingGeometry . 67 0.13 Deriving the Gravitational Field Equations . .. 68 0.14 HarmonicCoordinates . 71 0.14.1 ThelinearizedTheory . 71 0.15 TheBendingofLight. 77 0.16 PerfectFluids ........................... 84 0.17 Particle Orbits in Schwarzschild Metric . 88 0.18 AnIsotropicUniverse. 93 0.19 MoreontheSchwarzschild Geometry . 102 0.20 BlackHoleswithChargeandSpin. 110 0.21 Equivalence Principle, Fermions, and Fancy Formalism . .115 0.22 QuantizedFieldTheory . .121 3 4. Last Latexed: November 19, 2015 at 11:03 Joel A. Shapiro Note: This is being typed piecemeal in 2012 from handwritten notes in a red looseleaf marked 617 (1983) but may have originated in 1979 0.1 Introduction I am, myself, an elementary particle physicist, and my interest in general relativity has come from the growth of a field of quantum gravity. Because the gravitational inderactions of reasonably small objects are so weak, quantum gravity is a field almost entirely divorced from contact with reality in the form of direct confrontation with experiment. There are three areas of contact 1. In relativistic quantum mechanics, one usually formulates the physical quantities in terms of fields. A field is a physical degree of freedom, or variable, definded at each point of space and time. -
General Relativity
GENERALRELATIVITY t h i m o p r e i s1 17th April 2020 1 [email protected] CONTENTS 1 differential geomtry 3 1.1 Differentiable manifolds 3 1.2 The tangent space 4 1.2.1 Tangent vectors 6 1.3 Dual vectors and tensor 6 1.3.1 Tensor densities 8 1.4 Metric 11 1.5 Connections and covariant derivatives 13 1.5.1 Note on exponential map/Riemannian normal coordinates - TO DO 18 1.6 Geodesics 20 1.6.1 Equivalent deriavtion of the Geodesic Equation - Weinberg 22 1.6.2 Character of geodesic motion is sustained and proper time is extremal on geodesics 24 1.6.3 Another remark on geodesic equation using the principle of general covariance 26 1.6.4 On the parametrization of the path 27 1.7 An equivalent consideration of parallel transport, geodesics 29 1.7.1 Formal solution to the parallel transport equa- tion 31 1.8 Curvature 33 1.8.1 Torsion and metric connection 34 1.8.2 How to get from the connection coefficients to the connection-the metric connection 34 1.8.3 Conceptional flow of how to add structure on our mathematical constructs 36 1.8.4 The curvature 37 1.8.5 Independent components of the Riemann tensor and intuition for curvature 39 1.8.6 The Ricci tensor 41 1.8.7 The Einstein tensor 43 1.9 The Lie derivative 43 1.9.1 Pull-back and Push-forward 43 1.9.2 Connection between coordinate transformations and diffeomorphism 47 1.9.3 The Lie derivative 48 1.10 Symmetric Spaces 50 1.10.1 Killing vectors 50 1.10.2 Maximally Symmetric Spaces and their Unique- ness 54 iii iv co n t e n t s 1.10.3 Maximally symmetric spaces and their construc- tion -
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
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
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 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
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
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" 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.