Einstein Equation at Singularities
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Modified Theories of Relativistic Gravity
Modified Theories of Relativistic Gravity: Theoretical Foundations, Phenomenology, and Applications in Physical Cosmology by c David Wenjie Tian A thesis submitted to the School of Graduate Studies in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Theoretical Physics (Interdisciplinary Program) Faculty of Science Date of graduation: October 2016 Memorial University of Newfoundland March 2016 St. John’s Newfoundland and Labrador Abstract This thesis studies the theories and phenomenology of modified gravity, along with their applications in cosmology, astrophysics, and effective dark energy. This thesis is organized as follows. Chapter 1 reviews the fundamentals of relativistic gravity and cosmology, and Chapter 2 provides the required Co-authorship 2 2 Statement for Chapters 3 ∼ 6. Chapter 3 develops the L = f (R; Rc; Rm; Lm) class of modified gravity 2 µν 2 µανβ that allows for nonminimal matter-curvature couplings (Rc B RµνR , Rm B RµανβR ), derives the “co- herence condition” f 2 = f 2 = − f 2 =4 for the smooth limit to f (R; G; ) generalized Gauss-Bonnet R Rm Rc Lm gravity, and examines stress-energy-momentum conservation in more generic f (R; R1;:::; Rn; Lm) grav- ity. Chapter 4 proposes a unified formulation to derive the Friedmann equations from (non)equilibrium (eff) thermodynamics for modified gravities Rµν − Rgµν=2 = 8πGeffTµν , and applies this formulation to the Friedman-Robertson-Walker Universe governed by f (R), generalized Brans-Dicke, scalar-tensor-chameleon, quadratic, f (R; G) generalized Gauss-Bonnet and dynamical Chern-Simons gravities. Chapter 5 systemati- cally restudies the thermodynamics of the Universe in ΛCDM and modified gravities by requiring its com- patibility with the holographic-style gravitational equations, where possible solutions to the long-standing confusions regarding the temperature of the cosmological apparent horizon and the failure of the second law of thermodynamics are proposed. -
M-Eigenvalues of the Riemann Curvature Tensor of Conformally Flat Manifolds
Commun. Math. Res. Vol. 36, No. 3, pp. 336-353 doi: 10.4208/cmr.2020-0052 August 2020 M-Eigenvalues of the Riemann Curvature Tensor of Conformally Flat Manifolds Yun Miao1, Liqun Qi2 and Yimin Wei3,∗ 1 School of Mathematical Sciences, Fudan University, Shanghai 200433, P. R. of China. 2 Department of Applied Mathematics, the Hong Kong Polytechnic University, Hong Kong. 3 School of Mathematical Sciences and Shanghai Key Laboratory of Contemporary Applied Mathematics, Fudan University, Shanghai 200433, P. R. of China. Received 6 May 2020; Accepted 28 May 2020 Abstract. We investigate the M-eigenvalues of the Riemann curvature tensor in the higher dimensional conformally flat manifold. The expressions of M- eigenvalues and M-eigenvectors are presented in this paper. As a special case, M-eigenvalues of conformal flat Einstein manifold have also been discussed, and the conformal the invariance of M-eigentriple has been found. We also reveal the relationship between M-eigenvalue and sectional curvature of a Rie- mannian manifold. We prove that the M-eigenvalue can determine the Rie- mann curvature tensor uniquely. We also give an example to compute the M- eigentriple of de Sitter spacetime which is well-known in general relativity. AMS subject classifications: 15A48, 15A69, 65F10, 65H10, 65N22 Key words: M-eigenvalue, Riemann curvature tensor, Ricci tensor, conformal invariant, canonical form. ∗Corresponding author. Email addresses: [email protected], [email protected] (Y. Wei), [email protected] (Y. Miao), [email protected] (L. Qi) Y. Miao, L. Qi and Y. Wei / Commun. Math. Res., 36 (2020),pp.336-353 337 1 Introduction The eigenvalue problem is a very important topic in tensor computation [4]. -
New Class of Exact Solutions to Einstein-Maxwell-Dilaton Theory on Four-Dimensional Bianchi Type IX Geometry
New Class of Exact Solutions to Einstein-Maxwell-dilaton Theory on Four-dimensional Bianchi Type IX Geometry A Thesis Submitted to the College of Graduate and Postdoctoral Studies in Partial Fulfillment of the Requirements for the degree of Master of Science in the Department of Physics & Engineering Physics University of Saskatchewan Saskatoon By Bardia Hosseinzadeh Fahim c Bardia Hosseinzadeh Fahim, August 2020. All rights reserved. Permission to Use In presenting this thesis in partial fulfilment of the requirements for a Postgraduate degree from the University of Saskatchewan, I agree that the Libraries of this University may make it freely available for inspection. I further agree that permission for copying of this thesis in any manner, in whole or in part, for scholarly purposes may be granted by the professor or professors who supervised my thesis work or, in their absence, by the Head of the Department or the Dean of the College in which my thesis work was done. It is understood that any copying or publication or use of this thesis or parts thereof for financial gain shall not be allowed without my written permission. It is also understood that due recognition shall be given to me and to the University of Saskatchewan in any scholarly use which may be made of any material in my thesis. Requests for permission to copy or to make other use of material in this thesis in whole or part should be addressed to: Head of the Department of Physics & Engineering Physics 163 Physics Building 116 Science Place University of Saskatchewan Saskatoon, Saskatchewan Canada S7N 5E2 Or Dean College of Graduate and Postdoctoral Studies University of Saskatchewan 116 Thorvaldson Building, 110 Science Place Saskatoon, Saskatchewan S7N 5C9 Canada i Abstract We construct new classes of cosmological solution to the five dimensional Einstein-Maxwell-dilaton theory, that are non-stationary and almost conformally regular everywhere. -
Mathematics of General Relativity - Wikipedia, the Free Encyclopedia Page 1 of 11
Mathematics of general relativity - Wikipedia, the free encyclopedia Page 1 of 11 Mathematics of general relativity From Wikipedia, the free encyclopedia The mathematics of general relativity refers to various mathematical structures and General relativity techniques that are used in studying and formulating Albert Einstein's theory of general Introduction relativity. The main tools used in this geometrical theory of gravitation are tensor fields Mathematical formulation defined on a Lorentzian manifold representing spacetime. This article is a general description of the mathematics of general relativity. Resources Fundamental concepts Note: General relativity articles using tensors will use the abstract index Special relativity notation . Equivalence principle World line · Riemannian Contents geometry Phenomena 1 Why tensors? 2 Spacetime as a manifold Kepler problem · Lenses · 2.1 Local versus global structure Waves 3 Tensors in GR Frame-dragging · Geodetic 3.1 Symmetric and antisymmetric tensors effect 3.2 The metric tensor Event horizon · Singularity 3.3 Invariants Black hole 3.4 Tensor classifications Equations 4 Tensor fields in GR 5 Tensorial derivatives Linearized Gravity 5.1 Affine connections Post-Newtonian formalism 5.2 The covariant derivative Einstein field equations 5.3 The Lie derivative Friedmann equations 6 The Riemann curvature tensor ADM formalism 7 The energy-momentum tensor BSSN formalism 7.1 Energy conservation Advanced theories 8 The Einstein field equations 9 The geodesic equations Kaluza–Klein -
M-Eigenvalues of Riemann Curvature Tensor of Conformally Flat Manifolds
M-eigenvalues of Riemann Curvature Tensor of Conformally Flat Manifolds Yun Miao∗ Liqun Qi† Yimin Wei‡ August 7, 2018 Abstract We generalized Xiang, Qi and Wei’s results on the M-eigenvalues of Riemann curvature tensor to higher dimensional conformal flat manifolds. The expression of M-eigenvalues and M-eigenvectors are found in our paper. As a special case, M-eigenvalues of conformal flat Einstein manifold have also been discussed, and the conformal the invariance of M- eigentriple has been found. We also discussed the relationship between M-eigenvalue and sectional curvature of a Riemannian manifold. We proved that the M-eigenvalue can determine the Riemann curvature tensor uniquely and generalize the real M-eigenvalue to complex cases. In the last part of our paper, we give an example to compute the M-eigentriple of de Sitter spacetime which is well-known in general relativity. Keywords. M-eigenvalue, Riemann Curvature tensor, Ricci tensor, Conformal invariant, Canonical form. AMS subject classifications. 15A48, 15A69, 65F10, 65H10, 65N22. 1 Introduction The eigenvalue problem is a very important topic in tensor computation theories. Xiang, Qi and Wei [1, 2] considered the M-eigenvalue problem of elasticity tensor and discussed the relation between strong ellipticity condition of a tensor and it’s M-eigenvalues, and then extended the M-eigenvalue problem to Riemann curvature tensor. Qi, Dai and Han [12] introduced the M-eigenvalues for the elasticity tensor. The M-eigenvalues θ of the fourth-order tensor Eijkl are defined as follows. arXiv:1808.01882v1 [math.DG] 28 Jul 2018 j k l Eijkly x y = θxi, ∗E-mail: [email protected]. -
Sieving the Landscape of Gravity Theories
SISSA — Scuola Internazionale Superiore di Studi Avanzati Ph.D. Curriculum in Astrophysics Academic year 2013/2014 Sieving the Landscape of Gravity Theories From the Equivalence Principles to the Near-Planck Regime Thesis submitted for the degree of Doctor Philosophiæ Advisors: Candidate: Prof. Stefano Liberati Eolo Di Casola Prof. Sebastiano Sonego 22nd October 2014 Abstract This thesis focusses on three main aspects of the foundations of any theory of gravity where the gravitational field admits a geometric interpretation: (a) the principles of equivalence; (b) their role as selection rules in the landscape of extended theories of gravity; and (c) the possible modifications of the spacetime structure at a “mesoscopic” scale, due to underlying, microscopic-level, quantum- gravitational effects. The first result of the work is the introduction of a formal definition of the Gravitational Weak Equivalence Principle, which expresses the universality of free fall of test objects with non-negligible self-gravity, in a matter-free environ- ment. This principle extends the Galilean universality of free-fall world-lines for test bodies with negligible self-gravity (Weak Equivalence Principle). Second, we use the Gravitational Weak Equivalence Principle to build a sieve for some classes of extended theories of gravity, to rule out all models yielding non-universal free-fall motion for self-gravitating test bodies. When applied to metric theories of gravity in four spacetime dimensions, the method singles out General Relativity (both with and without the cosmological constant term), whereas in higher-dimensional scenarios the whole class of Lanczos–Lovelock gravity theories also passes the test. Finally, we focus on the traditional, manifold-based model of spacetime, and on how it could be modified, at a “mesoscopic” (experimentally attainable) level, by the presence of an underlying, sub-Planckian quantum regime. -
Invariant Construction of Cosmological Models Erik Jörgenfelt
Invariant Construction of Cosmological Models Erik Jörgenfelt Spring 2016 Thesis, 15hp Bachelor of physics, 180hp Department of physics Abstract Solutions to the Einstein field equations can be constructed from invariant objects. If a solution is found it is locally equivalent to all solutions constructed from the same set. Here the method is applied to spatially-homogeneous solutions with pre-determined algebras, under the assumption that the invariant frame field of standard vectors is orthogonal on the orbits. Solutions with orthogonal fluid flow of Bianchi type III are investigated and two classes of orthogonal dust (zero pressure) solutions are found, as well as one vacuum energy solution. Tilted dust solutions with non-vanishing vorticity of Bianchi types I–III are also investigated, and it is shown that no such solution exist given the assumptions. Sammanfattning Lösningar till Einsteins fältekvationer kan konstrueras från invarianta objekt. Om en lösning hittas är den lokalt ekvivalent till alla andra lösningar konstruerade från samma mängd. Här appliceras metoden på spatialt homogena lösningar med förbestämda algebror, under anta- gandet att den invarianta ramen med standard-vektorer är ortogonal på banorna. Lösningar med ortogonalt fluidflöde av Bianchi typ III undersöks och två klasser av dammlösningar (noll tryck) hittas, liksom en vakum-energilösning. Dammlösningar med lut och noll-skild vorticitet av Bianchi typ I–III undersöks också, och det visas att inga sådana lösningar existerar givet antagandena. Contents 1 Introduction 1 2 Preliminaries 3 2.1 Smooth Manifolds...................................3 2.2 The Tangent and Co-tangent Bundles........................4 2.3 Tensor Fields and Differential Forms.........................6 2.4 Lie Groups...................................... -
Nonsingular Schwarzschild-De Sitter Black Hole
Nonsingular Schwarzschild-de Sitter Black Hole Damien A. Easson Department of Physics, Arizona State University, Tempe, AZ 85287-1504 Abstract We combine notions of a maximal curvature scale in nature with that of a minimal curvature scale to construct a non-singular Schwarzschild-de Sitter black hole. We present an exact solution within the context of two-dimensional dilaton gravity. For a range of parameters the solution approaches Schwarzschild-de Sitter at large values of the radial coordinate, asymptotically approaching a de Sitter metric with constant minimal curvature, while approaching a maximal constant curvature smooth spacetime as the radial coordinate approaches zero. The spacetime is geodesically complete and generically has both a black hole horizon and a cosmological horizon. arXiv:1712.09455v2 [hep-th] 7 Nov 2018 [email protected] Contents 1 Introduction 1 2 Two Dimensional Gravity 2 3 Nonsingular SdS Black Hole Solution 3 3.1 Extrema Curvature Conjecture . 3 3.2 Interpolating potential construction . 5 3.3 Solving the equations of motion . 5 4 Conclusions 7 A Coordinates 8 1 Introduction There are few puzzles in theoretical physics more vexing than the enigmatic singularity on the interior of a black hole. This physical impossibility, is assumed to be conveniently hidden from the distant observer by an event horizon. According to Einstein's theory of General Relativity (GR), an observer falling into the black hole will inevitably encounter ever-increasing destructive tidal forces on his way towards the singularity, a fate guaranteed by the singularity theorems of Hawking and Penrose [1]. The singularity event occurs at extremely high curvatures, near the Planck scale. -
University of Amsterdam
University of Amsterdam MSc Physics Theoretical Physics Master Thesis The BTZ Black Hole In Metric and Chern-Simons Formulation by Eline van der Mast 0588105 July 2014 54 ECTS 02/2013 - 07/2014 Supervisor: Examiner: A. Castro, Dr. J. de Boer, Prof. Institute for Theoretical Physics Amsterdam ABSTRACT In this thesis, we review the BTZ black hole within the metric formulation of gen- eral relativity and Chern-Simons theory. We introduce the necessary concepts of general relativity and discuss three-dimensional gravity, specifically Anti-de Sitter vacuum space. Then, we review the identification process to get to the BTZ black hole solution, as well as an explicit quotienting formulation. The thermodynamic and geometric properties of the black hole are then discussed. The Brown-York stress tensor procedure is explained to give these properties physical meaning, with a correction by Balasubramanian and Kraus. In the second part of this thesis we introduce Chern-Simons theory and show its equivalence to three-dimensional gravity. The BTZ black hole is then shown to be defined through holonomies and thermodynamics. We find that the identifications in the metric formulation are equivalent to the holonomies in the Chern-Simons formulation; and the thermo- dynamics are the same. CONTENTS 1. Introduction :::::::::::::::::::::::::::::::::: 6 2. Introduction to Gravity ::::::::::::::::::::::::::: 8 2.1 General Relativity . .8 2.1.1 Differential Geometry . .9 2.1.2 The Riemann Tensor . .9 2.2 Gravity in Three Dimensions . 11 2.2.1 Degrees of Freedom . 11 3. Local and Global Aspects of Anti-de Sitter Space ::::::::::::: 13 3.1 Global Anti-de Sitter Space . -
Curvature Invariants for Charged and Rotating Black Holes
universe Article Curvature Invariants for Charged and Rotating Black Holes James Overduin 1,2,* , Max Coplan 1, Kielan Wilcomb 3 and Richard Conn Henry 2 1 Department of Physics, Astronomy and Geosciences, Towson University, Towson, MD 21252, USA; [email protected] 2 Department of Physics and Astronomy, Johns Hopkins University, Baltimore, MD 21218, USA; [email protected] 3 Department of Physics, University of California, San Diego, CA 92093, USA; [email protected] * Correspondence: [email protected] Received: 25 October 2019; Accepted: 21 January 2020; Published: 24 January 2020 Abstract: Riemann curvature invariants are important in general relativity because they encode the geometrical properties of spacetime in a manifestly coordinate-invariant way. Fourteen such invariants are required to characterize four-dimensional spacetime in general, and Zakhary and McIntosh showed that as many as seventeen can be required in certain degenerate cases. We calculate explicit expressions for all seventeen of these Zakhary–McIntosh curvature invariants for the Kerr–Newman metric that describes spacetime around black holes of the most general kind (those with mass, charge, and spin), and confirm that they are related by eight algebraic conditions (dubbed syzygies by Zakhary and McIntosh), which serve as a useful check on our results. Plots of these invariants show richer structure than is suggested by traditional (coordinate-dependent) textbook depictions, and may repay further investigation. Keywords: black holes; curvature invariants; general relativity PACS: 04.20.Jb; 04.70.Bw; 95.30.Sf 1. Introduction Quantities whose value is manifestly independent of coordinates are particularly useful in general relativity [1]. Riemann or curvature invariants, formed from the Riemann tensor and derivatives of the metric, are one example. -
Open Craig Price-Dissertation.Pdf
The Pennsylvania State University The Graduate School Eberly College of Science COVARIANT FORMULATIONS OF THE LIGHT-ATOM PROBLEM A Dissertation in Physics by Craig Chandler Price © 2018 Craig Chandler Price Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy August 2018 The dissertation of Craig Chandler Price was reviewed and approved∗ by the following: Nathan D. Gemelke Assistant Professor of Physics Dissertation Advisor, Chair of Committee David S. Weiss Professor of Physics Associate Head for Research Marcos Rigol Professor of Physics Zhiwen Liu Professor of Electrical Engineering Nitin Samarth Professor of Physics Department Head ∗Signatures are on file in the Graduate School. ii Abstract We explore the ramifications of the light-matter interaction of ultracold neutral atoms in a covariant, or coordinate free, approach in the quantum limit. To do so we describe the construction and initial characterization of a novel atom-optical system in which there is no obvious preferred coordinate system but is fed back upon itself to amplify coherent quantum excitations of any emergent spontaneous order of the system. We use weakly dissipative, spatially incoherent light that is modulated by amplified fluctuations of the phase profile of a Sagnac interferometer whose symmetry is broken by the action of a generalized Raman sideband cooling process. The cooling process is done in a limit of low probe intensity which effectively produces slow light and forms a platform to explore general relativistic analogues. The disordered potential landscape is populated with k-vector defects where the atomic density contracts with an inward velocity potentially faster than the speed of the slow light at that region. -
How to Derive the Kerr Metric by Cheating Quite a Bit; a Lecture Given to Mat 451 on April 23, 2009
HOW TO DERIVE THE KERR METRIC BY CHEATING QUITE A BIT; A LECTURE GIVEN TO MAT 451 ON APRIL 23, 2009 WILLIE W. WONG Abstract. In this set of notes, an unconventional method of deriving the Kerr metric is presented. 1. Introduction and the first ansatz In this note we give a heuristic derivation of the Kerr metric, in a way quite significantly different from the classical methods. This is in no way a formal write- up, so for a more rigorous derivation, and for references, please see the wonderful article by Roberto Bergamini and Stefano Viaggiu, “A novel derivation for Kerr metric in Papapetrou gauge.”1 The method described herein is inspired by Marc Mars’ paper “A spacetime characterization of the Kerr metric”2, and also by the author’s 2009 PhD dissertation. The question we seek to answer is: find a solution of the Einstein vacuum equa- tions that is stationary and axially symmetric. However, to actually answer the question, we will need to impose very significant, and not ab initio justifiable, con- straints. It is only with great hindsight (that we know the solution we seek already) that the constraints seem natural. On the other hand, we will try to argue that these constraints are not completely wild guesses: by following a particular coher- ent chain of thought, it may have been possible to have obtained the Kerr metric through this method with no prior knowledge of the metric. The principal argument here is thus: we first consider the Ernst potential on a stationary solution to the Einstein equations.