Notes on General Relativity
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A Mathematical Derivation of the General Relativistic Schwarzschild
A Mathematical Derivation of the General Relativistic Schwarzschild Metric An Honors thesis presented to the faculty of the Departments of Physics and Mathematics East Tennessee State University In partial fulfillment of the requirements for the Honors Scholar and Honors-in-Discipline Programs for a Bachelor of Science in Physics and Mathematics by David Simpson April 2007 Robert Gardner, Ph.D. Mark Giroux, Ph.D. Keywords: differential geometry, general relativity, Schwarzschild metric, black holes ABSTRACT The Mathematical Derivation of the General Relativistic Schwarzschild Metric by David Simpson We briefly discuss some underlying principles of special and general relativity with the focus on a more geometric interpretation. We outline Einstein’s Equations which describes the geometry of spacetime due to the influence of mass, and from there derive the Schwarzschild metric. The metric relies on the curvature of spacetime to provide a means of measuring invariant spacetime intervals around an isolated, static, and spherically symmetric mass M, which could represent a star or a black hole. In the derivation, we suggest a concise mathematical line of reasoning to evaluate the large number of cumbersome equations involved which was not found elsewhere in our survey of the literature. 2 CONTENTS ABSTRACT ................................. 2 1 Introduction to Relativity ...................... 4 1.1 Minkowski Space ....................... 6 1.2 What is a black hole? ..................... 11 1.3 Geodesics and Christoffel Symbols ............. 14 2 Einstein’s Field Equations and Requirements for a Solution .17 2.1 Einstein’s Field Equations .................. 20 3 Derivation of the Schwarzschild Metric .............. 21 3.1 Evaluation of the Christoffel Symbols .......... 25 3.2 Ricci Tensor Components ................. -
Thermodynamic Aspects of Gravity
SCUOLA INTERNAZIONALE SUPERIORE DI STUDI AVANZATI INTERNATIONAL SCHOOL FOR ADVANCED STUDIES Astrophysics Sector Thermodynamic Aspects of Gravity Thesis submitted for the degree of Doctor Philosophiæ SUPERVISOR: CANDIDATE: Prof. Stefano Liberati Goffredo Chirco October 2011 To Scimmia, Citto, Jesus & Chetty Acknowledgements I am heartily thankful to my supervisor, Stefano Liberati, for his incredible enthusiasm, his guidance, and his encouragement during these amazing PhD years in Trieste. I am indebted to the whole Astrophysics group and to many of my colleagues from the Astrophysics, the Astroparticle and the High Energy group, for what I have learned and for the strong and sincere support I have received in any respect. Thank you Chris, Lorenzo, Thomas, Silke, Valeria, Carlo, John, Barbara, Carmelo, Giulia, Stefano, Vic´e,Dottore, Conte, Luca, Dario, Chetan, Irene, for sharing time, discussions, ideas and cafeteria food. I owe my deepest gratitude to Eleonora, Raffaello,Valentina, Federico, Daniele, Lucaemm´eand to all the amazing people who made my days in SISSA so hard to forget. Lastly, I offer my regards and blessings to Eleonora and Lucia, for their coffees, croissants and love. Goffredo Chirco Abstract In this thesis we consider a scenario where gravitational dynamics emerges from the holographic hydrodynamics of some microscopic, quantum system living in a local Rindler wedge. We start by considering the area scal- ing properties of the entanglement entropy of a local Rindler horizon as a conceptually basic realization of the holographic principle. From the gen- eralized second law and the Bekenstein bound we derive the gravitational dynamics via the entropy balance approach developed in [Jacobson 1995]. -
Schwarzschild Solution to Einstein's General Relativity
Schwarzschild Solution to Einstein's General Relativity Carson Blinn May 17, 2017 Contents 1 Introduction 1 1.1 Tensor Notations . .1 1.2 Manifolds . .2 2 Lorentz Transformations 4 2.1 Characteristic equations . .4 2.2 Minkowski Space . .6 3 Derivation of the Einstein Field Equations 7 3.1 Calculus of Variations . .7 3.2 Einstein-Hilbert Action . .9 3.3 Calculation of the Variation of the Ricci Tenosr and Scalar . 10 3.4 The Einstein Equations . 11 4 Derivation of Schwarzschild Metric 12 4.1 Assumptions . 12 4.2 Deriving the Christoffel Symbols . 12 4.3 The Ricci Tensor . 15 4.4 The Ricci Scalar . 19 4.5 Substituting into the Einstein Equations . 19 4.6 Solving and substituting into the metric . 20 5 Conclusion 21 References 22 Abstract This paper is intended as a very brief review of General Relativity for those who do not want to skimp on the details of the mathemat- ics behind how the theory works. This paper mainly uses [2], [3], [4], and [6] as a basis, and in addition contains short references to more in-depth references such as [1], [5], [7], and [8] when more depth was needed. With an introduction to manifolds and notation, special rel- ativity can be constructed which are the relativistic equations of flat space-time. After flat space-time, the Lagrangian and calculus of vari- ations will be introduced to construct the Einstein-Hilbert action to derive the Einstein field equations. With the field equations at hand the Schwarzschild equation will fall out with a few assumptions. -
“Geodesic Principle” in General Relativity∗
A Remark About the “Geodesic Principle” in General Relativity∗ Version 3.0 David B. Malament Department of Logic and Philosophy of Science 3151 Social Science Plaza University of California, Irvine Irvine, CA 92697-5100 [email protected] 1 Introduction General relativity incorporates a number of basic principles that correlate space- time structure with physical objects and processes. Among them is the Geodesic Principle: Free massive point particles traverse timelike geodesics. One can think of it as a relativistic version of Newton’s first law of motion. It is often claimed that the geodesic principle can be recovered as a theorem in general relativity. Indeed, it is claimed that it is a consequence of Einstein’s ∗I am grateful to Robert Geroch for giving me the basic idea for the counterexample (proposition 3.2) that is the principal point of interest in this note. Thanks also to Harvey Brown, Erik Curiel, John Earman, David Garfinkle, John Manchak, Wayne Myrvold, John Norton, and Jim Weatherall for comments on an earlier draft. 1 ab equation (or of the conservation principle ∇aT = 0 that is, itself, a conse- quence of that equation). These claims are certainly correct, but it may be worth drawing attention to one small qualification. Though the geodesic prin- ciple can be recovered as theorem in general relativity, it is not a consequence of Einstein’s equation (or the conservation principle) alone. Other assumptions are needed to drive the theorems in question. One needs to put more in if one is to get the geodesic principle out. My goal in this short note is to make this claim precise (i.e., that other assumptions are needed). -
The Schwarzschild Metric and Applications 1
The Schwarzschild Metric and Applications 1 Analytic solutions of Einstein's equations are hard to come by. It's easier in situations that e hibit symmetries. 1916: Karl Schwarzschild sought the metric describing the static, spherically symmetric spacetime surrounding a spherically symmetric mass distribution. A static spacetime is one for which there exists a time coordinate t such that i' all the components of g are independent of t ii' the line element ds( is invariant under the transformation t -t A spacetime that satis+es (i) but not (ii' is called stationary. An example is a rotating azimuthally symmetric mass distribution. The metric for a static spacetime has the form where xi are the spatial coordinates and dl( is a time*independent spatial metric. -ross-terms dt dxi are missing because their presence would violate condition (ii'. 23ote: The Kerr metric, which describes the spacetime outside a rotating ( axisymmetric mass distribution, contains a term ∝ dt d.] To preser)e spherical symmetry& dl( can be distorted from the flat-space metric only in the radial direction. In 5at space, (1) r is the distance from the origin and (2) 6r( is the area of a sphere. Let's de+ne r such that (2) remains true but (1) can be violated. Then, A,xi' A,r) in cases of spherical symmetry. The Ricci tensor for this metric is diagonal, with components S/ 10.1 /rimes denote differentiation with respect to r. The region outside the spherically symmetric mass distribution is empty. 9 The vacuum Einstein equations are R = 0. To find A,r' and B,r'# (. -
New Exact Solutions of Relativistic Hydrodynamics for Longitudinally Expanding Fireballs
Article New Exact Solutions of Relativistic Hydrodynamics for Longitudinally Expanding Fireballs Tamás Csörg˝o 1,2,∗ ID , Gábor Kasza 2, Máté Csanád 3 ID and Zefang Jiang 4,5 ID 1 Wigner Research Centre for Physics, P.O. Box 49, H-1525 Budapest 114, Hungary 2 Faculty of Natural Sciences, Károly Róbert Campus, Eszterházy Károly University, H-3200 Gyöngyös, Mátrai út 36, Hungary; [email protected] 3 Department of Atomic Physics, Eötvös University, Pázmány P. s. 1/A, H-1117 Budapest, Hungary; [email protected] 4 Key Laboratory of Quark and Lepton Physics, Ministry of Education, Wuhan 430079, China; [email protected] 5 Institute of Particle Physics, Central China Normal University, Wuhan 430079, China * Correspondence: [email protected] Received: 4 May 2018; Accepted: 28 May 2018; Published: 1 June 2018 Abstract: We present new, exact, finite solutions of relativistic hydrodynamics for longitudinally expanding fireballs for arbitrary constant value of the speed of sound. These new solutions generalize earlier, longitudinally finite, exact solutions, from an unrealistic to a reasonable equation of state, characterized by a temperature independent (average) value of the speed of sound. Observables such as the rapidity density and the pseudorapidity density are evaluated analytically, resulting in simple and easy to fit formulae that can be matched to the high energy proton–proton and heavy ion collision data at RHIC and LHC. In the longitudinally boost-invariant limit, these new solutions approach the Hwa–Bjorken solution and the corresponding rapidity distributions approach a rapidity plateaux. Keywords: relativistic hydrodynamics; exact solution; quark-gluon plasma; longitudinal flow; rapidity distribution; pseudorapidity distribution 1. -
Line Element in Noncommutative Geometry
Line element in noncommutative geometry P. Martinetti G¨ottingenUniversit¨at Wroclaw, July 2009 . ? ? - & ? !? The line element p µ ν ds = gµν dx dx is mainly useful to measure distance Z y d(x; y) = inf ds: x If, for some quantum gravity reasons, [x µ; x ν ] 6= 0 is one losing the notion of distance ? (annoying then to speak of noncommutative geo-metry). ? - . ? !? The line element p µ ν ds = gµν dx dx & ? is mainly useful to measure distance Z y d(x; y) = inf ds: x If, for some quantum gravity reasons, [x µ; x ν ] 6= 0 is one losing the notion of distance ? (annoying then to speak of noncommutative geo-metry). ? - !? The line element p µ ν ds = gµν dx dx . & ? ? is mainly useful to measure distance Z y d(x; y) = inf ds: x If, for some quantum gravity reasons, [x µ; x ν ] 6= 0 is one losing the notion of distance ? (annoying then to speak of noncommutative geo-metry). ? - The line element p µ ν ds = gµν dx dx . & ? ? is mainly useful to measure distance Z y !? d(x; y) = inf ds: x If, for some quantum gravity reasons, [x µ; x ν ] 6= 0 is one losing the notion of distance ? (annoying then to speak of noncommutative geo-metry). The line element p µ ν ds = gµν dx dx . & ? ? is mainly useful to measure distance ? -Z y !? d(x; y) = inf ds: x If, for some quantum gravity reasons, [x µ; x ν ] 6= 0 is one losing the notion of distance ? (annoying then to speak of noncommutative geo-metry). -
Geodesic Spheres in Grassmann Manifolds
GEODESIC SPHERES IN GRASSMANN MANIFOLDS BY JOSEPH A. WOLF 1. Introduction Let G,(F) denote the Grassmann manifold consisting of all n-dimensional subspaces of a left /c-dimensional hermitian vectorspce F, where F is the real number field, the complex number field, or the algebra of real quater- nions. We view Cn, (1') tS t Riemnnian symmetric space in the usual way, and study the connected totally geodesic submanifolds B in which any two distinct elements have zero intersection as subspaces of F*. Our main result (Theorem 4 in 8) states that the submanifold B is a compact Riemannian symmetric spce of rank one, and gives the conditions under which it is a sphere. The rest of the paper is devoted to the classification (up to a global isometry of G,(F)) of those submanifolds B which ure isometric to spheres (Theorem 8 in 13). If B is not a sphere, then it is a real, complex, or quater- nionic projective space, or the Cyley projective plane; these submanifolds will be studied in a later paper [11]. The key to this study is the observation thut ny two elements of B, viewed as subspaces of F, are at a constant angle (isoclinic in the sense of Y.-C. Wong [12]). Chapter I is concerned with sets of pairwise isoclinic n-dimen- sional subspces of F, and we are able to extend Wong's structure theorem for such sets [12, Theorem 3.2, p. 25] to the complex numbers nd the qua- ternions, giving a unified and basis-free treatment (Theorem 1 in 4). -
Quantum Simulation of Rindler Transformations Carlos Sabín1*
Sabín EPJ Quantum Technology (2018)5:5 https://doi.org/10.1140/epjqt/s40507-018-0069-0 R E S E A R C H Open Access Quantum simulation of Rindler transformations Carlos Sabín1* *Correspondence: csl@iff.csic.es 1Instituto de Física Fundamental, Abstract CSIC, Madrid, Spain We show how to implement a Rindler transformation of coordinates with an embedded quantum simulator. A suitable mapping allows to realise the unphysical operation in the simulated dynamics by implementing a quantum gate on an enlarged quantum system. This enhances the versatility of embedded quantum simulators by extending the possible in-situ changes of reference frames to the non-inertial realm. Keywords: Quantum simulations; Relativistic physics 1 Introduction The original conception of a quantum simulator [1] as a device that implements an in- volved quantum dynamics in a more amenable quantum system has given rise to a wealth of experiments in a wide variety of quantum platforms such as cold atoms, trapped ions, superconducting circuits and photonics networks [2–5]. In parallel, an alternate approach to quantum simulations has been developed in the last years allowing to observe in the lab- oratory physical phenomena beyond experimental reach, such as Zitterbewegung [6, 7]or Klein paradox [8–10]. Moreover, along this vein a quantum simulator can also be used to implement an artificial dynamics that has only been conceived theoretically, such as the Majorana equation [11, 12] or even the action of a mathematical transformation such as charge conjugation [11, 12], time and spatial parity operations [13–15] or switching among particle statistics [16]. -
General Relativity Fall 2019 Lecture 13: Geodesic Deviation; Einstein field Equations
General Relativity Fall 2019 Lecture 13: Geodesic deviation; Einstein field equations Yacine Ali-Ha¨ımoud October 11th, 2019 GEODESIC DEVIATION The principle of equivalence states that one cannot distinguish a uniform gravitational field from being in an accelerated frame. However, tidal fields, i.e. gradients of gravitational fields, are indeed measurable. Here we will show that the Riemann tensor encodes tidal fields. Consider a fiducial free-falling observer, thus moving along a geodesic G. We set up Fermi normal coordinates in µ the vicinity of this geodesic, i.e. coordinates in which gµν = ηµν jG and ΓνσjG = 0. Events along the geodesic have coordinates (x0; xi) = (t; 0), where we denote by t the proper time of the fiducial observer. Now consider another free-falling observer, close enough from the fiducial observer that we can describe its position with the Fermi normal coordinates. We denote by τ the proper time of that second observer. In the Fermi normal coordinates, the spatial components of the geodesic equation for the second observer can be written as d2xi d dxi d2xi dxi d2t dxi dxµ dxν = (dt/dτ)−1 (dt/dτ)−1 = (dt/dτ)−2 − (dt/dτ)−3 = − Γi − Γ0 : (1) dt2 dτ dτ dτ 2 dτ dτ 2 µν µν dt dt dt The Christoffel symbols have to be evaluated along the geodesic of the second observer. If the second observer is close µ µ λ λ µ enough to the fiducial geodesic, we may Taylor-expand Γνσ around G, where they vanish: Γνσ(x ) ≈ x @λΓνσjG + 2 µ 0 µ O(x ). -
Unique Properties of the Geodesic Dome High
Printing: This poster is 48” wide by 36” Unique Properties of the Geodesic Dome high. It’s designed to be printed on a large-format printer. Verlaunte Hawkins, Timothy Szeltner || Michael Gallagher Washkewicz College of Engineering, Cleveland State University Customizing the Content: 1 Abstract 3 Benefits 4 Drawbacks The placeholders in this poster are • Among structures, domes carry the distinction of • Consider the dome in comparison to a rectangular • Domes are unable to be partitioned effectively containing a maximum amount of volume with the structure of equal height: into rooms, and the surface of the dome may be formatted for you. Type in the minimum amount of material required. Geodesic covered in windows, limiting privacy placeholders to add text, or click domes are a twentieth century development, in • Geodesic domes are exceedingly strong when which the members of the thin shell forming the considering both vertical and wind load • Numerous seams across the surface of the dome an icon to add a table, chart, dome are equilateral triangles. • 25% greater vertical load capacity present the problem of water and wind leakage; SmartArt graphic, picture or • 34% greater shear load capacity dampness within the dome cannot be removed • This union of the sphere and the triangle produces without some difficulty multimedia file. numerous benefits with regards to strength, • Domes are characterized by their “frequency”, the durability, efficiency, and sustainability of the number of struts between pentagonal sections • Acoustic properties of the dome reflect and To add or remove bullet points structure. However, the original desire for • Increasing the frequency of the dome closer amplify sound inside, further undermining privacy from text, click the Bullets button widespread residential, commercial, and industrial approximates a sphere use was hindered by other practical and aesthetic • Zoning laws may prevent construction in certain on the Home tab. -
ASSIGNMENTS Week 4 (F. Saueressig) Cosmology 13/14 (NWI-NM026C) Prof
ASSIGNMENTS Week 4 (F. Saueressig) Cosmology 13/14 (NWI-NM026C) Prof. A. Achterberg, Dr. S. Larsen and Dr. F. Saueressig Exercise 1: Flat space in spherical coordinates For constant time t the line element of special relativity reduces to the flat three-dimensional Euclidean metric Euclidean 2 SR 2 2 2 2 (ds ) = (ds ) =const = dx + dy + dz . (1) − |t Use the coordinate transformation x = r sin θ cos φ, y = r sin θ sin φ , z = r cos θ , (2) with θ [0,π] and φ [0, √2 π[ to express the line element in spherical coordinates. ∈ ∈ Exercise 2: Distances, areas and volumes For constant time the metric describing the spatial part of a homogeneous closed Friedmann- Robertson Walker (FRW) universe is given by 2 2 dr 2 2 2 2 ds = + r dθ + sin θ dφ , r [ 0 , a ] . (3) 1 (r/a)2 ∈ − The scale factor a(t) depends on time only so that, for t = const, it can be considered as a fixed number. Compute from this line element a) The proper circumference of the sphere around the equator. b) The proper distance from the center of the sphere up to the coordinate radius a. c) The proper area of the two-surface at r = a. d) The proper volume inside r = a. Compare your results to the ones expected from flat Euclidean space. Which of the quantities are modified by the property that the line element (3) is curved? Exercise 3: The metric is covariantly constant In the lecture we derived a formula for the Christoffel symbol Γα 1 gαβ g + g g .