DOCSLIB.ORG

Rotating Sources, Gravito-Magnetism, the Kerr Metric

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Rotating Sources, Gravito-Magnetism, the Kerr Metric Supplemental Lecture 6 Rotating Sources, Gravito-Magnetism and The Kerr Black Hole Abstract This lecture consists of several topics in general relativity dealing with rotating sources, gravito- magnetism and a rotating black hole described by the Kerr metric. We begin by studying slowly rotating sources, such as planets and stars where the gravitational fields are weak and linearized gravity applies. We find the metric for these sources which depends explicitly on their angular momentum J. We consider the motion of gyroscopes and point particles in these spaces and discover “frame dragging” and Lense-Thirring precession. Gravito-magnetism is also discovered in the context of gravity’s version of the Lorentz force law. These weak field results can also be obtained directly from special relativity and are consequences of the transformation laws of forces under boosts. However, general relativity allows us to go beyond linear, weak field physics, to strong gravity in which space time is highly curved. We turn to rotating black holes and we review the phenomenology of the Kerr metric. The physics of the “ergosphere”, the space time region between a surface of infinite redshift and an event horizon, is discussed. Two appendices consider rocket motion in the vicinity of a black hole and the exact redshift in strong but time independent fields. Appendix B illustrates the close connection between symmetries and conservation laws in general relativity. Keywords: Rotating Sources, Gravito-Magnetism, frame-dragging, Kerr Black Hole, linearized gravity, Einstein-Maxwell equations, Lense-Thirring precession, Gravity Probe B (GP-B). Introduction. Weak Field General Relativity. In addition to strong gravity, the textbook studied space times which are only slightly curved. There were several reasons for this emphasis, including, 1. compatibility of general relativity and Newton’s theory of gravitation in the limit of weak field, static, non-relativistic applications, and 2. gravitational waves. “Slightly curved” means that throughout a region of space time there exist coordinate systems where, = + , 1 (1.) where is the Minkowski metric, that readsℎ in a Cartesian�ℎ � ≪ coordinate system, = 1, = = = 1 with all other components vanishing. 00 11 22 33 Some care− must be used in this formalism because the decomposition Eq. 1 is not generally covariant [1,2,3]. There are two families of coordinate transformations which play an important role here: 1. Lorentz transformations (boosts) of special relativity, and 2. Infinitesimal general coordinate transformations. Under boosts in Minkowski space time,, = ′ � where the coefficients of the Lorentz transformation depend on the velocity between frames S and S’. The explicit formulas for were derived , discussed and applied extensively in the textbook. Under the action of a Lorentz boost the Minkowski metric is left invariant = ′ = while the dynamical gravitational field transforms as a second rank ∑tensor = in a static Minkowski space time.ℎ This fact was used in the ′ discussionsℎ of ∑gravitational ℎ waves in Chapter 12 of the textbook to view problems in linearized, weak field general relativity as problems of special relativity. In this approach is considered ( ) to be a field, analogous to of electrodynamics, which propagates in Minkowskiℎ space time and satisfies the principles of special relativity. Chapter 12 also considered the invariance of linearized general relativity under infinitesimal coordinate transformations, = + ( ) (2) ′ where ( ) is comparable in size to . It is easy to calculate the transformation law for the metric and deduce the transformation ℎlaws for (unchanged) and , ℎ = = + ′ � ′ �� − � � − �� ℎ� ′ = + + ℎ − − ⋯ + (3) ′ ≡ ℎ where raising and lowering of indices is done with the Minkowski metric, = . We learn from Eq. 3 that, ∑ = (4) ′ Again, viewing these formulas from theℎ perspectiveℎ − of static− Minkowski space time, Eq. 4 is analogous to a gauge transformation in electrodynamics: it leaves the physics unchanged but describes it with a transformed tensor . Furthermore, Chapter 12 derived the linear form of ′ the Einstein field equations, = ℎ = where is the Ricci tensor, 1 8 4 is the Ricci scalar, = , and − is 2the energy −-momentum tensor of the matter fields in space time. We found in∑ Chapter 12 that the field equations simplified if we defined = , where = . Furthermore, we found that if we choose the “Lorenzℎ� gauge” 1 ℎfor − 2, ℎ ℎ ∑ ℎ ℎ = 0 (5) � then the Einstein field equations reduce,∑ to linearℎ order in , ℎ� = (6) 16 4 □ℎ� − = = where □ is the wave operator, 2 . We used this equation to study 1 2 2 2 gravitational waves in Chapter□ 12 of∑ the textbook. The− choice∇ of the Lorenz gauge, Eq. 5, was essential to obtain a tractable, appealing wave equation, Eq. 6. As with most low order approximations to differential equations in physics, linearized gravity has several limitations [1,2,3]. We see from the formulas above (and will see in our applications below) that the linear formulation satisfies the linear superposition principle, just like electrodynamics. Therefore, the non-linearities and the foundations of general relativity, that all forms of energy-momentum experience universal gravitational attraction, are compromised. However, in linearized gravity we can compute the motion of test particles in the weak gravitational fields generated by masses whose motion is prescribed externally. In other words one typically does not calculate how the masses that generate the gravitational fields themselves move under their own gravitational fields. One treats sources of electric and magnetic fields in electrodynamics by this rule and we apply it here. This is how we illustrated gravitational wave generation in the textbook: we let describe two orbiting masses (“Neutron Stars”) and saw that their time varying quadrupole moment generates gravitational waves described by . In a quantitative calculation the full Einstein equations, = = , ℎmust� be 1 8 4 used everywhere and a fully consistent calculation results where − the2 dynamical − metric describes both the propagation of gravitational waves far from the source as well as the internal dynamics of the source itself. Far from the source the Einstein field equations reduce to linearized gravity. These limitations of the linearized gravity approach show up in Eq. 6 [1,2,3]. The Lorenz gauge condition reads = 0 which implies = 0 which represents energy- momentum conservation∑ in flat,ℎ� static Minkowski space∑ time. But the exact conservation law should read = 0, where is the covariant derivative, = + ∑+ , where the Christoffel symbols bring the curvature of space time into ∑the Γproblem. The∑ curvatureΓ may be strong in the vicinity of a radiating source and the hypothesis = + , 1, may not apply there. The full formalism of general relativity is requ ired. Or oneℎ can�ℎ, and� ≪frequently does, proceed phenomenologically and treat as an external source and only use Eq. 6 far from the source in analogy to radiation problems in electrodynamics. Slowly Rotating Stars, Frame-Dragging and Lense-Thirring Precession Now let’s apply this formalism to some simple but instructive situations [1,2,3]. What is the metric, to first order in , outside a rotating mass (a star or planet)? Although the source is rotating let’s suppose thatℎ there is no explicit time dependence. Examples might be a rigid sphere rotating at a constant angular velocity . Or perhaps the star only has cylindrical symmetry about a fixed axis of rotation. In either case, = 0 and = 0. The wave equation Eq. 6 reduces to Poisson’s equation, 0 0ℎ� ( ) = ( ) (7) 2 −16 4 ∇ ℎ� ⃗ ⃗ which was discussed and solved in the textbook in the context of electrodynamics, ( ) ( ) = (8) | | 4 �⃗ 3 4 ℎ� ⃗ ∫ ⃗−�⃗ Furthermore, recall from Chapter 12 that the simplest ( ) reads, ( ) = ( ) ( )⃗ ( ) (9) where ( ) is the proper mass density of the⃗ source ⃗ and ⃗ ( ) is⃗ its four velocity. If we apply this source ⃗ to non-relativistic stars and planets where ( )⃗ 1, then to first order = ( , ). And � ⃗ ⁄� ≪ ⁄ ≈ �⃗ = = = 0 (10) 00 2 0 where we neglect because it is a second order correction and the formalism ≈ here is accurate only to first order. The expressions for the metric follow, ( ) ( ) ( ) = ( ) = (11) | | | | 00 4 �⃗ 3 0 4 �⃗ 3 2 3 ℎ� ⃗ − ∫ ⃗−�⃗ ℎ� ⃗ − ∫ ⃗−�⃗ and ( ) 0. To bring out the similarities to problems of rotating charge distributions in electroℎ� dynamics,⃗ ≈ it is convenient to define, ( ) ( ) ( ) = ( ) = (12) | | | | �⃗ 3 4 �⃗ 3 2 Φ ⃗ − ∫ ⃗−�⃗ ⃗ − ∫ ⃗−�⃗ So, = ( ) = 0 (13) 00 4Φ 0 2 ℎ� ℎ� ⃗ ℎ� ≈ In order to retrieve the metric = + , we recall, from Chapter 12 of the textbook, that = , where = . Noteℎ that one can invert this expression by noting 1 ℎ� ℎ − 2 ℎ ℎ ∑ ℎ that = = , so = . From Eq. 13 we read off = = 1 00 , ℎ�so ∑ ℎ� −ℎ ℎ ℎ� − 2 ℎ� ℎ� ℎ� − ∑ ℎ� 00 ℎ� 1 1 = = 2 2 00 �00 00 � � ℎ 1ℎ − 1 ℎ ℎ = = = = 2 2 ℎ11 ℎ�11 − 11ℎ� ℎ� ℎ22 ℎ33 = (14) 1 ℎ01 2 This produces the metric to first order in | | 1, �⃗⁄ ≪ = 1 + + 1 ( + + ) (15) 2 2Φ 2 2 2 2Φ 2 2 2 2 2 � � ∑ − � − � This generalizes the weak field isotropic metric studied in the textbook to stationary rotating sources which produces the new term. 0 These expressions can be simplified further for a rotating uniform rigid sphere of radius R. Suppose that one samples the gravitational field far from the source, . Then the leading term
Load more

Recommended publications
  • MITOCW | 14. Linearized Gravity I: Principles and Static Limit
    MITOCW | 14. Linearized gravity I: Principles and static limit. [SQUEAKING] [RUSTLING] [CLICKING] SCOTT All right, so in today's recorded lecture, I would like to pick up where we started-- HUGHES: excuse me. I'd like to pick up where we stopped last time. So I discussed the Einstein field equations in the previous two lectures. I derived them first from the method that was used by Einstein in his original work on the subject. And then I laid out the way of coming to the Einstein field equations using an action principle, using what we call the Einstein Hilbert action. Both of them lead us to this remarkably simple equation, if you think about it in terms simply of the curvature tensor. This is saying that a particular version of the curvature. You start with the Riemann tensor. You trace over two indices. You reverse the trace such that this whole thing has zero divergence. And you simply equate that to the stress energy tensor with a coupling factor with a complex constant proportionality that ensures that this recovers the Newtonian limit. The Einstein Hilbert exercise demonstrated that this is in a very quantifiable way, the simplest possible way of developing a theory of gravity in this framework. The remainder of this course is going to be dedicated to solving this equation, and exploring the properties of the solutions that arise from this. And so let me continue the discussion I began at the end of the previous lecture. We are going to find it very useful to regard this as a set of differential equations for the spacetime metric given a source.
    [Show full text]
  • Linearized Einstein Field Equations
    General Relativity Fall 2019 Lecture 15: Linearized Einstein field equations Yacine Ali-Ha¨ımoud October 17th 2019 SUMMARY FROM PREVIOUS LECTURE We are considering nearly flat spacetimes with nearly globally Minkowski coordinates: gµν = ηµν + hµν , with jhµν j 1. Such coordinates are not unique. First, we can make Lorentz transformations and keep a µ ν globally-Minkowski coordinate system, with hµ0ν0 = Λ µ0 Λ ν0 hµν , so that hµν can be seen as a Lorentz tensor µ µ µ ν field on flat spacetime. Second, if we make small changes of coordinates, x ! x − ξ , with j@µξ j 1, the metric perturbation remains small and changes as hµν ! hµν + 2ξ(µ,ν). By analogy with electromagnetism, we can see these small coordinate changes as gauge transformations, leaving the Riemann tensor unchanged at linear order. Since we will linearize the relevant equations, we may work in Fourier space: each Fourier mode satisfies an independent equation. We denote by ~k the wavenumber and by k^ its direction and k its norm. We have decomposed the 10 independent components of the metric perturbation according to their transformation properties under spatial rotations: there are 4 independent \scalar" components, which can be taken, for instance, ^i ^i^j to be h00; k h0i; hii, and k k hij { or any 4 linearly independent combinations thereof. There are 2 independent ilm^ ilm^ ^j transverse \vector" components, each with 2 independent components: klh0m and klhmjk { these are proportional to the curl of h0i and to the curl of the divergence of hij, and are divergenceless (transverse to the ~ TT Fourier wavenumber k).
    [Show full text]
  • 9 Linearized Gravity and Gravitational Waves
    9 Linearized gravity and gravitational waves 9.1 Linearized gravity 9.1.1 Metric perturbation as tensor field 1 We are looking for small perturbations hab around the Minkowski metric ηab, gab = ηab + hab , hab 1 . (9.1) ≪ These perturbations may be caused either by the propagation of gravitational waves through a detector or by the gravitational potential of a star. In the first case, current experiments show that we should not hope for h larger than (h) 10−22. Keeping only terms linear in h O ∼ is therefore an excellent approximation. Choosing in the second case as application the final phase of the spiral-in of a neutron star binary system, deviations from Newtonian limit can become large. Hence one needs a systematic “post-Newtonian” expansion or even a numerical analysis to describe properly such cases. We choose a Cartesian coordinate system xa and ask ourselves which transformations are compatible with the splitting (9.1) of the metric. If we consider global (i.e. space-time inde- b ′a a b pendent) Lorentz transformations Λa, then x = Λb x . The metric tensor transform as ′ c d c d c d ′ c d gab = ΛaΛb gcd = ΛaΛb (ηcd + hcd)= ηab + ΛaΛb hcd = ηab + ΛaΛb hcd . (9.2) Thus Lorentz transformations respect the splitting (9.1) and the perturbation hab transforms as a rank-2 tensor on Minkowski space. We can view therefore hab as a symmetric rank-2 tensor field defined on Minkowski space that satisfies the linearized Einstein equations, similar as the photon field is a rank-1 tensor field fulfilling Maxwell’s equations.
    [Show full text]
  • The Newtonian and Relativistic Theory of Orbits and the Emission of Gravitational Waves
    108 The Open Astronomy Journal, 2011, 4, (Suppl 1-M7) 108-150 Open Access The Newtonian and Relativistic Theory of Orbits and the Emission of Gravitational Waves Mariafelicia De Laurentis1,2,* 1Dipartimento di Scienze Fisiche, Universitá di Napoli " Federico II",Compl. Univ. di Monte S. Angelo, Edificio G, Via Cinthia, I-80126, Napoli, Italy 2INFN Sez. di Napoli, Compl. Univ. di Monte S. Angelo, Edificio G, Via Cinthia, I-80126, Napoli, Italy Abstract: This review paper is devoted to the theory of orbits. We start with the discussion of the Newtonian problem of motion then we consider the relativistic problem of motion, in particular the post-Newtonian (PN) approximation and the further gravitomagnetic corrections. Finally by a classification of orbits in accordance with the conditions of motion, we calculate the gravitational waves luminosity for different types of stellar encounters and orbits. Keywords: Theroy of orbits, gravitational waves, post-newtonian (PN) approximation, LISA, transverse traceless, Elliptical orbits. 1. INTRODUCTION of General Relativity (GR). Indeed, the PN approximation method, developed by Einstein himself [1], Droste and de Dynamics of standard and compact astrophysical bodies could be aimed to unravelling the information contained in Sitter [2, 3] within one year after the publication of GR, led the gravitational wave (GW) signals. Several situations are to the predictions of i) the relativistic advance of perihelion of planets, ii) the gravitational redshift, iii) the deflection possible such as GWs emitted by coalescing compact binaries–systems of neutron stars (NS), black holes (BH) of light, iv) the relativistic precession of the Moon orbit, that driven into coalescence by emission of gravitational are the so–called "classical" tests of GR.
    [Show full text]
  • Using Standard Syste
    PHYSICAL REVIEW D, VOLUME 60, 103513 Preheating in generalized Einstein theories Anupam Mazumdar and Luı´s E. Mendes Astrophysics Group, Blackett Laboratory, Imperial College London SW7 2BZ, United Kingdom ͑Received 10 February 1999; published 26 October 1999͒ We study the preheating scenario in generalized Einstein theories, considering a class of such theories which are conformally equivalent to those of an extra field with a modified potential in the Einstein frame. Resonant creation of bosons from an oscillating inflaton has been studied before in the context of general relativity taking also into account the effect of metric perturbations in linearized gravity. As a natural generalization we include the dilatonic–Brans-Dicke field without any potential of its own and in particular we study the linear theory of perturbations including the metric perturbations in the longitudinal gauge. We show that there is an amplifi- cation of the perturbations in the dilaton–Brans-Dicke field on super-horizon scales (k!0) due to the fluc- tuations in the metric, thus leading to an oscillating Newton’s constant with very high frequency within the horizon and with growing amplitude outside the horizon. We briefly mention the entropy perturbations gen- erated by such fluctuations and also the possibility to excite the Kaluza-Klein modes in the theories where the dilatonic–Brans-Dicke field is interpreted as a homogeneous field appearing due to the dimensional reduction from higher-dimensional theories. ͓S0556-2821͑99͒01820-2͔ PACS number͑s͒: 98.80.Cq I. INTRODUCTION to variations in the Newtonian gravitation ‘‘constant’’ G, and introduces a new coupling constant ␻, with general relativity One of the most important epochs in the history of the recovered in the limit 1/␻!0.
    [Show full text]
  • Gravitational Radiation in F(R) Gravity: a Geometric Approach
    Gravitational Radiation in f(R) Gravity: A Geometric Approach Adam Scott Kelleher A dissertation submitted to the faculty of the University of North Carolina at Chapel Hill in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Physics and Astronomy. Chapel Hill 2013 Approved by: Laura Mersini-Houghton Y. Jack Ng Charles Evans Dmitri Khveshchenko Ryan Rohm Hugon Karvovski Abstract ADAM SCOTT KELLEHER: Gravitational Radiation in f(R) Gravity: A Geometric Approach. (Under the direction of Laura Mersini-Houghton.) I summarize experimental and theoretical constraints on gravity theories. I explore metric f(R) gravity, and explore scalar field theory analogs. I present a different kind of mechanism to raise the effective scalar mass in f(R) gravity in environments with particular ranges of background scalar curvatures, and thus suppress scalar effects on solar system curvature scales, while allowing scalar effects at different curvature scales. I review the post-Newtonian and post-Minkowskian mathemat- ical machinery for General Relativity, and generalize these expansions to metric f(R) gravity up to second order in small parameters. ii Dedicated to Bill Walsh, a true brother. iii Acknowledgments I would like to thank my brother for his support, Bart Dunlap and Greg Herschlag for useful conversations, and my advisor Laura Mersini-Houghton for her guidance. I thank the Department of Physics and Astronomy at UNC Chapel Hill for supporting this work. iv Table of Contents List of Figures ........................................... viii List of Abbreviations and Symbols .............................. ix 1 Introduction .......................................... 1 2 Properties of Gravity Theories .............................. 4 2.1 Equivalence Principles .
    [Show full text]
  • On Nearly Newtonian Potentials and Their Implications to Astrophysics
    galaxies Article On Nearly Newtonian Potentials and Their Implications to Astrophysics Abraao J. S. Capistrano 1,2 ID 1 Federal University of Latin American Integration, Natural Sciences Interdisciplinary Center, Itaipu Technological Park, P.o.b: 2123, 85867-670 Foz do iguaçu-PR, Brazil; [email protected] 2 Casimiro Montenegro Filho Astronomy Center, Itaipu Technological Park, 85867-900 Foz do iguaçu-PR, Brazil Received: 28 February 2018; Accepted: 27 March 2018; Published: 17 April 2018 Abstract: We review the concept of the slow motion problem in General relativity. We discuss how the understanding of this process may imprint influence on the explanation of astrophysical problems. Keywords: general relativity; gravitational field; astrophysics 1. Introduction Since the publication of the General relativity theory (GRT) in 1915 by Einstein, the problem of motion has been a paramount issue to solve, even with all the triumphs of GRT: the prediction of the deviation of light when it passes through a sufficient matter source like a star, or on the explanation of the advance of perihelion of Mercury, or even the prediction of gravitational waves, recently discovered by direct evidence [1]. The answers firstly came up with a published paper of Einstein, Hoffman and Infeld [2] in 1938, and almost 25 years later, a better discussion on this theme was given with the publication of two important books by Fock [3] and Infeld and Plebanski [4]. In different ways, they showed essentially that the Einstein field equations give rise to the equations of motion. This is quite the contrary to what happens in the Newtonian theory where the equations of field and motion are at first sight well posed uncorrelated concepts.
    [Show full text]
  • Tensor Gauge Fields in Arbitrary Representations of GL(D,R)
    CORE Metadata, citation and similar papers at core.ac.uk Provided by CERN Document Server ULB–TH–02/23 hep-th/0208058 Tensor gauge fields in arbitrary representations of GL(D; R) : duality & Poincar´e lemma Xavier Bekaert and Nicolas Boulanger1 Facult´e des Sciences, Universit´e Libre de Bruxelles, Campus Plaine C.P. 231, B{1050 Bruxelles, Belgium [email protected] [email protected] Abstract Using a mathematical framework which provides a generalization of the de Rham complex (well-designed for p-form gauge fields), we have studied the gauge struc- ture and duality properties of theories for free gauge fields transforming in arbi- trary irreducible representations of GL(D; R). We have proven a generalization of the Poincar´e lemma which enables us to solve the above-mentioned problems in a systematic and unified way. 1“Chercheur F.R.I.A.”, Belgium 1 Introduction The surge of interest in string field theories has refocused attention on the old problem of formulating covariant field theories of particles carrying arbitrary representation of the Lorentz group. These fields appear as massive excitations of string (for spin S>2). It is believed that in a particular phase of M-theory, all such excitations become massless. The covariant formulation of massless gauge fields in arbitrary representations of the Lorentz group has been completed for D = 4 [1]. However, the generalization of this formulation to arbitrary values of D is a difficult problem since the case D = 4 is a very special one, as all the irreps of the little group SO(2) are totally symmetric.
    [Show full text]
  • The Quantization of Linear Gravitational Perturbations and the Hadamard Condition
    The Quantization of Linear Gravitational Perturbations and the Hadamard Condition David Stephen Hunt A thesis submitted for the degree of PhD University of York Department of Mathematics September 2012 Abstract The quantum field theory describing linear gravitational perturbations is important from a cosmological viewpoint, in particular when formulated on de Sitter spacetime, which is used in inflationary models. There is currently an ongoing controversy pertaining to the existence of a de Sitter invariant vacuum state for free gravitons. This thesis is a mathematically rigorous study of the theory and all constructions are performed in as general a setting as is possible, which allows us to then specialise to a particular spacetime when required. In particular, to study the case of de Sitter spacetime with a view to resolving the aforementioned controversy. The main results include the full construction of the classical phase space of the linearized Einstein system on a background cosmological vacuum spacetime, which includes proving when various gauge choices can be made. In particular, we prove that within a normal neighbourhood of any Cauchy surface, in a globally hyperbolic spacetime, one may pass to the synchronous gauge. We also consider the transverse-traceless gauge but show that there is a topological obstruction to achieving this, which rules out its general use. In constructing the phase space it is necessary to obtain a weakly non-degenerate symplectic product. We prove that this can be achieved for the case that the background spacetime admits a compact Cauchy surface by using results from the Arnowitt-Deser-Misner (ADM) formalism, specifically the initial data splittings due to Moncrief.
    [Show full text]
  • New Journal of Physics the Open–Access Journal for Physics
    New Journal of Physics The open–access journal for physics The basics of gravitational wave theory Eanna´ E´ Flanagan1 and Scott A Hughes2 1 Center for Radiophysics and Space Research, Cornell University, Ithaca, NY 14853, USA 2 Department of Physics and Center for Space Research, Massachusetts Institute of Technology, Cambridge, MA 02139, USA E-mail: fl[email protected] and [email protected] New Journal of Physics 7 (2005) 204 Received 11 January 2005 Published 29 September 2005 Online at http://www.njp.org/ doi:10.1088/1367-2630/7/1/204 Abstract. Einstein’s special theory of relativity revolutionized physics by teaching us that space and time are not separate entities, but join as ‘spacetime’. His general theory of relativity further taught us that spacetime is not just a stage on which dynamics takes place, but is a participant: the field equation of general relativity connects matter dynamics to the curvature of spacetime. Curvature is responsible for gravity, carrying us beyond the Newtonian conception of gravity that had been in place for the previous two and a half centuries. Much research in gravitation since then has explored and clarified the consequences of this revolution; the notion of dynamical spacetime is now firmly established in the toolkit of modern physics. Indeed, this notion is so well established that we may now contemplate using spacetime as a tool for other sciences. One aspect of dynamical spacetime—its radiative character, ‘gravitational radiation’—will inaugurate entirely new techniques for observing violent astrophysical processes. Over the next 100 years, much of this subject’s excitement will come from learning how to exploit spacetime as a tool for astronomy.
    [Show full text]
  • Variational Symmetries and Conservation Laws in Linearized Gravity Balraj Menon University of Central Arkansas, [email protected]
    Journal of the Arkansas Academy of Science Volume 60 Article 13 2006 Variational Symmetries and Conservation Laws in Linearized Gravity Balraj Menon University of Central Arkansas, [email protected] Follow this and additional works at: http://scholarworks.uark.edu/jaas Part of the Cosmology, Relativity, and Gravity Commons Recommended Citation Menon, Balraj (2006) "Variational Symmetries and Conservation Laws in Linearized Gravity," Journal of the Arkansas Academy of Science: Vol. 60 , Article 13. Available at: http://scholarworks.uark.edu/jaas/vol60/iss1/13 This article is available for use under the Creative Commons license: Attribution-NoDerivatives 4.0 International (CC BY-ND 4.0). Users are able to read, download, copy, print, distribute, search, link to the full texts of these articles, or use them for any other lawful purpose, without asking prior permission from the publisher or the author. This Article is brought to you for free and open access by ScholarWorks@UARK. It has been accepted for inclusion in Journal of the Arkansas Academy of Science by an authorized editor of ScholarWorks@UARK. For more information, please contact [email protected], [email protected]. Journal of the Arkansas Academy of Science, Vol. 60 [2006], Art. 13 ? <iri<iiioii<iik>y iiiifiLirits <«oq v^oiiSGi v<irion j_i«iWa illJ_iiDC3rizcu vji<iviij' BalrajMenon Department ofPhysics and Astronomy, 201 Donaghey Avenue, University ofCentral Arkansas, Conway, AR 72035 Correspondence: [email protected] — Abstract. The methods of symmetry group analysis are applied to the action functional oflinearized gravity to derive necessarj conditions for the existence of variational symmetries. Two classes of variational symmetries oflinearized gravity are discussed, anc the local conservation laws associated withthese variational symmetries are presented by applying Noether's theorem.
    [Show full text]
  • Gravitation in the Weak-Field Limit
    Massachusetts Institute of Technology Department of Physics Physics 8.962 Spring 2000 Gravitation in the Weak-Field Limit c 2000 Edmund Bertschinger. All rights reserved. 1 Introduction In special relativity, electromagnetism is described by a one-form field Aµ(x) in flat spacetime. Similarly, in the weak-field limit gravitation is described by a symmetric tensor field hµν(x) in flat spacetime. Pursuing the analogy can lead us to many insights about GR. These notes detail linearized GR, discussing particle motion via Hamiltonian dynamics, the gravitational field equations, the transverse gauge (giving the closest thing to an inertial frame in GR), gauge transformations, motion in accelerated and rotating frames, Mach’s principle, and more. Linear theory is also useful for most practical computations in general relativity. Linear theory suffices for nearly all experimental applications of general relativity per- formed to date, including the solar system tests (light deflection, perihelion precession, and Shapiro time delay measurements), gravitational lensing, and gravitational wave detection. The Hulse-Taylor binary pulsar offers some tests of gravity beyond lineaer theory (Taylor et al 1992), as do (in principle) cosmological tests of space curvature. Some of this material is found in Thorne et al (1986) and some in Bertschinger (1996) but much of it is new. The notation differs slightly from chapter 4 of my Les Houches lectures (Bertschinger 1996); in particular, φ and ψ are swapped there, and hij in those notes is denoted sij here (eq. 11 below). Throughout this set of notes, the Minkowski metric ηµν is used to raise and lower indices. In this set of notes we refer to gravity as a field in flat spacetime as opposed to the manifestation of curvature in spacetime.
    [Show full text]