Linearized Gravity
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Kaluza-Klein Gravity, Concentrating on the General Rel- Ativity, Rather Than Particle Physics Side of the Subject
Kaluza-Klein Gravity J. M. Overduin Department of Physics and Astronomy, University of Victoria, P.O. Box 3055, Victoria, British Columbia, Canada, V8W 3P6 and P. S. Wesson Department of Physics, University of Waterloo, Ontario, Canada N2L 3G1 and Gravity Probe-B, Hansen Physics Laboratories, Stanford University, Stanford, California, U.S.A. 94305 Abstract We review higher-dimensional unified theories from the general relativity, rather than the particle physics side. Three distinct approaches to the subject are identi- fied and contrasted: compactified, projective and noncompactified. We discuss the cosmological and astrophysical implications of extra dimensions, and conclude that none of the three approaches can be ruled out on observational grounds at the present time. arXiv:gr-qc/9805018v1 7 May 1998 Preprint submitted to Elsevier Preprint 3 February 2008 1 Introduction Kaluza’s [1] achievement was to show that five-dimensional general relativity contains both Einstein’s four-dimensional theory of gravity and Maxwell’s the- ory of electromagnetism. He however imposed a somewhat artificial restriction (the cylinder condition) on the coordinates, essentially barring the fifth one a priori from making a direct appearance in the laws of physics. Klein’s [2] con- tribution was to make this restriction less artificial by suggesting a plausible physical basis for it in compactification of the fifth dimension. This idea was enthusiastically received by unified-field theorists, and when the time came to include the strong and weak forces by extending Kaluza’s mechanism to higher dimensions, it was assumed that these too would be compact. This line of thinking has led through eleven-dimensional supergravity theories in the 1980s to the current favorite contenders for a possible “theory of everything,” ten-dimensional superstrings. -
Unification of Gravity and Quantum Theory Adam Daniels Old Dominion University, [email protected]
Old Dominion University ODU Digital Commons Faculty-Sponsored Student Research Electrical & Computer Engineering 2017 Unification of Gravity and Quantum Theory Adam Daniels Old Dominion University, [email protected] Follow this and additional works at: https://digitalcommons.odu.edu/engineering_students Part of the Elementary Particles and Fields and String Theory Commons, Engineering Physics Commons, and the Quantum Physics Commons Repository Citation Daniels, Adam, "Unification of Gravity and Quantum Theory" (2017). Faculty-Sponsored Student Research. 1. https://digitalcommons.odu.edu/engineering_students/1 This Report is brought to you for free and open access by the Electrical & Computer Engineering at ODU Digital Commons. It has been accepted for inclusion in Faculty-Sponsored Student Research by an authorized administrator of ODU Digital Commons. For more information, please contact [email protected]. Unification of Gravity and Quantum Theory Adam D. Daniels [email protected] Electrical and Computer Engineering Department, Old Dominion University Norfolk, Virginia, United States Abstract- An overview of the four fundamental forces of objects falling on earth. Newton’s insight was that the force that physics as described by the Standard Model (SM) and prevalent governs matter here on Earth was the same force governing the unifying theories beyond it is provided. Background knowledge matter in space. Another critical step forward in unification was of the particles governing the fundamental forces is provided, accomplished in the 1860s when James C. Maxwell wrote down as it will be useful in understanding the way in which the his famous Maxwell’s Equations, showing that electricity and unification efforts of particle physics has evolved, either from magnetism were just two facets of a more fundamental the SM, or apart from it. -
Post-Newtonian Approximation
Post-Newtonian gravity and gravitational-wave astronomy Polarization waveforms in the SSB reference frame Relativistic binary systems Effective one-body formalism Post-Newtonian Approximation Piotr Jaranowski Faculty of Physcis, University of Bia lystok,Poland 01.07.2013 P. Jaranowski School of Gravitational Waves, 01{05.07.2013, Warsaw Post-Newtonian gravity and gravitational-wave astronomy Polarization waveforms in the SSB reference frame Relativistic binary systems Effective one-body formalism 1 Post-Newtonian gravity and gravitational-wave astronomy 2 Polarization waveforms in the SSB reference frame 3 Relativistic binary systems Leading-order waveforms (Newtonian binary dynamics) Leading-order waveforms without radiation-reaction effects Leading-order waveforms with radiation-reaction effects Post-Newtonian corrections Post-Newtonian spin-dependent effects 4 Effective one-body formalism EOB-improved 3PN-accurate Hamiltonian Usage of Pad´eapproximants EOB flexibility parameters P. Jaranowski School of Gravitational Waves, 01{05.07.2013, Warsaw Post-Newtonian gravity and gravitational-wave astronomy Polarization waveforms in the SSB reference frame Relativistic binary systems Effective one-body formalism 1 Post-Newtonian gravity and gravitational-wave astronomy 2 Polarization waveforms in the SSB reference frame 3 Relativistic binary systems Leading-order waveforms (Newtonian binary dynamics) Leading-order waveforms without radiation-reaction effects Leading-order waveforms with radiation-reaction effects Post-Newtonian corrections Post-Newtonian spin-dependent effects 4 Effective one-body formalism EOB-improved 3PN-accurate Hamiltonian Usage of Pad´eapproximants EOB flexibility parameters P. Jaranowski School of Gravitational Waves, 01{05.07.2013, Warsaw Relativistic binary systems exist in nature, they comprise compact objects: neutron stars or black holes. These systems emit gravitational waves, which experimenters try to detect within the LIGO/VIRGO/GEO600 projects. -
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. -
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). -
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. -
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. -
Einstein's Gravitational Field
Einstein’s gravitational field Abstract: There exists some confusion, as evidenced in the literature, regarding the nature the gravitational field in Einstein’s General Theory of Relativity. It is argued here that this confusion is a result of a change in interpretation of the gravitational field. Einstein identified the existence of gravity with the inertial motion of accelerating bodies (i.e. bodies in free-fall) whereas contemporary physicists identify the existence of gravity with space-time curvature (i.e. tidal forces). The interpretation of gravity as a curvature in space-time is an interpretation Einstein did not agree with. 1 Author: Peter M. Brown e-mail: [email protected] 2 INTRODUCTION Einstein’s General Theory of Relativity (EGR) has been credited as the greatest intellectual achievement of the 20th Century. This accomplishment is reflected in Time Magazine’s December 31, 1999 issue 1, which declares Einstein the Person of the Century. Indeed, Einstein is often taken as the model of genius for his work in relativity. It is widely assumed that, according to Einstein’s general theory of relativity, gravitation is a curvature in space-time. There is a well- accepted definition of space-time curvature. As stated by Thorne 2 space-time curvature and tidal gravity are the same thing expressed in different languages, the former in the language of relativity, the later in the language of Newtonian gravity. However one of the main tenants of general relativity is the Principle of Equivalence: A uniform gravitational field is equivalent to a uniformly accelerating frame of reference. This implies that one can create a uniform gravitational field simply by changing one’s frame of reference from an inertial frame of reference to an accelerating frame, which is rather difficult idea to accept. -
Principle of Relativity and Inertial Dragging
ThePrincipleofRelativityandInertialDragging By ØyvindG.Grøn OsloUniversityCollege,Departmentofengineering,St.OlavsPl.4,0130Oslo, Norway Email: [email protected] Mach’s principle and the principle of relativity have been discussed by H. I. HartmanandC.Nissim-Sabatinthisjournal.Severalphenomenaweresaidtoviolate the principle of relativity as applied to rotating motion. These claims have recently been contested. However, in neither of these articles have the general relativistic phenomenonofinertialdraggingbeeninvoked,andnocalculationhavebeenoffered byeithersidetosubstantiatetheirclaims.HereIdiscussthepossiblevalidityofthe principleofrelativityforrotatingmotionwithinthecontextofthegeneraltheoryof relativity, and point out the significance of inertial dragging in this connection. Although my main points are of a qualitative nature, I also provide the necessary calculationstodemonstratehowthesepointscomeoutasconsequencesofthegeneral theoryofrelativity. 1 1.Introduction H. I. Hartman and C. Nissim-Sabat 1 have argued that “one cannot ascribe all pertinentobservationssolelytorelativemotionbetweenasystemandtheuniverse”. They consider an UR-scenario in which a bucket with water is atrest in a rotating universe,andaBR-scenariowherethebucketrotatesinanon-rotatinguniverseand givefiveexamplestoshowthatthesesituationsarenotphysicallyequivalent,i.e.that theprincipleofrelativityisnotvalidforrotationalmotion. When Einstein 2 presented the general theory of relativity he emphasized the importanceofthegeneralprincipleofrelativity.Inasectiontitled“TheNeedforan -
5D Kaluza-Klein Theories - a Brief Review
5D Kaluza-Klein theories - a brief review Andr´eMorgado Patr´ıcio, no 67898 Departamento de F´ısica, Instituto Superior T´ecnico, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal (Dated: 21 de Novembro de 2013) We review Kaluza-Klein theory in five dimensions from the General Relativity side. Kaluza's original idea is examined and two distinct approaches to the subject are presented and contrasted: compactified and noncompactified theories. We also discuss some cosmological implications of the noncompactified theory at the end of this paper. ^ ^ 1 ^ I. INTRODUCTION where GAB ≡ RAB − 2 g^ABR is the Einstein tensor. Inspired by the ties between Minkowki's 4D space- For the metric, we identify the αβ-part ofg ^AB with time and Maxwell's EM unification, Nordstr¨om[3] in 1914 gαβ, the α4-part as the electromagnetic potential Aα and and Kaluza[4] in 1921 showed that 5D general relativ- g^44 with a scalar field φ, parametrizing it as follows: ity contains both Einstein's 4D gravity and Maxwell's g + κ2φ2A A κφ2A EM. However, they imposed an artificial restriction of no g^ (x; y) = αβ α β α ; (4) AB κφ2A φ2 dependence on the fifth coordinate (cylinder condition). β Klein[5], in 1926, suggested a physical basis to avoid this where κ is a multiplicative factor. If we identifyp it in problem in the compactification of the fifth dimension, terms of the 4D gravitational constant by κ = 4 πG, idea now used in higher-dimensional generalisations to then, using the metric4 and applying the cylinder con- include weak and strong interactions. -
Gravity, Orbital Motion, and Relativity
Gravity, Orbital Motion,& Relativity Early Astronomy Early Times • As far as we know, humans have always been interested in the motions of objects in the sky. • Not only did early humans navigate by means of the sky, but the motions of objects in the sky predicted the changing of the seasons, etc. • There were many early attempts both to describe and explain the motions of stars and planets in the sky. • All were unsatisfactory, for one reason or another. The Earth-Centered Universe • A geocentric (Earth-centered) solar system is often credited to Ptolemy, an Alexandrian Greek, although the idea is very old. • Ptolemy’s solar system could be made to fit the observational data pretty well, but only by becoming very complicated. Copernicus’ Solar System • The Polish cleric Copernicus proposed a heliocentric (Sun centered) solar system in the 1500’s. Objections to Copernicus How could Earth be moving at enormous speeds when we don’t feel it? . (Copernicus didn’t know about inertia.) Why can’t we detect Earth’s motion against the background stars (stellar parallax)? Copernicus’ model did not fit the observational data very well. Galileo • Galileo Galilei - February15,1564 – January 8, 1642 • Galileo became convinced that Copernicus was correct by observations of the Sun, Venus, and the moons of Jupiter using the newly-invented telescope. • Perhaps Galileo was motivated to understand inertia by his desire to understand and defend Copernicus’ ideas. Orbital Motion Tycho and Kepler • In the late 1500’s, a Danish nobleman named Tycho Brahe set out to make the most accurate measurements of planetary motions to date, in order to validate his own ideas of planetary motion. -
Quantum Gravity, Effective Fields and String Theory
Quantum gravity, effective fields and string theory Niels Emil Jannik Bjerrum-Bohr The Niels Bohr Institute University of Copenhagen arXiv:hep-th/0410097v1 10 Oct 2004 Thesis submitted for the degree of Doctor of Philosophy in Physics at the Niels Bohr Institute, University of Copenhagen. 28th July 2 Abstract In this thesis we will look into some of the various aspects of treating general relativity as a quantum theory. The thesis falls in three parts. First we briefly study how gen- eral relativity can be consistently quantized as an effective field theory, and we focus on the concrete results of such a treatment. As a key achievement of the investigations we present our calculations of the long-range low-energy leading quantum corrections to both the Schwarzschild and Kerr metrics. The leading quantum corrections to the pure gravitational potential between two sources are also calculated, both in the mixed theory of scalar QED and quantum gravity and in the pure gravitational theory. Another part of the thesis deals with the (Kawai-Lewellen-Tye) string theory gauge/gravity relations. Both theories are treated as effective field theories, and we investigate if the KLT oper- ator mapping is extendable to the case of higher derivative operators. The constraints, imposed by the KLT-mapping on the effective coupling constants, are also investigated. The KLT relations are generalized, taking the effective field theory viewpoint, and it is noticed that some remarkable tree-level amplitude relations exist between the field the- ory operators. Finally we look at effective quantum gravity treated from the perspective of taking the limit of infinitely many spatial dimensions.