DOCSLIB.ORG

The Post-Newtonian Approximation in Massive Gravity

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Post-Newtonian Approximation in Massive Gravity The Post-Newtonian Approximation in Massive Gravity Sohan Vartak CERN Summer Student Programme 2015 Abstract We briefly review the post-Newtonian expansion as a way to yield the approximate equa- tions of motion from a metric theory of gravity in the slow-motion, weak-field limit. Then, the parameterized post-Newtonian (PPN) formalism is introduced, as it provides a method of characterizing metric theories of gravity (again, in the slow-motion, weak-field limit). Finally, the field equations for Lorentz-violating massive gravity are derived from the Lagrangian; these equations will allow for the future determination of the PPN parameters. 1 Introduction Thus far, the predictions of general relativity (GR) have been in very good agreement with exper- imental observations. The problem with GR is that it does not seem compatible with a quantum theory of gravity. Therefore, many alternative theories have been proposed which may be more eas- ily quantizable; in particular, we are interested in Lorentz-violating theories. We first introduce the post-Newtonian expansion, which yields the approximate equations of motion in the slow-motion, weak-field limit. Following this, we review the parameterized post-Newtonian formalism, which al- lows for the classification of a general metric theory of gravity based on the values of 10 parameters which appear in a post-Newtonian expansion of the metric. Since metric theories differ only in the way that matter (and possible other fields) generate the metric, these parameters allow one to characterize theories according to their physical predictions (such as Lorentz violation, momentum conservation violation, and non-linearity in the superposition principle). Finally, Lorentz-violating massive gravity is briefly explored. The Lagrangian for a specific theory is used to derive the field equations, which may later be utilized in the computation of the PPN parameters. We use the conventions of Blas and Sibiryakov [3]. 2 The Post-Newtonian Expansion NOTE: This section follows the discussion of Weinberg[1]. Suppose we have a system of slowly moving particles bound by gravitational forces. Taking the slow-motion, weak-field limit, we can expand in powers of the velocity and determine the equations of motion to one order higher than that given by Newtonian mechanics. This is the post-Newtonian expansion. The equations of motion are given by the geodesic equation: d2xµ dxν dxλ + Γµ = 0 (1) dτ 2 νλ dτ dτ where τ denotes the proper time. This implies that we can compute the accelerations using 1 d2xi dxν dxλ dxν dxλ dxi = −Γi + Γ0 : (2) dt2 νλ dt dt νλ dt dt dt We must compute the Christoffel symbols to various orders so that we can obtain our post- Newtonian approximation. We treat the metric tensor gµν as consisting of the background Minkowski metric ηµν with the corrections being expandable in powers of the velocity. Thus we have 2 4 g00 = 1 + g00 + g00 + ::: (3) 2 4 gij = −δij + gij + gij + ::: (4) 1 3 g0i = g0i + g0i + ::: (5) N N where g µν denotes the term of order v in gµν. We can then apply the formula 1 Γµ = gµρ (g + g − g ) (6) νλ 2 ρν,λ ρλ,ν νλ,ρ to determine the Christoffel symbols. We see now that we must determine the metric in order to carry out this computation. The Ricci tensor is also expanded in a series and computed in terms of the Christoffel symbols, allowing us to relate it to the metric: 2 1 2 R = − r2g (7) 00 2 00 2 2 2 2 4 1 4 1 @ g 1 2 @ g 1 2 2 R = − r2g + 00 + g 00 − r2g (8) 00 2 00 2 @t2 2 ij @xixj 2 00 3 1 3 R = − r2g (9) 0i 2 0i 2 1 2 R = − r2g : (10) ij 2 ij At this point, we make use of the Einstein equations to relate the metric components to the energy-momentum tensor (which is also expanded in a series). The resulting equations can be used to solve for the components of the metric and express it in terms of various potentials. The metric can then be used to derive the post-Newtonian equations of motion. 3 The Parameterized Post-Newtonian Formalism The form of the metric up to post-Newtonian order in a general theory of gravity can be summa- rized by 10 parameters (α1; α2; α3; β; γ; ζ1; ζ2; ζ3; ζ4; ξ) when the metric is given in terms of specific potential functions. This formalism is discussed in more depth by Will[2]. The parameters appear as follows: 2 g00 = 1 − 2U + 2βU + 2ξΦW − (2γ + 2 + α3 + ζ1 − 2ξ)Φ1 − 2(3γ − 2β + 1 + ζ2 + ξ)Φ2 − 2(1 + ζ3)Φ3 − 2(3γ + 3ζ4 − 2ξ)Φ4 + (ζ1 − 2ξ)A (11) 2 gij = −(1 + 2γU)δij (12) 1 1 g = (4γ + 3 + α − α + ζ − 2ξ)V + (1 + α + ζ + 2ξ)W : (13) 0i 2 1 2 1 i 2 2 1 i Physically, γ represents the amount of space curvature produced by a unit rest mass. β repre- sents non-linearity in the superposition principle for gravity. ξ represents the presence of preferred- location effects. α1;2;3 represent the presence of preferred-frame effects, and ζ1;2;3;4 and α3 represent the presence of violation of total momentum conservation. Note that α3 thus contributes to two different physical effects. In general relativity, the only non-zero parameters are γ = β = 1. The potentials appearing in the metric all have the form Z ρ(y)f F (x) = G d3y : (14) N jx − yj The correspondences F $ f are given by i Φ1 $ vivi Φ3 $ Π U $ 1 Vi $ v (15) p (v (x − y) )2 v (x − y )(xi − yi) Φ $ U Φ $ A $ i i W $ j j j (16) 2 4 ρ jx − yj2 i jx − yj2 Z (x − y) (y − z) (x − z) Φ $ d3zρ(z) j j − j (17) W jx − yj2 jx − zj jy − zj Note that for U, Φ1;2;3;4, and Vi, we have F;ii = −4πGN ρf: (18) We also define the superpotential Z 3 χ = −GN d yρjx − yj (19) which satisfies χ;ii = −2U (20) χ;0i = Vi − Wi: (21) 4 Lorentz-Violating Massive Gravity As their name suggests, massive gravity theories endow the graviton with mass. We examine a specific form of Lorentz-violating massive gravity as introduced by Blas and Sibiryakov[3]. Lorentz invariance is broken down to the subgroup of spatial rotations via the introduction of four scalar 0 a fields, φ ; φ where a = 1; 2; 3. The Lagrangian (including Einstein-Hilbert and matter terms LEH and Lm, respectively) is given by 1 1 0 2 42 L = R + Lm + 4 (@µφ ) − µ0 16πG 8µ0 3 ! κ0 X − (@ φ0)2 − µ4 P µν@ φa@ φa + 3µ4 4µ4 µ 0 µ ν 0 a 1 X 2 − P µν@ φa@ φb + µ4δab 8µ4 µ ν a;b !2 κ X + P µν@ φa@ φa + 3µ4 : (22) 8µ4 µ ν a We denote the massive gravity Lagrangian by LS = L − (LEH + Lm) and define 0 0 @µφ @νφ Pµν = gµν − λρ 0 0 : (23) g @λφ @ρφ p We first vary the action S = R L −gd4x with respect to the metric and find R δL 1 µνp 4 δS = δgµν − 2 gµνL δg −gd x. Thus, the condition that the action be stationary (δS = 0) implies δL 1 − g L = 0: (24) δgµν 2 µν Computing, we find δLS κ0 X 1 X = − (@ φ0)2 − µ4 Kaa − P αβ@ φa@ φb + µ4δab Kab δgµν 4µ4 α 0 µν 4µ4 α ν µν 0 a a;b ! ! κ X X + P αβ@ φa@ φa + 3µ4 Kbb : (25) 4µ4 α β µν a b We have defined δgσρ δgσα δgρα δgαβ Kab ≡ − u uρ + u uσ + uσuρu u @ φa@ φb (26) µν δgµν δgµν α δgµν α α β δgµν σ ρ @ φ0 where we have defined u ≡ pµ with X ≡ gαβ@ φ0@ φ0. Note that a and b are not tensor µ X α β indices, but merely labeling indices; thus, the summation convention does not apply to them. We can now define our energy-momentum tensor as δL 1 T S ≡ −2 S − g L : (27) µν δgµν 2 µν S Thus, the field equations ultimately take the form m S Gµν = 8πG Tµν + Tµν (28) 1 m where Gµν ≡ Rµν − 2 gµνR is the familiar Einstein tensor and Tµν denotes the matter energy- momentum tensor. Using the Euler-Lagrange equations @ @L − @L = 0 and noting that the second term vanishes µ @(@µφ) @φ for all φ fields, we find for φ0 4 ! 1 κ0 X (@ φ0)2 − µ4 @ δµβ@ φ0 − @ δµα@ φ0 P βλ@ φa@ φa + 3µ4 2µ4 α 0 µ β 2µ4 µ α β λ 0 0 a ! κ0 X 1 − (@ φ0)2 − µ4 p @ (−gµβuν − gµνuβ + 2uµuβuν)@ φa@ φb 4µ4 α 0 µ β ν 0 a X 1 X 1 − (P αβ@ φa@ φb + µ4δab) p @ (−gµλuν − gµνuλ + 2uµuλuν)@ φa@ φb 4µ4 α β µ λ ν a;b X ! ! κ X X 1 + P αβ@ φa@ φa + 3µ4 p @ (−gµλuν − gµνuλ + 2uµuλuν)@ φb@ φb = 0: (29) 4µ4 α β µ λ ν a b X For φi, we find κ0 0 2 4 µν i − 4 (@λφ ) − µ0 @µP @νφ 2µ0 1 X − (P λρ@ φa@ φb + µ4δab)@ δia[P µν@ φb + δab(P µν@ φi)] 2µ4 λ ρ µ ν ν a;b ib µν a ab µν i +δ [P @νφ + δ (P @νφ )] ! κ X + P λρ@ φa@ φa + 3µ4 @ P µν@ φi = 0: (30) 2µ4 λ ρ µ ν a 5 Conclusion We have found the field equations for Lorentz-violating massive gravity in the form investigated by Blas and Sibiryakov.
Load more

Recommended publications
  • 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 .................
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Introduction to General Relativity
    INTRODUCTION TO GENERAL RELATIVITY Gerard 't Hooft Institute for Theoretical Physics Utrecht University and Spinoza Institute Postbox 80.195 3508 TD Utrecht, the Netherlands e-mail: [email protected] internet: http://www.phys.uu.nl/~thooft/ Version November 2010 1 Prologue General relativity is a beautiful scheme for describing the gravitational ¯eld and the equations it obeys. Nowadays this theory is often used as a prototype for other, more intricate constructions to describe forces between elementary particles or other branches of fundamental physics. This is why in an introduction to general relativity it is of importance to separate as clearly as possible the various ingredients that together give shape to this paradigm. After explaining the physical motivations we ¯rst introduce curved coordinates, then add to this the notion of an a±ne connection ¯eld and only as a later step add to that the metric ¯eld. One then sees clearly how space and time get more and more structure, until ¯nally all we have to do is deduce Einstein's ¯eld equations. These notes materialized when I was asked to present some lectures on General Rela- tivity. Small changes were made over the years. I decided to make them freely available on the web, via my home page. Some readers expressed their irritation over the fact that after 12 pages I switch notation: the i in the time components of vectors disappears, and the metric becomes the ¡ + + + metric. Why this \inconsistency" in the notation? There were two reasons for this. The transition is made where we proceed from special relativity to general relativity.
    [Show full text]
  • General Relativity Fall 2019 Lecture 11: the Riemann Tensor
    General Relativity Fall 2019 Lecture 11: The Riemann tensor Yacine Ali-Ha¨ımoud October 8th 2019 The Riemann tensor quantifies the curvature of spacetime, as we will see in this lecture and the next. RIEMANN TENSOR: BASIC PROPERTIES α γ Definition { Given any vector field V , r[αrβ]V is a tensor field. Let us compute its components in some coordinate system: σ σ λ σ σ λ r[µrν]V = @[µ(rν]V ) − Γ[µν]rλV + Γλ[µrν]V σ σ λ σ λ λ ρ = @[µ(@ν]V + Γν]λV ) + Γλ[µ @ν]V + Γν]ρV 1 = @ Γσ + Γσ Γρ V λ ≡ Rσ V λ; (1) [µ ν]λ ρ[µ ν]λ 2 λµν where all partial derivatives of V µ cancel out after antisymmetrization. σ Since the left-hand side is a tensor field and V is a vector field, we conclude that R λµν is a tensor field as well { this is the tensor division theorem, which I encourage you to think about on your own. You can also check that explicitly from the transformation law of Christoffel symbols. This is the Riemann tensor, which measures the non-commutation of second derivatives of vector fields { remember that second derivatives of scalar fields do commute, by assumption. It is completely determined by the metric, and is linear in its second derivatives. Expression in LICS { In a LICS the Christoffel symbols vanish but not their derivatives. Let us compute the latter: 1 1 @ Γσ = @ gσδ (@ g + @ g − @ g ) = ησδ (@ @ g + @ @ g − @ @ g ) ; (2) µ νλ 2 µ ν λδ λ νδ δ νλ 2 µ ν λδ µ λ νδ µ δ νλ since the first derivatives of the metric components (thus of its inverse as well) vanish in a LICS.
    [Show full text]
  • JOHN EARMAN* and CLARK GL YMUURT the GRAVITATIONAL RED SHIFT AS a TEST of GENERAL RELATIVITY: HISTORY and ANALYSIS
    JOHN EARMAN* and CLARK GL YMUURT THE GRAVITATIONAL RED SHIFT AS A TEST OF GENERAL RELATIVITY: HISTORY AND ANALYSIS CHARLES St. John, who was in 1921 the most widely respected student of the Fraunhofer lines in the solar spectra, began his contribution to a symposium in Nncure on Einstein’s theories of relativity with the following statement: The agreement of the observed advance of Mercury’s perihelion and of the eclipse results of the British expeditions of 1919 with the deductions from the Einstein law of gravitation gives an increased importance to the observations on the displacements of the absorption lines in the solar spectrum relative to terrestrial sources, as the evidence on this deduction from the Einstein theory is at present contradictory. Particular interest, moreover, attaches to such observations, inasmuch as the mathematical physicists are not in agreement as to the validity of this deduction, and solar observations must eventually furnish the criterion.’ St. John’s statement touches on some of the reasons why the history of the red shift provides such a fascinating case study for those interested in the scientific reception of Einstein’s general theory of relativity. In contrast to the other two ‘classical tests’, the weight of the early observations was not in favor of Einstein’s red shift formula, and the reaction of the scientific community to the threat of disconfirmation reveals much more about the contemporary scientific views of Einstein’s theory. The last sentence of St. John’s statement points to another factor that both complicates and heightens the interest of the situation: in contrast to Einstein’s deductions of the advance of Mercury’s perihelion and of the bending of light, considerable doubt existed as to whether or not the general theory did entail a red shift for the solar spectrum.
    [Show full text]
  • The Theory of Relativity and Applications: a Simple Introduction
    The Downtown Review Volume 5 Issue 1 Article 3 December 2018 The Theory of Relativity and Applications: A Simple Introduction Ellen Rea Cleveland State University Follow this and additional works at: https://engagedscholarship.csuohio.edu/tdr Part of the Engineering Commons, and the Physical Sciences and Mathematics Commons How does access to this work benefit ou?y Let us know! Recommended Citation Rea, Ellen. "The Theory of Relativity and Applications: A Simple Introduction." The Downtown Review. Vol. 5. Iss. 1 (2018) . Available at: https://engagedscholarship.csuohio.edu/tdr/vol5/iss1/3 This Article is brought to you for free and open access by the Student Scholarship at EngagedScholarship@CSU. It has been accepted for inclusion in The Downtown Review by an authorized editor of EngagedScholarship@CSU. For more information, please contact [email protected]. Rea: The Theory of Relativity and Applications What if I told you that time can speed up and slow down? What if I told you that everything you think you know about gravity is a lie? When Albert Einstein presented his theory of relativity to the world in the early 20th century, he was proposing just that. And what’s more? He’s been proven correct. Einstein’s theory has two parts: special relativity, which deals with inertial reference frames and general relativity, which deals with the curvature of space- time. A surface level study of the theory and its consequences followed by a look at some of its applications will provide an introduction to one of the most influential scientific discoveries of the last century.
    [Show full text]
  • Teleparallelism Arxiv:1506.03654V1 [Physics.Pop-Ph] 11 Jun 2015
    Teleparallelism A New Way to Think the Gravitational Interactiony At the time it celebrates one century of existence, general relativity | Einstein's theory for gravitation | is given a companion theory: the so-called teleparallel gravity, or teleparallelism for short. This new theory is fully equivalent to general relativity in what concerns physical results, but is deeply different from the con- ceptual point of view. Its characteristics make of teleparallel gravity an appealing theory, which provides an entirely new way to think the gravitational interaction. R. Aldrovandi and J. G. Pereira Instituto de F´ısica Te´orica Universidade Estadual Paulista S~aoPaulo, Brazil arXiv:1506.03654v1 [physics.pop-ph] 11 Jun 2015 y English translation of the Portuguese version published in Ci^enciaHoje 55 (326), 32 (2015). 1 Gravitation is universal One of the most intriguing properties of the gravitational interaction is its universality. This hallmark states that all particles of nature feel gravity the same, independently of their masses and of the matter they are constituted. If the initial conditions of a motion are the same, all particles will follow the same trajectory when submitted to a gravitational field. The origin of universality is related to the concept of mass. In principle, there should exist two different kinds of mass: the inertial mass mi and the gravitational mass mg. The inertial mass would describe the resistance a particle shows whenever one attempts to change its state of motion, whereas the gravitational mass would describe how a particle reacts to the presence of a gravitational field. Particles with different relations mg=mi, therefore, should feel gravity differently when submitted to a given gravitational field.
    [Show full text]
  • Albert Einstein and Relativity
    The Himalayan Physics, Vol.1, No.1, May 2010 Albert Einstein and Relativity Kamal B Khatri Department of Physics, PN Campus, Pokhara, Email: [email protected] Albert Einstein was born in Germany in 1879.In his the follower of Mach and his nascent concept helped life, Einstein spent his most time in Germany, Italy, him to enter the world of relativity. Switzerland and USA.He is also a Nobel laureate and worked mostly in theoretical physics. Einstein In 1905, Einstein propounded the “Theory of is best known for his theories of special and general Special Relativity”. This theory shows the observed relativity. I will not be wrong if I say Einstein a independence of the speed of light on the observer’s deep thinker, a philosopher and a real physicist. The state of motion. Einstein deduced from his concept philosophies of Henri Poincare, Ernst Mach and of special relativity the twentieth century’s best David Hume infl uenced Einstein’s scientifi c and known equation, E = m c2.This equation suggests philosophical outlook. that tiny amounts of mass can be converted into huge amounts of energy which in deed, became the Einstein at the age of 4, his father showed him a boon for the development of nuclear power. pocket compass, and Einstein realized that there must be something causing the needle to move, despite Einstein realized that the principle of special relativity the apparent ‘empty space’. This shows Einstein’s could be extended to gravitational fi elds. Since curiosity to the space from his childhood. The space Einstein believed that the laws of physics were local, of our intuitive understanding is the 3-dimensional described by local fi elds, he concluded from this that Euclidean space.
    [Show full text]
  • 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.
    [Show full text]
  • 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
    [Show full text]