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Reference Frames and Equations of Motion in the First Ppn Approximation of Scalar-Tensor Theories of Gravity

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REFERENCE FRAMES AND EQUATIONS OF MOTION IN THE FIRST PPN APPROXIMATION OF SCALAR-TENSOR THEORIES OF GRAVITY A Dissertation presented to the Faculty of the Graduate School University of Missouri-Columbia In Partial Ful¯llment of the Requirements for the Degree of Doctor of Philosophy by IGOR VLASOV Dr. Sergei Kopeikin, Dissertation Supervisor MAY 2006 ACKNOWLEDGMENTS I am very grateful to my advisor Dr. Sergei Kopeikin for his endless energy, patience and positive attitude. Without his continuous encouragement and consultations this work would have never been completed. I am thankful to the faculty and sta® of the Department of Physics and Astronomy for a very worm and friendly atmosphere. I am also indebted to my family and friends whose love, care and support are always with me. ii Contents ACKNOWLEDGEMENTS ii LIST OF TABLES vi LIST OF ILLUSTRATIONS vii Chapter 1 Notations 1 2 Introduction 13 2.1 General Outline of the Dissertation . 13 2.2 Motivations and Historical Background . 14 3 Statement of the Problem 21 3.1 Field Equations in the Scalar-Tensor Theory of Gravity . 21 3.2 The Tensor of Energy-Momentum . 23 3.3 Basic Principles of the Post-Newtonian Approximations . 25 3.4 The Gauge Conditions and the Residual Gauge Freedom . 29 3.5 The Reduced Field Equations . 30 4 Global PPN Coordinate System 33 4.1 Dynamic and Kinematic Properties of the Global Coordinates . 33 4.2 The Metric Tensor and the Scalar Field in the Global Coordinates . 38 5 Multipolar Decomposition of the Metric Tensor and the Scalar Field in the Global Coordinates 41 5.1 General Description of Multipole Moments . 41 iii 5.2 Thorne-Blanchet-Damour Active Multipole Moments . 44 5.3 Thorne-Blanchet-Damour Scalar Multipole Moments . 47 5.4 Thorne-Blanchet-Damour Conformal Multipole Moments . 48 5.5 Post-Newtonian Conservation Laws . 50 6 Local PPN Coordinate System 53 6.1 Dynamic and Kinematic Properties of the Local Coordinates . 53 6.2 The Metric Tensor and the Scalar Field in the Local Coordinates . 56 6.2.1 The Scalar Field: Internal and External Solutions . 58 6.2.2 The Metric Tensor: Internal Solution . 58 6.2.3 The Metric Tensor: External Solution . 60 6.2.4 The Metric Tensor: The Coupling Terms . 66 6.3 Multipolar Decomposition of the Body's Gravitational Field in the Local Coordinates . 67 7 Parameterized Post-Newtonian Transformation from Local to Global Co- ordinates 73 7.1 Preliminary Remarks . 73 7.2 General Structure of the Coordinate Transformation . 76 7.3 Transformation of the Coordinate Bases . 79 8 Matching of the Post-Newtonian Expansions of the Metric Tensor and the Scalar Field 82 8.1 Historical Background . 82 8.2 Method of the Matched Asymptotic Expansions in the PPN Formalism . 85 8.3 Transformation of Gravitational Potentials from the Local to Global Coor- dinates . 88 8.3.1 Transformation of the Internal Potentials . 88 8.3.2 Transformation of the External Potentials . 93 8.4 Matching of the Scalar Field . 94 8.5 Matching of the Metric Tensor . 95 8.5.1 Matching of g00(t; x) andg ^®¯(u; w) in the Newtonian approximation 95 2 8.5.2 Matching of gij(t; x) andg ^®¯(u; w) to the order of O(² )...... 97 3 8.5.3 Matching of g0i(t; x) andg ^®¯(u; w) to the order of O(² )...... 100 4 8.5.4 Matching of g00(t; x) andg ^®¯(u; w) to the order of O(² )...... 103 iv 8.6 Final Form of the PPN Coordinate Transformation from Local to Global Coordinates . 108 9 Translational Equations of Motion of Extended Bodies 110 9.1 Introduction . 110 9.2 Macroscopic post-Newtonian Equations of Motion . 114 9.3 De¯nitions of Mass, the Center of Mass and the Linear Momentum of an Extended Body in the N-body System . 115 9.4 Translational Equation of Motion in Local Coordinates . 120 9.5 Equation of Translational Motion in Global Coordinates . 126 10 Rotational Equations of Motion of Extended Bodies 129 10.1 Post-Newtonian De¯nition of the Angular Momentum of the Body . 129 10.2 Equations of Rotational Motion in Local Coordinates . 130 11 Motion of Spherically-Symmetric and Rigidly Rotating Bodies 134 11.1 De¯nition of a Spherically-Symmetric and Rigidly Rotating Body . 134 11.2 Coordinate Transformation of Multipole Moments . 138 11.3 Multipolar Decomposition of Gravitational Potentials in Global Coordinates 140 11.4 Translational Equations of Motion . 141 11.5 Rotational Equations of Motion . 150 12 Conclusion 152 A Solution of the Laplace Equation for Scalar, Vector and Tensor Fields 153 B The Christo®el Symbols and the Riemann Tensor 157 B.1 The Christo®el Symbols . 157 B.2 The Riemann Tensor of External Gravitational Field in Local Coordinates 160 C Comparison with the Klioner-So®el Approach to Reference Frames in the PPN Formalism 161 BIBLIOGRAPHY 165 VITA 172 v List of Tables Table page 1.1 Particular Conventions and Symbols . 2 vi List of Figures Figure page 4.1 Astronomical N-body System . 34 4.2 Hierarchy of Coordinate Frames . 35 6.1 Two Hypersurfaces of Constant Time . 71 8.1 Lie transfer . 89 8.2 Two World Lines . 98 vii Chapter 1 Notations Greek indices ®; ¯; γ; ::: run from 0 to 3 and mark space-time components of four-dimensional objects. Roman indices i; j; k; ::: run from 1 to 3 and denote space components of three- dimensional objects. Repeated indices mean the Einstein summation rule, for instance, ® 0 1 2 3 k 1 2 3 A B® = A B0 + A B1 + A B2 + A B3, and Tk = T 1 + T 2 + T 3, etc. Minkowski metric is denoted ´®¯ = diag(¡1; +1; +1; +1). Kronecker symbol (the unit matrix) is denoted ±ij = diag(1; 1; 1). Levi-Civita fully antisymmetric symbol is "ijk such that "123 = +1. Kronecker symbol is used to raise and lower Roman indices. Complete metric tensor g®¯ is used to raise and lower the Greek indices in exact tensor equations whereas the Minkowski metric ´®¯ is employed for raising and lowering indices in the post- Newtonian approximations Parentheses surrounding a group of Roman indices mean symmetrization, for example, A(ij) = (1=2)(Aij + Aji). Brackets around two Roman indices denote antisymmetrization, that is A[ij] = (1=2)(Aij ¡ Aji). Angle brackets surrounding a group of Roman indices denote the symmetric trace-free (STF) part of the corresponding three-dimensional object, for instance, 1 1 1 1 A = A ¡ ± A ;A = A ¡ ± A ¡ ± A ¡ ± A : <ij> (ij) 3 ij kk <ijk> (ijk) 5 ij kpp 5 jk ipp 5 ik jpp We also use multi-index notations, for example, AL = Ai1i2:::il ;BP ¡1 = Bi1i2:::ip¡1 ;D<L> = D<i1i2:::il> : 1 Sum over multi-indices is understood as L i1i2:::il bL¡1 bi1i2:::il¡1 ALQ = Ai1i2:::il Q ;PaL¡1T = Pai1i2:::il¡1 T : ® ¡1 Comma denotes a partial derivative, for example, Á;® = @Á=@x , where Á;0 = c @Á=@t; Á;i = i ® @Á=@x ; and semicolon T ;¯ denotes a covariant derivative. L-order partial derivative with respect to spatial coordinates is denoted @L = @i1 :::@il . Other conventions are introduced as they appear in the text. We summarize these particular conventions and notations in the following table for the convenience of the readers: Table 1.1: Particular Conventions and Symbols Symbol Description Equation(s) g¹º physical (Jordan-Fierz frame) metric (3.1.4) tensor g~¹º conformal (Einstein frame) metric ten- (3.1.6) sor g determinant of g¹º (3.1.1) g~ determinant ofg ~¹º (3.4.2) ´¹º Minkowski (flat) metric tensor (3.3.4) ® ¡¹º Christo®el symbols (3.1.3) R¹º Ricci tensor (3.1.2) R Ricci scalar (3.1.1) 2 Symbol Description Equation(s) ~ R¹º conformal Ricci tensor (3.1.7) T¹º energy-momentum tensor of matter (3.1.2) T trace of the energy-momentum tensor (3.1.2) Á scalar ¯eld (3.1.1) Á0 background value of the scalar ¯eld Á (3.3.1) ³ dimensionless perturbation of the (3.3.1) scalar ¯eld θ(Á) coupling function of the scalar ¯eld (3.1.1) g Laplace-Beltrami operator (3.1.3) D'Alembert operator in the Minkowski (3.5.5) space-time ½ density of matter in the co-moving (3.2.1) frame ½¤ invariant (Fock) density of matter (3.3.18) ¦ internal energy of matter in the co- (3.2.1) moving frame ¼¹º tensor of (anisotropic) stresses of mat- (3.2.1) ter u® 4-velocity of matter (3.2.1) 3 Symbol Description Equation(s) vi 3-dimensional velocity of matter in the (3.3.15) global frame ! asymptotic value of the coupling func- (3.3.2) tion θ(Á) !0 asymptotic value of the derivative of (3.3.2) the coupling function θ(Á) c ultimate speed of general and special (3.1.1) theories of relativity ² small dimensional parameter, ² = 1=c (3.3.4) h¹º metric tensor perturbation, g¹º ¡ ´¹º (4.1.1) (n) n h ¹º metric tensor perturbation of order ² (3.3.4) in the post-Newtonian expansion of the metric tensor (2) N shorthand notation for h 00 (3.3.9) (4) L shorthand notation for h 00 (3.3.9) (1) Ni shorthand notation for h 0i (3.3.9) (3) Li shorthand notation for h 0i (3.3.9) (2) Hij shorthand notation for h ij (3.3.9) (2) H shorthand notation for h kk (3.3.9) 4 Symbol Description Equation(s) N;~ L~ shorthand notations for perturbations (5.4.1) of conformal metricg ~¹º γ "space-curvature" PPN parameter (3.5.1) ¯ "non-linearity" PPN parameter (3.5.2) ´ Nordtvedt parameter, ´ = 4¯ ¡ γ ¡ 3 (5.3.1) G observed value of the universal gravita- (3.5.4) tional constant G bare value of the universal gravitational (3.3.3), (3.5.4) constant x® = (x0; xi) global coordinates with x0 = ct and xi ´ x w® = (w0; wi) local coordinates with w0 = cu and wi ´ w U Newtonian gravitational potential in (4.2.1) the global frame U (A) Newtonian gravitational potential of (4.2.7) body A in the global frame Ui vector potential in the global frame (4.2.4) (A) Ui vector potential of body A in the global (4.2.7) frame 5 Symbol Description Equation(s) Â; ©1;:::; ©4 various special gravitational potentials (4.2.2),
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  • RELATIVISTIC GRAVITY and the ORIGIN of INERTIA and INERTIAL MASS K Tsarouchas

    RELATIVISTIC GRAVITY and the ORIGIN of INERTIA and INERTIAL MASS K Tsarouchas

    RELATIVISTIC GRAVITY AND THE ORIGIN OF INERTIA AND INERTIAL MASS K Tsarouchas To cite this version: K Tsarouchas. RELATIVISTIC GRAVITY AND THE ORIGIN OF INERTIA AND INERTIAL MASS. 2021. hal-01474982v5 HAL Id: hal-01474982 https://hal.archives-ouvertes.fr/hal-01474982v5 Preprint submitted on 3 Feb 2021 (v5), last revised 11 Jul 2021 (v6) HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Distributed under a Creative Commons Attribution| 4.0 International License Relativistic Gravity and the Origin of Inertia and Inertial Mass K. I. Tsarouchas School of Mechanical Engineering National Technical University of Athens, Greece E-mail-1: [email protected] - E-mail-2: [email protected] Abstract If equilibrium is to be a frame-independent condition, it is necessary the gravitational force to have precisely the same transformation law as that of the Lorentz-force. Therefore, gravity should be described by a gravitomagnetic theory with equations which have the same mathematical form as those of the electromagnetic theory, with the gravitational mass as a Lorentz invariant. Using this gravitomagnetic theory, in order to ensure the relativity of all kinds of translatory motion, we accept the principle of covariance and the equivalence principle and thus we prove that, 1.