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GEM: the User Manual Understanding Spacetime Splittings and Their Relationships1

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GEM: the User Manual Understanding Spacetime Splittings and Their Relationships1 GEM: the User Manual Understanding Spacetime Splittings and Their Relationships1 Robert T. Jantzen Paolo Carini and Donato Bini June 15, 2021 1This manuscript is preliminary and incomplete and is made available with the understanding that it will not be cited or reproduced without the permission of the authors. Abstract A comprehensive review of the various approaches to splitting spacetime into space-plus-time, establishing a common framework based on test observer families which is best referred to as gravitoelectromagnetism. Contents Preface vi 1 Introduction 1 1.1 Motivation: Local Special Relativity plus Rotating Coordinates . 1 1.2 Why bother? . 4 1.3 Starting vocabulary . 6 1.4 Historical background . 8 1.5 Orthogonalization in the Lorentzian plane . 11 1.6 Notation and conventions . 14 2 The congruence point of view and the measurement process 17 2.1 Algebra . 17 2.1.1 Observer orthogonal decomposition . 18 2.1.2 Observer-adapted frames . 21 2.1.3 Relative kinematics: algebra . 23 2.1.4 Splitting along parametrized spacetime curves and test particle worldlines . 26 2.1.5 Addition of velocities and the aberration map . 28 2.2 Derivatives . 29 2.2.1 Natural derivatives . 29 2.2.2 Covariant derivatives . 30 2.2.3 Kinematical quantities . 31 2.2.4 Splitting the exterior derivative . 33 2.2.5 Splitting the differential form divergence operator . 35 2.2.6 Spatial vector analysis . 35 2.2.7 Ordinary and Co-rotating Fermi-Walker derivatives . 37 2.2.8 Relation between Lie and Fermi-Walker temporal derivatives . 39 2.2.9 Total spatial covariant derivatives . 43 2.2.10 Splitting the total covariant derivative . 46 2.3 Observer-adapted frame derivatives . 47 2.3.1 Natural frame derivatives . 47 2.3.2 Splitting the connection coefficients . 48 2.3.3 Observer-adapted connection components . 49 2.3.4 Splitting covariant derivatives . 50 2.3.5 Observer-adapted components of total spatial covariant derivatives . 52 2.4 Relative kinematics: applications . 55 2.4.1 Splitting the acceleration equation . 55 2.4.2 Analogy with electromagnetism: gravitoelectromagnetism . 57 2.4.3 Maxwell-like equations . 58 2.4.4 Splitting the spin transport equation . 60 2.4.5 Relative Fermi-Walker transport and gyro precession . 62 2.4.6 The Schiff Precession Formula . 65 2.4.7 The relative angular velocity as a boost derivative . 67 i 2.4.8 Relative kinematics: transformation of spatial gravitational fields . 68 2.5 Spatial curvature and torsion . 70 2.5.1 Definitions . 70 2.5.2 Algebraic symmetries . 70 2.5.3 Symmetry-obeying spatial curvature . 72 2.5.4 Spatial Ricci tensors and scalar curvatures . 73 2.5.5 Pair interchange symmetry . 73 2.5.6 Spatial covariant exterior derivative . 74 2.6 The symmetrized curl operator for symmetric spatial 2-tensors . 76 2.7 Splitting spacetime curvature . 77 2.7.1 Splitting definitions . 77 2.7.2 Spacetime duality and curvature . 78 2.7.3 Evaluation of splitting fields . 79 2.7.4 Maxwell-like equations . 81 2.8 Mixed commutation formulas . 82 2.8.1 Splitting the Ricci identities . 82 2.8.2 Commuting $(u)u and r(u) ................................ 83 2.9 Splitting the Bianchi identities of the second kind . 84 2.9.1 Spacetime identities . 84 2.9.2 Spatial identities . 87 2.10 \Time without space defines space without time" and vice versa . 88 3 The slicing and threading points of view 89 3.1 Introduction . 89 3.2 Algebra . 89 3.2.1 The nonlinear reference frame . 89 3.2.2 Measurement and the lapse function . 90 3.2.3 The Shift . 92 3.2.4 Computational frames and the reference decomposition . 93 3.2.5 Decomposing the metric . 94 3.2.6 Relationship between the reference and observer decompositions . 96 3.2.7 The slicing, threading and reference representations . 97 3.2.8 Transformation between slicing and threading points of view . 98 3.2.9 So far: . 99 3.3 Derivatives . 101 3.3.1 Evolution . 101 3.3.2 Natural time derivatives . 101 3.3.3 Natural spatial derivatives . 102 3.3.4 Gauge transformations of the nonlinear reference frame . 103 3.3.5 Observer-adapted frame structure functions and kinematical quantities . 105 3.3.6 Spatial covariant derivative . 106 3.3.7 Spatial vector analysis . 107 3.3.8 Partially-observer-adapted frames: connection components . 108 3.3.9 Total spatial covariant derivatives . 109 3.3.10 Spatial gravitational forces . 111 3.3.11 Second-order acceleration equation . 112 3.3.12 The spin transport equation . 113 3.3.13 Transformation of spatial gravitational fields . 114 3.4 Spatial curvature . 115 3.5 Initial value problem? . 115 3.5.1 Hypersurface and slicing points of view . 115 3.5.2 Thin sandwich problem . 116 3.5.3 Congruence and threading points of view . 116 ii 3.5.4 Perfect fluids . 117 4 Maxwell's equations 118 4.1 Introduction . 118 4.2 Splitting the electromagnetic field . 118 4.2.1 Congruence point of view . 118 4.2.2 Slicing and threading points of view . 119 4.2.3 Observer Boost . 120 4.2.4 Reference representation (Landau-Lifshitz-Hanni) . 120 4.3 Splitting the 4-current . 122 4.4 Splitting Maxwell's equations . ..
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