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{chapter}{Preface}{vi} {chapter} {1}Introduction {1} {1.1}Motivation: Local Special Relativity plus Rotating Coordinates}{1} {1.2}Why bother?}{3} {1.3}Starting vocabulary}{5} {1.4}Historical background}{6} {1.5}Notation and conventions}{10} {chapter} {2}The congruence point of view and the measurement process}{12} {2.1}Algebra}{12} {2.1.1}Observer orthogonal decomposition}{13} {2.1.2}Observer-adapted frames}{16} {2.1.3}Relative kinematics: algebra}{18} {2.1.4}Splitting along a parametrized spacetime curve}{21} {2.1.5}Splitting along a test particle worldline}{21} {2.1.6}Addition of velocities and the aberration map}{22} {2.2}Derivatives}{23} {2.2.1}Natural derivatives}{23} {2.2.2}Covariant derivatives}{24} {2.2.3}Kinematical quantities}{25} {2.2.4}Splitting the exterior derivative}{27} {2.2.5}Splitting the differential form divergence operator}{29} {2.2.6}Spatial vector analysis}{29} {2.2.7}Ordinary and Co-rotating Fermi-Walker derivatives}{31} {2.2.8}Relation between Lie and Fermi-Walker temporal derivatives}{33} {2.2.9}Total spatial covariant derivatives}{37} {2.2.10}Splitting the total covariant derivative}{40} {2.3}Observer-adapted frame derivatives}{41} {2.3.1}Natural frame derivatives}{41} {2.3.2}Splitting the connection coefficients}{42} {2.3.3}Observer-adapted connection components}{43} {2.3.4}Splitting covariant derivatives}{44} {2.3.5}Observer-adapted components of total spatial covariant derivatives}{46} {2.4}Relative kinematics: applications}{49} {2.4.1}Splitting the acceleration equation}{49} {2.4.2}Analogy with electromagnetism: gravitoelectromagnetism}{51} {2.4.3}Maxwell-like equations}{52} {2.4.4}Splitting the spin transport equation}{54} {2.4.5}Relative Fermi-Walker transport and gyro precession}{56} {2.4.6}The Schiff Precession Formula}{59} {2.4.7}The relative angular velocity as a boost derivative}{61} {2.4.8}Relative kinematics: transformation of spatial gravitational fields}{62} {2.5}Spatial curvature and torsion}{64} {2.5.1}Definitions}{64} {2.5.2}Algebraic symmetries}{64} {2.5.3}Symmetry-obeying spatial curvature}{66} {2.5.4}Spatial Ricci tensors and scalar curvatures}{67} {2.5.5}Pair interchange symmetry}{67} {2.5.6}Spatial covariant exterior derivative}{68} {2.6}The symmetrized curl operator for symmetric spatial 2-tensors}{70} {2.7}Splitting spacetime curvature}{71} {2.7.1}Splitting definitions}{71} {2.7.2}Spacetime duality and curvature}{72} {2.7.3}Evaluation of splitting fields}{73} {2.7.4}Maxwell-like equations}{75} {2.8}Mixed commutation formulas}{76} {2.8.1}Splitting the Ricci identities}{76} {2.8.2}Commuting $\hbox {\pit \char '44}(u)_{\hbox {$u$}}$ and $\nabla (u)$}{77} {2.9}Splitting the Bianchi identities of the second kind}{78} {2.9.1}Spacetime identities}{78} {2.9.2}Spatial identities}{81} {2.10}``Time without space defines space without time" and vice versa}{82} {chapter} {3}The slicing and threading points of view}{83} {3.1}Introduction}{83} {3.2}Algebra}{83} {3.2.1}The nonlinear reference frame}{83} {3.2.2}Measurement and the lapse function}{84} {3.2.3}The Shift}{86} {3.2.4}Computational frames and the reference decomposition}{87} {3.2.5}Decomposing the metric}{88} {3.2.6}Relationship between the reference and observer decompositions}{90} {3.2.7}The slicing, threading and reference representations}{91} {3.2.8}Transformation between slicing and threading points of view}{92} {3.2.9}So far:}{93} {3.3}Derivatives}{95} {3.3.1}Evolution}{95} {3.3.2}Natural time derivatives}{95} {3.3.3}Natural spatial derivatives}{96} {3.3.4}Gauge transformations of the nonlinear reference frame}{97} {3.3.5}Observer-adapted frame structure functions and kinematical quantities}{99} {3.3.6}Spatial covariant derivative}{100} {3.3.7}Spatial vector analysis}{101} {3.3.8}Partially-observer-adapted frames: connection components}{102} {3.3.9}Total spatial covariant derivatives}{103} {3.3.10}Spatial gravitational forces}{105} {3.3.11}Second-order acceleration equation}{106} {3.3.12}The spin transport equation}{107} {3.3.13}Transformation of spatial gravitational fields}{107} {3.4}Spatial curvature}{109} {3.5}Initial value problem?}{109} {3.5.1}Hypersurface and slicing points of view}{109} {3.5.2}Thin sandwich problem}{110} {3.5.3}Congruence and threading points of view}{110} {3.5.4}Perfect fluids}{111} {chapter} {4}Maxwell's equations}{112} {4.1}Introduction}{112} {4.2}Splitting the electromagnetic field}{112} {4.2.1}Congruence point of view}{112} {4.2.2}Slicing and threading points of view}{113} {4.2.3}Observer Boost}{114} {4.2.4}Reference representation (Landau-Lifshitz-Hanni)}{114} {4.3}Splitting the 4-current}{116} {4.4}Splitting Maxwell's equations}{117} {4.4.1}Congruence point of view}{117} {4.4.2}Slicing and threading points of view}{118} {4.5}Vector potential}{120} {4.6}Wave equations}{121} {4.7}Computational 3-space representations}{122} {4.8}Lines of force}{122} {chapter} {5}Stationary spacetimes}{123} {5.1}Stationary nonlinear reference frame}{123} {5.2}Synchronization gap and Sagnac effect}{124} {5.3}Rotating spatial Cartesian coordinates in flat spacetime}{127} {5.4}Stationary axially-symmetric case: rotating Minkowski, G{\accent "7F o}del and Kerr spacetimes}{129} {chapter} {6}Perturbation problems}{133} {6.1}Linearization about an orthogonal nonlinear reference frame}{133} {6.2}Post-Newtonian approximation}{133} {6.3}The Newtonian limit}{133} {6.4}Friedmann-Robertson-Walker Perturbations}{133} {chapter} {A}Formulas from differential geometry}{1} {A.1}Manifold}{1} {A.2}Frame and dual frame}{1} {A.3}Linear transformations}{2} {A.4}Change of frame}{3} {A.5}Metric}{3} {A.6}Connection}{4} {A.7}Metric connection}{5} {A.8}Curvature}{5} {A.9}Total covariant derivative}{6} {A.10}Parallel transport and geodesics}{7} {A.11}Generalized Kronecker deltas}{7} {A.12}Symmetrization/antisymmetrization}{7} {A.13}Exterior product}{8} {A.14}Hodge star duality operation}{8} {A.15}Exterior derivative}{9} {A.16}Differential form divergence operator}{10} {A.17}DeRham Laplacian}{10} {A.18}Covariant exterior derivative}{11} {A.19}Ricci identities}{12} {A.20}Bianchi identities of the first and second kind}{12} {A.21}Ricci Tensor and Scalar Curvature}{12} {A.22}Contracted Bianchi Identities of the Second Kind and the Weyl Tensor}{13} {A.23}$n=3$ Structure Functions and Orthonormal Frame Connection Components}{14} {A.24}Lie derivative}{15} {chapter} {References}{1}