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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}