\contentsline {chapter}{Preface}{vi} \contentsline {chapter}{\numberline {1}Introduction}{1} \contentsline {section}{\numberline {1.1}Motivation: Local Special Relativity plus Rotating Coordinates}{1} \contentsline {section}{\numberline {1.2}Why bother?}{3} \contentsline {section}{\numberline {1.3}Starting vocabulary}{5} \contentsline {section}{\numberline {1.4}Historical background}{6} \contentsline {section}{\numberline {1.5}Notation and conventions}{10} \contentsline {chapter}{\numberline {2}The congruence point of view and the measurement process}{12} \contentsline {section}{\numberline {2.1}Algebra}{12} \contentsline {subsection}{\numberline {2.1.1}Observer orthogonal decomposition}{13} \contentsline {subsection}{\numberline {2.1.2}Observer-adapted frames}{16} \contentsline {subsection}{\numberline {2.1.3}Relative kinematics: algebra}{18} \contentsline {subsection}{\numberline {2.1.4}Splitting along a parametrized spacetime curve}{21} \contentsline {subsection}{\numberline {2.1.5}Splitting along a test particle worldline}{21} \contentsline {subsection}{\numberline {2.1.6}Addition of velocities and the aberration map}{22} \contentsline {section}{\numberline {2.2}Derivatives}{23} \contentsline {subsection}{\numberline {2.2.1}Natural derivatives}{23} \contentsline {subsection}{\numberline {2.2.2}Covariant derivatives}{24} \contentsline {subsection}{\numberline {2.2.3}Kinematical quantities}{25} \contentsline {subsection}{\numberline {2.2.4}Splitting the exterior derivative}{27} \contentsline {subsection}{\numberline {2.2.5}Splitting the differential form divergence operator}{29} \contentsline {subsection}{\numberline {2.2.6}Spatial vector analysis}{29} \contentsline {subsection}{\numberline {2.2.7}Ordinary and Co-rotating Fermi-Walker derivatives}{31} \contentsline {subsection}{\numberline {2.2.8}Relation between Lie and Fermi-Walker temporal derivatives}{33} \contentsline {subsection}{\numberline {2.2.9}Total spatial covariant derivatives}{37} \contentsline {subsection}{\numberline {2.2.10}Splitting the total covariant derivative}{40} \contentsline {section}{\numberline {2.3}Observer-adapted frame derivatives}{41} \contentsline {subsection}{\numberline {2.3.1}Natural frame derivatives}{41} \contentsline {subsection}{\numberline {2.3.2}Splitting the connection coefficients}{42} \contentsline {subsection}{\numberline {2.3.3}Observer-adapted connection components}{43} \contentsline {subsection}{\numberline {2.3.4}Splitting covariant derivatives}{44} \contentsline {subsection}{\numberline {2.3.5}Observer-adapted components of total spatial covariant derivatives}{46} \contentsline {section}{\numberline {2.4}Relative kinematics: applications}{49} \contentsline {subsection}{\numberline {2.4.1}Splitting the acceleration equation}{49} \contentsline {subsection}{\numberline {2.4.2}Analogy with electromagnetism: gravitoelectromagnetism}{51} \contentsline {subsection}{\numberline {2.4.3}Maxwell-like equations}{52} \contentsline {subsection}{\numberline {2.4.4}Splitting the spin transport equation}{54} \contentsline {subsection}{\numberline {2.4.5}Relative Fermi-Walker transport and gyro precession}{56} \contentsline {subsection}{\numberline {2.4.6}The Schiff Precession Formula}{59} \contentsline {subsection}{\numberline {2.4.7}The relative angular velocity as a boost derivative}{61} \contentsline {subsection}{\numberline {2.4.8}Relative kinematics: transformation of spatial gravitational fields}{62} \contentsline {section}{\numberline {2.5}Spatial curvature and torsion}{64} \contentsline {subsection}{\numberline {2.5.1}Definitions}{64} \contentsline {subsection}{\numberline {2.5.2}Algebraic symmetries}{64} \contentsline {subsection}{\numberline {2.5.3}Symmetry-obeying spatial curvature}{66} \contentsline {subsection}{\numberline {2.5.4}Spatial Ricci tensors and scalar curvatures}{67} \contentsline {subsection}{\numberline {2.5.5}Pair interchange symmetry}{67} \contentsline {subsection}{\numberline {2.5.6}Spatial covariant exterior derivative}{68} \contentsline {section}{\numberline {2.6}The symmetrized curl operator for symmetric spatial 2-tensors}{70} \contentsline {section}{\numberline {2.7}Splitting spacetime curvature}{71} \contentsline {subsection}{\numberline {2.7.1}Splitting definitions}{71} \contentsline {subsection}{\numberline {2.7.2}Spacetime duality and curvature}{72} \contentsline {subsection}{\numberline {2.7.3}Evaluation of splitting fields}{73} \contentsline {subsection}{\numberline {2.7.4}Maxwell-like equations}{75} \contentsline {section}{\numberline {2.8}Mixed commutation formulas}{76} \contentsline {subsection}{\numberline {2.8.1}Splitting the Ricci identities}{76} \contentsline {subsection}{\numberline {2.8.2}Commuting $\hbox {\it \char '44}(u)_{\hbox {$u$}}$ and $\nabla (u)$}{77} \contentsline {section}{\numberline {2.9}Splitting the Bianchi identities of the second kind}{78} \contentsline {subsection}{\numberline {2.9.1}Spacetime identities}{78} \contentsline {subsection}{\numberline {2.9.2}Spatial identities}{81} \contentsline {section}{\numberline {2.10}Time without space defines space without time" and vice versa}{82} \contentsline {chapter}{\numberline {3}The slicing and threading points of view}{83} \contentsline {section}{\numberline {3.1}Introduction}{83} \contentsline {section}{\numberline {3.2}Algebra}{83} \contentsline {subsection}{\numberline {3.2.1}The nonlinear reference frame}{83} \contentsline {subsection}{\numberline {3.2.2}Measurement and the lapse function}{84} \contentsline {subsection}{\numberline {3.2.3}The Shift}{86} \contentsline {subsection}{\numberline {3.2.4}Computational frames and the reference decomposition}{87} \contentsline {subsection}{\numberline {3.2.5}Decomposing the metric}{88} \contentsline {subsection}{\numberline {3.2.6}Relationship between the reference and observer decompositions}{90} \contentsline {subsection}{\numberline {3.2.7}The slicing, threading and reference representations}{91} \contentsline {subsection}{\numberline {3.2.8}Transformation between slicing and threading points of view}{92} \contentsline {subsection}{\numberline {3.2.9}So far:}{93} \contentsline {section}{\numberline {3.3}Derivatives}{95} \contentsline {subsection}{\numberline {3.3.1}Evolution}{95} \contentsline {subsection}{\numberline {3.3.2}Natural time derivatives}{95} \contentsline {subsection}{\numberline {3.3.3}Natural spatial derivatives}{96} \contentsline {subsection}{\numberline {3.3.4}Gauge transformations of the nonlinear reference frame}{97} \contentsline {subsection}{\numberline {3.3.5}Observer-adapted frame structure functions and kinematical quantities}{99} \contentsline {subsection}{\numberline {3.3.6}Spatial covariant derivative}{100} \contentsline {subsection}{\numberline {3.3.7}Spatial vector analysis}{101} \contentsline {subsection}{\numberline {3.3.8}Partially-observer-adapted frames: connection components}{102} \contentsline {subsection}{\numberline {3.3.9}Total spatial covariant derivatives}{103} \contentsline {subsection}{\numberline {3.3.10}Spatial gravitational forces}{105} \contentsline {subsection}{\numberline {3.3.11}Second-order acceleration equation}{106} \contentsline {subsection}{\numberline {3.3.12}The spin transport equation}{107} \contentsline {subsection}{\numberline {3.3.13}Transformation of spatial gravitational fields}{107} \contentsline {section}{\numberline {3.4}Spatial curvature}{109} \contentsline {section}{\numberline {3.5}Initial value problem?}{109} \contentsline {subsection}{\numberline {3.5.1}Hypersurface and slicing points of view}{109} \contentsline {subsection}{\numberline {3.5.2}Thin sandwich problem}{110} \contentsline {subsection}{\numberline {3.5.3}Congruence and threading points of view}{110} \contentsline {subsection}{\numberline {3.5.4}Perfect fluids}{111} \contentsline {chapter}{\numberline {4}Maxwell's equations}{112} \contentsline {section}{\numberline {4.1}Introduction}{112} \contentsline {section}{\numberline {4.2}Splitting the electromagnetic field}{112} \contentsline {subsection}{\numberline {4.2.1}Congruence point of view}{112} \contentsline {subsection}{\numberline {4.2.2}Slicing and threading points of view}{113} \contentsline {subsection}{\numberline {4.2.3}Observer Boost}{114} \contentsline {subsection}{\numberline {4.2.4}Reference representation (Landau-Lifshitz-Hanni)}{114} \contentsline {section}{\numberline {4.3}Splitting the 4-current}{116} \contentsline {section}{\numberline {4.4}Splitting Maxwell's equations}{117} \contentsline {subsection}{\numberline {4.4.1}Congruence point of view}{117} \contentsline {subsection}{\numberline {4.4.2}Slicing and threading points of view}{118} \contentsline {section}{\numberline {4.5}Vector potential}{120} \contentsline {section}{\numberline {4.6}Wave equations}{121} \contentsline {section}{\numberline {4.7}Computational 3-space representations}{122} \contentsline {section}{\numberline {4.8}Lines of force}{122} \contentsline {chapter}{\numberline {5}Stationary spacetimes}{123} \contentsline {section}{\numberline {5.1}Stationary nonlinear reference frame}{123} \contentsline {section}{\numberline {5.2}Synchronization gap and Sagnac effect}{124} \contentsline {section}{\numberline {5.3}Rotating spatial Cartesian coordinates in flat spacetime}{127} \contentsline {section}{\numberline {5.4}Stationary axially-symmetric case: rotating Minkowski, G\"odel and Kerr spacetimes}{129} \contentsline {chapter}{\numberline {6}Perturbation problems}{133} \contentsline {section}{\numberline {6.1}Linearization about an orthogonal nonlinear reference frame}{133} \contentsline {section}{\numberline {6.2}Post-Newtonian approximation}{133} \contentsline {section}{\numberline {6.3}The Newtonian limit}{133} \contentsline {section}{\numberline {6.4}Friedmann-Robertson-Walker Perturbations}{133} \contentsline {chapter}{\numberline {A}Formulas from differential geometry}{1} \contentsline {subsection}{\numberline {A.1}Manifold}{1} \contentsline {subsection}{\numberline {A.2}Frame and dual frame}{1} \contentsline {subsection}{\numberline {A.3}Linear transformations}{2} \contentsline {subsection}{\numberline {A.4}Change of frame}{3} \contentsline {subsection}{\numberline {A.5}Metric}{3} \contentsline {subsection}{\numberline {A.6}Unit volume $n$-form}{4} \contentsline {subsection}{\numberline {A.7}Connection}{4} \contentsline {subsection}{\numberline {A.8}Metric connection}{5} \contentsline {subsection}{\numberline {A.9}Curvature}{6} \contentsline {subsection}{\numberline {A.10}Total covariant derivative}{6} \contentsline {subsection}{\numberline {A.11}Parallel transport and geodesics}{7} \contentsline {subsection}{\numberline {A.12}Generalized Kronecker deltas}{7} \contentsline {subsection}{\numberline {A.13}Symmetrization/antisymmetrization}{8} \contentsline {subsection}{\numberline {A.14}Exterior product}{8} \contentsline {subsection}{\numberline {A.15}Hodge star duality operation}{8} \contentsline {subsection}{\numberline {A.16}Complex Duality Operation}{9} \contentsline {subsection}{\numberline {A.17}Exterior derivative}{9} \contentsline {subsection}{\numberline {A.18}Differential form divergence operator}{10} \contentsline {subsection}{\numberline {A.19}De Rham Laplacian}{11} \contentsline {subsection}{\numberline {A.20}Covariant exterior derivative}{12} \contentsline {subsection}{\numberline {A.21}Ricci identities}{13} \contentsline {subsection}{\numberline {A.22}Bianchi identities of the first and second kind}{13} \contentsline {subsection}{\numberline {A.23}Ricci Tensor and Scalar Curvature}{13} \contentsline {subsection}{\numberline {A.24}Contracted Bianchi Identities of the Second Kind and the Weyl Tensor}{14} \contentsline {subsection}{\numberline {A.25}$n=3$ Structure Functions and Orthonormal Frame Connection Components}{15} \contentsline {subsection}{\numberline {A.26}Lie derivative}{16} \contentsline {chapter}{References}{1} \contentsline {section}{\numberline {A.1}final page}{11}