Understanding Spacetime Splittings and Their Relationships

Robert T Jantzen, Paolo Carini, Donato Bini

This is a latex document under preparation.


Table of Contents

{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}

21-nov-1997: robert.jantzen@villanova.edu