Coordinate Systems

3 Coordinate Systems

3.1 Riemann normal coordinates
  3.1.1 Definition and coordinate transformation
  3.1.2 Metric near $x ′$
3.2 Fermi normal coordinates
  3.2.1 Fermi–Walker transport
  3.2.2 Tetrad and dual tetrad on $γ$
  3.2.3 Fermi normal coordinates
  3.2.4 Coordinate displacements near $γ$
  3.2.5 Metric near $γ$
  3.2.6 Thorne–Hartle coordinates
3.3 Retarded coordinates
  3.3.1 Geometrical elements
  3.3.2 Definition of the retarded coordinates
  3.3.3 The scalar field $r(x)$ and the vector field $kα(x)$
  3.3.4 Frame components of tensor fields on the world line
  3.3.5 Coordinate displacements near $γ$
  3.3.6 Metric near $γ$
  3.3.7 Transformation to angular coordinates
  3.3.8 Specialization to $aμ = 0 = Rμν$
3.4 Transformation between Fermi and retarded coordinates; advanced point
  3.4.1 From retarded to Fermi coordinates
  3.4.2 From Fermi to retarded coordinates
  3.4.3 Transformation of the tetrads at $x$
  3.4.4 Advanced point