FWIW (I'm probably the only one ever likely to read this), in Ch. 7.0 Lightman, Press, Price, and Teukolsky's Problem Book in Relativity and Gravitation puts th

FWIW (I'm probably the only one ever likely to read this), in Ch. 7.0 Lightman, Press, Price, and Teukolsky's Problem Book in Relativity and Gravitation puts the previous comment in reverse (which I love and will steal). The entire brief section quoted below is glorious in its economy of English, too. Forgive the lazy \latex anyone who actually sees this, including future me. "If u is the tangent vector to a curve, a tensor Q is said to be parallel propagated along the curve if \nabla_u Q = 0. If the tangent vector is itself parallel propagated, \nabla_u u = 0 (tangent vector "covariantly constant") the curve is a geodesic, the generalization of a straight line in flat space. If x^\alpha(\lambda) is the geodesic (with u^\alpha = dx^\alpha / d\lambda) then the components of the geodesic equation are 0 = (\nabla_u u)^\mu = \frac{du^\mu}{d\lambda} + u^\alpha u^\beta \Gamma^\mu_{\alpha\beta}. "Here \lambda must be an affine parameter along the curve; for non-null curves this means \lambda must be proportional to the proper length. "If a curve is timelike, u is its tangent vector, and a := \nabla_u u = Du/d\tau, then a vector V is said to be Fermi-Walker transported along u if \nabla_u V = (u \otimes a - a \otimes u) \cdot V." See also problem 7.11 and its solution. Also of interest is Matthias Blau et al. 2006, "Fermi coordinates and Penrose limits". doi:10.1088/0264-9381/23/11/020 hep-th/0603109 which adapts Fermi coordinates to null geodesics. (abs. "(Fermi coordinates are direct measures of geodesic distance in space-time)... We describe in some detail the construction of Fermi coordinates" §4, "We now come to the general construction of Fermi coordinates associated to a null geodesic \gamma in a space-time with Lorentzian metric g_munu. Along \gamma we introduce a parallel transported pseudo-orthonormal frame ... Fermi coordinate are uniquely determined by a choice of pseudo-orthonormal frame along the null geodesic \gamma" "For many (in particular more advanced) purposes it is useful to rephrase the above construction of Fermi coordinates in terms of the Synge world function").