On the geodesic segment that links to we introduce an orthonormal basis that is parallel transported on the geodesic. The frame indices , , …, run from 0 to 3 and the basis vectors satisfy
(You will have noticed that we use sans-serif symbols for the frame indices. This is to distinguish them from another set of frame indices that will appear below. The frame indices introduced here run from 0 to 3; those to be introduced later will run from 1 to 3.)
Any vector field on can be decomposed in the basis : , and the vector’s frame components are given by . If is parallel transported on the geodesic, then the coefficients are constants. The vector at can then be expressed as , orparallel propagator: it takes a vector at and parallel-transports it to along the unique geodesic that links these points.
Similarly, we find that
The relation can also be expressed as , and this reveals that
The action of the parallel propagator on tensors of arbitrary rank is easy to figure out. For example, suppose that the dual vector is parallel transported on . Then the frame components are constants, and the dual vector at can be expressed as , or
Because the basis vectors are parallel transported on , they satisfy at and at . This immediately implies that the parallel propagators must satisfy
Eq. (5.5) and the completeness relations of Eqs. (5.2) or (5.4) imply that
and at coincidence we have , or . The coincidence limit for can then be obtained from Synge’s rule, and an additional application of the rule gives . Our results are
Living Rev. Relativity 14, (2011), 7
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