2.3 Parallel propagator
2.3.1 Tetrad on
On the geodesic 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 frame vectors satisfy
where is the Minkowski metric (which we shall use to raise and lower frame
indices). We have the completeness relations
and we define a dual tetrad by
this is also parallel transported on . In terms of the dual tetrad the completeness relations take the form
and it is easy to show that the tetrad and its dual satisfy and . Equations (68, 69,
70, 71) hold everywhere on . In particular, with an appropriate change of notation they hold at and
; for example, is the metric at .
2.3.2 Definition and properties of the parallel propagator
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 , or
The object is the parallel propagator: It takes a vector at and parallel-transports it to
along the unique geodesic that links these points.
Similarly, we find that
and we see that performs the inverse operation: It takes a vector at and
parallel-transports it back to . Clearly,
and these relations formally express the fact that is the inverse of .
The relation can also be expressed as , and this reveals that
The ordering of the indices, and the ordering of the arguments, are therefore arbitrary.
The action of the parallel propagator on tensors of arbitrary ranks 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
It is therefore the inverse propagator that takes a dual vector at and parallel-transports it to .
As another example, it is easy to show that a tensor at obtained by parallel transport from
must be given by
Here we need two occurrences of the parallel propagator, one for each tensorial index. Because the metric
tensor is covariantly constant, it is automatically parallel transported on , and a special case of
Equation (77) is therefore .
Because the basis vectors are parallel transported on , they satisfy at
and at . This immediately implies that the parallel propagators must satisfy
Another useful property of the parallel propagator follows from the fact that if is tangent to
the geodesic connecting to , then . Using Equations (55) and (56), this observation
gives us the relations
2.3.3 Coincidence limits
Equation (72) and the completeness relations of Equations (69) or (71) imply that
Other coincidence limits are obtained by differentiation of Equations (78). For example, the relation
implies , and at coincidence we have
the second result was obtained by applying Synge’s rule on the first result. Further differentiation
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