## 5 Parallel propagator

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

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 (5.1) – (5.4) hold everywhere on . In particular, with an appropriate change of notation they hold at and ; for example, is the metric at .

(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.)

### 5.2 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 arbitrary.

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

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 Eq. (5.10) is .

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 Eqs. (3.3) and (3.4), this observation gives us the relations

### 5.3 Coincidence limits

Eq. (5.5) and the completeness relations of Eqs. (5.2) or (5.4) imply that

Other coincidence limits are obtained by differentiation of Eqs. (5.11). 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 gives

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