### 2.2 Coincidence limits

It is useful to determine the limiting behaviour of the bitensors as approaches . We introduce the notation

to designate the limit of any bitensor as approaches ; this is called the coincidence limit of the bitensor. We assume that the coincidence limit is a unique tensorial function of the base point , independent of the direction in which the limit is taken. In other words, if the limit is computed by letting after evaluating as a function of on a specified geodesic , it is assumed that the answer does not depend on the choice of geodesic.

#### 2.2.1 Computation of coincidence limits

From Equations (53, 55, 56) we already have

Additional results are obtained by repeated differentiation of the relations (57) and (58). For example, Equation (57) implies , or after using Equation (55). From the assumption stated in the preceding paragraph, becomes independent of in the limit , and we arrive at . By very similar calculations we obtain all other coincidence limits for the second derivatives of the world function. The results are
From these relations we infer that , so that , where was defined in Equation (61).

To generate coincidence limits of bitensors involving primed indices, it is efficient to invoke Synge’s rule,

in which “” designates any combination of primed and unprimed indices; this rule will be established below. For example, according to Synge’s rule we have , and since the coincidence limit of is zero, this gives us , as was stated in Equation (63). Similarly, . The results of Equation (63) can thus all be generated from the known result for .

The coincidence limits of Equation (63) were derived from the relation . We now differentiate this twice more and obtain . At coincidence we have

or if we recognize that the operations of raising or lowering indices and taking the limit commute. Noting the symmetries of , this gives us , or , or . Since the last factor is zero, we arrive at

The last three results were derived from by employing Synge’s rule.

We now differentiate the relation three times and obtain

At coincidence this reduces to . To simplify the third term we differentiate Ricci’s identity with respect to and then take the coincidence limit. This gives us . The same manipulations on the second term give . Using the identity and the symmetries of the Riemann tensor, it is then easy to show that . Gathering the results, we obtain , and Synge’s rule allows us to generalize this to any combination of primed and unprimed indices. Our final results are

#### 2.2.2 Derivation of Synge’s rule

We begin with any bitensor in which is a multi-index that represents any number of unprimed indices, and a multi-index that represents any number of primed indices. (It does not matter whether the primed and unprimed indices are segregated or mixed.) On the geodesic that links to we introduce an ordinary tensor where is a multi-index that contains the same number of indices as . This tensor is arbitrary, but we assume that it is parallel transported on ; this means that it satisfies at . Similarly, we introduce an ordinary tensor in which contains the same number of indices as . This tensor is arbitrary, but we assume that it is parallel transported on ; at it satisfies . With , , and we form a biscalar defined by

Having specified the geodesic that links to , we can consider to be a function of and . If is not much larger than (so that is not far from ), we can express as

Alternatively,

and these two expressions give

because the left-hand side is the limit of when . The partial derivative of with respect to is equal to , and in the limit this becomes . Similarly, the partial derivative of with respect to is , and in the limit this becomes . Finally, , and its derivative with respect to is . Gathering the results we find that

and the final statement of Synge’s rule,

follows from the fact that the tensors and , and the direction of the selected geodesic , are all arbitrary. Equation (67) reduces to Equation (64) when is substituted in place of .