6 Expansion of bitensors near coincidence

6.1 General method

We would like to express a bitensor near coincidence as an expansion in powers of , the closest analogue in curved spacetime to the flat-spacetime quantity . For concreteness we shall consider the case of rank-2 bitensor, and for the moment we will assume that the tensorial indices all refer to the base point .

The expansion we seek is of the form

in which the “expansion coefficients” , , and are all ordinary tensors at ; this last tensor is symmetric in the pair of indices and , and measures the size of a typical component of .

To find the expansion coefficients we differentiate Eq. (6.1) repeatedly and take coincidence limits. Equation (6.1) immediately implies . After one differentiation we obtain , and at coincidence this reduces to . Taking the coincidence limit after two differentiations yields . The expansion coefficients are therefore

These results are to be substituted into Eq. (6.1), and this gives us to second order in .

Suppose now that the bitensor is , with one index referring to and the other to . The previous procedure can be applied directly if we introduce an auxiliary bitensor whose indices all refer to the point . Then can be expanded as in Eq. (6.1), and the original bitensor is reconstructed as , or

The expansion coefficients can be obtained from the coincidence limits of and its derivatives. It is convenient, however, to express them directly in terms of the original bitensor by substituting the relation and its derivatives. After using the results of Eq. (5.13) – (5.15) we find
The only difference with respect to Eq. (6.3) is the presence of a Riemann-tensor term in .

Suppose finally that the bitensor to be expanded is , whose indices all refer to . Much as we did before, we introduce an auxiliary bitensor whose indices all refer to , we expand as in Eq. (6.1), and we then reconstruct the original bitensor. This gives us

and the expansion coefficients are now
This differs from Eq. (6.4) by the presence of an additional Riemann-tensor term in .

6.2 Special cases

We now apply the general expansion method developed in the preceding subsection to the bitensors , , and . In the first instance we have , , and . In the second instance we have , , and . In the third instance we have , , and . This gives us the expansions

Taking the trace of the last equation returns , or
where was shown in Section 3.4 to describe the expansion of the congruence of geodesics that emanate from . Equation (6.10) reveals that timelike geodesics are focused if the Ricci tensor is nonzero and the strong energy condition holds: when we see that is smaller than 3, the value it would take in flat spacetime.

The expansion method can easily be extended to bitensors of other tensorial ranks. In particular, it can be adapted to give expansions of the first derivatives of the parallel propagator. The expansions

and thus easy to establish, and they will be needed in part III of this review.

6.3 Expansion of tensors

The expansion method can also be applied to ordinary tensor fields. For concreteness, suppose that we wish to express a rank-2 tensor at a point in terms of its values (and that of its covariant derivatives) at a neighbouring point . The tensor can be written as an expansion in powers of and in this case we have

If the tensor field is parallel transported on the geodesic that links to , then Eq. (6.12) reduces to Eq. (5.10). The extension of this formula to tensors of other ranks is obvious.

To derive this result we express , the restriction of the tensor field on , in terms of its tetrad components . Recall from Section 5.1 that is an orthonormal basis that is parallel transported on ; recall also that the affine parameter ranges from (its value at ) to (its value at ). We have , , and can be expressed in terms of quantities at by straightforward Taylor expansion. Since, for example,

where we have used Eq. (3.4), we arrive at Eq. (6.12) after involving Eq. (5.6).