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 bitensor’s indices all refer to the base point .
The expansion we seek is of the form
To find the expansion coefficients we differentiate Equation (83) repeatedly and take coincidence limits. Equation (83) 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
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 Equation (83), and the original bitensor is reconstructed as , or
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 Equation (83), and we then reconstruct the original bitensor. This gives us
We now apply the general expansion method developed in the preceding Section 2.4.1 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
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
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
To derive this result we express , the restriction of the tensor field on , in terms of its tetrad components . Recall from Section 2.3.1 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 Equation (56), we arrive at Equation (93) after involving Equation (73).
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