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
in which the “expansion coefficients”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
and thus easy to establish, and they will be needed in Section 4 of this review.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
).