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.
From Equations (53, 55
, 56
) we already have
To generate coincidence limits of bitensors involving primed indices, it is efficient to invoke Synge’s rule,
in which “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
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
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