On the geodesic segment that links
to
we introduce an orthonormal basis
that is
parallel transported on the geodesic. The frame indices
,
, …, run from 0 to 3 and the basis vectors
satisfy
(You will have noticed that we use sans-serif symbols for the frame indices. This is to distinguish them from another set of frame indices that will appear below. The frame indices introduced here run from 0 to 3; those to be introduced later will run from 1 to 3.)
Any vector field on
can be decomposed in the basis
:
, and the vector’s frame
components are given by
. If
is parallel transported on the geodesic, then the
coefficients
are constants. The vector at
can then be expressed as
, or
Similarly, we find that
and we see that The relation can also be expressed as
, and this reveals that
The action of the parallel propagator on tensors of arbitrary rank is easy to figure out. For example,
suppose that the dual vector is parallel transported on
. Then the frame components
are constants, and the dual vector at
can be expressed as
, or
Because the basis vectors are parallel transported on , they satisfy
at
and
at
. This immediately implies that the parallel propagators must satisfy
Eq. (5.5) and the completeness relations of Eqs. (5.2
) or (5.4
) imply that
and at coincidence we have , or
. The coincidence limit for
can then be obtained from Synge’s rule, and an additional application of the rule gives
. Our results are
http://www.livingreviews.org/lrr-2011-7 |
Living Rev. Relativity 14, (2011), 7
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