The retarded coordinates of refer to a point
on
that is linked to
by a future-directed
null geodesic (see Figure 8
). We refer to this
point as
’s retarded
point, and to tensors at
we assign indices
,
, etc. We let
be the retarded coordinates of
, with
denoting the value of
’s proper-time parameter at
,
representing the
affine-parameter distance from
to
along the null geodesic, and
denoting a unit vector (
)
that determines the direction of the geodesic. The retarded
coordinates are defined by
and
. Finally, we denote by
the tetrad at
that is obtained by parallel
transport of
on the null geodesic.
The reader not interested in following the details of this discussion can be informed that
Our final task will be to define, along with the
retarded and simultaneous points, an advanced
point on the world line
(see Figure 8
). This is taken on in
Section 3.4.4. Throughout this section we
shall set
, where
is the rotation
tensor defined by Equation (138
) - the tetrad vectors
will be assumed to be Fermi-Walker transported on
.
Quantities at can be
related to quantities at
by Taylor
expansion along the world line
. To implement this strategy
we must first find an expression for
. (Although
we use the same notation, this should not be confused with the van
Vleck determinant introduced in Section 2.5.)
Consider the function of the
proper-time parameter
defined by
in which is kept fixed and in which
is an arbitrary point on the world line. We have
that
and
, and
can ultimately be obtained by expressing
as
and expanding in powers of
. Formally,
where overdots (or a number within brackets)
indicate repeated differentiation with respect to . We have
We now express all of this in retarded
coordinates by invoking the expansion of Equation (88) for
(as well as additional expansions for the higher
derivatives of the world function, obtained by further
differentiation of this result) and the relation
first derived in
Equation (144
). With a degree of
accuracy sufficient for our purposes we obtain
Collecting our results we obtain
which can readily be solved for expressed as an expansion in powers of
. The final result is
To obtain relations between the spatial coordinates we consider the functions
in which is fixed and
is an arbitrary point on
. We have that the
retarded coordinates are given by
, while
the Fermi coordinates are given instead by
. This last expression can be expanded
in powers of
, producing
with
Collecting our results we obtain
The techniques developed in the preceding
Section 3.4.2 can easily be adapted to the
task of relating the retarded coordinates of to its Fermi normal coordinates. Here we use
as the reference point and express all quantities at
as Taylor expansions about
.
We begin by considering the function
of the proper-time parameter
on
. We have that
and
, and
is now obtained by
expressing
as
and expanding
in powers of
. Using the fact that
, we have
Expressions for the derivatives of evaluated at
can be constructed from
results derived previously in Section 3.4.1:
it suffices to replace all primed indices by barred indices and
then substitute the relation
that
follows immediately from Equation (116
). This gives
after recalling that . Solving for
as an expansion in powers of
returns
An expression for can be
obtained by expanding
in powers of
. We have
and substitution of our previous results gives
for the retarded distance betweenFinally, the retarded coordinates can be related to the Fermi coordinates
by expanding
in powers of
, so that
Results from the preceding Section 3.4.2 can again be imported with mild alterations, and we find
Recall that we have constructed two sets of
basis vectors at . The first set is the tetrad
that is obtained by parallel transport of
on the spacelike geodesic that links
to the simultaneous point
. The second
set is the tetrad
that is obtained by parallel
transport of
on the null geodesic that
links
to the retarded point
. Since each
tetrad forms a complete set of basis vectors, each member of
can be decomposed in the tetrad
, and correspondingly, each member of
can be decomposed in the tetrad
. These decompositions are worked out in this
Section. For this purpose we shall consider the functions
in which is a fixed point in a
neighbourhood of
,
is an arbitrary
point on the world line, and
is the
parallel propagator on the unique geodesic that links
to
. We have
,
,
, and
.
We begin with the decomposition of in the tetrad
associated
with the retarded point
. This decomposition will be
expressed in the retarded coordinates as an expansion in powers of
. As in Section 3.2.1 we express quantities at
in terms of quantities at
by expanding in powers of
. We have
with
with
We now turn to the decomposition of in the tetrad
associated
with the simultaneous point
. This decomposition will be
expressed in the Fermi normal coordinates as an expansion in powers
of
. Here, as in Section 3.2.2, we shall express quantities
at
in terms of quantities at
. We begin with
and we evaluate the derivatives of at
. To accomplish this we rely
on our previous results (replacing primed indices with barred
indices), on the expansions of Equation (92
), and on the
decomposition of
in the tetrads at
and
. This gives
in which we substitute
It will prove convenient to introduce on the
world line, along with the retarded and simultaneous points, an
advanced point associated with the
field point . The advanced point will be denoted
, with
denoting the value of the proper-time
parameter at
; to tensors at this point we assign
indices
,
, etc. The advanced point is
linked to
by a past-directed
null geodesic (refer back to Figure 8
), and it can be
located by solving
together with the
requirement that
be a future-directed null
vector. The affine-parameter distance between
and
along the null geodesic is given by
We wish first to find an expression for
in terms of the retarded coordinates of
. For this purpose we
define
and re-introduce the function
first considered in Section 3.4.2.
We have that
, and
can ultimately be obtained by expressing
as
and expanding in powers of
. Recalling that
, we
have
Using the expressions for the derivatives of that were first obtained in Section 3.4.1,
we write this as
Solving for as an expansion in powers of
, we obtain
Our next task is to derive an expression for the
advanced distance . For this purpose we observe that
, which we can expand in powers of
. This gives
which then becomes
After substituting Equation (229) for
and witnessing a number of cancellations, we arrive
at the simple expression
From Equations (166), (167
), and (229
) we deduce that the
gradient of the advanced time
is given by