The Fermi normal coordinates of refer to a point on that is related to by a spacelike geodesic that intersects orthogonally; see Figure 8. We refer to this point as ’s simultaneous point, and to tensors at we assign indices , , etc. We let be the Fermi normal coordinates of , with denoting the value of ’s proper-time parameter at , representing the proper distance from to along the spacelike geodesic, and denoting a unit vector () that determines the direction of the geodesic. The Fermi normal coordinates are defined by and . Finally, we denote by the tetrad at that is obtained by parallel transport of on the spacelike geodesic.

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 who does not wish to follow the details of this discussion can be informed that: (i) our results concerning the transformation from the retarded coordinates to the Fermi normal coordinates are contained in Eqs. (11.1) – (11.3) below; (ii) our results concerning the transformation from the Fermi normal coordinates to the retarded coordinates are contained in Eqs. (11.4) – (11.6); (iii) the decomposition of each member of in the tetrad is given in retarded coordinates by Eqs. (11.7) and (11.8); and (iv) the decomposition of each member of in the tetrad is given in Fermi normal coordinates by Eqs. (11.9) and (11.10).

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 11.4.

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 7.)

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 Eq. (6.7) 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 Eq. (10.7). 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

where we show explicitly that all frame components are evaluated at the retarded point .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 subsection 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 11.1: it suffices to replace all primed indices by barred indices and then substitute the relation that follows immediately from Eq. (9.5). This gives

after recalling that . Solving for as an expansion in powers of returns

in which we emphasize that all frame components are evaluated at the simultaneous point .An expression for can be obtained by expanding in powers of . We have

and substitution of our previous results gives

for the retarded distance between and .Finally, the retarded coordinates can be related to the Fermi coordinates by expanding in powers of , so that

Results from the preceding subsection 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 subsection. 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 9.1 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 9.2, 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 Eq. (6.11), 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

and we shall call this the advanced distance between and the world line. Notice that is a positive quantity.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 11.2. 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 11.1, we write this as

Solving for as an expansion in powers of , we obtain

in which all frame components are evaluated at the retarded point .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 Eq. (11.12) for and witnessing a number of cancellations, we arrive at the simple expression

From Eqs. (10.29), (10.30), and (11.12) we deduce that the gradient of the advanced time is given by

where the expansion in powers of was truncated to a sufficient number of terms. Similarly, Eqs. (10.30), (10.31), and (11.13) imply that the gradient of the advanced distance is given by where and were first introduced in Eqs. (10.24) and (10.25), respectively. We emphasize that in Eqs. (11.14) and (11.15), all frame components are evaluated at the retarded point .

Living Rev. Relativity 14, (2011), 7
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