We introduce the same geometrical elements as in Section 9: we have a timelike curve described by relations , its normalized tangent vector , and its acceleration vector . We also have an orthonormal triad that is FW transported on the world line according to

where are the frame components of the acceleration vector. Finally, we have a dual tetrad , with and . The tetrad and its dual give rise to the completeness relations which are the same as in Eq. (9.4).The Fermi normal coordinates of Section 9 were constructed on the basis of a spacelike geodesic connecting a field point to the world line. The retarded coordinates are based instead on a null geodesic going from the world line to the field point. We thus let be within the normal convex neighbourhood of , be the unique future-directed null geodesic that goes from the world line to , and be the point at which intersects the world line, with denoting the value of the proper-time parameter at this point.

From the tetrad at we obtain another tetrad at by parallel transport on . By raising the frame index and lowering the vectorial index we also obtain a dual tetrad at : and . The metric at can be then be expressed as

and the parallel propagator from to is given by

The quasi-Cartesian version of the retarded coordinates are defined by

the last statement indicates that and are linked by a null geodesic. From the fact that is a null vector we obtain and is a positive quantity by virtue of the fact that is a future-directed null geodesic – this makes past-directed. In flat spacetime, , and in a Lorentz frame that is momentarily comoving with the world line, ; with the speed of light set equal to unity, is also the spatial distance between and as measured in this frame. In curved spacetime, the quantity can still be called the retarded distance between the point and the world line. Another consequence of Eq. (10.5) is that where is a unit spatial vector that satisfies .A straightforward calculation reveals that under a displacement of the point , the retarded coordinates change according to

where is a future-directed null vector at that is tangent to the geodesic . To obtain these results we must keep in mind that a displacement of typically induces a simultaneous displacement of because the new points and must also be linked by a null geodesic. We therefore have , and the first relation of Eq. (10.8) follows from the fact that a displacement along the world line is described by .

If we keep linked to by the relation , then the quantity

can be viewed as an ordinary scalar field defined in a neighbourhood of . We can compute the gradient of by finding how changes under a displacement of (which again induces a displacement of ). The result isSimilarly, we can view

as an ordinary vector field, which is tangent to the congruence of null geodesics that emanate from . It is easy to check that this vector satisfies the identities from which we also obtain . From this last result and Eq. (10.10) we deduce the important relation In addition, combining the general statement – cf. Eq. (5.12) – with Eq. (10.7) gives the vector at is therefore obtained by parallel transport of on . From this and Eq. (10.4) we get the alternative expression which confirms that is a future-directed null vector field (recall that is a unit vector).The covariant derivative of can be computed by finding how the vector changes under a displacement of . (It is in fact easier to calculate first how changes, and then substitute our previous expression for .) The result is

From this we infer that satisfies the geodesic equation in affine-parameter form, , and Eq. (10.13) informs us that the affine parameter is in fact . A displacement along a member of the congruence is therefore given by . Specializing to retarded coordinates, and using Eqs. (10.8) and (10.12), we find that this statement becomes and , which integrate to and , respectively, with still denoting a constant unit vector. We have found that the congruence of null geodesics emanating from is described by in the retarded coordinates. Here, the two angles () serve to parameterize the unit vector , which is independent of .Eq. (10.16) also implies that the expansion of the congruence is given by

Using Eq. (6.10), we find that this becomes , or after using Eq. (10.7). Here, , , and are the frame components of the Ricci tensor evaluated at . This result confirms that the congruence is singular at , because diverges as in this limit; the caustic coincides with the point .Finally, we infer from Eq. (10.16) that is hypersurface orthogonal. This, together with the property that satisfies the geodesic equation in affine-parameter form, implies that there exists a scalar field such that

This scalar field was already identified in Eq. (10.8): it is numerically equal to the proper-time parameter of the world line at . We conclude that the geodesics to which is tangent are the generators of the null cone . As Eq. (10.17) indicates, a specific generator is selected by choosing a direction (which can be parameterized by two angles ), and is an affine parameter on each generator. The geometrical meaning of the retarded coordinates is now completely clear; it is illustrated in Figure 7.The metric at in the retarded coordinates will be expressed in terms of frame components of vectors and tensors evaluated on the world line . For example, if is the acceleration vector at , then as we have seen,

are the frame components of the acceleration at proper time .Similarly,

are the frame components of the Riemann tensor evaluated on . From these we form the useful combinations in which the quantities depend on the angles only – they are independent of and .We have previously introduced the frame components of the Ricci tensor in Eq. (10.19). The identity

follows easily from Eqs. (10.23) – (10.25) and the definition of the Ricci tensor.In Section 9 we saw that the frame components of a given tensor were also the components of this tensor (evaluated on the world line) in the Fermi normal coordinates. We should not expect this property to be true also in the case of the retarded coordinates: the frame components of a tensor are not to be identified with the components of this tensor in the retarded coordinates. The reason is that the retarded coordinates are in fact singular on the world line. As we shall see, they give rise to a metric that possesses a directional ambiguity at . (This can easily be seen in Minkowski spacetime by performing the coordinate transformation .) Components of tensors are therefore not defined on the world line, although they are perfectly well defined for . Frame components, on the other hand, are well defined both off and on the world line, and working with them will eliminate any difficulty associated with the singular nature of the retarded coordinates.

The changes in the quasi-Cartesian retarded coordinates under a displacement of are given by Eq. (10.8). In these we substitute the standard expansions for and , as given by Eqs. (6.7) and (6.8), as well as Eqs. (10.7) and (10.14). After a straightforward (but fairly lengthy) calculation, we obtain the following expressions for the coordinate displacements:

Notice that the result for is exact, but that is expressed as an expansion in powers of .These results can also be expressed in the form of gradients of the retarded coordinates:

Notice that Eq. (10.29) follows immediately from Eqs. (10.15) and (10.20). From Eq. (10.30) and the identity we also infer where we have used the facts that and ; these last results were derived in Eqs. (10.24) and (10.25). It may be checked that Eq. (10.31) agrees with Eq. (10.10).

It is straightforward (but fairly tedious) to invert the relations of Eqs. (10.27) and (10.28) and solve for and . The results are

These relations, when specialized to the retarded coordinates, give us the components of the dual tetrad at . The metric is then computed by using the completeness relations of Eq. (10.3). We find where . We see (as was pointed out in Section 10.4) that the metric possesses a directional ambiguity on the world line: the metric at still depends on the vector that specifies the direction to the point . The retarded coordinates are therefore singular on the world line, and tensor components cannot be defined on .By setting in Eqs. (10.34) – (10.36) we obtain the metric of flat spacetime in the retarded coordinates. This we express as

In spite of the directional ambiguity, the metric of flat spacetime has a unit determinant everywhere, and it is easily inverted: The inverse metric also is ambiguous on the world line.To invert the curved-spacetime metric of Eqs. (10.34) – (10.36) we express it as and treat as a perturbation. The inverse metric is then , or

The results for and are exact, and they follow from the general relations and that are derived from Eqs. (10.13) and (10.20).The metric determinant is computed from , which gives

where we have substituted the identity of Eq. (10.26). Comparison with Eq. (10.19) gives us the interesting relation , where is the expansion of the generators of the null cones .

Because the vector satisfies , it can be parameterized by two angles . A canonical choice for the parameterization is . It is then convenient to perform a coordinate transformation from to , using the relations . (Recall from Section 10.3 that the angles are constant on the generators of the null cones , and that is an affine parameter on these generators. The relations therefore describe the behaviour of the generators.) The differential form of the coordinate transformation is

where the transformation matrix satisfies the identity .We introduce the quantities

which act as a (nonphysical) metric in the subspace spanned by the angular coordinates. In the canonical parameterization, . We use the inverse of , denoted , to raise upper-case Latin indices. We then define the new object which satisfies the identities The second result follows from the fact that both sides are simultaneously symmetric in and , orthogonal to and , and have the same trace.From the preceding results we establish that the transformation from to is accomplished by

while the transformation from to is accomplished by With these transformation rules it is easy to show that in the angular coordinates, the metric is given by The results , , and are exact, and they follow from the fact that in the retarded coordinates, and .The nonvanishing components of the inverse metric are

The results , , and are exact, and they follow from the same reasoning as before.Finally, we note that in the angular coordinates, the metric determinant is given by

where is the determinant of ; in the canonical parameterization, .

In this subsection we specialize our previous results to a situation where is a geodesic on which the Ricci tensor vanishes. We therefore set everywhere on .

We have seen in Section 9.6 that when the Ricci tensor vanishes on , all frame components of the Riemann tensor can be expressed in terms of the symmetric-tracefree tensors and . The relations are , , and . These can be substituted into Eqs. (10.23) – (10.25) to give

In these expressions the dependence on retarded time is contained in and , while the angular dependence is encoded in the unit vector .It is convenient to introduce the irreducible quantities

These are all orthogonal to : and . In terms of these Eqs. (10.59) – (10.61) becomeWhen Eqs. (10.67) – (10.69) are substituted into the metric tensor of Eqs. (10.34) – (10.36) – in which is set equal to zero – we obtain the compact expressions

The metric becomes after transforming to angular coordinates using the rules of Eq. (10.48). Here we have introduced the projections It may be noted that the inverse relations are , , , and , where was introduced in Eq. (10.46).

Living Rev. Relativity 14, (2011), 7
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