10 Retarded coordinates
10.1 Geometrical elements
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
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
10.2 Definition of the retarded coordinates
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
10.3 The scalar field and the vector field
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 is
Similarly, 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
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
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
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: Retarded coordinates of a point relative to a world line . The retarded time
selects a particular null cone, the unit vector selects a particular generator of this null
cone, and the retarded distance selects a particular point on this generator. This figure is identical
to Figure 4.
10.4 Frame components of tensor fields on the world line
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 .
are the frame components of the Riemann tensor evaluated on . From these we form the useful
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
10.5 Coordinate displacements near
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).
10.6 Metric near
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
10.7 Transformation to angular coordinates
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
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
Finally, we note that in the angular coordinates, the metric determinant is given by
where is the determinant of ; in the canonical parameterization, .
10.8 Specialization to
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) become
When 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
It may be noted that the inverse relations are , , , and
, where was introduced in Eq. (10.46).