

3.3 Retarded coordinates
3.3.1
Geometrical
elements
We introduce the same geometrical elements as
in Section 3.2: We have a timelike curve
described by relations
, its normalized
tangent vector
, and its acceleration vector
. We also have an orthonormal triad
that is transported on the world line according
to
where
are the frame components of the
acceleration vector and
is
a prescribed rotation tensor. Here the triad is not Fermi-Walker transported: For added
generality we allow the spatial vectors to rotate as they are
transported on the world line. While
will be set to
zero in most sections of this paper, the freedom to perform such a
rotation can be useful and will be exploited in Section 5.4. It is easy to check that
Equation (138) is compatible with
the requirement that the tetrad
be
orthonormal everywhere on
. 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 Equation (115).
The Fermi normal coordinates of Section 3.2 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
3.3.2
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 Equation (142) is that
where
is a 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
Equation (145) follows from the fact
that a displacement along the world line is described by
.
3.3.3
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 Equation (147) we deduce the
important relation
In addition, combining the general statement
(cf. Equation (79)) with
Equation (144) gives
the vector at
is therefore obtained by parallel
transport of
on
. From this and Equation (141) 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
first calculate 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
Equation (150) 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 Equations (145) and (149), 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
.
Equation (153) also implies that the
expansion of the congruence is given by
Using the expansion for
given by Equation (91), we find that this
becomes
, or
after using Equation (144). 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 Equation (153) 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 Equation (145): 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 Equation (154) 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.
3.3.4
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
.
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 Equation (156). The identity
follows easily from Equations (160, 161, 162) and the definition of
the Ricci tensor.
In Section 3.2 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.
3.3.5
Coordinate
displacements near 
The changes in the quasi-Cartesian retarded
coordinates under a displacement of
are given by
Equation (145). In these we
substitute the standard expansions for
and
, as given by Equations (88) and (89), as well as
Equations (144) and (151). 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 Equation (166) follows immediately
from Equations (152) and (157). From
Equation (167) and the identity
we also infer
where we have used the facts that
and
; these last results were derived in
Equations (161) and (162). It may be checked
that Equation (168) agrees with
Equation (147).
3.3.6
Metric near 
It is straightforward (but fairly tedious) to
invert the relations of Equations (164) and (165) 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 Equation (140). We find
with
We see (as was pointed out in Section 3.3.4)
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
Equations (171, 172, 173) 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
Equations (171, 172, 173) 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
Equations (150) and (157).
The metric determinant is computed from
, which gives
where we have substituted the identity of Equation (163). Comparison with
Equation (156) then gives us the
interesting relation
, where
is the expansion of the generators of
the null cones
.
3.3.7
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 3.3.3
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 takes the form of
with
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,
.
3.3.8
Specialization to

In this section we specialize our previous
results to a situation where
is a geodesic on which the
Ricci tensor vanishes. We therefore set
everywhere on
, and for simplicity we also set
to zero.
We have seen in Section 3.2.6 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
Equations (160, 161, 162) 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 Equations (196, 197, 198) become
When Equations (204, 205, 206) are substituted into
the metric tensor of Equations (171, 172, 173) - in which
and
are both set equal to zero - we obtain
the compact expressions
The metric becomes
after transforming to angular coordinates using the rules of
Equation (185). Here we have
introduced the projections
It may be noted that the inverse relations are
,
,
, and
, where
was introduced in Equation (183).

