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
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
The quasi-Cartesian version of the retarded coordinates are defined by
the last statement indicates that A straightforward calculation reveals that under a displacement of the point , the retarded
coordinates change according to
If we keep linked to
by the relation
, then the quantity
Similarly, we can view
as an ordinary vector field, which is tangent to the congruence of null geodesics that emanate from 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
Eq. (10.16) also implies that the expansion of the congruence is given by
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
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,
Similarly,
are the frame components of the Riemann tensor evaluated on We have previously introduced the frame components of the Ricci tensor in Eq. (10.19). The identity
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:
These results can also be expressed in the form of gradients of the retarded coordinates:
Notice that Eq. (10.29
It is straightforward (but fairly tedious) to invert the relations of Eqs. (10.27) and (10.28
) and solve for
and
. The results are
By setting in Eqs. (10.34
) – (10.36
) we obtain the metric of flat spacetime in the
retarded coordinates. This we express as
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 metric determinant is computed from , which gives
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
We introduce the quantities
which act as a (nonphysical) metric in the subspace spanned by the angular coordinates. In the canonical parameterization, From the preceding results we establish that the transformation from to
is accomplished
by
The nonvanishing components of the inverse metric are
The resultsFinally, we note that in the angular coordinates, the metric determinant is given by
where
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
It is convenient to introduce the irreducible quantities
These are all orthogonal to 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
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Living Rev. Relativity 14, (2011), 7
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