Let be the world line of a point particle
in a curved spacetime. It is described by parametric relations
in which
is proper time. Its tangent
vector is
and its acceleration is
; we shall also encounter
.
On we erect an orthonormal basis that
consists of the four-velocity
and three spatial vectors
labelled by a frame index
. These
vectors satisfy the relations
,
, and
.
We take the spatial vectors to be Fermi-Walker transported on the
world line:
, where
Consider a point in a neighbourhood of
the world line
. We assume that
is sufficiently close to the world line that a unique geodesic
links
to any neighbouring point
on
. The two-point function
, known as
Synge’s world function [55
], is numerically
equal to half the squared geodesic distance between
and
; it is positive if
and
are spacelike related, negative if they are timelike related, and
is zero if
and
are linked by a null geodesic. We denote its gradient
by
, and
gives a meaningful notion of a separation vector
(pointing from
to
).
To construct a coordinate system in this
neighbourhood we locate the unique point on
which is linked to
by a future-directed
null geodesic (this geodesic is directed from
to
); I shall refer to
as the retarded point
associated with
, and
will be called the retarded time. To tensors at
we assign indices
,
, …; this will distinguish them from tensors at a
generic point
on the world line, to which we have
assigned indices
,
, …. We have
, and
is a null vector that can be
interpreted as the separation between
and
.
To tensors at we assign indices
,
, …. These tensors will be decomposed in
a tetrad
that is constructed as follows: Given
we locate its associated retarded point
on the world line, as well as the null geodesic that
links these two points; we then take the tetrad
at
and parallel transport it to
along the null geodesic to obtain
.