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 , 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 .
The retarded coordinates of the point are , where are the frame components of the separation vector. They come with a straightforward interpretation (see Figure 4). The invariant quantityretarded distance between and the world line. The unit vector
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 .
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