This effect is examined via the equations of motion of the particle which describe
its local behavior, and they can be obtained only if one keeps an instantaneous
record in the immediate neighborhood of the particle. For this purpose one
constructs a three-dimensional hypersurface around the world-line of the particle,
or the world-tube, which is generated by a small sphere surrounding the particle as
time varies. In terms of Synge's world function, the generating sphere of radius e,
as time varies, produces a hypersphere defined by
2
a= (2-130)
;o, = -en(i)a' i, (2-131)
alza' = 0, (2-132)
where n') (I = 2, 3) denotes spatial basis vectors which are orthogonal to each
other and span the hypersurface orthogonal to the world-line of the particle,
)Jn(j)a/, 6, (2-133)
n(j)aa' = 0, (2-134)
and fi represents a set of direction cosines which satisfy
ii = 1. (2-135)
In terms of i one can specify the direction relative to na of an arbitrary unit
vector which is perpendicular to the world-line at z. Then, starting in the direction
of this arbitrary vector, one constructs a geodesic emanating from z extending out
to a fixed distance C to a point x. The coordinates of x will depend on the direction
cosines i and on the proper time r which is the parameter for the point z.