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.