One may introduce a bi-scalar of geodetic interval, which is of fundamental
importance in the study of the non-local properties of spacetime. It is defined as
the magnitude of the invariant distance between x and z as measured along the
geodesic joining them. Denoting it by s(x, z), one may express its basic properties
in the equations
ab a'b'
gS;a;b S;a'S;b' = 1, (2-34)
lims = 0, (2-35)
X-Z
where the signature of the metric is taken as (-1, +1, +1, +1) (compare this
with Dirac's convention in Section 2.1). The interval between x and z is said to
be spacelike when the sign is + and timelike when the sign is in Eq. (2-34).
However, the bi-scalar itself is taken non-negative. When s = 0, the locus of points
x define the light cone through z.
Geodesics joining x and z may not necessarily be unique, and the bi-scalar of
geodetic interval can be multiple-valued. However, there will be a region in which
the geodetic interval is single valued, and our attention is confined to this region
in developing our argument: the geodetic interval in this single-valued region can
serve as the structural element of covariant expansion techniques later. And in
order to avoid Ii i~1i I! point" problems, instead of s, it will be more convenient to
work with the quantity, which is known as Synge's world function [13],
S- =s2, (2-36)
2
which satisfies
11 a'b'
t abq ;;b 'bqa ;a':;b' = a, (2-37)
2 2
lima = 0, (2-38)
where he interval is said o be spacelike wih sign and imelike wih sign.
where the interval is said to be spacelike with + sign and timelike with sign.