Eqs. (2-57)-(2-59) have the unique solution
6(x, z) = g/2(X)g1/2() -1(z, x), (2-60)
where
9 = 9ab (2-61)
A local vector density Ab, of weight w transforms into the local vector Aa along the
geodesic from z to x by parallel displacement in the manner
A, w-gab'Ab,. (2-62)
The transformation by parallel displacement can be extended to the general case.
A bi-scalar of fundamental importance in the theory of geodesics is the Van
Vleck determinant, given by
A -g- 1 ;ab' (2-63)
where
= -I9ab' (2-64)
with the property
g(x,z) g/2(x)g1/2(z) =g(z,x). (2-65)
Differentiating Eq. (2-37) repeatedly and using Eq. (2-63), one can show that
A-1 (A"7) = 4. (2-66)
Also important is the expansion of this determinant, known to be
A R a'b'(;a';b' + O(s3). (2-67)
6