Using a, a bi-tensor Ta'b', whose indices all refer to the same point z, can be
expanded about z in the covariant form
1
Ta/'b = Aa/b/ + Aa/b/'c/'Uc + Aa'/b/cd'/;c' ;d' + O(S3), (2-39)
2
where the expansion coefficients Aa'b', Aa/bc'/, Aa'b'c'd/, etc. are ordinary local tensors
at z. These coefficients can be determined in terms of the covariant derivatives of
Ta/b' :
Aa/b/ = lim Ta/b/, (2-40)
x-z
Aab/c/ = lim Ta'b';c' Aa'b/;c (2-41)
X-Z
Aa'b'c'd' = lim Ta'b/;c/'d Aa'b';c'd' Aa'b'c';d' Aa'b'd';c'. (2-42)
x-z
A particular example of such expansions to note is
J;a'b' ga'b' ,+ R 'c'b'd' c' d'; + ( 0S3). (2-43)
3
One can develop the expansions to higher orders and obtain further
a'b'c' (Ra'c'b'+ Rad'b'c/ ;,d' +0(s2), (2-44)
1
;a'b'c'd' (Ra'cb'd' + Ra'd'b'c') + 0(s). (2-45)
3
For expanding a bi-tensor whose indices do not all refer to the same point,
for example Tab', one introduces a device called the bi-vector of geodetic paral-
lel displacement and denotes it by gab'(x, z). This bi-vector has the significant
geometrical interpretation in the defining equations
gab';cgdd 0, (2-46)
gab';c'gcd';d' = 0, (2-47)
lim gab' gab' or lim gab' 6ab.
X-z X-z
(2-48)