Our aim here is to find the quartic order expansions for Fermi normal coordinates,
which are derived from the inverse transformations of the quartic order expansions
of XA in s and A. XA are determined to the quartic order in s and A when
we evaluate the integral in Eq. (4-100) along with Eqs. (4-105)-(4-107). The
contributions from the order terms O(s3/ 4) or O[(s, A)3/)" 4] can be disregarded
since through the integration via Eq. (4-100) these become higher than quartic.
Substituting Eqs. (4-105)-(4-107) into Eq. (4-100) and performing the integral,
we obtain
XA(s, 1, A 2, 3)
AS+i nA +K
0 o(K)
SBC,D Uo CD + 3 0C 0D 2 ( BK)
+3Djs (^B K) L B ( K (C L (D M
-24 C, DE 0 C D 00 fs4 C+ 4D 10 3 ()ABK)
+6uDuE 2 (K) K (L) A
+(4 B K) L)( (L D ) () E
+ [(o(K)A ) (OL) ) L (M)M () (N)
+0[(s, A)5/R4]. (4-108)
The parameter s for a temporal measure along F becomes the time coordinate
TFN and the parameters A' for spatial measures along F(I) become the spatial
coordinates X1N (I = 1, 2, 3) when Eq. (4-108) is inverted and solved for s and A'.