of geodesics XA(s, A1, A2, A3), which will be inverted later to give Fermi normal
coordinates: here s is a parameter for a temporal measure along F and becomes
the time coordinate TFN via the inversion, and A' (I = 1, 2, 3) are parameters
for spatial measures along F(I) and become the spatial coordinates XIN via the
inversion. By combining the integral solutions of the geodesic equations (4-97)-
(4-99) we obtain
XA(s,Al,A2,A3) dsA ds I + dA' ds
r= ( )s )
+dA dA (4-100)
I=1 () J=-1
where A = (A1, A2, A3), and the subscripts outside () mean that these variables
are held fixed while the partial differentiations are performed with respect to the
others.
To evaluate the above integral, one need find proper expansions of each
integrand in terms of s and A. From Eqs. (4-97)-(4-99) we have
(8 ) -0 BC X-
BC, D u + 2 BC, DE BCD
a(/ ) (-BCU(I)o
8( X=o
iBC, D (I)U BC, DE (I)B Ci D E
S(1
+0(_k3/R4), (4-102)