Substituting these linear approximations for s and A' on the right hand sides
of Eqs. (4-109) and (4-110), we finally obtain the expressions of Fermi normal
coordinates, written in terms of the initial coordinates XA, precisely up to the
quartic order
TFN = -UA [XA + ABCDBCXD + A BCDEXXXCDE + O(X5/R4),
(4-113)
XN A'I L A + ABCDX BC D + (A BCDEX B_" X + o(X5/R4),
(4-114)
where
ABCD PQ, R V B TT RD + 37TQB rRchPD + 376RBhPChQD
b o
+hPBhQchRD) (4-115)
BCDE PQ,RS (7 PB7Q CRDTSE + 47r QB rRr SDhPE
ABCDE 4 P
+6rRB rSchPDhQE + 47RBhPchQDhRE
+hPBhQchRDhSE) (4-116)
with 7AB being the time-projection tensor and hAB being the space-projection
tensor, which are defined as
7AB -U oo (4-117)
hAB A (I) AB Au (4-118)
ho(I)noB A o 0oB,
respectively.1
1 The identity provided in Eq. (4-118) is obvious from the local tetrad, or lo-
cal vierbein: vAP) {uA, uj, n2, n ) ) where P e (0,1, 2, 3) is the label for