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