Then the inverse-Lorentz boost must be the identity transformation,
oA T A-A "A, (4-123)
A TFN o
A -A AA XA 6Ai, (4-124)
FN o
and the whole transformation in Eq. (4-120) is characterized by cubic or higher
order corrections between the two coordinate expressions. Also, we can specify
A A (4-125)
IBC,D o =HABCD + HACBD HBCD, (-125)
BFC,DE o = 3 (HABCDE + HACBDE HBCDE) (4-126)
using the components of the metric perturbations taken from Eqs. (4-50) and
(4-51). Then, with iU, nI) C, D and FBDE o specified, the transformation in
Eq. (4-120) is completely determined to be
S= TN KLXXF NpN + O(X N/R4), (4-127)
S FN 1N 68 KL FN N
1 1 K L
X XN SCKXFNP~N + 'KLXFNX FNN
6 3
1 1
SKXFNPFNTFN + t'KLXFNXFNXFNTFN
6 3
1 1
SKLX KNX K N 2 KLMX NXN XM X N
~ KLFN FNPFN 1 i FN FN FN FN
24 2
-2MK MLXKFNX pN + O(X5 N 4), (4-128)
63 MK LFNFNPFN FN
where T TTHZ, X' XT HZ and PFN XFN + YN + ZFN. In Appendix C it is
verified that the coordinate transformation via Eqs. (4-127) and (4-128) properly
converts the metric of Fermi normal geometry into that of THZ normal geometry
with the help of some properties on the Riemann tensors for vacuum spacetime.
Finally, in order to express the THZ coordinates XA in terms of the back-
ground coordinates x" = (t, r, 0, 0), we combine Eqs. (4-113) and (4-114) with
(4-127) and (4-128) along with (4-83). The final result is