Following the same procedure as in Eqs. (4-101)-(4-108), Eq. (4-119) develops
into
XA (TFN, FN)
A A K)XK
U TEN + "o(KXFN
1 FA [D B C D r3 i I C Dm2 XK
- 6 ABC, D 0oT#N +' .) 0TNXFKN
-+" C D 7cK xL + B C D xK xL M
+3,, -) )no(L)Uo TFN FN FN + o(K) o(L) o(M) FN FN FN
1 A BCDE4
24 BC,DE oLo o EFN .+ ), D o C D ETNXFKN
+6ro(K)nO(L) Uo FNXFNXFN
+ ,R -)rC D (E TFNX K LNX M
B C D E rK rL rM rN
+o(K) o(L) o(M) o(N) XFN FN FN FN
+O(X#N/T4), (4-120)
where the subscript o denotes the evaluation at the location of the particle, and the
quantities uA, nA A FcD o and FAC,DE o are evaluated in the coordinates XA. If
we identify XA with THZ normal coordinates, then this equation exactly tells us
how Fermi normal coordinates transform into THZ normal coordinates.
The linear part of the above transformation implies the inverse-Lorentz boost
and is responsible for the relationship between the metrics of the two geometries at
the location of the particle. The two metrics are in fact both Minkowskian there,
(4-121)
9(THZ)ABo g- (FN)AB o -7AB.
According to Ref. [24], this relationship must be satisfied via
A'B' AB OxA' OXB'
9(THZ) (FN) 8XAN AXBN.
FN EBN
(4-122)