coordinates, its square is expressed as
p2 XXI x 2 + y2 + Z2. (4-142)
Substituting Eqs. (4-130) along with (4-132)-(4-135) into this equation and
simplifying the algebra, we obtain
P2 -hABA A + lhAE iABC, D BXC D E TAC ABCD BDE XE
3 o 3 o
+ hAF ABC,DE B C D E F _- TAC ABCD, E BXDXEXFX
12 o 12 o
1 TACT EF ABCD,E B D F G G
4 o
+ 7ACR7 E7GH -ABCD,E XBXDXFXGXH + 0(6/R4), (4-143)
6 0
where XA represents the initial normal coordinates and is constructed from the
background coordinates x" = (t, r, 0, 0) via Eqs. (4-131) together with (4-85),
(4-136) and (4-137). Also, 7AB and hAB are computed via Eqs. (4-117) and (4-118)
along with (4-138), and ABC, D RABCD BC, DE and RABCD, E are evaluated
using Eqs. (4-93)-(4-96) along with (4-91) and (4-92). The actual expression of p2
in the background coordinates x" = (t, r, 0, 0) would be very lengthy to the order
specified in Eq. (4-143), and here we specify it only to the cubic order in order to