with p > 0 and p+ q > 2.
(ii) The coordinates satisfy the de Donder gauge condition
OAQAB = O(Xn),
where gAB /gAB
The metric perturbation in THZ coordinates is described as [18]
gAB
TIAB + HAB
rIAB + 2HAB + 3HAB + O(p4/R4), p/R 0,
with
3HABdXAdXB
-SjXI XJ(dT2 + 6KLdXKdXL)
+ KPQ3I XPXIdTdXK
3
-20 XXJXK 2PIX dTdXK
+ XIJPQeQK PK X 1 pi2 BQ.XP dX'dXJ
(4-50)
-SIJKXIXXK (dT2 + 6KLdXKdXL)
3
2
+ KPQBQIJXPXI JdTdXK
3(p/ dXdXJ,
+O(p4/R4 IJjdXdXj,
where rlAB is the flat Minkowski metric, CIJK is the flat space Levi-Civita tensor,
p = (X2 + 2 + Z2)1/2, and the indices I, J, K, L, P and Q are spatial and
raised and lowered with the three dimensional flat space metric 61j while the
index 0 denotes the time component. The external multiple moments are spatial,
symmetric, tracefree tensors and are defined in terms of the Riemann tensor
(4-48)
2HABdXAdXB
(4-49)
and
(4-51)