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)