evaluated on F as

 EJ RoloJ, (4-52)

 B3 = iPQRpQjo, (4-53)
 2

 SIJK [VKRoIo STj (4-54)

 IJK IPQVKRPQJO]STF, (4-55)

where STF means to take the symmetric, tracefree part with respect to the

spatial indices I, J, ... and the dot denotes differentiation of the multiple

moment with respect to T along F. Dimensionally, S j ~ BIJ 0(1/R2) and

SIJK U31JK SIJ J ~ 0(1/Rt3). The fact that all of the external multiple
moments are tracefree comes from the assumption that the background geometry is

a vacuum solution of the Einstein equations.

 The THZ coordinates are a special kind of harmonic (or de Donder) coordi-

nates. We may express the perturbed field by defining

 HAB AB AB, (4-56)

where gAB ggAB. A coordinate system is harmonic if and only if

 AHAB = 0. (4-57)

Zhang [21] provides an expansion of gAB for an arbitrary solution of the vacuum

Einstein equations in THZ coordinates, in his equation (3.26). In the leading lower

order terms, the metric perturbation HAB in this expansion is described as [18]


HAB 2 AB + 3HAB + O(p4/R4),


(4-58)