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)