how = 0, (3-11)
sin 08oe(sin Ohte + 8444)> = 0, (3-12)
sin 08o(sin Ohro + 84&,4) = 0, (3-13)
as the mode-independent expression of the RW gauge. Now we can transform this
description into GHP language. It is a relatively straightforward process now to write the
tetrad components of the metric perturbation (hiz, hi,, etc.) in terms of the coordinate
components of the metric perturbation (htt, h,,, etc.) and invert the relations. With
this knowledge in hand, it becomes evident that Equations 3-10 and 3-11 are simply
combinations of
Amm =0 and hm = 0.
The effect of these conditions is to remove the spin-weight +2 pieces from the metric
perturbation. After a quick look at the coordinate form of the a and 8' operators, we note
that Equations 3-12 and 3-13 are combinations of
B'hlm + c7l;.. = 0 and allt, + B'hm = 0,
which restricts the form of the spin-weight +1 parts of the metric perturbation. Note that
the essence of the RW gauge lies in the fact that all of the information about gravitational
radiation gets pushed into the spin-1 components of the metric perturbation.
In this language, it is natural to generalize these conditions to more general type
D spacetimes on the basis of spin-weight considerations. The spirit of the RW gauge
-II- -_ -r ;that we keep the requirement that no spin 2 components enter the metric
perturbation. The requirement on the spin 1 components is easily generalizable by putting
in pieces proportional to -r and -r' which both vanish in the Schwarzschild background.