this subject in OsI Ilpter 3. Continuing in our cartoon language (Equation 1-4), we now
consider the two sectors of the metric perturbation independently, writing
00v dd yodd
d 0 0 vodd Uodd ( 0
odd ?Ll todd todd
odd ?Ll todd todd
and
81 82 even Deven
even 82 'U3 U]
even even even 84 even
even .. even even S4-
The final step before appealing to the Einstein equations consists of choosing a gauge.
Equation 1-3 is invariant under the transformation
hub i ab ( @)~ab = ab a~b + b a, (1-12)
where (a is an arbitrary vector and 4~ is the Lie derivative. Taking the odd-parity sector
as an example, the R;- ear--Wheeler gauge vector takes the form
(" = (0, 0, AeAB BY~) Im)3
where A is a function chosen so that the odd parity part of the metric perturbation 1-10
takes the form
0 0 0 v dd
0 0 O r
0 00 0
vodd ?L" 0 0.
Similar simplifications arise in the even-parity sector.