A Second-order expansions of the Ricci tensor
We present here various expansions used in solving the second-order Einstein equation in
Section 22.4. We require an expansion of the second-order Ricci tensor
, defined by
where
is the trace-reversed metric perturbation, and an expansion of a certain piece of
.
Specifically, we require an expansion of
in powers of the Fermi radial coordinate
, where for
a function
,
consists of
with the acceleration
set to zero. We write
where the second superscript index in parentheses denotes the power of
. Making use of the expansion of
, obtained by setting the acceleration to zero in the results for
found in Section 22.3, one finds
and
and
Next, we require an analogous expansion of
, where
is defined for
any
by setting the acceleration to zero in
. The coefficients of the
and
terms in
this expansion can be found in Section 22.4; the coefficient of
will be given here. For
compactness, we define this coefficient to be
. The
-component of this quantity is given by
The
-component is given by
This can be decomposed into irreducible STF pieces via the identities
which follow from Eqs. (B.3) and (B.7), and which lead to
The
-component is given by
Again, this can be decomposed, using the identities
which lead to