## 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