Equations 4-5 require
l"(hab ~(a;b)) = 0, (4-8)
where the semicolon denotes the covariant derivative. In terms of components this reads
2P61 = hul,
(4-9)
(P + p)(m + (B + ')ll = him,
(P + P)(m~ + (8' + 7')(: = him.
Similarly, for the trace condition in Equation 4-6 to be satisfied by the gauge transformed
metric, we require
a'(m + am + (P' + pl)(1 + (P + P)(n = hmm. (4-10)
Any extra gauge transformation that satisfies l"~((;b) = 0-solves the homogeneous form
of Equation 4-9preserves the four 1- h gauge conditions in Equations 4-5. This is what
is meant by residual gauge freedom. We will explicitly use this residual gauge freedom to
impose the 1- h and trace conditions simultaneously, thus establishing the IRG. We will
find that some gauge freedom still remains, as explained in Section 4.3.
Now, we turn our attention to the general case of type II background spacetimes.
4.2 Imposing the IRG in type II
In order to show that residual gauge freedom can be used to impose the IRG, we
need to solve for the residual gauge freedom as well as examine any perturbed Einstein
equation that might impede the imposition of the trace condition of the IRG. For this, we
turn to a coordinate-free integration method develop by Held. Rather than give a detailed
explanation, we present the basics and refer the interested reader to the literature for an
in-depth account [45, 46].