in which the first form indicates that the equation is real, while the second form and its
complex conjugate (which follow from the fact that Pp = p2 and Pp = p2) is the one we
will use to integrate the equation below. If we had not made use of the Goldberg-Sachs
theorem, there would be terms such as o-phmm appearing in Equation 4-12 and our
argument would not hold. We immediately see that 1 = 0 is necessary to satisfy the
trace condition in Equation 4-6. Next we turn our attention to the question of whether
the condition SiI = 0, is sufficient to impose Equation 4-6 using residual gauge freedom.
In order to address this question we will integrate SiI = 0 and the residual gauge
vector, given by the homogeneous form of Equations 4-9. Full integration of the
homogeneous form of Equations 4-9 is carried out in Appendix C, but we will work
through the integration of SiI = 0 here to illustrate the method. We begin by rewriting
Equation 4-12, with the help of Pp = p2 and its complex conjugate, as:
{(9 -,c 2p)( + p p)}h = p'29 kmm ,,)= 0. (4-13)
Integrating once gives
and another integration leads to
hmm= ao lbo(p + p). (4-15)
p 2
However, hmm is, by definition, a real quantity, so we add the complex conjugate and use
be to represent a real quantity in the second term. The final result is that
hmm = aoP + aoP + bo(p + p). (4-16)
p p