is no surprise, then, that the resulting background geometry possesses enough spherical
symmetry to allow for a straightforward treatment of the problem. It can he directly
verified that such a procedure would remove the 8 dependence in Equations 6-81-684 and
allow for a matching on r = constant surfaces (which are round 2-spheres in this case).
Because this approach fails to shed new light on the situation in the full K~err spacetime,
we will not follow it here. Instead, we will focus on Equations 6-66-6-69, which we know
to be correct.
Let's review the situation. We have established that the metric perturbation in
Equations 6-66-669 is a perturbation towards another K~err solution with differing
mass. Furthermore, we previously established that 6M~ = pE (Equation 6-26). The
problem is that we are currently unable to perform the matching. In practice, the relevant
portion of the spacetime is the exterior where gravitational radiation and the non-radiated
multipoles are observed far away from the source. Because of this, we contend that
considerations from the K~omar formula and Wald's theorem together provide the correct
perturbation in the exterior spacetime, independently of any matching considerations.
Thus our result is likely useful in the EAIRI problem even though we lack the metric
perturbation everywhere in the spacetime. Moreover, the perturbation is still simple to
interpret and .I-i-.npind'' ;cally flat, so it is amenable to some analysis.
This being the case, we remark that mass perturbations of the K~err background
remain confined to the s = 0 sector of the perturbation. It is likely that this is true in
general (at least in type D), but a general proof of this remains elusive. Furthermore,
contrary to what one might expect in the K~err spacetime, the mass perturbation does
not mix spherical harmonic -modes, but is purely -e = 0. We now turn our attention to
angular momentum perturbations.
6.2.2 Angular Momentum Perturbations
Our lack of success in matching mass perturbations extends to angular momentum
perturbations in precisely the same way, though the expressions involved are more