has a solution given by
6M =[] (6-81)
[0]~ =9 (6-82)
[P] [Ca], (6-83)
dS a2a Sin2 8
-[0], (6-84)
d# (T2 82 a2
which is again easily seen to reduce to the Schwarzschild result in the appropriate limit.
From these equations we can see clearly the issues involved in choosing a matching
surface. First, because the left sides of Equations 6-81-684 are all constant, this must be
reflected in the right sides as well, which currently exhibit dependence on both r and 0.
Presumably, some choice of r = r (0) will enforce this, though it is currently unclear what
that choice might be. Note that because of this, r = constant surfaces do not appear to be
good for matching.
What we have encountered appears to be an instance of a longstanding problem
with matching the K~err solution to a source [74, 75]. Namely, there is no known matter
solution that correctly reproduces the multiple structure of the full K~err geometry. In our
problem, we're trying to force the issue by specifying both the metric and the source. On
the other hand, because we're not matching the entire source, which includes quadrupole
and higher moments, but only the non-radiated multipoles that merely take us from one
K~err solution to the next, it is not clear that the matching (in this instance) should fail.
Though we are unable to perform the matching here, we maintain that nothing forbids it.
Most authors faced with this issue turn to the -lei-- rotation" approximation and
keep only terms linear in a. In this approximation the K~err metric can be viewed as the
first order perturbation of the Schwarzschild solution to the K~err solution. That is, the
background is given by Schwarzschild plus a term identical to that in Equation 6-59. It