which possesses only axial symmetry, the situation is immensely more complicated. This
issue will be discussed below.
Once we've agreed on a E,, fulfilling our first matching condition requires us to
simply equate the components of the metric (on E,). In other words,
[9a 9 |4 Rb, =0 (6-4)
where |4, indicates the restriction to E,. The only (slight) complication that arises here is
ensuring that there is enough freedom in the metric perturbation to perform the matching.
This will generally require performing a gauge transformation on the interior and exterior
spacetimes. Although this introduces some gauge dependence into the problem, the end
result 6M~ or 6a is in fact gauge invariant, as we will see below.
Imposing the second condition is a bit more involved because of the presence of
the source. By choosing a good matching surface, E,, we can effectively -on! I. out"
the angular dependence of the source. If, for example, E, is a 2-sphere, we can use the
completeness relations to write the angular delta function according to
= 0 m= -
Similar relations hold for complete sets of functions on different closed 2-surfaces. The
source now consists solely of a radial delta function. To handle this, we impose the
perturbed Einstein equations as, for example,
lim Sab dT b dr (6-6)
e-> ro-e o-
where Sab denotes the perturbed Einstein tensor and l~b denotes the stress-energy tensor
of the source and ro is the location of E, as seen from both sides. For a delta function
source due a particle of mass p in a circular equatorial orbit of the K~err spacetime,
lab s 6 rO)b COS 8)6(4 Ot), (6-7)