remaining spin-0 jump conditions all yield the same result. With the source given by the
a 0 limit of Equation 6-7, we have for the tetrad components of the relevant objects:
km ro 2M Yoo (6-44)
& Pr 21Yoo (6-45)
mm 2
fi 8xpl-E
16r 3, s(r ro)6(cos 0)6(4 Ot), (6-46)
a rofo
with all the 6M~ dependence replaced according to Equation 6-43. Imposing Equation 6-6
then leads to
-6_ (cos 8)6(~ Rt),
orJ
2p(i + j) 8xp~E -
ioYo(,)= mx/,O)em0 ) (6-47)
= 0 m= -
where we've decomposed the angular delta functions according to Equation 6-5. We
can eliminate the sum on the right side of Equation 6-48 by multiplying both sides by
Yoo(0, ~), integrating over the sphere and exploiting the orthogonality of the spherical
harmonics. The result is that
(4r) 1/21E
P= -(6-48)
z+j
where we've used Yoo(B, 4) = Yoo(B, 4) = (4xr)-1/2. Finally, We have
[a] =-(6-49)
ro 2M~
sM = pE. (6-50)
These equations complete our construction of the matched spacetime. Note that the
above only restricts the difference between a~ on the interior and exterior. If we recall
Equation 6-41, we see that the same is generally true of Q(r) as well if we drop the
requirements of regularity in the interior and .I--phllcl~ flatness in the exterior.