which represents the largest class of gaugfe transformations consistent with form
invariance. This requirement also restricts
S(t, 4) = pt + S(4),
(6-74)
while stationarity again necessitates
P(t) = a~t.
(6-75)
Next we turn our attention to the matching problem.
In order to clarify the issues involved in the matching problem, we'll take a look
at the matching conditions themselves. Suppose we've chosen some E,, but have yet to
specify it explicitly. That is, we have not yet written (or imposed) r = something. The full
set of matching conditions now take the form
(6-76)
(6-77)
(6-78)
(6-79)
htt [Ca](p2 + 2rM~) + 2[Plamr sin2 0 2r61M = 0,
44 : 2 [ca]amr [P] (a2a COS2 H 2 2 + 2) + 2amr)
+2amr dS- 2ar61M = 0,
h,, dQ r6M~
her Q = 0,
& : a2T Sin2 ObM~ (a2 COS2 2 2 2) + 2amr) = 0i, (6-80)
where a = T2 2M~r + a2 and 752 = 2 + 2 COS2 8 aS before and we have imposed the
condition in Equation 6-79 in the others. Note that this reduces to the Schwarzschild
result in Equations 6-39-641 by taking a 0 and setting r = ro. This set of equations