496
G. Kamath
the constants c, b work to
4c = 1 ±
√
2,
2b =
m 2 + M
1 ±
√
2
(26)
with the parameter m 2 hitherto free, as a solution to
4c
2 m
4
+ (4c + 1) m
2 M +
1
4
7M
2
+
−3 ± 2
√
2
J 2
l 2
= 0
(27)
Clearly, m = 0 is not a root of this equation as it would imply that
J
2
= 7
−3 ∓ 2
√
2
M
2 l
2
(28)
Thus it would seem that there are no θ -dependent solutions to (19) with the metric
given by (18). That this conclusion is wrong is easily checked from a Frobenius
approach to the differential equation; as an alternative therefore to Whittaker’s
method for Eq. (17), the sequel suggests a way out may consist in solving Maxwell’s
equations as first-order partial differential equations with constant coefficients in
curved space following Moses [5] and Plebanski [8]. To this end we define following
Ref. [8],
D a =
√ −gf
0a , H =
√ −gf
21 , E a = f a0 , B = f 21
(29)
to obtain
D i,i = −
4π
c
j
0
D 1,0 + H ,2 =
4π
c
j
1
(30)
D 2,0 − H ,1 =
4π
c
j
2
−B ,0 + E 2,1 − E 1,2 = 0
as Maxwell’s equations, noting that in Cartesian coordinates (see Eq. (43) below)
√ −g = 1. The definitions