48
2 Bivector Formalism
= −
i e 2 r 4
2 (r 4 + a 2 z 2 ) 2 k
c;p N cp k b
= −
e 2 a z r 5
(r 4 + a 2 z 2 ) 3 k b .
(2.215)
Putting (2.211), and (2.213) with (2.215), into (2.207) we finally arrive at
G ,d k
d
+
3 r
r 2 + i a z
G =
e 2 r 5 (r 2 − i a z)
(r 4 + a 2 z 2 ) 3 .
(2.216)
Substituting
G = G(r, z) −
e 2 r 4
(r 2 + i a z) 3 (r 2 − i a z)
,
(2.217)
we have the following equation for G:
G ,d k
d
+
3 r
r 2 + i a z
G = 0 .
(2.218)
The general solution of this equation is
G =
m(w) r 3
(r 2 + i a z) 3 with w =
z
r
,
(2.219)
where m is an arbitrary function of its argument. For asymptotic spherical symmetry
we must have
m = constant .
(2.220)
Combining (2.217) and (2.219) we obtain the function G in (2.205), namely,
G =
m r 3
(r 2 + i a z) 3 −
e 2 r 4
(r 2 + i a z) 3 (r 2 − i a z)
,
(2.221)
with m, e constants.
References
1. L. Mariot, C. R. Acad. Sci. 238, 2055 (1954)
2. I. Robinson, J. Math. Phys. 2, 290 (1961)
3. I. Robinson, A. Trautman, J. Math. Phys. 24, 1425 (1983)
4. L.P. Eisenhart, Riemannian Geometry (Princeton University Press, Princeton, 1966)
5. I. Robinson, A. Schild, J. Math. Phys. 2, 484 (1963)