8
Run-Away Reissner–Nordström Particle
Abstract
In the absence of external fields a Reissner–Nordström particle of mass m and
charge e for which m and e 2 are small of first order performs run-away motion.
Since in particular m 2 < e 2 there is no event horizon associated with this object
and so we refer to it as a Reissner–Nordström particle. A Reissner–Nordström
particle of mass m and magnetic monopole moment g for which m and g 2 are
small of first order behaves in precisely the same way.
8.1
Robinson–Trautman Solutions of the Einstein–Maxwell
Equations
It follows from Sect. 8.2 of the previous chapter that the relevant form of line
element in the absence of external fields is given by (2.1)–(2.5) with the functions
α, β, a, b vanishing. The result is a Robinson–Trautman [1] line element
ds
2
= −r
2 p
−2 (dx
2
+ dy
2 ) + 2 du dr + c du
2
= −(ϑ
(1) )
2
− (ϑ
(2) )
2
+ 2 ϑ
(3) ϑ
(4)
= g (a)(b) ϑ
(a) ϑ
(b) ,
(8.1)
with the basis 1-forms taken to be
ϑ
(1)
= r p
−1 dx , ϑ
(2)
= r p
−1 dy , ϑ
(3)
= dr +
1
2
c du , ϑ
(4)
= du ,
(8.2)
and the Robinson–Trautman potential 1-form
A = e(u)
1
r
+ w
du .
(8.3)
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
P. A. Hogan, D. Puetzfeld, Frontiers in General Relativity, Lecture Notes
in Physics 984, https://doi.org/10.1007/978-3-030-69370-1_8
187