3
Hypothetical Objects in Electromagnetism
and Gravity
Abstract
I. The Maxwell field of a charged light-like particle with a non-geodesic world
line (a light-like analogue of the Liénard–Wiechert field) can be constructed
utilising the Minkowskian geometry in the neighbourhood of such a world line. In
the process a fundamental question regarding the existence of a special parameter
along the world line has to be addressed.
II. In the original generalisation of the Schwarzschild black hole with the
introduction of a variable mass by Vaidya, the variable mass depends upon a
parameter which has a simple geometrical origin. This parameter can also be
identified from the geometry in the case of a Kerr black hole. With the mass
and angular momentum in this case depending upon this parameter a rotating
generalisation of the Vaidya space-time emerges.
3.1
Part I: A Light-Like Charge
From the point of view of Minkowskian geometry the Coulomb field is a solution
of Maxwell’s equations on Minkowskian space-time which is singular on a timelike geodesic (the history of a point charge). The Liénard–Wiechert field is the
generalisation of the Coulomb field which is singular on a non-geodesic world
line (the history of an accelerated point charge). When considering a hypothetical
charged particle moving with the speed of light there is a Maxwell field available to
describe it which is singular on a null geodesic and is a spin-off from the Robinson–
Trautman solutions of the Einstein–Maxwell field equations. To describe it we begin
with the Minkowskian line element in coordinates X i = (T , X, Y, Z):
ds
2
= η ij dX
i dX
j
= dT
2
− dX
2
− dY
2
− dZ
2 .
(3.1)
© 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_3
51
Hypothetical Objects in Electromagnetism
and Gravity
Abstract
I. The Maxwell field of a charged light-like particle with a non-geodesic world
line (a light-like analogue of the Liénard–Wiechert field) can be constructed
utilising the Minkowskian geometry in the neighbourhood of such a world line. In
the process a fundamental question regarding the existence of a special parameter
along the world line has to be addressed.
II. In the original generalisation of the Schwarzschild black hole with the
introduction of a variable mass by Vaidya, the variable mass depends upon a
parameter which has a simple geometrical origin. This parameter can also be
identified from the geometry in the case of a Kerr black hole. With the mass
and angular momentum in this case depending upon this parameter a rotating
generalisation of the Vaidya space-time emerges.
3.1
Part I: A Light-Like Charge
From the point of view of Minkowskian geometry the Coulomb field is a solution
of Maxwell’s equations on Minkowskian space-time which is singular on a timelike geodesic (the history of a point charge). The Liénard–Wiechert field is the
generalisation of the Coulomb field which is singular on a non-geodesic world
line (the history of an accelerated point charge). When considering a hypothetical
charged particle moving with the speed of light there is a Maxwell field available to
describe it which is singular on a null geodesic and is a spin-off from the Robinson–
Trautman solutions of the Einstein–Maxwell field equations. To describe it we begin
with the Minkowskian line element in coordinates X i = (T , X, Y, Z):
ds
2
= η ij dX
i dX
j
= dT
2
− dX
2
− dY
2
− dZ
2 .
(3.1)
© 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_3
51