3.3 Maxwell Field of a Non-Geodesic Charge
57
with
¯
q( ¯
ξ , ¯
η, ¯
u) =
1
2
¯
a
1 ( ¯
u) ¯
η
¯
ξ
2
−
1
3
¯
η
2
+
1
2
¯
a
2 ( ¯
u) ¯
ξ
¯
η
2
−
1
3
¯
ξ
2
,
(3.40)
and
¯
h 0 =
∂ 2 ¯
q
∂ ¯
ξ∂ ¯
η
.
(3.41)
We see that ¯
a 0 and ¯
a 3 do not appear in (3.40). However from (3.35) we have A 0 =
A 3 and then (3.37) gives ¯
a 0 = ¯
a 3 . If we use ¯
u given by (3.34) then ¯
v 0 − ¯
v 3 = 1
(in general we have ¯
v 0 − ¯
v 3 = constant = 0). Now the orthogonality of ¯
v i and ¯
a i
yields ¯
a 0 = ¯
a 3 = ¯
v 1 ¯
a 1 + ¯
v 2 ¯
a 2 .
3.3
Maxwell Field of a Charge with Non-geodesic World Line
Guided by the work of Robinson and Trautman [3, 4] on solutions of the vacuum
Einstein–Maxwell field equations, and requiring the solution of Maxwell’s equations for the electromagnetic field of a charge having a non-geodesic light-like world
line to specialise to the case of a charge having a geodesic light-like world line in
Sect. 3.1 above, we look for a potential 1-form to describe the Maxwell field in the
non-geodesic case given by
A = e
1
¯
r
+ G( ¯
ξ, ¯
η, ¯
u)
d ¯
u ,
(3.42)
with G( ¯
ξ , ¯
η, ¯
u) to be determined in order to satisfy Maxwell’s vacuum field
equations. The following basis 1-forms are suggested by the form of the line element
(3.39):
¯
ϑ
(1)
= ¯
r
d ¯
ξ +
∂ ¯
q
∂ ¯
η
d ¯
u
, ¯
ϑ
(2)
= ¯
r
d ¯
η +
∂ ¯
q
∂ ¯
ξ
d ¯
u
,
¯
ϑ
(3)
= d ¯
r − ¯
h 0 ¯
r d ¯
u and ¯
ϑ
(4)
= d ¯
u .
(3.43)
The candidate for Maxwell 2-form is the exterior derivative of (3.42):
F = dA = −
e
¯
r 2 d ¯
r ∧ d ¯
u + e
∂G
∂ ¯
ξ
d ¯
ξ ∧ d ¯
u + e
∂G
∂ ¯
η
d ¯
η ∧ d ¯
u
= −
e
¯
r 2 ϑ
(3)
∧ ϑ
(4)
+
e
¯
r
∂G
∂ ¯
ξ
ϑ
(1)
∧ ϑ
(4)
+
e
¯
r
∂G
∂ ¯
η
ϑ
(2)
∧ ϑ
(4) . (3.44)
57
with
¯
q( ¯
ξ , ¯
η, ¯
u) =
1
2
¯
a
1 ( ¯
u) ¯
η
¯
ξ
2
−
1
3
¯
η
2
+
1
2
¯
a
2 ( ¯
u) ¯
ξ
¯
η
2
−
1
3
¯
ξ
2
,
(3.40)
and
¯
h 0 =
∂ 2 ¯
q
∂ ¯
ξ∂ ¯
η
.
(3.41)
We see that ¯
a 0 and ¯
a 3 do not appear in (3.40). However from (3.35) we have A 0 =
A 3 and then (3.37) gives ¯
a 0 = ¯
a 3 . If we use ¯
u given by (3.34) then ¯
v 0 − ¯
v 3 = 1
(in general we have ¯
v 0 − ¯
v 3 = constant = 0). Now the orthogonality of ¯
v i and ¯
a i
yields ¯
a 0 = ¯
a 3 = ¯
v 1 ¯
a 1 + ¯
v 2 ¯
a 2 .
3.3
Maxwell Field of a Charge with Non-geodesic World Line
Guided by the work of Robinson and Trautman [3, 4] on solutions of the vacuum
Einstein–Maxwell field equations, and requiring the solution of Maxwell’s equations for the electromagnetic field of a charge having a non-geodesic light-like world
line to specialise to the case of a charge having a geodesic light-like world line in
Sect. 3.1 above, we look for a potential 1-form to describe the Maxwell field in the
non-geodesic case given by
A = e
1
¯
r
+ G( ¯
ξ, ¯
η, ¯
u)
d ¯
u ,
(3.42)
with G( ¯
ξ , ¯
η, ¯
u) to be determined in order to satisfy Maxwell’s vacuum field
equations. The following basis 1-forms are suggested by the form of the line element
(3.39):
¯
ϑ
(1)
= ¯
r
d ¯
ξ +
∂ ¯
q
∂ ¯
η
d ¯
u
, ¯
ϑ
(2)
= ¯
r
d ¯
η +
∂ ¯
q
∂ ¯
ξ
d ¯
u
,
¯
ϑ
(3)
= d ¯
r − ¯
h 0 ¯
r d ¯
u and ¯
ϑ
(4)
= d ¯
u .
(3.43)
The candidate for Maxwell 2-form is the exterior derivative of (3.42):
F = dA = −
e
¯
r 2 d ¯
r ∧ d ¯
u + e
∂G
∂ ¯
ξ
d ¯
ξ ∧ d ¯
u + e
∂G
∂ ¯
η
d ¯
η ∧ d ¯
u
= −
e
¯
r 2 ϑ
(3)
∧ ϑ
(4)
+
e
¯
r
∂G
∂ ¯
ξ
ϑ
(1)
∧ ϑ
(4)
+
e
¯
r
∂G
∂ ¯
η
ϑ
(2)
∧ ϑ
(4) . (3.44)