3.2 Non-Geodesic Null World Line
53
The exterior derivative of this 1-form is the 2-form
F = dA = −
e
r 2 dr ∧ du = −
e
r 2 ϑ
(3)
∧ ϑ
(4) ,
(3.10)
and its Hodge dual is the 2-form
∗ F =
e
r 2 ϑ
1
∧ ϑ
2
= e dξ ∧ dη .
(3.11)
It is clear that the exterior derivative of this 2-form vanishes,
d
∗ F = 0 ,
(3.12)
and thus F is a Maxwell field. In other words the potential 1-form (3.9) gives rise
to the Maxwell field of a charged particle travelling with the speed of light.
3.2
Geometry Based on a Non-geodesic Null World Line
Following the description above of a light-like analogue of the Coulomb field, we
now turn our attention to the construction of a light-like analogue of the Liénard–
Wiechert field in which the charge has a non-geodesic null world line and its
electromagnetic field specialises to the Maxwell field given above when the world
line of the charge is a null geodesic [1]. The parametric equations of the nongeodesic null world line are now
X
i
= w
i (u) with v
i
=
dw i
du
and v
i v i = 0 .
(3.13)
We define
a
i
=
dv i
du
⇒ a
i v i = 0 .
(3.14)
If a i = 0 then the world line (3.13) is a null geodesic with u an affine parameter
along it while if a i = λ(u) v i , for some function λ(u), then the world line is a null
geodesic but u is not an affine parameter along it. We will exclude these cases from
now on. We now generalise the position 4-vector (3.3) to the position 4-vector of a
point of Minkowskian space-time relative to the world line (3.13) by writing
X
i
= w
i (u) + r k
i ,
(3.15)