In classical electromagnetic theory, the electromagnetic field due to any number of electrons moving in any manner is determined by a theorem which expresses the scalar and vector potentials of the field in terms of the positions and velocities of the electrons. The theorem may be stated thus: Denoting by
t
¯
the instant at which radiation was emitted from an electron
e
so as to reach a point P (
x, y, z
), at the instant
t
, by (
x´
¯
,
y´
¯
,
z´
¯
) the co-ordinates of the electron at the instant
t
¯
, by
r
¯
the distance between the points (
x´
¯
,
y´
¯
,
z´
¯
) and (
x, y, z
) and by (
v
x
,
v
y
,
v
z
) the components of velocity of the electron at the instant
t
¯
, then the four-vector of the electromagnetic potential at P, due to the electron
e
, is (
Φ
0
,
Φ
1
,
Φ
2
,
Φ
3
) = (
e
/
s
, -
ev
x
/
s
,
ev
y
/
s
,
ev
z
/
s
), (1) where
s
=
r
¯
+ {(
x´
¯
-
x
)
v
x
+ (
y´
¯
-
y
)
v
y
+ (
z´
¯
-
z
)
v
z
}/
c
. The object of the present paper is to study the extension of this theorem to electromagnetic field which contain gravitating masses, so that the metric of space-time is no longer Galilean. It is obvious at the outset that there will be difficulty in making such an extension, because the quantities occurring in formula (1) cannot readily be generalised to non-Galilean space-time; the quantities
r
¯
and
s
, in fact, belong essentially to action-at-a-distance theories, and therefore if a formula exists which expresses the electromagnetic potential in a gravitational field in terms of the electric charges and their motions, it must be altogether different in type form the formula (1) above.
Maxwell's Equations, Stokes' Theorem, and the Conservation of Charge
