60
3 Hypothetical Objects in Electromagnetism and Gravity
with G given by (3.58). In these coordinates if ¯
k i = δ
i
3 then ¯
k i = δ 4
i , confirming
that ¯
k i ¯
k i = 0, and we easily find that
¯
F ij ¯
k
j
=
e
r 2
¯
k i and
∗ ¯
F ij ¯
k
j
= 0 .
(3.59)
It follows from these equations that the Maxwell field is algebraically general
with ¯
k i a principal null direction. Also for large values of ¯
r the Maxwell field
becomes algebraically special (radiative) with degenerate principal null direction
¯
k i which then represents the direction in Minkowskian space-time of the history of
the electromagnetic radiation emitted by the charged particle on account of the fact
that its world line is not a geodesic.
Many years ago Synge [6] suggested a model of a charged particle having a nongeodesic world line in Minkowskian space-time. Synge chose a 4-potential A i , in
coordinates X i , which mimics the 4-potential for the Liénard–Wiechert field in the
time-like case by taking
A
i
=
e
r
v
i
⇒ A = A i dX
i
=
e
r
v i dX
i ,
(3.60)
with r given via (3.15). One can verify (see [1] or [6]) that the 2-form F = dA
satisfies Maxwell’s equations d ∗ F = 0 and that the Maxwell field F is algebraically
special (radiative) for all values of r > 0. However the Maxwell field vanishes if the
world line r = 0 is a geodesic. To see this we note that (3.15) in effect gives u, r, k i
as functions of X i . Taking the partial derivative of (3.15) with respect to X j results
in
δ
i
j = v
i u ,j + k
i r ,j + r k
i
,j .
(3.61)
Multiplying this by v i (using v i v i = 0 and v i k i = 1) yields
v j = r ,j + r v i k
i
,j = r ,j − r a i k
i u ,j = r ,j − r h 0 u ,j ,
(3.62)
with h 0 given by (3.20). Thus we have the 1-form
v j dX
j
= dr − r h 0 du .
(3.63)
Hence Synge’s potential 1-form can be written
A = −e h 0 du + e d(log r) .
(3.64)
If the world line r = 0 is a geodesic then a i = λ(u) v i and so h 0 = a i k i = λ(u).
In this case A is an exact differential and therefore the corresponding Maxwell field
vanishes.
3 Hypothetical Objects in Electromagnetism and Gravity
with G given by (3.58). In these coordinates if ¯
k i = δ
i
3 then ¯
k i = δ 4
i , confirming
that ¯
k i ¯
k i = 0, and we easily find that
¯
F ij ¯
k
j
=
e
r 2
¯
k i and
∗ ¯
F ij ¯
k
j
= 0 .
(3.59)
It follows from these equations that the Maxwell field is algebraically general
with ¯
k i a principal null direction. Also for large values of ¯
r the Maxwell field
becomes algebraically special (radiative) with degenerate principal null direction
¯
k i which then represents the direction in Minkowskian space-time of the history of
the electromagnetic radiation emitted by the charged particle on account of the fact
that its world line is not a geodesic.
Many years ago Synge [6] suggested a model of a charged particle having a nongeodesic world line in Minkowskian space-time. Synge chose a 4-potential A i , in
coordinates X i , which mimics the 4-potential for the Liénard–Wiechert field in the
time-like case by taking
A
i
=
e
r
v
i
⇒ A = A i dX
i
=
e
r
v i dX
i ,
(3.60)
with r given via (3.15). One can verify (see [1] or [6]) that the 2-form F = dA
satisfies Maxwell’s equations d ∗ F = 0 and that the Maxwell field F is algebraically
special (radiative) for all values of r > 0. However the Maxwell field vanishes if the
world line r = 0 is a geodesic. To see this we note that (3.15) in effect gives u, r, k i
as functions of X i . Taking the partial derivative of (3.15) with respect to X j results
in
δ
i
j = v
i u ,j + k
i r ,j + r k
i
,j .
(3.61)
Multiplying this by v i (using v i v i = 0 and v i k i = 1) yields
v j = r ,j + r v i k
i
,j = r ,j − r a i k
i u ,j = r ,j − r h 0 u ,j ,
(3.62)
with h 0 given by (3.20). Thus we have the 1-form
v j dX
j
= dr − r h 0 du .
(3.63)
Hence Synge’s potential 1-form can be written
A = −e h 0 du + e d(log r) .
(3.64)
If the world line r = 0 is a geodesic then a i = λ(u) v i and so h 0 = a i k i = λ(u).
In this case A is an exact differential and therefore the corresponding Maxwell field
vanishes.