Given the path of a particle of charge q located at space-time coordinate x µ q
( τ ) for invariant time τ , the Li´ A
ν at coordinate x
µ
,
µ ∈ { 0 , 1 , 2 , 3 } , induced by the motion of the charge can be shown to be
4 π
A
ν q
V
ν
=
R σ V
σ
(1) where the expression on the right is evaluated at the retarded time τ
0 defined by
R
σ
R
σ
= 0 (2) and where the argument of the potential relative to the location of the charge is
R
σ
= x
σ − x
σ q
( τ ) .
(3)
To find the electromagnetic field F µν at the coordinate x ρ resulting from the expression for the four potential exhibited in equation (1) consider the relative position four vector R
σ
= R
σ
( x
ρ
, τ ) as a function of x
ρ and τ . Since the path of the particle, x µ q
( τ ) , intersects the past lightcone of the event x ρ once we can regard the invariant time, τ = τ ( x ρ ) , as a function of x ρ . Equation (3) provides
∂
µ
R
σ
=
∂R σ
∂x µ
=
∂x σ
∂x µ
− d x σ q d τ
∂ τ
∂x µ
= δ
µ
σ − V
σ
∂ τ
∂x µ
(4) involving a Kronecker delta and a four velocity of the charged particle denoted by
V
σ
= d x
σ q d τ
.
Again, taking the partial derivative of equation (2) with respect to x
µ yields
(5)
( ∂
µ
R
σ
) R
σ
+ R
σ
( ∂
µ
R
σ
) = 2 R
σ
( ∂
µ
R
σ
) = 0 (6) from which, with an abbreviated from of equation (4) , one may conclude that
R
σ
δ
σ
µ
− V
σ
∂
µ
τ = R
µ
− R
σ
V
σ
∂
µ
τ = 0 (7) resulting in equivalent expressions for the four gradient of the invariant time
∂
µ
τ =
R
µ
R
σ
V σ
, that is,
∂ τ
∂x
µ
=
R µ
R σ V
σ
.
(8)
1

Taking the partial derivative of the four potential below will require the derivative
∂
µ
R
ρ
= δ
ρ
µ
∂ τ
− V
ρ
∂x
µ
= δ
ρ
µ
R µ
− V
ρ
R σ V
σ
(9) that follows from the use of equations (4) and (8) , written with altered indices.
Also required will be a derivative of the four velocity V ν = V ν ( τ ) in equation (5) , treated as a function of τ ( x ρ ) , and again using equation (8)
∂
µ
V
ν
=
∂V
ν
∂x
µ
= d V
ν d τ
∂τ
∂x
µ
ν
R
µ
R σ V
σ
=
R
µ
V
˙ ν
R σ V
σ
(10) where V
˙
µ refers to d V µ / d τ , the derivative with respect to the invariant time.
Evaluating the partial derivative of the four potential A
ν in equation (1) reveals
4 q
π
∂
µ
A
ν
=
=
∂ µ V ν
R σ V
σ
−
V ν
( R σ V
σ
)
2
[ R
ρ
( ∂
µ
V
ρ
) + ( ∂
µ
R
ρ
) V
ρ
]
R
µ
V
˙ ν
( R σ V
σ
) 2
−
V
ν
( R σ V
σ
) 2
"
R
ρ
R
µ
V
˙ ρ
R λ V
λ
+ δ
µ
ρ
R
µ
− V
ρ
R λ V
λ
V
ρ
(11)
#
=
R
µ
V
˙ ν
( R σ V
σ
) 2
−
R
ρ
V
˙ ρ
( R σ V
σ
) 3
R
µ
V
ν −
V
µ
V
ν
( R σ V
σ
) 2
+
V
ρ
V
ρ
( R σ V
σ
) 3
R
µ
V
ν
.
The third term will not contribute to F
µν
, being symmetric in indices µ and ν .
Noting that the four velocity of the particle V σ = γ ( c, ~ ) obeys
V
ρ
V
ρ
= γ
2
( c
2 − ~
2
) = γ
2 c
2
γ
− 2
= c
2
(12) equations (11) and (12) indicate that the electromagnetic field may be written
F
µν
= ∂
µ
A
ν − ∂
ν
A
µ
= F
µν acc
µν
+ F vel
(13) where the acceleration and velocity elements of the electromagnetic tensor are
4 π F
µν acc
= q
( R σ V
σ
) 2
( R
µ
V
˙ ν − R
ν
V
˙ µ
) − q R ρ V
˙
ρ
( R σ V
σ
) 3
( R
µ
V
ν − R
ν
V
µ
) (14)
4 π F
µν vel
= q c 2
( R σ V
σ
) 3
( R
µ
V
ν − R
ν
V
µ
) (15) with all quantities here being evaluated at the retarded time given by equation (2) .
In the case where the particle is at rest with V σ = ( c, ~ the radiative part of the field is zero F
µν acc
F i 0 vel
= 0 ; the non-radiative part indicates here that the electric field gives the expected inverse square law and that the magnetic field F i j vel is zero.
References
1. Classical Electrodynamics (third edition), J. D. Jackson, John Wiley, 1998.
2. Classical Theory of Fields, E. M. Lifshitz & L. D. Landau, Pergamon, 1962.
3. Classical Field Theory, Francis E. Low, John Wiley, 1997.
2