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1.6 Retarded, singular, and radiative electromagnetic fields of a point electric charge

The retarded solution to Equation (13View Equation) is
∫ A α(x) = e G α+μ(x,z)uμ dτ, (22 ) γ
where the integration is over the world line of the point electric charge. Because the retarded solution is the physically relevant solution to the wave equation, it will not be necessary to put a label ‘ret’ on the vector potential.

From the vector potential we form the electromagnetic field tensor F αβ, which we decompose in the tetrad α α (e0,ea) introduced at the end of Section 1.5. We then express the frame components of the field tensor in retarded coordinates, in the form of an expansion in powers of r. This gives

F (u,r,Ωa ) ≡ F (x )eα (x )eβ(x) a0 αβ a 0 = -eΩ − e-(a − aΩb Ω ) + 1-eR ΩbΩcΩ − 1e (5R Ωb + R ΩbΩc ) r2 a r a b a 3 b0c0 a 6 a0b0 ab0c -1- ( b c ) 1- 1- b tail + 12 e 5R00 + RbcΩ Ω + R Ωa + 3eRa0 − 6eRab Ω + Fa0 + 𝒪 (r), (23 ) a α β Fab(u,r,Ω ) ≡ Fαβ(x )ea (x )eb(x) e- 1- c = r (aaΩb − Ωaab) + 2 e(Ra0bc − Rb0ac + Ra0c0Ωb − ΩaRb0c0 )Ω 1 − --e(Ra0 Ωb − ΩaRb0 ) + F taaibl+ 𝒪 (r), (24 ) 2
F tail= Fta′il′(x′)eα′uβ′, F tail= F tai′l′(x′)eα′eβ′ (25 ) a0 α β a ab αβ a b
are the frame components of the “tail part” of the field, which is given by
tail ′ ∫ u− ′ μ F α′β′(x) = 2e ∇ [α′G+ β′]μ(x,z )u dτ. (26 ) −∞
In these expressions, all tensors (or their frame components) are evaluated at the retarded point x′ ≡ z(u) associated with x; for example, aa ≡ aa(u) ≡ aα′eα′ a. The tail part of the electromagnetic field tensor is written as an integral over the portion of the world line that corresponds to the interval − ∞ < τ ≤ u− ≡ u − 0+; this represents the past history of the particle. The integral is cut short at u − to avoid the singular behaviour of the retarded Green’s function when z(τ) coincides with ′ x; the portion of the Green’s function involved in the tail integral is smooth, and the singularity at coincidence is completely accounted for by the other terms in Equations (23View Equation) and (24View Equation).

The expansion of F αβ(x) near the world line does indeed reveal many singular terms. We first recognize terms that diverge when r → 0; for example the Coulomb field Fa0 diverges as − 2 r when we approach the world line. But there are also terms that, though they stay bounded in the limit, possess a directional ambiguity at r = 0; for example Fab contains a term proportional to Ra0bcΩc whose limit depends on the direction of approach.

This singularity structure is perfectly reproduced by the singular field S F αβ obtained from the potential

∫ A α(x) = e G α (x,z)uμ dτ, (27 ) S γ Sμ
where α GS μ(x,z) is the singular Green’s function of Equation (14View Equation). Near the world line the singular field is given by
F S(u, r,Ωa) ≡ F S (x )eα(x)eβ(x) a0 αβ a 0 -e e-( b ) 2- 1- b c 1- ( b b c) = r2Ωa − r aa − abΩ Ωa − 3e˙aa + 3 eRb0c0Ω Ω Ωa − 6e 5Ra0b0Ω + Rab0cΩ Ω 1 ( b c ) 1 b + --e 5R00 + RbcΩ Ω + R Ωa − --eRabΩ + 𝒪 (r), (28 ) S a S12 α β 6 F ab(u, r,Ω ) ≡ F αβ(x )ea(x)eb(x) e 1 c = --(aaΩb − Ωaab ) + -e(Ra0bc − Rb0ac + Ra0c0Ωb − ΩaRb0c0)Ω r 2 − 1e(Ra0 Ωb − ΩaRb0 ) + 𝒪 (r). (29 ) 2
Comparison of these expressions with Equations (23View Equation) and (24View Equation) does indeed reveal that all singular terms are shared by both fields.

The difference between the retarded and singular fields defines the radiative field R F αβ(x). Its frame components are

F R = 2-e˙aa + 1eRa0 + Ftail+ 𝒪(r), (30 ) a0 3 3 a0 F Rab = F taaibl+ 𝒪 (r), (31 )
and at ′ x the radiative field becomes
( ) R ( ) 2 γ′ 1 γ′ δ′ tail Fα′β′ = 2eu[α′ g β′]γ′ + uβ′]uγ′ 3-˙a + 3-R δ′u + Fα′β′, (32 )
where ′ ′ ˙aγ = Da γ∕dτ is the rate of change of the acceleration vector, and where the tail term was given by Equation (26View Equation). We see that FR (x) αβ is a smooth tensor field, even on the world line.
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