18 Motion of an electric charge

18.1 Dynamics of a point electric charge

A point particle carries an electric charge e and moves on a world line γ described by relations μ z (λ), in which λ is an arbitrary parameter. The particle generates a vector potential A α(x) and an electromagnetic field F αβ(x) = ∇ αAβ − ∇ βA α. The dynamics of the entire system is governed by the action

S = Sfield + Sparticle + Sinteraction, (18.1 )
where Sfield is an action functional for a free electromagnetic field in a spacetime with metric gαβ, S particle is the action of a free particle moving on a world line γ in this spacetime, and S interaction is an interaction term that couples the field to the particle.

The field action is given by

∫ √ --- Sfield = − -1-- FαβF αβ − gd4x, (18.2 ) 16π
where the integration is over all of spacetime. The particle action is
∫ S = − m dτ, (18.3 ) particle γ
where m is the bare mass of the particle and ∘ ------------- dτ = − gμν(z)z˙μz˙νdλ is the differential of proper time along the world line; we use an overdot to indicate differentiation with respect to the parameter λ. Finally, the interaction term is given by
∫ ∫ √ --- Sinteraction = e A μ(z)˙zμdλ = e Aα(x )g αμ(x,z )z˙μδ4(x, z) − gd4xd λ. (18.4 ) γ
Notice that both Sparticle and Sinteraction are invariant under a reparameterization ′ λ → λ (λ ) of the world line.

Demanding that the total action be stationary under a variation δA α(x) of the vector potential yields Maxwell’s equations

Fαβ = 4πjα (18.5 ) ;β
with a current density jα(x) defined by
∫ α α μ j (x) = e γ g μ(x,z)˙z δ4(x,z)dλ. (18.6 )
These equations determine the electromagnetic field Fαβ once the motion of the electric charge is specified. On the other hand, demanding that the total action be stationary under a variation δzμ(λ) of the world line yields the equations of motion
Du-μ- μ ν m dτ = eF ν(z)u (18.7 )
for the electric charge. We have adopted τ as the parameter on the world line, and introduced the four-velocity μ μ u (τ ) := dz ∕d τ.

The electromagnetic field Fαβ is invariant under a gauge transformation of the form A α → A α + ∇ αΛ, in which Λ(x) is an arbitrary scalar function. This function can always be chosen so that the vector potential satisfies the Lorenz gauge condition,

α ∇ αA = 0. (18.8 )
Under this condition the Maxwell equations of Eq. (18.5View Equation) reduce to a wave equation for the vector potential,
α α β α □A − R βA = − 4 πj , (18.9 )
where αβ □ = g ∇ α∇ β is the wave operator and α R β is the Ricci tensor. Having adopted τ as the parameter on the world line, we can re-express the current density of Eq. (18.6View Equation) as
∫ α α μ j (x) = e γ g μ(x,z)u δ4(x,z)dτ, (18.10 )
and we shall use Eqs. (18.9View Equation) and (18.10View Equation) to determine the electromagnetic field of a point electric charge. The motion of the particle is in principle determined by Eq. (18.7View Equation), but because the vector potential obtained from Eq. (18.9View Equation) is singular on the world line, these equations have only formal validity. Before we can make sense of them we will have to analyze the field’s singularity structure near the world line. The calculations to be carried out parallel closely those presented in Section 17 for the case of a scalar charge; the details will therefore be kept to a minimum and the reader is referred to Section 17 for additional information.

18.2 Retarded potential near the world line

The retarded solution to Eq. (18.9View Equation) is α ∫ α ′ β′ ′√ --′4 ′ A (x) = G +β′(x, x)j (x ) g d x, where α ′ G +β′(x, x ) is the retarded Green’s function introduced in Section 15. After substitution of Eq. (18.10View Equation) we obtain

∫ A α(x) = e G +αμ(x,z)u μdτ, (18.11 ) γ
in which zμ(τ ) gives the description of the world line γ and u μ(τ) = dzμ∕dτ. Because the retarded Green’s function is defined globally in the entire spacetime, Eq. (18.11View Equation) applies to any field point x.

We now specialize Eq. (18.11View Equation) to a point x close to the world line. We let 𝒩 (x) be the normal convex neighbourhood of this point, and we assume that the world line traverses 𝒩 (x); refer back to Figure 9View Image. As in Section 17.2 we let τ < and τ > be the values of the proper-time parameter at which γ enters and leaves 𝒩 (x), respectively. Then Eq. (18.11View Equation) can be expressed as

∫ τ< ∫ τ> ∫ ∞ A α(x) = e G +αμ(x,z )u μdτ + e G +αμ(x,z)u μdτ + e G+αμ(x,z)uμdτ. −∞ τ< τ>

The third integration vanishes because x is then in the past of z(τ), and G α+μ(x,z) = 0. For the second integration, x is the normal convex neighbourhood of z(τ ), and the retarded Green’s function can be expressed in the Hadamard form produced in Section 15.2. This gives

∫ τ> ∫ τ> ∫ τ> G α (x,z)uμdτ = U α (x,z)uμδ (σ )dτ + V α (x, z)uμ𝜃 (− σ )d τ, τ< + μ τ< μ + τ< μ +

and to evaluate this we let x ′ := z(u) be the retarded point associated with x; these points are related by ′ σ (x,x ) = 0 and α′ r := σα′u is the retarded distance between x and the world line. To perform the first integration we change variables from τ to σ, noticing that σ increases as z(τ) passes through x ′; the integral evaluates to ′ U αβ′uβ ∕r. The second integration is cut off at τ = u by the step function, and we obtain our final expression for the vector potential of a point electric charge:

∫ u ∫ τ< A α(x) = eU α′(x,x′)uβ′ + e V α(x, z)uμdτ + e G α(x,z)u μdτ. (18.12 ) r β τ< μ −∞ +μ
This expression applies to a point x sufficiently close to the world line that there exists a nonempty intersection between 𝒩 (x) and γ.

18.3 Electromagnetic field in retarded coordinates

When we differentiate the vector potential of Eq. (18.12View Equation) we must keep in mind that a variation in x induces a variation in ′ x, because the new points x + δx and ′ ′ x + δx must also be linked by a null geodesic. Taking this into account, we find that the gradient of the vector potential is given by

e β′ e β′ e( β′ γ′ β′) β′ tail ∇ βA α(x) = − -2U αβ′u ∂βr + --Uαβ′;βu + -- Uαβ′;γ′u u + U αβ′a ∂βu + eVαβ′u ∂βu + A αβ(1(8x.)1,3 ) r r r
where the “tail integral” is defined by
∫ u ∫ τ< Ataαiβl(x ) = e ∇ βVαμ(x,z)u μdτ + e ∇ βG+ αμ(x,z)uμd τ τ< − ∞ ∫ u− = e ∇ βG+ αμ(x,z)uμd τ. (18.14 ) − ∞
The second form of the definition, in which we integrate the gradient of the retarded Green’s function from τ = − ∞ to τ = u − := u − 0+ to avoid the singular behaviour of the retarded Green’s function at σ = 0, is equivalent to the first form.

We shall now expand F αβ = ∇ αA β − ∇βA α in powers of r, and express the result in terms of the retarded coordinates (u,r,Ωa ) introduced in Section 10. It will be convenient to decompose the electromagnetic field in the tetrad (eα,eα) 0 a that is obtained by parallel transport of α′ α′ (u ,ea ) on the null geodesic that links x to ′ x := z(u); this construction is detailed in Section 10. We recall from Eq. (10.4View Equation) that the parallel propagator can be expressed as ′ ′ ′ gαα = uα e0α + eαa eaα. The expansion relies on Eq. (10.29View Equation) for ∂αu, Eq. (10.31View Equation) for ∂αr, and we shall need

[ ] β′ α′ 1 2( a a b) 3 Uαβ′u = g α uα′ + 12r R00 + 2R0aΩ + Rab Ω Ω uα′ + O(r ) , (18.15 )
which follows from Eq. (15.10View Equation) and the relation ′ ′ ′ σα = − r(u α + Ωae αa ) first encountered in Eq. (10.7View Equation). We shall also need the expansions
[ ] U ′ uβ′ = − 1-rgα′gβ′ R ′ ′ + R ′ ′Ωc − 1(R ′ + R ′Ωc )u ′ + O (r) (18.16 ) αβ ;β 2 α β α0β0 α0β c 3 β 0 βc α
and
[ ] β′ γ′ β′ α′ 1 b 1 ( b) 2 Uαβ′;γ′u u + U αβ′a = g α aα′ + -rR α′0b0Ω − -r R00 + R0b Ω uα′ + O(r ) (18.17 ) 2 6
that follow from Eqs. (15.10View Equation) – (15.12View Equation). And finally, we shall need
[ ] β′ 1 α′ 1 Vαβ′u = − 2g α R α′0 − 6Ru α′ + O (r) , (18.18 )
a relation that was first established in Eq. (15.14View Equation).

Collecting all these results gives

a α β Fa0 (u, r,Ω ) := Fαβ(x)ea(x )e0 (x ) e- e( b ) 1- b c 1-( b b c) = r2Ωa − r aa − abΩ Ωa + 3eRb0c0Ω Ω Ωa − 6e 5Ra0b0Ω + Rab0cΩ Ω 1 ( ) 1 1 + --e 5R00 + RbcΩbΩc + R Ωa + -eRa0 − --eRabΩb + F taa0il+ O (r), (18.19 ) 12 β 3 6 Fab (u, r,Ωa) := Fαβ(x)eαa(x )eb (x ) e( ) 1 ( ) c = --aaΩb − Ωaab + --e Ra0bc − Rb0ac + Ra0c0Ωb − ΩaRb0c0 Ω r ( 2 ) − 1e Ra0 Ωb − ΩaRb0 + F taabil+ O (r), (18.20 ) 2
where
tail tail ′ α′ β′ tail tail ′ α′ β′ F a0 = Fα′β′(x )ea u , Fab = F α′β′(x)ea eb (18.21 )
are the frame components of the tail integral; this is obtained from Eq. (18.14View Equation) evaluated at x′:
∫ u− F tai′l′(x′) = 2e ∇ ′G ′ (x′,z)uμdτ. (18.22 ) αβ −∞ [α +β]μ
It should be emphasized that in Eqs. (18.19View Equation) and (18.20View Equation), all frame components are evaluated at the retarded point ′ x := z (u ) associated with x; for example, α′ aa := aa(u) := aα′ea. It is clear from these equations that the electromagnetic field Fαβ(x) is singular on the world line.

18.4 Electromagnetic field in Fermi normal coordinates

We now wish to express the electromagnetic field in the Fermi normal coordinates of Section 9; as before those will be denoted (t,s,ωa). The translation will be carried out as in Section 17.4, and we will decompose the field in the tetrad α α (¯e0,¯ea ) that is obtained by parallel transport of ¯α α¯ (u ,ea ) on the spacelike geodesic that links x to the simultaneous point ¯x := z (t).

Our first task is to decompose Fαβ(x) in the tetrad (¯eα0,¯eαa), thereby defining F¯a0 := F αβ¯eαa¯eβ0 and F¯ab := Fαβ ¯eα¯eβ a b. For this purpose we use Eqs. (11.7View Equation), (11.8View Equation), (18.19View Equation), and (18.20View Equation) to obtain

( ) F¯a0 = e-Ωa − e-aa − abΩbΩa + 1eabΩbaa + 1e˙a0Ωa − 5eRa0b0Ωb + 1eRb0c0ΩbΩc Ωa r2 r 2 2 6 3 1- b c 1--( b c ) 1- 1- b ¯tail + 3eRab0cΩ Ω + 12e 5R00 + Rbc Ω Ω + R Ωa + 3eRa0 − 6 eRabΩ + F a0 + O (r)
and
1 ( ) 1 ( ) 1 ( ) ¯Fab = -e Ωa ˙ab − ˙aaΩb + --e Ra0bc − Rb0ac Ωc − -e Ra0 Ωb − ΩaRb0 + ¯Ftaabil+ O (r), 2 2 2

where all frame components are still evaluated at x′, except for

¯ F¯ta0ail:= Fαta¯βi¯l(¯x)e¯αau ¯β, F¯taaibl:= Fαta¯i¯βl (¯x)e¯αaeβb ,

which are evaluated at ¯x.

We must still translate these results into the Fermi normal coordinates a (t,s,ω ). For this we involve Eqs. (11.4View Equation), (11.5View Equation), and (11.6View Equation), and we recycle some computations that were first carried out in Section 17.4. After some algebra, we arrive at

F¯a0 (t,s,ωa ) := F αβ(x)¯eαa(x)¯eβ0(x) e e( ) 3 3 ( ) 3 2 = -- ωa − ---aa + abωb ωa + -eabωbaa + -e abωb 2ωa + -e˙a0ωa + --e˙aa s2 2s 4 8 8 3 − 2-eR ωb − 1eR ωbωc ω + -1e(5R + R ωbωc + R )ω 3 a0b0 6 b0c0 a 12 00 bc a 1 1 b tail + 3-eRa0 − 6eRab ω + ¯Fa0 + O (s), (18.23 ) ¯ a α β Fab (t,s,ω ) := F αβ(x)¯ea(x)¯eb(x) 1- ( ) 1- ( ) c 1-( ) = 2 e ωaa˙b − ˙aaωb + 2 e Ra0bc − Rb0ac ω − 2e Ra0 ωb − ωaRb0 ¯ tail + Fab + O (s), (18.24 )
where all frame components are now evaluated at ¯x := z (t); for example, aa := aa(t) := a¯αe¯αa.

Our next task is to compute the averages of F¯a0 and F¯ab over S (t,s ), a two-surface of constant t and s. These are defined by

⟨ ⟩ 1 ∮ a ⟨ ⟩ 1 ∮ a F¯a0 (t,s) = -- ¯Fa0(t,s,ω )d𝒜, F¯ab (t,s) = -- ¯Fab(t,s,ω )d𝒜, (18.25 ) 𝒜 S(t,s) 𝒜 S(t,s)
where d𝒜 is the element of surface area on S(t,s), and ∮ 𝒜 = d𝒜. Using the methods developed in Section 17.4, we find
⟨ ¯ ⟩ 2e- 2- 1- ¯tail Fa0 = − 3saa + 3 e˙aa + 3eRa0 + F a0 + O (s), (18.26 ) ⟨ ¯ ⟩ ¯ tail Fab = Fab + O(s). (18.27 )
The averaged field is singular on the world line, but we nevertheless take the formal limit s → 0 of the expressions displayed in Eqs. (18.26View Equation) and (18.27View Equation). In the limit the tetrad (¯eα,¯eα) 0 a becomes α¯ ¯α (u ,ea), and we can easily reconstruct the field at ¯x from its frame components. We thus obtain
⟨ ⟩ ( ) ( )( ) F ¯ = lim − 4e- u[¯αa¯ + 2eu [¯α g ¯ + u¯u ¯γ 2˙a¯γ + 1-R¯γ¯u¯δ + Fta¯il, (18.28 ) ¯αβ s→0 3s β] β]¯γ β] 3 3 δ ¯αβ
where the tail term can be copied from Eq. (18.22View Equation),
∫ t− Fta¯il(¯x) = 2e ∇ [α¯G+ ¯β]μ(¯x,z)uμd τ. (18.29 ) ¯αβ −∞
The tensors appearing in Eq. (18.28View Equation) all refer to ¯x := z(t), which now stands for an arbitrary point on the world line γ.

18.5 Singular and regular fields

The singular vector potential

∫ AαS (x ) = e GSαμ(x, z)uμdτ (18.30 ) γ
is the (unphysical) solution to Eqs. (18.9View Equation) and (18.10View Equation) that is obtained by adopting the singular Green’s function of Eq. (15.24View Equation) instead of the retarded Green’s function. We will see that the singular field S Fαβ reproduces the singular behaviour of the retarded solution, and that the difference, R S Fαβ = Fαβ − Fαβ, is smooth on the world line.

To evaluate the integral of Eq. (18.30View Equation) we assume once more that x is sufficiently close to γ that the world line traverses 𝒩 (x); refer back to Figure 9View Image. As before we let τ < and τ > be the values of the proper-time parameter at which γ enters and leaves 𝒩 (x ), respectively. Then Eq. (18.30View Equation) becomes

∫ ∫ ∫ α τ< α μ τ> α μ ∞ α μ AS (x ) = e GSμ(x, z)u dτ + e G Sμ(x,z)u d τ + e G Sμ(x,z)u dτ. − ∞ τ< τ>

The first integration vanishes because x is then in the chronological future of z(τ), and α G Sμ (x, z) = 0 by Eq. (15.27View Equation). Similarly, the third integration vanishes because x is then in the chronological past of z(τ). For the second integration, x is the normal convex neighbourhood of z(τ), the singular Green’s function can be expressed in the Hadamard form of Eq. (15.33View Equation), and we have

∫ ∫ ∫ τ> α μ 1 τ> α μ 1 τ> α μ G Sμ(x,z)u dτ = -- U μ(x, z)u δ+(σ )dτ + -- U μ(x,z)u δ− (σ )dτ τ< 2 τ<∫ τ 2 τ< 1- > α μ − 2 τ V μ(x,z)u 𝜃(σ)dτ. <
To evaluate these we let x ′ := z(u) and x′′ := z(v) be the retarded and advanced points associated with x, respectively. To perform the first integration we change variables from τ to σ, noticing that σ increases as z(τ ) passes through x′; the integral evaluates to U α ′uβ′∕r β. We do the same for the second integration, but we notice now that σ decreases as z(τ) passes through ′′ x; the integral evaluates to α β′′ U β′′u ∕radv, where α′′ radv := − σα′′u is the advanced distance between x and the world line. The third integration is restricted to the interval u ≤ τ ≤ v by the step function, and we obtain the expression
∫ v α -e- α β′ --e-- α β′′ 1- α μ A S(x) = 2rU β′u + 2radvU β′′u − 2e u V μ(x,z )u dτ (18.31 )
for the singular vector potential.

Differentiation of Eq. (18.31View Equation) yields

∇ AS (x) = − -e-U ′uβ′∂ r − ---e--U ′′uβ′′∂ r + -e-U ′ uβ′ β α 2r2 α β β 2radv2 αβ β adv 2r αβ ;β e-( β′ γ′ β′) --e-- β′′ + 2r U αβ′;γ′u u + Uαβ′a ∂βu + 2r Uαβ′′;βu ( ) adv + --e-- U αβ′′;γ′′uβ′′uγ′′ + U αβ′′aβ′′ ∂βv + 1eVαβ′uβ′∂βu 2radv ∫ 2 1 β′′ 1 v μ − 2eVαβ′′u ∂βv − 2e ∇βV αμ(x,z)u d τ, (18.32 ) u
and we would like to express this as an expansion in powers of r. For this we will rely on results already established in Section 18.3, as well as additional expansions that will involve the advanced point x ′′. We recall that a relation between retarded and advanced times was worked out in Eq. (11.12View Equation), that an expression for the advanced distance was displayed in Eq. (11.13View Equation), and that Eqs. (11.14View Equation) and (11.15View Equation) give expansions for ∂αv and ∂αradv, respectively.

To derive an expansion for U αβ′′uβ′′ we follow the general method of Section 11.4 and introduce the functions U (τ ) := U (x,z)uμ α αμ. We have that

β′′ ˙ ′ 1-¨ ′2 ( ′3) Uαβ′′u := U α(v) = Uα(u) + U α(u)Δ + 2U α(u)Δ + O Δ ,

where overdots indicate differentiation with respect to τ, and Δ ′ := v − u. The leading term β′ U α(u) := Uαβ′u was worked out in Eq. (18.15View Equation), and the derivatives of U α(τ) are given by

[ ] β′ γ′ β′ α′ 1 b 1 ( b) 2 U˙α(u) = Uαβ′;γ′u u + U αβ′a = g α aα′ + 2rR α′0b0Ω − 6r R00 + R0bΩ uα′ + O(r )

and

[ ] ¨ β′ γ′ δ′ ( β′ γ′ β′γ′) β′ α′ 1- Uα(u ) = U αβ′;γ′δ′u u u + Uαβ′;γ′ 2a u + u a + Uαβ′˙a = g α ˙aα′ + 6R00u α′ + O (r) ,

according to Eqs. (18.17View Equation) and (15.12View Equation). Combining these results together with Eq. (11.12View Equation) for Δ ′ gives

[ β′′ α′ ( b) 2 2 b Uαβ′′u = g α uα′ + 2r 1 − rabΩ aα′ + 2r ˙aα′ + r R α′0b0Ω ] -1- 2( a a b) 3 + 12 r R00 − 2R0aΩ + RabΩ Ω uα′ + O (r ) , (18.33 )
which should be compared with Eq. (18.15View Equation). It should be emphasized that in Eq. (18.33View Equation) and all equations below, all frame components are evaluated at the retarded point x′, and not at the advanced point. The preceding computation gives us also an expansion for
U αβ′′;γ′′uβ′′u γ′′ + U αβ′′aβ′′ := ˙Uα(v ) = U˙α(u) + ¨Uα(u )Δ′ + O(Δ ′2),

which becomes

′′ ′′ ′′ ′[ 1 1 ( ) ] U αβ′′;γ′′uβ uγ + U αβ′′aβ = gαα aα′ + 2r˙aα′ + -rR α′0b0Ωb + --r R00 − R0bΩb uα′ + O (r2) ,(18.34 ) 2 6
and which should be compared with Eq. (18.17View Equation).

We proceed similarly to derive an expansion for U ′′ u β′′ αβ ;β. Here we introduce the functions μ U αβ(τ) := U αμ;βu and express β′′ Uαβ′′;βu as ˙ ′ ′2 Uαβ (v ) = U αβ(u) + Uαβ(u )Δ + O(Δ ). The leading term ′ U αβ(u) := Uαβ′;βuβ was computed in Eq. (18.16View Equation), and

[ ] U˙αβ (u ) = Uαβ′;βγ′uβ′u γ′ + U αβ′;βaβ′ = 1gα′αgβ′ R α′0β′0 − 1-uα′Rβ′0 + O(r) 2 β 3

follows from Eq. (15.11View Equation). Combining these results together with Eq. (11.12View Equation) for Δ′ gives

[ ] β′′ 1- α′ β′ c 1( c) Uαβ′′;βu = 2 rg αg β R α′0β′0 − R α′0β′cΩ − 3 R β′0 − R β′cΩ uα′ + O (r) , (18.35 )
and this should be compared with Eq. (18.16View Equation). The last expansion we shall need is
′′ 1 ′[ 1 ] Vαβ′′uβ = − --gαα Rα′0 − -Ru α′ + O(r) , (18.36 ) 2 6
which follows at once from Eq. (18.18View Equation).

It is now a straightforward (but still tedious) matter to substitute these expansions into Eq. (18.32View Equation) to obtain the projections of the singular electromagnetic field S S S F αβ = ∇ αA β − ∇ βA α in the same tetrad (eα0,eαa) that was employed in Section 18.3. This gives

FS (u,r,Ωa) := F S(x )eα(x)eβ(x) a0 αβ a( 0 ) ( ) = -eΩa − e-aa − abΩb Ωa − 2-e˙aa + 1eRb0c0ΩbΩc Ωa − 1-e 5Ra0b0Ωb + Rab0cΩbΩc r2 r 3 3 6 1--( b c ) 1- b + 12e 5R00 + RbcΩ Ω + R Ωa − 6eRab Ω + O (r), (18.37 ) S a S α β Fab(u,r,Ω ) := Fαβ(x )ea(x)eb(x) e( ) 1- ( ) c = r aa Ωb − Ωaab + 2 e Ra0bc − Rb0ac + Ra0c0Ωb − ΩaRb0c0 Ω 1 ( ) − -e Ra0 Ωb − ΩaRb0 + O(r), (18.38 ) 2
in which all frame components are evaluated at the retarded point x′. Comparison of these expressions with Eqs. (18.19View Equation) and (18.20View Equation) reveals that the retarded and singular fields share the same singularity structure.

The difference between the retarded field of Eqs. (18.19View Equation), (18.20View Equation) and the singular field of Eqs. (18.37View Equation), (18.38View Equation) defines the regular field R F αβ(x). Its tetrad components are

R 2 1 tail F a0 = -e˙aa + -eRa0 + Fa0 + O(r), (18.39 ) R 3tail 3 F ab = F ab + O (r), (18.40 )
and we see that F R αβ is a regular tensor field on the world line. There is therefore no obstacle in evaluating the regular field directly at ′ x = x, where the tetrad α α (e0,ea) becomes α′ α′ (u ,ea ). Reconstructing the field at x′ from its frame components, we obtain
( ) R ′ ( ) 2-γ′ 1- γ′ δ′ tail Fα′β′(x ) = 2eu[α′gβ′]γ′ + uβ′]uγ′ 3 ˙a + 3R δ′u + Fα′β′, (18.41 )
where the tail term can be copied from Eq. (18.22View Equation),
∫ − tail ′ u ′ μ F α′β′(x ) = 2e ∇ [α′G+ β′]μ(x ,z)u dτ. (18.42 ) −∞
The tensors appearing in Eq. (18.41View Equation) all refer to the retarded point x′ := z(u ), which now stands for an arbitrary point on the world line γ.

18.6 Equations of motion

The retarded field Fαβ of a point electric charge is singular on the world line, and this behaviour makes it difficult to understand how the field is supposed to act on the particle and exert a force. The field’s singularity structure was analyzed in Sections 18.3 and 18.4, and in Section 18.5 it was shown to originate from the singular field S Fαβ; the regular field R S Fα β = Fαβ − Fαβ was then shown to be regular on the world line.

To make sense of the retarded field’s action on the particle we follow the discussion of Section 17.6 and temporarily picture the electric charge as a spherical hollow shell; the shell’s radius is s0 in Fermi normal coordinates, and it is independent of the angles contained in the unit vector ωa. The net force acting at proper time τ on this shell is proportional to the average of a F αβ(τ, s0,ω ) over the shell’s surface. Assuming that the field on the shell is equal to the field of a point particle evaluated at s = s0, and ignoring terms that disappear in the limit s0 → 0, we obtain from Eq. (18.28View Equation)

⟨ ⟩ ( )( ) e Fμν u ν = − (δm )aμ + e2 gμν + uμu ν 2˙aν + 1R νλuλ + eFμtaνiluν, (18.43 ) 3 3
where
2 δm := lim 2e-- (18.44 ) s0→0 3s0
is formally a divergent quantity and
∫ τ− eF tailu ν = 2e2uν ∇ G ′(z (τ ),z(τ′))u λ′dτ′ (18.45 ) μν − ∞ [μ + ν]λ
is the tail part of the force; all tensors in Eq. (18.43View Equation) are evaluated at an arbitrary point z(τ) on the world line.

Substituting Eqs. (18.43View Equation) and (18.45View Equation) into Eq. (18.7View Equation) gives rise to the equations of motion

( ) ∫ τ− (m + δm )aμ = e2(δμ + uμuν) 2-˙aν + 1R ν uλ + 2e2u ν ∇ [μG ν]′(z(τ),z(τ′))uλ′dτ′(18.46 ) ν 3 3 λ − ∞ +λ
for the electric charge, with m denoting the (also formally divergent) bare mass of the particle. We see that m and δm combine in Eq. (18.46View Equation) to form the particle’s observed mass mobs, which is finite and gives a true measure of the particle’s inertia. All diverging quantities have thus disappeared into the procedure of mass renormalization.

We must confess, as we did in the case of the scalar self-force, that the derivation of the equations of motion outlined above returns the wrong expression for the self-energy of a spherical shell of electric charge. We obtained δm = 2e2∕(3s0), while the correct expression is δm = e2∕(2s0); we are wrong by a factor of 4∕3. As before we believe that this discrepancy originates in a previously stated assumption, that the field on the shell (as produced by the shell itself) is equal to the field of a point particle evaluated at s = s0. We believe that this assumption is in fact wrong, and that a calculation of the field actually produced by a spherical shell would return the correct expression for δm. We also believe, however, that except for the diverging terms that determine δm, the difference between the shell’s field and the particle’s field should vanish in the limit s0 → 0. Our conclusion is therefore that while our expression for δm is admittedly incorrect, the statement of the equations of motion is reliable.

Apart from the term proportional to δm, the averaged force of Eq. (18.43View Equation) has exactly the same form as the force that arises from the regular field of Eq. (18.41View Equation), which we express as

( ) R ν 2( ) 2-ν 1- ν λ tail ν eFμνu = e gμν + u μuν 3a˙ + 3 R λu + eF μν u . (18.47 )
The force acting on the point particle can therefore be thought of as originating from the regular field, while the singular field simply contributes to the particle’s inertia. After mass renormalization, Eq. (18.46View Equation) is equivalent to the statement
ma μ = eF Rμν(z)uν, (18.48 )
where we have dropped the superfluous label “obs” on the particle’s observed mass.

For the final expression of the equations of motion we follow the discussion of Section 17.6 and allow an external force feμxt to act on the particle, and we replace, on the right-hand side of the equations, the acceleration vector by f μ ∕m ext. This produces

Du μ ( )( 2 Df ν 1 ) ∫ τ− ( ) ′ m -----= feμxt + e2 δμν + uμu ν ------ext+ -R νλu λ + 2e2u ν ∇ [μG +νλ]′ z(τ),z(τ′) uλd(1τ8′,.49 ) dτ 3m dτ 3 −∞
in which m denotes the observed inertial mass of the electric charge and all tensors are evaluated at z(τ), the current position of the particle on the world line; the primed indices in the tail integral refer to the point ′ z(τ ), which represents a prior position. We recall that the integration must be cut short at τ′ = τ− := τ − 0+ to avoid the singular behaviour of the retarded Green’s function at coincidence; this procedure was justified at the beginning of Section 18.3. Equation (18.49View Equation) was first derived (without the Ricci-tensor term) by Bryce S. DeWitt and Robert W. Brehme in 1960 [54], and then corrected by J.M. Hobbs in 1968 [95]. An alternative derivation was produced by Theodore C. Quinn and Robert M. Wald in 1997 [150Jump To The Next Citation Point]. In a subsequent publication [151Jump To The Next Citation Point], Quinn and Wald proved that the total work done by the electromagnetic self-force matches the energy radiated away by the particle.
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