Go to previous page Go up Go to next page

5.2 Motion of an electric charge

5.2.1 Dynamics of a point electric charge

A point particle carries an electric charge e and moves on a world line γ described by relations zμ(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, (433 )
where Sfield is an action functional for a free electromagnetic field in a spacetime with metric gαβ, Sparticle is the action of a free particle moving on a world line γ in this spacetime, and Sinteraction is an interaction term that couples the field to the particle.

The field action is given by

∫ -1-- αβ√ --- 4 S field = − 16π FαβF − gd x, (434 )
where the integration is over all of spacetime. The particle action is
∫ Sparticle = − m dτ, (435 ) γ
where m is the bare mass of the particle and ∘ ------------- dτ = − gμν(z)z˙μz˙ν dλ 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
∫ ∫ μ α μ √ --- 4 Sinteraction = e A μ(z)˙z dλ = e Aα (x )gμ (x, z)˙z δ4(x,z) − gd xdλ. (436 ) 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 α δA (x) of the vector potential yields Maxwell’s equations

Fαβ = 4πjα (437 ) ;β
with a current density jα(x) defined by
∫ j α(x) = e gαμ(x,z)˙zμδ4(x,z)d λ. (438 ) γ
These equations determine the electromagnetic field F αβ, once the motion of the electric charge is specified. On the other hand, demanding that the total action be stationary under a variation μ δz (λ) of the world line yields the equations of motion for the electric charge,
Du--μ μ ν m dτ = eF ν(z)u . (439 )
We have adopted τ as the parameter on the world line, and introduced the four-velocity μ μ u (τ ) ≡ dz ∕dτ.

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. (440 )
Under this condition the Maxwell equations of Equation (437View Equation) reduce to a wave equation for the vector potential,
α α β α □A − R βA = − 4 πj , (441 )
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 Equation (438View Equation) as
∫ α α μ j (x ) = e γ gμ (x, z)u δ4(x,z)d τ, (442 )
and we shall use Equations (441View Equation) and (442View Equation) to determine the electromagnetic field of a point electric charge. The motion of the particle is in principle determined by Equation (439View Equation), but because the vector potential obtained from Equation (441View 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 5.1 for the case of a scalar charge; the details will therefore be kept to a minimum and the reader is referred to Section 5.1 for additional information.

5.2.2 Retarded potential near the world line

The retarded solution to Equation (441View Equation) is ∫ √ -- A α(x) = G α+β′(x, x′)jβ′(x′) g′d4x ′, where G +αβ′(x,x′) is the retarded Green’s function introduced in Section 4.4. After substitution of Equation (442View Equation) we obtain

∫ A α(x) = e G α+μ(x,z)uμ dτ, (443 ) 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, Equation (443View Equation) applies to any field point x.

We now specialize Equation (443View 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 5.1.2 we let τ< and τ> be the values of the proper-time parameter at which γ enters and leaves 𝒩 (x ), respectively. Then Equation (443View Equation) can be expressed as

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

The third integration vanishes because x is then in the past of z(τ), and G +αμ(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 4.4.2. This gives

∫ τ> ∫ τ> ∫ τ> G α (x,z)uμ dτ = Uα (x,z)uμδ (σ)dτ + V α(x,z)u μθ (− σ)dτ, 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:

e ′ ∫ u ∫ τ< A α(x) = --Uαβ′(x, x′)u β + e V αμ(x,z)uμ dτ + e G α+μ(x,z)uμ dτ. (444 ) r τ< −∞
This expression applies to a point x sufficiently close to the world line that there exists a nonempty intersection between 𝒩 (x) and γ.

5.2.3 Electromagnetic field in retarded coordinates

When we differentiate the vector potential of Equation (444View 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

∇βA α(x) = e- β′ e- β′ e-( β′ γ′ β′) β′ tail − r2U αβ′u ∂βr + rUαβ′;βu + r U αβ′;γ′u u + Uαβ′a ∂ βu + eVαβ′u ∂βu + A αβ(x), (445 )
where the “tail integral” is defined by
∫ u ∫ τ< Ataαiβl(x) = e ∇ βV αμ(x,z)uμ dτ + e ∇ βG+ αμ(x,z)uμ dτ τ< − ∞ ∫ u− = e ∇βG+ αμ(x, z)uμ dτ. (446 ) −∞
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 Fα β = ∇ αAβ − ∇ βA α in powers of r, and express the result in terms of the retarded coordinates (u,r,Ωa ) introduced in Section 3.3. It will be convenient to decompose the electromagnetic field in the tetrad α α (e0,ea) that is obtained by parallel transport of α′ α′ (u ,e a ) on the null geodesic that links x to ′ x ≡ z(u); this construction is detailed in Section 3.3. Note that throughout this section we set ωab = 0, where ωab is the rotation tensor defined by Equation (138View Equation): The tetrad vectors eα′ a are taken to be Fermi–Walker transported on γ. We recall from Equation (141View Equation) that the parallel propagator can be expressed as gα′= uα′e0+ eα′ea α α a α. The expansion relies on Equation (166View Equation) for ∂αu, Equation (168View Equation) for ∂αr, and we shall need

[ ( ) ] U αβ′uβ′ = gα′α uα′ +-1-r2 R00 + 2R0aΩa + Rab ΩaΩb uα ′ + 𝒪 (r3) , (447 ) 12
which follows from Equation (323View Equation) and the relation σα′ = − r(uα′ + Ωae α′) a first encountered in Equation (144View Equation). We shall also need the expansions
′ 1 ′ ′[ 1 ] U αβ′;βuβ = − -rgααgββ Rα′0β′0 + R α′0β′cΩc − -(R β′0 + R β′cΩc )uα′ + 𝒪 (r) (448 ) 2 3
and
[ ] β′ γ′ β′ α′ 1- b 1- ( b) 2 Uαβ′;γ′u u + U αβ′a = g α aα′ + 2rR α′0b0Ω − 6r R00 + R0bΩ u α′ + 𝒪 (r ) (449 )
that follow from Equations (323View Equation, 324View Equation, 325View Equation). And finally, we shall need
[ ] β′ 1 α′ 1 Vαβ′u = − 2-g α R α′0 − 6-Ru α′ + 𝒪 (r) , (450 )
a relation that was first established in Equation (327View 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 + 3 eRb0c0Ω Ω Ωa − 6e 5Ra0b0Ω + Rab0cΩ Ω 1 ( ) 1 1 + ---e 5R00 + RbcΩbΩc + R Ωa + -eRa0 − -eRab Ωb + Fata0il+ 𝒪 (r), (451 ) 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 − 1-e(Ra0 Ωb − ΩaRb0 ) + F taaibl+ 𝒪 (r), (452 ) 2
where
tail tail ′ α′ β′ tail tail ′ α′ β′ F a0 = Fα′β′(x )ea u , Fab = F α′β′(x)ea eb (453 )
are the frame components of the tail integral; this is obtained from Equation (446View Equation) evaluated at x′:
∫ u− Ftail(x′) = 2e ∇ ′G ′ (x′,z)uμ dτ. (454 ) α′β′ −∞ [α +β ]μ
It should be emphasized that in Equations (451View Equation) and (452View Equation), all frame components are evaluated at the retarded point x′ ≡ z(u) associated with x; for example, ′ aa ≡ aa(u) ≡ aα′eαa. It is clear from these equations that the electromagnetic field Fαβ(x) is singular on the world line.

5.2.4 Electromagnetic field in Fermi normal coordinates

We now wish to express the electromagnetic field in the Fermi normal coordinates of Section 3.2; as before those will be denoted (t,s, ωa). The translation will be carried out as in Section 5.1.4, and we will decompose the field in the tetrad (¯eα,¯eα ) 0 a that is obtained by parallel transport of (u¯α,eα¯) a 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 β ¯Fa0 ≡ Fαβ ¯eαa¯e0 and ¯Fab ≡ Fαβ¯eαa ¯eβb. For this purpose we use Equations (224View Equation, 225View Equation) and (451View Equation, 452View Equation) to obtain

( ) F¯a0 = e-Ωa − e- aa − abΩbΩa + 1eabΩbaa + 1ea˙0Ωa − 5eRa0b0Ωb + 1eRb0c0ΩbΩc Ωa r2 r 2 2 6 3 1- b c -1- ( b c ) 1- 1- b ¯tail + 3eRab0cΩ Ω + 12e 5R00 + RbcΩ Ω + R Ωa + 3eRa0 − 6eRabΩ + F a0 + 𝒪 (r)
and
1 1 1 F¯ab = -e(Ωa ˙ab − ˙aaΩb) + -e (Ra0bc − Rb0ac)Ωc − -e (Ra0Ωb − ΩaRb0 ) + ¯Ftaabil+ 𝒪 (r), 2 2 2

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

¯F tail ≡ F tai¯l(¯x)e¯αu ¯β, F¯tail≡ F ta¯il(¯x)e¯αe¯β, a0 α¯β a ab ¯αβ a b

which are evaluated at ¯x.

We must still translate these results into the Fermi normal coordinates (t,s,ωa ). For this we involve Equations (221View Equation, 222View Equation, 223View Equation), and we recycle some computations that were first carried out in Section 5.1.4. After some algebra, we arrive at

¯ a α β Fa0(t,s, ω ) ≡ Fαβ(x )e¯a (x)¯e0(x) -e e-( b ) 3- b 3-( b)2 3- 2- = s2ωa − 2s aa + abω ωa + 4eabω aa + 8e abω ωa + 8 e˙a0ωa + 3 e˙aa 2 1 1 ( ) − -eRa0b0ωb − --eRb0c0ωb ωcωa + ---e 5R00 + Rbc ωbωc + R ωa 3 6 12 + 1-eRa0 − 1-eRabωb + F¯tail+ 𝒪 (s), (455 ) 3 6 a0 ¯Fab(t,s, ωa) ≡ Fαβ(x )e¯α (x)¯eβ(x) a b = 1e (ωa˙ab − ˙aaωb) + 1e (Ra0bc − Rb0ac)ωc − 1e(Ra0 ωb − ωaRb0) + F¯taabil+ 𝒪 (s), 2 2 2 (456 )
where all frame components are now evaluated at ¯x ≡ z(t); for example, ¯α aa ≡ aa(t) ≡ a¯αea.

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

∮ ∮ ⟨ ¯F ⟩(t,s) = 1- F¯ (t,s,ωa )d𝒜, ⟨F¯ ⟩(t,s) = 1- F¯ (t,s,ωa) d𝒜, (457 ) a0 𝒜 S(t,s) a0 ab 𝒜 S(t,s)ab
where d𝒜 is the element of surface area on S(t,s), and ∮ 𝒜 = d𝒜. Using the methods developed in Section 5.1.4, we find
⟨ ⟩ 2e 2 1 F¯a0 = − --aa + --e˙aa + -eRa0 + F¯taa0il+ 𝒪 (s), (458 ) ⟨ ⟩ 3s 3 3 ¯Fab = ¯Ftaabil+ 𝒪 (s). (459 )
The averaged field is singular on the world line, but we nevertheless take the formal limit s → 0 of the expressions displayed in Equations (458View Equation) and (459View Equation). In the limit the tetrad α α (¯e0,¯ea) becomes (uα¯, e¯αa), and we can easily reconstruct the field at ¯x from its frame components. We thus obtain
( ) ( ) ⟨ ⟩ 4e ( ) 2 ¯γ 1 ¯γ ¯δ tail F¯α¯β = lsi→m0 − 3s- u[¯αa¯β] + 2eu [¯α g¯β]¯γ + uβ¯]u¯γ 3 ˙a + 3R ¯δu + F ¯α¯β , (460 )
where the tail term can be copied from Equation (454View Equation),
∫ t− Fαta¯βi¯l(¯x) = 2e ∇ [¯αG+ ¯β]μ(¯x,z )u μdτ. (461 ) −∞
The tensors appearing in Equation (460View Equation) all refer to ¯x ≡ z (t), which now stands for an arbitrary point on the world line γ.

5.2.5 Singular and radiative fields

The singular vector potential

∫ α α μ A S(x) = e G Sμ(x,z )u dτ (462 ) γ
is the (unphysical) solution to Equations (441View Equation) and (442View Equation) that is obtained by adopting the singular Green’s function of Equation (335View Equation) instead of the retarded Green’s function. We will see that the singular field S F αβ reproduces the singular behaviour of the retarded solution, and that the difference, F R = Fαβ − F S αβ αβ, is smooth on the world line.

To evaluate the integral of Equation (462View 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 Equation (462View Equation) becomes

∫ ∫ ∫ α τ< α μ τ> α μ ∞ α μ AS(x) = e G Sμ(x,z )u dτ + e G Sμ(x,z )u dτ + 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 G Sαμ(x,z) = 0 by Equation (338View 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 Equation (344View Equation), and we have

∫ τ> G Sαμ(x,z )u μdτ = τ< 1 ∫ τ> 1 ∫ τ> 1∫ τ> -- U αμ(x, z)uμδ+(σ )dτ + -- U αμ(x,z)uμδ− (σ )dτ − -- V αμ(x,z)uμθ(σ )dτ. 2 τ< 2 τ< 2 τ<
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 α β′ U β′u ∕r. 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 Uαβ′′uβ′′∕radv, where radv ≡ − σ α′′uα′′ 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 A α(x) = -e-U α′uβ′ +--e-- Uα ′′uβ′′ − 1e V α(x, z)uμd τ (463 ) S 2r β 2radv β 2 u μ
for the singular vector potential.

Differentiation of Equation (463View Equation) yields

S e β′ e β′′ e β′ ∇ βAα(x ) = − 2r2-Uαβ′u ∂βr − 2r---2U αβ′′u ∂βradv + 2rU αβ′;βu e ( ′ ′ adv ′) e ′′ + --- U αβ′;γ′u βuγ + Uαβ′aβ ∂ βu + -----Uαβ′′;βuβ 2r 2radv --e--( β′′ γ′′ β′′) 1- β′ + 2radv Uαβ′′;γ′′u u + U αβ′′a ∂ βv + 2eV αβ′u ∂βu 1 1 ∫ v − -eV αβ′′uβ′′∂βv − -e ∇ βVαμ(x,z )uμdτ, (464 ) 2 2 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 5.2.3, 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 Equation (229View Equation), that an expression for the advanced distance was displayed in Equation (230View Equation), and that Equations (231View Equation) and (232View Equation) give expansions for ∂αv and ∂αradv, respectively.

To derive an expansion for β′′ U αβ′′u we follow the general method of Section 3.4.4 and introduce the functions Uα (τ ) ≡ Uαμ(x,z)u μ. We have that

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

where overdots indicate differentiation with respect to τ, and Δ ′ ≡ v − u. The leading term β′ U α(u) ≡ Uα β′u was worked out in Equation (447View Equation), and the derivatives of U α(τ) are given by

[ ( ) ] U˙α(u) = U αβ′;γ′uβ′uγ′ + Uαβ′aβ′ = g αα′ aα′ + 1rR α′0b0Ωb − 1-r R00 + R0b Ωb u α′ + 𝒪 (r2) 2 6

and

′ ′ ′ ( ′ ′ ′ ′) ′ ′[ 1 ] ¨Uα(u ) = U αβ′;γ′δ′uβ uγu δ + Uαβ′;γ′ 2aβ uγ + uβ aγ + Uαβ′˙aβ = gαα a˙α′ +-R00u α′ + 𝒪(r) , 6

according to Equations (449View Equation) and (325View Equation). Combining these results together with Equation (229View Equation) for ′ Δ gives

[ U ′′uβ′′ = gα′ u ′ + 2r (1 − ra Ωb) a ′ + 2r2˙a ′ + r2R ′ Ωb αβ α α b α α α 0b0 1 ( ) ] + --r2 R00 − 2R0aΩa + RabΩa Ωb u α′ + 𝒪 (r3) , (465 ) 12
which should be compared with Equation (447View Equation). It should be emphasized that in Equation (465View 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
β′′ γ′′ β′′ ′ ′2 Uαβ′′;γ′′u u + Uα β′′a ≡ ˙Uα(v) = U˙α(u) + ¨Uα(u )Δ + 𝒪 (Δ ),

which becomes

[ ] U ′′ ′′uβ′′uγ′′ + U ′′aβ′′ = gα′ a ′ + 2ra˙′ + 1rR ′ Ωb + 1r(R − R Ωb) u ′ + 𝒪 (r2) , αβ ;γ αβ α α α 2 α 0b0 6 00 0b α (466 )
and which should be compared with Equation (449View Equation).

We proceed similarly to derive an expansion for U αβ′′;βu β′′. Here we introduce the functions μ U αβ(τ) ≡ Uαμ;βu and express ′′ β′′ U αβ ;βu as ˙ ′ ′2 U αβ(v) = Uαβ(u ) + U αβ(u)Δ + 𝒪 (Δ ). The leading term β′ U αβ(u) ≡ Uα β′;βu was computed in Equation (448View Equation), and

′ ′ ′ 1 ′ ′[ 1 ] ˙Uαβ(u) = U αβ′;βγ′u βuγ + Uαβ′;βa β = --gααgββ R α′0β′0 − -uα′R β′0 + 𝒪 (r) 2 3

follows from Equation (324View Equation). Combining these results together with Equation (229View Equation) for Δ ′ gives

[ ] U αβ′′;βu β′′ = 1rgα′αgβ′ Rα′0β′0 − R α′0β′cΩc − 1(R β′0 − R β′cΩc )uα′ + 𝒪(r) , (467 ) 2 β 3
and this should be compared with Equation (448View Equation). The last expansion we shall need is
[ ] Vαβ′′u β′′ = − 1gα′ Rα′0 − 1Ru α′ + 𝒪 (r) , (468 ) 2 α 6
which follows at once from Equation (450View Equation).

It is now a straightforward (but still tedious) matter to substitute these expansions into Equation (464View Equation) to obtain the projections of the singular electromagnetic field F S = ∇ αAS − ∇ βAS αβ β α in the same tetrad (eα,eα) 0 a that was employed in Section 5.2.3. This gives

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), (469 ) 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), (470 ) 2
in which all frame components are evaluated at the retarded point x′. Comparison of these expressions with Equations (451View Equation) and (452View Equation) reveals that the retarded and singular fields share the same singularity structure.

The difference between the retarded field of Equations (451View Equation, 452View Equation) and the singular field of Equations (469View Equation, 470View Equation) defines the radiative field F Rαβ(x). Its tetrad components are

R 2- 1- tail F a0 = 3 e˙aa + 3eRa0 + Fa0 + 𝒪(r), (471 ) F R = F tail+ 𝒪 (r), (472 ) ab ab
and we see that F Rαβ is a smooth tensor field on the world line. There is therefore no obstacle in evaluating the radiative field directly at x = x ′, where the tetrad (eα ,eα) 0 a becomes (u α′,eα′) a. Reconstructing the field at ′ x from its frame components, we obtain
( ) R ′ ( ) 2 γ′ 1 γ′ δ′ tail F α′β′(x) = 2eu [α′ gβ′]γ′ + uβ′]uγ′ 3a˙ + 3R δ′u + F α′β′, (473 )
where the tail term can be copied from Equation (454View Equation),
∫ u− Fta′il′(x′) = 2e ∇ [α′G+ β′]μ(x′,z)uμ dτ. (474 ) α β −∞
The tensors appearing in Equation (473View Equation) all refer to the retarded point ′ x ≡ z(u), which now stands for an arbitrary point on the world line γ.

5.2.6 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 5.2.3 and 5.2.4, and in Section 5.2.5 it was shown to originate from the singular field S Fαβ; the radiative field R S Fαβ = F αβ − Fαβ was then shown to be smooth on the world line.

To make sense of the retarded field’s action on the particle we follow the discussion of Section 5.1.6 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. This was worked out at the end of Section 5.2.4, and ignoring terms that disappear in the limit s0 → 0, we obtain

( ) e ⟨Fμν⟩u ν = − (δm )aμ + e2 (g μν + u μuν) 2a˙ν + 1R νλu λ + eF tμaiνl u ν, (475 ) 3 3
where
2e2 δm ≡ slim→0 ---- (476 ) 0 3s0
is formally a divergent quantity and
∫ τ− tail ν 2 ν ′ ′ λ′ ′ eF μν u = 2e u −∞∇ [μG+ ν]λ (z(τ),z(τ )) u dτ (477 )
is the tail part of the force; all tensors in Equation (475View Equation) are evaluated at an arbitrary point z(τ) on the world line.

Substituting Equations (475View Equation) and (477View Equation) into Equation (439View Equation) gives rise to the equations of motion for the electric charge

( ) ∫ − μ 2 μ μ 2 ν 1 ν λ 2 τ [μ ν] ′ λ′ ′ (m + δm )a = e (δ ν + u uν) 3a˙ + 3R λu + 2e uν ∇ G +λ′ (z(τ),z(τ ))u dτ , (478 ) −∞
with m denoting the (also formally divergent) bare mass of the particle. We see that m and δm combine in Equation (478View 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.

Apart from the term proportional to δm, the averaged force of Equation (475View Equation) has exactly the same form as the force that arises from the radiative field of Equation (473View Equation), which we express as

( 2 1 ) eFRμνuν = e2 (gμν + u μuν) --˙aν + -R νλu λ + eF tμaiνl u ν. (479 ) 3 3
The force acting on the point particle can therefore be thought of as originating from the (smooth) radiative field, while the singular field simply contributes to the particle’s inertia. After mass renormalization, Equation (478View Equation) is equivalent to the statement
ma = eF R (z)uν, (480 ) μ μν
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 5.1.6 and allow an external force μ fext to act on the particle, and we replace, on the right-hand side of the equations, the acceleration vector by μ fext∕m. This produces

μ ( ν ) ∫ τ− m Du---= f μ + e2(δμ + u μuν) -2--Df-ext-+ 1R ν uλ + 2e2u ν ∇ [μG ν]′ (z(τ),z(τ′))uλ′ dτ′, dτ ext ν 3m dτ 3 λ − ∞ +λ (481 )
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 5.2.3. Equation (481View Equation) was first derived (without the Ricci-tensor term) by Bryce S. DeWitt and Robert W. Brehme in 1960 [24Jump To The Next Citation Point], and then corrected by J.M. Hobbs in 1968 [29]. An alternative derivation was produced by Theodore C. Quinn and Robert M. Wald in 1997 [49Jump To The Next Citation Point]. In a subsequent publication [50Jump 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.
  Go to previous page Go up Go to next page