17 Motion of a scalar charge

17.1 Dynamics of a point scalar charge

A point particle carries a scalar charge q and moves on a world line γ described by relations zμ(λ ), in which λ is an arbitrary parameter. The particle generates a scalar potential Φ (x ) and a field Φ α(x) := ∇ αΦ (x ). The dynamics of the entire system is governed by the action

S = Sfield + Sparticle + Sinteraction, (17.1 )
where Sfield is an action functional for a free scalar 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

∫ ( )√ --- Sfield = − -1- g αβΦαΦ β + ξR Φ2 − gd4x, (17.2 ) 8π
where the integration is over all of spacetime; the field is coupled to the Ricci scalar R by an arbitrary constant ξ. The particle action is
∫ Sparticle = − m0 dτ, (17.3 ) γ
where m 0 is the bare mass of the particle and dτ = ∘ −-g--(z)˙zμz˙νd λ μν is the differential of proper time along the world line; we use an overdot on μ z (λ ) to indicate differentiation with respect to the parameter λ. Finally, the interaction term is given by
∫ ∫ √ --- Sinteraction = q Φ(z)dτ = q Φ(x )δ4(x,z ) − gd4xd τ. (17.4 ) 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 δΦ (x) of the field configuration yields the wave equation

( ) □ − ξR Φ(x) = − 4π μ(x) (17.5 )
for the scalar potential, with a charge density μ(x) defined by
∫ μ (x) = q δ4(x,z)dτ. (17.6 ) γ
These equations determine the field Φα(x ) once the motion of the scalar 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
μ m (τ)Du---= q(gμν + uμuν)Φ (z) (17.7 ) dτ ν
for the scalar charge. We have here adopted τ as the parameter on the world line, and introduced the four-velocity μ μ u (τ ) := dz ∕dτ. The dynamical mass that appears in Eq. (17.7View Equation) is defined by m (τ) := m0 − q Φ(z), which can also be expressed in differential form as
dm-- μ dτ = − qΦ μ(z)u . (17.8 )
It should be clear that Eqs. (17.7View Equation) and (17.8View Equation) are valid only in a formal sense, because the scalar potential obtained from Eqs. (17.5View Equation) and (17.6View Equation) diverges on the world line. Before we can make sense of these equations we have to analyze the field’s singularity structure near the world line.

17.2 Retarded potential near the world line

The retarded solution to Eq. (17.5View Equation) is ∫ √ -- Φ (x) = G+ (x,x′)μ(x′) g′d4x ′, where G+ (x,x ′) is the retarded Green’s function introduced in Section 14. After substitution of Eq. (17.6View Equation) we obtain

∫ Φ (x) = q γ G+ (x,z )d τ, (17.9 )
in which z(τ) gives the description of the world line γ. Because the retarded Green’s function is defined globally in the entire spacetime, Eq. (17.9View Equation) applies to any field point x.
View Image

Figure 9: The region within the dashed boundary represents the normal convex neighbourhood of the point x. The world line γ enters the neighbourhood at proper time τ< and exits at proper time τ>. Also shown are the retarded point z (u) and the advanced point z (v ).

We now specialize Eq. (17.9View Equation) to a point x near the world line; see Figure 9View Image. We let 𝒩 (x) be the normal convex neighbourhood of this point, and we assume that the world line traverses 𝒩 (x). Let τ< be the value of the proper-time parameter at which γ enters 𝒩 (x) from the past, and let τ> be its value when the world line leaves 𝒩 (x). Then Eq. (17.9View Equation) can be broken up into the three integrals

∫ τ< ∫ τ> ∫ ∞ Φ(x ) = q G+ (x,z)dτ + q G+ (x,z )d τ + q G+ (x, z)dτ. −∞ τ< τ>

The third integration vanishes because x is then in the past of z(τ), and G+ (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 14.2. This gives

∫ τ> ∫ τ> ∫ τ> τ< G+ (x,z )d τ = τ< U(x,z )δ+ (σ)dτ + τ< V(x, z)𝜃+(− σ)dτ,

and to evaluate this we refer back to Section 10 and let x′ := z(u) be the retarded point associated with x; these points are related by ′ σ(x,x ) = 0 and α′ r := σ α′u is the retarded distance between x and the world line. We resume the index convention of Section 10: to tensors at x we assign indices α, β, etc.; to tensors at x′ we assign indices α′, β ′, etc.; and to tensors at a generic point z(τ ) on the world line we assign indices μ, ν, etc.

To perform the first integration we change variables from τ to σ, noticing that σ increases as z (τ) passes through x′. The change of σ on the world line is given by dσ := σ (x,z + dz) − σ(x,z) = σ μuμdτ, and we find that the first integral evaluates to U(x, z)∕(σμuμ) with z identified with x ′. The second integration is cut off at τ = u by the step function, and we obtain our final expression for the retarded potential of a point scalar charge:

∫ ∫ q- ′ u τ< Φ (x ) = rU (x,x ) + q V(x, z)dτ + q G+ (x,z)dτ. (17.10 ) 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.

17.3 Field of a scalar charge in retarded coordinates

When we differentiate the potential of Eq. (17.10View 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 – you may refer back to Section 10.2 for a detailed discussion. This means, for example, that the total variation of U (x,x′) is δU = U(x + δx, x′ + δx ′) − U (x, x′) = U;α δxα + U;α′u α′δu. The gradient of the scalar potential is therefore given by

-q ′ q- ′ q- ′ α′ ′ tail Φ α(x) = − r2U (x,x )∂αr + r U;α(x,x ) + rU;α′(x, x)u ∂αu + qV(x, x)∂αu + Φα (x), (17.11 )
where the “tail integral” is defined by
∫ u ∫ τ< Φtαail(x ) = q ∇ αV (x, z)dτ + q ∇ αG+ (x,z)d τ τ< −∞ ∫ u− = q ∇ αG+ (x,z)dτ. (17.12 ) − ∞
In the second form of the definition we integrate ∇ G (x,z ) α + from τ = − ∞ to almost τ = u, but we cut the integration short at − + τ = u := u − 0 to avoid the singular behaviour of the retarded Green’s function at σ = 0. This limiting procedure gives rise to the first form of the definition, with the advantage that the integral need not be broken up into contributions that refer to 𝒩 (x) and its complement, respectively.

We shall now expand Φ α(x) in powers of r, and express the results in terms of the retarded coordinates a (u,r,Ω ) introduced in Section 10. It will be convenient to decompose Φα(x ) in the tetrad (eα0,eαa) that is obtained by parallel transport of ′ ′ (uα ,eαa ) on the null geodesic that links x to x ′ := z(u); this construction is detailed in Section 10. The expansion relies on Eq. (10.29View Equation) for ∂αu, Eq. (10.31View Equation) for ∂ r α, and we shall need

′ -1- 2( a a b) 3 U(x,x ) = 1 + 12 r R00 + 2R0a Ω + RabΩ Ω + O (r ), (17.13 )
which follows from Eq. (14.10View Equation) and the relation α′ α′ a α′ σ = − r(u + Ω ea ) first encountered in Eq. (10.7View Equation); recall that
α′ β′ α′β′ α′β′ R00 (u) = R α′β′u u , R0a(u ) = R α′β′u ea , Rab (u) = Rα′β′ea eb

are frame components of the Ricci tensor evaluated at ′ x. We shall also need the expansions

′ 1- α′( b) 2 U;α (x, x) = 6rg α R α′0 + R α′bΩ + O (r ) (17.14 )
and
( ) U;α′(x, x′)u α′ = − 1-r R00 + R0aΩa + O(r2) (17.15 ) 6
which follow from Eqs. (14.11View Equation); recall from Eq. (10.4View Equation) that the parallel propagator can be expressed as gα′ = uα ′e0 + eα′ea α α a α. And finally, we shall need
1 ( ) V (x,x′) = --- 1 − 6ξ R + O (r), (17.16 ) 12
a relation that was first established in Eq. (14.13View Equation); here R := R (u) is the Ricci scalar evaluated at x ′.

Collecting all these results gives

Φ (u, r,Ωa) := Φ (x )eα (x ) 0 α 0 = qa Ωa + 1qR Ωa Ωb + -1-(1 − 6ξ)qR + Φtail+ O (r), (17.17 ) r a 2 a0b0 12 0 Φa (u, r,Ωa) := Φα(x )eαa (x ) q q 1 1 ( ) = − -2Ωa − --abΩbΩa − -qRb0c0ΩbΩc Ωa − --q Ra0b0Ωb − Rab0cΩbΩc r r 3 6 + 1-q[R − R ΩbΩc − (1 − 6ξ)R ]Ω + 1-q(R + R Ωb ) + Φtail+ O (r), (17.18 ) 12 00 bc a 6 a0 ab a
where a = a ′eα′ a α a are the frame components of the acceleration vector,
Ra0b0(u) = Rα′γ′β′δ′eαa′uγ′eβ′u δ′, Rab0c(u ) = R α′γ′β′δ′eαa′eγ′u β′eδc′ b b

are frame components of the Riemann tensor evaluated at x′, and

′ ′ Φt0ail(u) = Φtαail′ (x′)u α, Φtaail(u) = Φtaαil′ (x′)eαa (17.19 )
are the frame components of the tail integral evaluated at ′ x. Equations (17.17View Equation) and (17.18View Equation) show clearly that Φ α(x) is singular on the world line: the field diverges as −2 r when r → 0, and many of the terms that stay bounded in the limit depend on Ωa and therefore possess a directional ambiguity at r = 0.

17.4 Field of a scalar charge in Fermi normal coordinates

The gradient of the scalar potential can also be expressed in the Fermi normal coordinates of Section 9. To effect this translation we make ¯x := z (t) the new reference point on the world line. We resume here the notation of Section 11 and assign indices ¯α, ¯ β, …to tensors at ¯x. The Fermi normal coordinates are denoted (t,s,ωa ), and we let (¯eα,¯eα) 0 a be the tetrad at x that is obtained by parallel transport of (uα¯, e¯α) a on the spacelike geodesic that links x to ¯x.

Our first task is to decompose Φ α(x) in the tetrad (¯eα0,¯eαa), thereby defining ¯Φ0 := Φ α¯eα0 and ¯Φa := Φα ¯eαa. For this purpose we use Eqs. (11.7View Equation), (11.8View Equation), (17.17View Equation), and (17.18View Equation) to obtain

¯ [ 2 ] [ ( b) a 1- 2 a 1-2 a b 3] Φ0 = 1 + O(r ) Φ0 + r 1 − abΩ a + 2 ra˙ + 2r R 0b0Ω + O (r ) Φa 1 1 = − -q˙aaΩa + --(1 − 6ξ)qR + ¯Φt0ail+ O(r) 2 12
and
[ ] [ ] ¯Φa = δba + 1-r2abaa − 1r2Rba0cΩc + O (r3) Φb + raa + O (r2) Φ0 2 2 q- q- b 1- b 1- b c 1- b 1- b c = − r2Ωa − r abΩ Ωa + 2qabΩ aa − 3qRb0c0Ω Ω Ωa − 6 qRa0b0Ω − 3qRab0cΩ Ω 1 [ b c ] 1 ( b) tail + --q R00 − Rbc Ω Ω − (1 − 6ξ)R Ωa + --q Ra0 + RabΩ + ¯Φa + O (r), 12 6
where all frame components are still evaluated at x′, except for ¯Φt0ail and ¯Φtaail which are evaluated at ¯x.

We must still translate these results into the Fermi normal coordinates (t,s,ωa ). For this we involve Eqs. (11.4View Equation), (11.5View Equation), and (11.6View Equation), from which we deduce, for example,

1 1 1 3 b 3 b 15( b)2 3 1 -2Ωa = -2ωa + --aa − ---abω ωa − --abω aa + --- abω ωa + -a˙0ωa − -˙aa r s 2s 2s 4 8 8 3 + ˙abωbωa + 1-Ra0b0ωb − 1Rb0c0ωbωc ωa − 1Rab0cωbωc + O (s) 6 2 3
and
( ) 1abΩbΩa = 1-abωbωa + 1abωbaa − 3-abωb 2ωa − 1˙a0ωa − a˙bωb ωa + O(s), r s 2 2 2

in which all frame components (on the right-hand side of these relations) are now evaluated at ¯x; to obtain the second relation we expressed aa(u) as aa (t) − s˙aa(t) + O (s2) since according to Eq. (11.4View Equation), u = t − s + O (s2).

Collecting these results yields

¯Φ (t,s,ωa) := Φ (x)¯eα(x) 0 α 0 = − 1qa˙ωa + 1-(1 − 6ξ)qR + ¯Φtail+ O(s), (17.20 ) 2 a 12 0 ¯Φa(t,s,ωa) := Φ α(x)¯eαa(x) q q ( ) 3 3 ( ) 1 1 = − -2ωa − --- aa − abωbωa + --qabωbaa − --q abωb 2ωa + --q˙a0ωa + -qa˙a s 2s 4 8 8 3 − 1qR ωb + 1qR ωbωcω + 1-q[R − R ωbωc − (1 − 6ξ)R]ω 3 a0b0 6 b0c0 a 12 00 bc a 1 ( b) tail + 6q Ra0 + Rabω + ¯Φ a + O (s ). (17.21 )
In these expressions, aa(t) = a¯αe¯α a are the frame components of the acceleration vector evaluated at ¯x, ˙a (t) = ˙a u¯α 0 ¯α and ˙a (t) = a˙ e¯α a ¯α a are frame components of its covariant derivative, α¯ ¯γ ¯β ¯δ Ra0b0(t) = Rα¯¯γβ¯δ¯eau ebu are frame components of the Riemann tensor evaluated at ¯x,
¯α ¯β α¯ ¯β ¯α ¯β R00 (t) = R ¯α¯βu u , R0a (t) = R ¯α¯βu e a, Rab(t) = R ¯α¯βeaeb

are frame components of the Ricci tensor, and R(t) is the Ricci scalar evaluated at ¯x. Finally, we have that

Φ¯ta0il(t) = Φta¯αil(¯x)u¯α, ¯Φtaail(t) = Φta¯αil(x¯)e¯αa (17.22 )
are the frame components of the tail integral – see Eq. (17.12View Equation) – evaluated at ¯x := z(t).

We shall now compute the averages of ¯Φ0 and ¯Φa over S (t,s), a two-surface of constant t and s; these will represent the mean value of the field at a fixed proper distance away from the world line, as measured in a reference frame that is momentarily comoving with the particle. The two-surface is charted by angles A 𝜃 (A = 1,2) and it is described, in the Fermi normal coordinates, by the parametric relations a a A xˆ = sω (𝜃 ); a canonical choice of parameterization is ωa = (sin𝜃 cosϕ,sin 𝜃sinϕ, cos𝜃). Introducing the transformation matrices ωaA := ∂ωa∕∂ 𝜃A, we find from Eq. (9.16View Equation) that the induced metric on S (t,s) is given by

2 2[ 1 2 3] A B ds = s ωAB − 3s RAB + O (s )d 𝜃 d𝜃 , (17.23 )
where ωAB := δabωa ωb A B is the metric of the unit two-sphere, and where RAB := Racbdωaωc ωbωd A B depends on t and the angles 𝜃A. From this we infer that the element of surface area is given by
2[ 1- 2 c a b 3 ] d𝒜 = s 1 − 6 s R acb(t)ω ω + O(s ) dω, (17.24 )
where ∘ ---------2 dω = det[ωAB ]d 𝜃 is an element of solid angle – in the canonical parameterization, dω = sin 𝜃d𝜃d ϕ. Integration of Eq. (17.24View Equation) produces the total surface area of S(t,s), and 𝒜 = 4πs2 [1 − -1s2Rabab + O(s3)] 18.

The averaged fields are defined by

∮ ∮ ⟨ ¯ ⟩ 1- ¯ A ⟨ ¯ ⟩ 1- ¯ A Φ0 (t,s) = 𝒜 S(t,s)Φ0 (t,s,𝜃 )d𝒜, Φa (t,s) = 𝒜 S (t,s)Φa (t,s,𝜃 )d𝒜, (17.25 )
where the quantities to be integrated are scalar functions of the Fermi normal coordinates. The results
∮ ∮ ∮ 1-- a -1- a b 1- ab -1- a b c 4π ω dω = 0, 4π ω ω dω = 3 δ , 4π ω ω ω dω = 0, (17.26 )
are easy to establish, and we obtain
⟨¯ ⟩ 1-- ¯tail Φ0 = 12(1 − 6ξ)qR + Φ0 + O(s), (17.27 ) ⟨ ⟩ q 1 1 tail ¯Φa = − --aa + --q˙aa + -qRa0 + Φ¯a + O (s). (17.28 ) 3s 3 6
The averaged field is still singular on the world line. Regardless, we shall take the formal limit s → 0 of the expressions displayed in Eqs. (17.27View Equation) and (17.28View Equation). In the limit the tetrad (e¯α0,¯eαa) reduces to (u¯α,eα¯) a, and we can reconstruct the field at ¯x by invoking the completeness relations δ¯α = − u¯αu + eα¯ea ¯β ¯β a ¯β. We thus obtain
⟨ ⟩ ( ) ( )( ) Φ ¯α = lim − -q- a¯α − -1-(1 − 6ξ)qRu ¯α + q g ¯ + u¯αu ¯ 1˙a¯β + 1-R¯β¯γu¯γ + Φtail, (17.29 ) s→0 3s 12 ¯αβ β 3 6 ¯α
where the tail integral can be copied from Eq. (17.12View Equation),
∫ t− Φtail(¯x) = q ∇ ¯αG+ (¯x,z)dτ. (17.30 ) α¯ −∞
The tensors appearing in Eq. (17.29View Equation) all refer to ¯x := z(t), which now stands for an arbitrary point on the world line γ.

17.5 Singular and regular fields

The singular potential

∫ ΦS (x) = q GS (x,z)dτ (17.31 ) γ
is the (unphysical) solution to Eqs. (17.5View Equation) and (17.6View Equation) that is obtained by adopting the singular Green’s function of Eq. (14.30View Equation) instead of the retarded Green’s function. As we shall see, the resulting singular field S Φα(x ) reproduces the singular behaviour of the retarded solution; the difference, ΦRα(x) = Φ α(x) − ΦSα(x ), is smooth on the world line.

To evaluate the integral of Eq. (17.31View 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. (17.31View Equation) can be broken up into the three integrals

∫ ∫ ∫ S τ< τ> ∞ Φ (x ) = q GS(x,z )d τ + q GS (x, z)dτ + q GS(x, z)dτ. −∞ τ< τ>

The first integration vanishes because x is then in the chronological future of z(τ), and GS (x,z) = 0 by Eq. (14.21View 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. (14.32View Equation), and we have

∫ ∫ ∫ ∫ τ> 1 τ> 1 τ> 1 τ> GS (x, z)dτ = 2- U (x, z)δ+(σ)dτ + 2- U (x,z)δ− (σ)dτ − 2- V (x,z)𝜃(σ)dτ. τ< τ< τ< τ<

To evaluate these we re-introduce the retarded point x′ := z(u) and let x′′ := z(v) be the advanced point associated with x; we recall from Section 11.4 that these points are related by ′′ σ (x, x ) = 0 and that α′′ radv := − σ α′′u is the advanced 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 (x,x ′)∕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 ′′ U (x,x )∕radv. The third integration is restricted to the interval u ≤ τ ≤ v by the step function, and we obtain our final expression for the singular potential of a point scalar charge:

q q 1 ∫ v ΦS (x ) = --U (x,x′) + -----U (x, x′′) − -q V (x,z)dτ. (17.32 ) 2r 2radv 2 u
We observe that ΦS(x ) depends on the state of motion of the scalar charge between the retarded time u and the advanced time v; contrary to what was found in Section 17.2 for the retarded potential, there is no dependence on the particle’s remote past.

We use the techniques of Section 17.3 to differentiate the potential of Eq. (17.32View Equation). We find

q q q q ′ ΦSα(x) = − --2U (x,x′)∂αr − -----2U (x,x′′)∂αradv + ---U;α(x,x′) + --U;α′(x, x′)u α∂ αu 2r 2radv 2r 2r + --q--U (x,x ′′) + --q--U ′′(x, x′′)uα′′∂ v + 1-qV (x,x′)∂ u − 1qV (x,x ′′)∂ v 2radv ;α 2radv ;α α 2 α 2 α 1 ∫ v − -q ∇ αV (x,z )d τ, (17.33 ) 2 u
and we would like to express this as an expansion in powers of r. For this we shall rely on results already established in Section 17.3, as well as additional expansions that will involve the advanced point x′′. Those we develop now.

We recall first 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 (x,x ) we follow the general method of Section 11.4 and define a function U (τ) := U (x,z(τ )) of the proper-time parameter on γ. We have that

1 ( ) U (x, x′′) := U (v) = U (u + Δ ′) = U (u) + ˙U(u)Δ ′ +-¨U (u)Δ ′2 + O Δ ′3 , 2

where overdots indicate differentiation with respect to τ, and where ′ Δ := v − u. The leading term ′ U (u) := U(x, x) was worked out in Eq. (17.13View Equation), and the derivatives of U (τ) are given by

( ) U˙(u) = U;α′uα′ = − 1r R00 + R0a Ωa + O (r2) 6

and

¨ ′′ α′ β′ ′ α′ 1- U(u ) = U;α β u u + U;α a = 6R00 + O (r),

according to Eqs. (17.15View Equation) and (14.11View Equation). Combining these results together with Eq. (11.12View Equation) for Δ ′ gives

′′ -1- 2( a a b) 3 U (x,x ) = 1 + 12 r R00 − 2R0a Ω + RabΩ Ω + O (r ), (17.34 )
which should be compared with Eq. (17.13View Equation). It should be emphasized that in Eq. (17.34View Equation) and all equations below, the frame components of the Ricci tensor are evaluated at the retarded point x ′ := z(u), and not at the advanced point. The preceding computation gives us also an expansion for U;α′′u α′′ := U˙(v) = U˙(u) + U¨(u)Δ ′ + O (Δ ′2). This becomes
′′ α′′ 1- ( a) 2 U;α′′(x, x )u = 6 r R00 − R0a Ω + O (r ), (17.35 )
which should be compared with Eq. (17.15View Equation).

We proceed similarly to derive an expansion for U;α(x, x′′). Here we introduce the functions U α(τ) := U;α (x, z(τ)) and express U;α(x,x′′) as Uα(v) = U α(u) + ˙Uα(u )Δ ′ + O(Δ ′2). The leading term U α(u) := U;α(x,x′) was computed in Eq. (17.14View Equation), and

′ 1 ′ U˙α(u) = U;αβ′u β = − --gααR α′0 + O(r) 6

follows from Eq. (14.11View Equation). Combining these results together with Eq. (11.12View Equation) for ′ Δ gives

′′ 1- α′( b) 2 U;α(x,x ) = − 6rg α R α′0 − R α′bΩ + O (r ), (17.36 )
and this should be compared with Eq. (17.14View Equation).

The last expansion we shall need is

′′ -1-( ) V (x,x ) = 12 1 − 6ξ R + O (r), (17.37 )
which follows at once from Eq. (17.16View Equation) and the fact that ′′ ′ V (x,x ) − V (x,x ) = O (r); the Ricci scalar is evaluated at the retarded point ′ x.

It is now a straightforward (but tedious) matter to substitute these expansions (all of them!) into Eq. (17.33View Equation) and obtain the projections of the singular field ΦS (x) α in the same tetrad (eα,eα) 0 a that was employed in Section 17.3. This gives

ΦS (u, r,Ωa) := ΦS(x )e α(x) 0 α 0 = qaaΩa + 1qRa0b0Ωa Ωb + O (r), (17.38 ) r 2 ΦSa(u, r,Ωa) := ΦSα(x )e αa(x) q q 1 1 1 ( ) = − -2Ωa − -abΩbΩa − -q˙aa − --qRb0c0ΩbΩcΩa − -q Ra0b0Ωb − Rab0cΩb Ωc r r 3 3 6 + 1-q[R − R ΩbΩc − (1 − 6ξ)R ]Ω + 1-qR Ωb, (17.39 ) 12 00 bc a 6 ab
in which all frame components are evaluated at the retarded point x′ := z (u ). Comparison of these expressions with Eqs. (17.17View Equation) and (17.18View Equation) reveals that the retarded and singular fields share the same singularity structure.

The difference between the retarded field of Eqs. (17.17View Equation), (17.18View Equation) and the singular field of Eqs. (17.38View Equation), (17.39View Equation) defines the regular field ΦR (x ) α. Its frame components are

R 1 tail Φ 0 = 12-(1 − 6ξ)qR + Φ 0 + O (r), (17.40 ) 1 1 ΦRa = --q˙aa + -qRa0 + Φtaail+ O (r), (17.41 ) 3 6
and we see that ΦRα(x) is a regular vector field on the world line. There is therefore no obstacle in evaluating the regular field directly at x = x′, where the tetrad (eα,eα) 0 a becomes (uα′,eα′) a. Reconstructing the field at ′ x from its frame components, we obtain
1 ( )( 1 ′ 1 ′ ′) ΦRα′(x′) = − ---(1 − 6 ξ)qRu α′ + q gα′β′ + uα′uβ′-a˙β + -R βγ′u γ + Φtαa′il, (17.42 ) 12 3 6
where the tail term can be copied from Eq. (17.12View Equation),
∫ u− Φtaαil′ (x′) = q ∇ α′G+ (x ′,z)dτ. (17.43 ) − ∞
The tensors appearing in Eq. (17.42View Equation) all refer to the retarded point x′ := z(u ), which now stands for an arbitrary point on the world line γ.

17.6 Equations of motion

The retarded field Φ (x) α of a point scalar 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 affect its motion. The field’s singularity structure was analyzed in Sections 17.3 and 17.4, and in Section 17.5 it was shown to originate from the singular field ΦSα(x); the regular field ΦRα(x) = Φ α(x) − ΦSα(x) was then shown to be regular on the world line.

To make sense of the retarded field’s action on the particle we temporarily model the scalar charge not as a point particle, but as a small hollow shell that appears spherical when observed in a reference frame that is momentarily comoving with the particle; 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 hollow shell is the average of qΦα(τ,s0,ωa ) over the surface of the shell. 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. (17.29View Equation)

⟨ ⟩ ( )( ) q Φ μ = − (δm )aμ − 1-(1 − 6ξ)q2Ru μ + q2 gμν + u μuν 1a˙ν + 1R νλu λ + qΦtaμil, (17.44 ) 12 3 6
where
2 δm := lim -q-- (17.45 ) s0→0 3s0
is formally a divergent quantity and
∫ τ− ( ) qΦtail = q2 ∇ μG+ z (τ ),z(τ′) dτ ′ (17.46 ) μ −∞
is the tail part of the force; all tensors in Eq. (17.44View Equation) are evaluated at an arbitrary point z(τ) on the world line.

Substituting Eqs. (17.44View Equation) and (17.46View Equation) into Eq. (17.7View Equation) gives rise to the equations of motion

[ ∫ τ− ] ( μ 2( μ μ ) 1-ν 1- ν λ ν ( ′ ) ′ m + δm )a = q δ ν + u uν 3 ˙a + 6 R λu + − ∞ ∇ G+ z(τ),z(τ ) dτ (17.47 )
for the scalar charge, with m := m0 − qΦ(z) denoting the (also formally divergent) dynamical mass of the particle. We see that m and δm combine in Eq. (17.47View Equation) to form the particle’s observed mass mobs, which is taken to be finite and to give a true measure of the particle’s inertia. All diverging quantities have thus disappeared into the process of mass renormalization. Substituting Eqs. (17.44View Equation) and (17.46View Equation) into Eq. (17.8View Equation), in which we replace m by mobs = m + δm, returns an expression for the rate of change of the observed mass,
∫ − dmobs 1 2 2 μ τ ( ′ ) ′ -d-τ--= − 12(1 − 6ξ)q R − q u ∇ μG+ z(τ),z(τ ) dτ . (17.48 ) − ∞
That the observed mass is not conserved is a remarkable property of the dynamics of a scalar charge in a curved spacetime. Physically, this corresponds to the fact that in a spacetime with a time-dependent metric, a scalar charge radiates monopole waves and the radiated energy comes at the expense of the particle’s inertial mass.

We must confess that the derivation of the equations of motion outlined above returns the wrong expression for the self-energy of a spherical shell of scalar charge. We obtained δm = q2∕(3s0), while the correct expression is δm = q2∕(2s0); we are wrong by a factor of 2∕3. 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 field of Eq. (17.44View Equation) has exactly the same form as the regular field of Eq. (17.42View Equation), which we re-express as

( ) qΦR = − 1-(1 − 6ξ)q2Ru + q2(g + u u ) 1-˙aν + 1R ν uλ + q Φtail. (17.49 ) μ 12 μ μν μ ν 3 6 λ μ
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, Eqs. (17.47View Equation) and (17.48View Equation) are equivalent to the statements
μ ( μν μ ν) R dm-- μ R ma = q g + u u Φν(z), d τ = − qu Φμ(z), (17.50 )
where we have dropped the superfluous label “obs” on the particle’s observed mass. Another argument in support of the claim that the motion of the particle should be affected by the regular field only was presented in Section 14.5.

The equations of motion displayed in Eqs. (17.47View Equation) and (17.48View Equation) are third-order differential equations for the functions zμ(τ). It is well known that such a system of equations admits many unphysical solutions, such as runaway situations in which the particle’s acceleration increases exponentially with τ, even in the absence of any external force [56, 101]. And indeed, our equations of motion do not yet incorporate an external force which presumably is mostly responsible for the particle’s acceleration. Both defects can be cured in one stroke. We shall take the point of view, the only admissible one in a classical treatment, that a point particle is merely an idealization for an extended object whose internal structure – the details of its charge distribution – can be considered to be irrelevant. This view automatically implies that our equations are meant to provide only an approximate description of the object’s motion. It can then be shown [112, 70] that within the context of this approximation, it is consistent to replace, on the right-hand side of the equations of motion, any occurrence of the acceleration vector by μ f ext∕m, where μ fext is the external force acting on the particle. Because feμxt is a prescribed quantity, differentiation of the external force does not produce higher derivatives of the functions zμ(τ), and the equations of motion are properly of the second order.

We shall strengthen this conclusion in Part V of the review, when we consider the motion of an extended body in a curved external spacetime. While the discussion there will concern the gravitational self-force, many of the lessons learned in Part V apply just as well to the case of a scalar (or electric) charge. And the main lesson is this: It is natural – indeed it is an imperative – to view an equation of motion such as Eq. (17.47View Equation) as an expansion of the acceleration in powers of 2 q, and it is therefore appropriate – indeed imperative – to insert the zeroth-order expression for ν ˙a within the term of order q2. The resulting expression for the acceleration is then valid up to correction terms of order q4. Omitting these error terms, we shall write, in final analysis, the equations of motion in the form

μ ( )[ ν ∫ τ− ( ) ] m Du---= f μext + q2 δμν + uμu ν -1--Df-ext-+ 1-Rνλu λ + ∇ νG+ z(τ),z(τ ′) dτ′ (17.51 ) dτ 3m dτ 6 −∞
and
dm 1 ∫ τ− ( ) ----= − ---(1 − 6 ξ)q2R − q2uμ ∇μG+ z(τ),z(τ′) dτ′, (17.52 ) dτ 12 −∞
where m denotes the observed inertial mass of the scalar charge, and where all tensors are evaluated at z(τ). We recall that the tail 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 17.3. Equations (17.51View Equation) and (17.52View Equation) were first derived by Theodore C. Quinn in 2000 [149]. In his paper Quinn also establishes that the total work done by the scalar self-force matches the amount of energy radiated away by the particle.
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