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1.8 Motion of a scalar charge in curved spacetime

The dynamics of a point scalar charge can be formulated in a way that stays fairly close to the electromagnetic theory. The particle’s charge q produces a scalar field Φ (x), which satisfies a wave equation
(□ − ξR) Φ = − 4π μ (34 )
that is very similar to Equation (13View Equation). Here, R is the spacetime’s Ricci scalar, and ξ is an arbitrary coupling constant; the scalar charge density μ (x) is given by a four-dimensional Dirac functional supported on the particle’s world line γ. The retarded solution to the wave equation is
∫ Φ (x) = q G+ (x,z) dτ, (35 ) γ
where G+ (x,z) is the retarded Green’s function associated with Equation (34View Equation). The field exerts a force on the particle, whose equations of motion are
ma μ = q(gμν + uμu ν)∇ νΦ, (36 )
where m is the particle’s mass; this equation is very similar to the Lorentz-force law. But the dynamics of a scalar charge comes with a twist: If Equations (34View Equation) and (36View Equation) are to follow from a variational principle, the particle’s mass should not be expected to be a constant of the motion. It is found instead to satisfy the differential equation
dm--= − quμ∇ μΦ, (37 ) dτ
and in general m will vary with proper time. This phenomenon is linked to the fact that a scalar field has zero spin: The particle can radiate monopole waves and the radiated energy can come at the expense of the rest mass.

The scalar field of Equation (35View Equation) diverges on the world line, and its singular part ΦS (x ) must be removed before Equations (36View Equation) and (37View Equation) can be evaluated. This procedure produces the radiative field ΦR (x), and it is this field (which satisfies the homogeneous wave equation) that gives rise to a self-force. The gradient of the radiative field takes the form of

( ) 1 1 ν 1 ν λ tail ∇ μΦR = − 12-(1 − 6ξ)qRu μ + q(gμν + uμuν) 3-˙a + 6R λu + Φμ (38 )
when it is evaluated of the world line. The last term is the tail integral
∫ − tail τ ′ ′ Φμ = q ∇ μG+ (z(τ),z(τ )) dτ, (39 ) − ∞
and this brings the dependence on the particle’s past.

Substitution of Equation (38View Equation) into Equations (36View Equation) and (37View Equation) gives the equations of motion of a point scalar charge. (At this stage I introduce an external force fμ ext and reduce the order of the differential equation.) The acceleration is given by

[ ν ∫ τ− ] ma μ = f μ + q2(δμ + u μu ) -1-Df-ext+ 1R ν uλ + ∇ νG (z(τ),z(τ′)) dτ ′ , (40 ) ext ν ν 3m dτ 6 λ −∞ +
and the mass changes according to
∫ τ− dm--= − 1-(1 − 6ξ)q2R − q2u μ ∇ μG+ (z(τ),z(τ′)) dτ′. (41 ) dτ 12 − ∞
These equations were first derived by Quinn [48Jump To The Next Citation Point]2.

In flat spacetime the Ricci-tensor term and the tail integral disappear, and Equation (40View Equation) takes the form of Equation (5View Equation) with 2 q ∕(3m ) replacing the factor of 2 2e ∕(3m ). In this simple case Equation (41View Equation) reduces to dm ∕dτ = 0 and the mass is in fact a constant. This property remains true in a conformally-flat spacetime when the wave equation is conformally invariant (ξ = 1∕6): In this case the Green’s function possesses only a light-cone part, and the right-hand side of Equation (41View Equation) vanishes. In generic situations the mass of a point scalar charge will vary with proper time.

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