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1.7 Motion of an electric charge in curved spacetime

I have argued in Section 1.4 that the self-force acting on a point electric charge is produced by the radiative field, and that the charge’s equations of motion should take the form of ma μ = fμext+ eFRμνuν, where feμxt is an external force also acting on the particle. Substituting Equation (32View Equation) gives
( 2 Df ν 1 ) ∫ τ− ν] ′ ma μ = fμext + e2(δμν + uμuν) ------ext+ -R νλu λ + 2e2u ν ∇ [μG +λ′ (z(τ),z(τ ′))uλ d τ′, (33 ) 3m dτ 3 −∞
in which 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 a point z(τ′) which represents a prior position; the integration is cut short at ′ − + τ = τ ≡ τ − 0 to avoid the singular behaviour of the retarded Green’s function at coincidence. To get Equation (33View Equation) I have reduced the order of the differential equation by replacing ˙aν with m −1f˙eνxt on the right-hand side; this procedure was explained at the end of Section 1.2.

Equation (33View Equation) is the result that was first derived by DeWitt and Brehme [24Jump To The Next Citation Point] and later corrected by Hobbs [29Jump To The Next Citation Point]. (The original equation did not include the Ricci-tensor term.) In flat spacetime the Ricci tensor is zero, the tail integral disappears (because the Green’s function vanishes everywhere within the domain of integration), and Equation (33View Equation) reduces to Dirac’s result of Equation (5View Equation). In curved spacetime the self-force does not vanish even when the electric charge is moving freely, in the absence of an external force: It is then given by the tail integral, which represents radiation emitted earlier and coming back to the particle after interacting with the spacetime curvature. This delayed action implies that, in general, the self-force is nonlocal in time: It depends not only on the current state of motion of the particle, but also on its past history. Lest this behaviour should seem mysterious, it may help to keep in mind that the physical process that leads to Equation (33View Equation) is simply an interaction between the charge and a free electromagnetic field R Fαβ; it is this field that carries the information about the charge’s past.


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