Radiation reaction in flat spacetime

1.2 Radiation reaction in flat spacetime

Let us first consider the relatively simple and well-understood case of a point electric charge $e$ moving in flat spacetime [52

, 30

, 56

, 47

]. The charge produces an electromagnetic vector potential $Aα$ that satisfies the wave equation

together with the Lorenz gauge condition $∂ Aα = 0 α$ . (On page 294 in [30

] Jackson explains why the term “Lorenz gauge” is preferable to “Lorentz gauge”.) The vector $α j$ is the charge’s current density, which is formally written in terms of a four-dimensional Dirac functional supported on the charge’s world line: The density is zero everywhere, except at the particle’s position where it is infinite. For concreteness we will imagine that the particle moves around a centre (perhaps another charge, which is taken to be fixed) and that it emits outgoing radiation. We expect that the charge will undergo a radiation reaction and that it will spiral down toward the centre. This effect must be accounted for by the equations of motion, and these must therefore include the action of the charge’s own field, which is the only available agent that could be responsible for the radiation reaction. We seek to determine this self-force acting on the particle.

An immediate difficulty presents itself: The vector potential, and also the electromagnetic field tensor, diverge on the particle’s world line, because the field of a point charge is necessarily infinite at the charge’s position. This behaviour makes it most difficult to decide how the field is supposed to act on the particle.

Difficult but not impossible. To find a way around this problem I note first that the situation considered here, in which the radiation is propagating outward and the charge is spiraling inward, breaks the time-reversal invariance of Maxwell’s theory. A specific time direction was adopted when, among all possible solutions to the wave equation, we chose $A αret$ , the retarded solution, as the physically-relevant solution. Choosing instead the advanced solution $Aαadv$ would produce a time-reversed picture in which the radiation is propagating inward and the charge is spiraling outward. Alternatively, choosing the linear superposition

would restore time-reversal invariance: Outgoing and incoming radiation would be present in equal amounts, there would be no net loss nor gain of energy by the system, and the charge would not undergo any radiation reaction. In Equation (2

) the subscript ‘S’ stands for ‘symmetric’, as the vector potential depends symmetrically upon future and past.

My second key observation is that while the potential of Equation (2 ) does not exert a force on the charged particle, it is just as singular as the retarded potential in the vicinity of the world line. This follows from the fact that $Aα ret$ , $A α adv$ , and $A α S$ all satisfy Equation (1 ), whose source term is infinite on the world line. So while the wave-zone behaviours of these solutions are very different (with the retarded solution describing outgoing waves, the advanced solution describing incoming waves, and the symmetric solution describing standing waves), the three vector potentials share the same singular behaviour near the world line – all three electromagnetic fields are dominated by the particle’s Coulomb field and the different asymptotic conditions make no difference close to the particle. This observation gives us an alternative interpretation for the subscript ‘S’: It stands for ‘singular’ as well as ‘symmetric’.

Because $A αS$ is just as singular as $Aαret$ , removing it from the retarded solution gives rise to a potential that is well behaved in a neighbourhood of the world line. And because $A αS$ is known not to affect the motion of the charged particle, this new potential must be entirely responsible for the radiation reaction. We therefore introduce the new potential

and postulate that it, and it alone, exerts a force on the particle. The subscript ‘R’ stands for ‘regular’, because $α A R$ is nonsingular on the world line. This property can be directly inferred from the fact that the regular potential satisfies the homogeneous version of Equation (1

), $□A α = 0 R$ ; there is no singular source to produce a singular behaviour on the world line. Since $A α R$ satisfies the homogeneous wave equation, it can be thought of as a free radiation field, and the subscript ‘R’ could also stand for ‘radiative’.

The self-action of the charge’s own field is now clarified: A singular potential $A αS$ can be removed from the retarded potential and shown not to affect the motion of the particle. (Establishing this last statement requires a careful analysis that is presented in the bulk of the paper; what really happens is that the singular field contributes to the particle’s inertia and renormalizes its mass.) What remains is a well-behaved potential $A αR$ that must be solely responsible for the radiation reaction. From the radiative potential we form an electromagnetic field tensor $FαRβ = ∂αARβ − ∂βARα$ , and we take the particle’s equations of motion to be

where $μ μ u = dz ∕d τ$ is the charge’s four-velocity ( $μ z (τ)$ gives the description of the world line and $τ$ is proper time), $μ μ a = du ∕dτ$ its acceleration, $m$ its (renormalized) mass, and $μ f ext$ an external force also acting on the particle. Calculation of the radiative field yields the more concrete expression

in which the second-term is the self-force that is responsible for the radiation reaction. We observe that the self-force is proportional to $e2$ , it is orthogonal to the four-velocity, and it depends on the rate of change of the external force. This is the result that was first derived by Dirac [25

]¹ .