A point particle carries an electric charge and moves on a world line described by relations , in which is an arbitrary parameter. The particle generates a vector potential and an electromagnetic field . The dynamics of the entire system is governed by the action
The field action is given by
Demanding that the total action be stationary under a variation of the vector potential yields Maxwell’s equations
The electromagnetic field is invariant under a gauge transformation of the form , in which is an arbitrary scalar function. This function can always be chosen so that the vector potential satisfies the Lorenz gauge condition,
The retarded solution to Eq. (18.9) is , where is the retarded Green’s function introduced in Section 15. After substitution of Eq. (18.10) we obtain
We now specialize Eq. (18.11) to a point close to the world line. We let be the normal convex neighbourhood of this point, and we assume that the world line traverses ; refer back to Figure 9. As in Section 17.2 we let and be the values of the proper-time parameter at which enters and leaves , respectively. Then Eq. (18.11) can be expressed as
The third integration vanishes because is then in the past of , and . For the second integration, is the normal convex neighbourhood of , and the retarded Green’s function can be expressed in the Hadamard form produced in Section 15.2. This gives
and to evaluate this we let be the retarded point associated with ; these points are related by and is the retarded distance between and the world line. To perform the first integration we change variables from to , noticing that increases as passes through ; the integral evaluates to . The second integration is cut off at by the step function, and we obtain our final expression for the vector potential of a point electric charge:
When we differentiate the vector potential of Eq. (18.12) we must keep in mind that a variation in induces a variation in , because the new points and must also be linked by a null geodesic. Taking this into account, we find that the gradient of the vector potential is given by
We shall now expand in powers of , and express the result in terms of the retarded coordinates introduced in Section 10. It will be convenient to decompose the electromagnetic field in the tetrad that is obtained by parallel transport of on the null geodesic that links to ; this construction is detailed in Section 10. We recall from Eq. (10.4) that the parallel propagator can be expressed as . The expansion relies on Eq. (10.29) for , Eq. (10.31) for , and we shall need
Collecting all these results gives
We now wish to express the electromagnetic field in the Fermi normal coordinates of Section 9; as before those will be denoted . The translation will be carried out as in Section 17.4, and we will decompose the field in the tetrad that is obtained by parallel transport of on the spacelike geodesic that links to the simultaneous point .
Our first task is to decompose in the tetrad , thereby defining and . For this purpose we use Eqs. (11.7), (11.8), (18.19), and (18.20) to obtain
where all frame components are still evaluated at , except for
which are evaluated at .
We must still translate these results into the Fermi normal coordinates . For this we involve Eqs. (11.4), (11.5), and (11.6), and we recycle some computations that were first carried out in Section 17.4. After some algebra, we arrive at
Our next task is to compute the averages of and over , a two-surface of constant and . These are defined by
The singular vector potential
To evaluate the integral of Eq. (18.30) we assume once more that is sufficiently close to that the world line traverses ; refer back to Figure 9. As before we let and be the values of the proper-time parameter at which enters and leaves , respectively. Then Eq. (18.30) becomes
The first integration vanishes because is then in the chronological future of , and by Eq. (15.27). Similarly, the third integration vanishes because is then in the chronological past of . For the second integration, is the normal convex neighbourhood of , the singular Green’s function can be expressed in the Hadamard form of Eq. (15.33), and we have
Differentiation of Eq. (18.31) yields
To derive an expansion for we follow the general method of Section 11.4 and introduce the functions . We have that
where overdots indicate differentiation with respect to , and . The leading term was worked out in Eq. (18.15), and the derivatives of are given by
according to Eqs. (18.17) and (15.12). Combining these results together with Eq. (11.12) for gives
We proceed similarly to derive an expansion for . Here we introduce the functions and express as . The leading term was computed in Eq. (18.16), and
follows from Eq. (15.11). Combining these results together with Eq. (11.12) for gives
It is now a straightforward (but still tedious) matter to substitute these expansions into Eq. (18.32) to obtain the projections of the singular electromagnetic field in the same tetrad that was employed in Section 18.3. This gives
The difference between the retarded field of Eqs. (18.19), (18.20) and the singular field of Eqs. (18.37), (18.38) defines the regular field . Its tetrad components are
The retarded field 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 18.3 and 18.4, and in Section 18.5 it was shown to originate from the singular field ; the regular field was then shown to be regular on the world line.
To make sense of the retarded field’s action on the particle we follow the discussion of Section 17.6 and temporarily picture the electric charge as a spherical hollow shell; the shell’s radius is in Fermi normal coordinates, and it is independent of the angles contained in the unit vector . The net force acting at proper time on this shell is proportional to the average of over the shell’s surface. Assuming that the field on the shell is equal to the field of a point particle evaluated at , and ignoring terms that disappear in the limit , we obtain from Eq. (18.28)
Substituting Eqs. (18.43) and (18.45) into Eq. (18.7) gives rise to the equations of motion
We must confess, as we did in the case of the scalar self-force, that the derivation of the equations of motion outlined above returns the wrong expression for the self-energy of a spherical shell of electric charge. We obtained , while the correct expression is ; we are wrong by a factor of . As before 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 . 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 . We also believe, however, that except for the diverging terms that determine , the difference between the shell’s field and the particle’s field should vanish in the limit . Our conclusion is therefore that while our expression for is admittedly incorrect, the statement of the equations of motion is reliable.
Apart from the term proportional to , the averaged force of Eq. (18.43) has exactly the same form as the force that arises from the regular field of Eq. (18.41), which we express as
For the final expression of the equations of motion we follow the discussion of Section 17.6 and allow an external force to act on the particle, and we replace, on the right-hand side of the equations, the acceleration vector by . This produces, and then corrected by J.M. Hobbs in 1968 . An alternative derivation was produced by Theodore C. Quinn and Robert M. Wald in 1997 . In a subsequent publication , Quinn and Wald proved that the total work done by the electromagnetic self-force matches the energy radiated away by the particle.
Living Rev. Relativity 14, (2011), 7
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