## 18 Motion of an electric charge

### 18.1 Dynamics of a point electric charge

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

where is an action functional for a free electromagnetic field in a spacetime with metric , is the action of a free particle moving on a world line in this spacetime, and is an interaction term that couples the field to the particle.

The field action is given by

where the integration is over all of spacetime. The particle action is
where is the bare mass of the particle and is the differential of proper time along the world line; we use an overdot to indicate differentiation with respect to the parameter . Finally, the interaction term is given by
Notice that both and are invariant under a reparameterization of the world line.

Demanding that the total action be stationary under a variation of the vector potential yields Maxwell’s equations

with a current density defined by
These equations determine the electromagnetic field once the motion of the electric charge is specified. On the other hand, demanding that the total action be stationary under a variation of the world line yields the equations of motion
for the electric charge. We have adopted as the parameter on the world line, and introduced the four-velocity .

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,

Under this condition the Maxwell equations of Eq. (18.5) reduce to a wave equation for the vector potential,
where is the wave operator and is the Ricci tensor. Having adopted as the parameter on the world line, we can re-express the current density of Eq. (18.6) as
and we shall use Eqs. (18.9) and (18.10) to determine the electromagnetic field of a point electric charge. The motion of the particle is in principle determined by Eq. (18.7), but because the vector potential obtained from Eq. (18.9) is singular on the world line, these equations have only formal validity. Before we can make sense of them we will have to analyze the field’s singularity structure near the world line. The calculations to be carried out parallel closely those presented in Section 17 for the case of a scalar charge; the details will therefore be kept to a minimum and the reader is referred to Section 17 for additional information.

### 18.2 Retarded potential near the world line

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

in which gives the description of the world line and . Because the retarded Green’s function is defined globally in the entire spacetime, Eq. (18.11) applies to any field point .

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:

This expression applies to a point sufficiently close to the world line that there exists a nonempty intersection between and .

### 18.3 Electromagnetic field in retarded coordinates

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

where the “tail integral” is defined by
The second form of the definition, in which we integrate the gradient of the retarded Green’s function from to to avoid the singular behaviour of the retarded Green’s function at , is equivalent to the first form.

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

which follows from Eq. (15.10) and the relation first encountered in Eq. (10.7). We shall also need the expansions
and
that follow from Eqs. (15.10) – (15.12). And finally, we shall need
a relation that was first established in Eq. (15.14).

Collecting all these results gives

where
are the frame components of the tail integral; this is obtained from Eq. (18.14) evaluated at :
It should be emphasized that in Eqs. (18.19) and (18.20), all frame components are evaluated at the retarded point associated with ; for example, . It is clear from these equations that the electromagnetic field is singular on the world line.

### 18.4 Electromagnetic field in Fermi normal coordinates

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

and

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

where all frame components are now evaluated at ; for example, .

Our next task is to compute the averages of and over , a two-surface of constant and . These are defined by

where is the element of surface area on , and . Using the methods developed in Section 17.4, we find
The averaged field is singular on the world line, but we nevertheless take the formal limit of the expressions displayed in Eqs. (18.26) and (18.27). In the limit the tetrad becomes , and we can easily reconstruct the field at from its frame components. We thus obtain
where the tail term can be copied from Eq. (18.22),
The tensors appearing in Eq. (18.28) all refer to , which now stands for an arbitrary point on the world line .

### 18.5 Singular and regular fields

The singular vector potential

is the (unphysical) solution to Eqs. (18.9) and (18.10) that is obtained by adopting the singular Green’s function of Eq. (15.24) instead of the retarded Green’s function. We will see that the singular field reproduces the singular behaviour of the retarded solution, and that the difference, , is smooth on the world line.

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

To evaluate these we let and be the retarded and advanced points associated with , respectively. To perform the first integration we change variables from to , noticing that increases as passes through ; the integral evaluates to . We do the same for the second integration, but we notice now that decreases as passes through ; the integral evaluates to , where is the advanced distance between and the world line. The third integration is restricted to the interval by the step function, and we obtain the expression
for the singular vector potential.

Differentiation of Eq. (18.31) yields

and we would like to express this as an expansion in powers of . For this we will rely on results already established in Section 18.3, as well as additional expansions that will involve the advanced point . We recall that a relation between retarded and advanced times was worked out in Eq. (11.12), that an expression for the advanced distance was displayed in Eq. (11.13), and that Eqs. (11.14) and (11.15) give expansions for and , respectively.

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

and

according to Eqs. (18.17) and (15.12). Combining these results together with Eq. (11.12) for gives

which should be compared with Eq. (18.15). It should be emphasized that in Eq. (18.33) and all equations below, all frame components are evaluated at the retarded point , and not at the advanced point. The preceding computation gives us also an expansion for

which becomes

and which should be compared with Eq. (18.17).

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

and this should be compared with Eq. (18.16). The last expansion we shall need is
which follows at once from Eq. (18.18).

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

in which all frame components are evaluated at the retarded point . Comparison of these expressions with Eqs. (18.19) and (18.20) reveals that the retarded and singular fields share the same singularity structure.

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

and we see that is a regular tensor field on the world line. There is therefore no obstacle in evaluating the regular field directly at , where the tetrad becomes . Reconstructing the field at from its frame components, we obtain
where the tail term can be copied from Eq. (18.22),
The tensors appearing in Eq. (18.41) all refer to the retarded point , which now stands for an arbitrary point on the world line .

### 18.6 Equations of motion

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)

where
is formally a divergent quantity and
is the tail part of the force; all tensors in Eq. (18.43) are evaluated at an arbitrary point on the world line.

Substituting Eqs. (18.43) and (18.45) into Eq. (18.7) gives rise to the equations of motion

for the electric charge, with denoting the (also formally divergent) bare mass of the particle. We see that and combine in Eq. (18.46) to form the particle’s observed mass , which is finite and gives a true measure of the particle’s inertia. All diverging quantities have thus disappeared into the procedure of mass renormalization.

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

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, Eq. (18.46) is equivalent to the statement
where we have dropped the superfluous label “obs” on the particle’s observed mass.

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

in which denotes the observed inertial mass of the electric charge and all tensors are evaluated at , the current position of the particle on the world line; the primed indices in the tail integral refer to the point , which represents a prior position. We recall that the integration must be cut short at to avoid the singular behaviour of the retarded Green’s function at coincidence; this procedure was justified at the beginning of Section 18.3. Equation (18.49) was first derived (without the Ricci-tensor term) by Bryce S. DeWitt and Robert W. Brehme in 1960 [54], and then corrected by J.M. Hobbs in 1968 [95]. An alternative derivation was produced by Theodore C. Quinn and Robert M. Wald in 1997 [150]. In a subsequent publication [151], Quinn and Wald proved that the total work done by the electromagnetic self-force matches the energy radiated away by the particle.