## Abstract

We present a partial differential equation describing the electromagnetic potentials around a charge distribution undergoing rigid motion at constant proper acceleration, and obtain a set of solutions to this equation. These solutions are used to find the self-force exactly in a chosen case. The electromagnetic self-force for a spherical shell of charge of proper radius *R* undergoing rigid motion at constant proper acceleration *a*_{0} is, to high-order approximation, , and this is conjectured to be exact.

## 1. Introduction

The problem of self-force, and the related problem of radiation reaction, has a long history; see Rohrlich [1] for a brief review and Spohn [2] for a longer survey. It is currently studied both because it is important in the extreme physical conditions now realized in experiments [3] and because it continues to raise questions about the consistency and correct treatment of electromagnetism [4,5].

Lorentz [6,7], Abraham [8], Poincaré [9,10] and Schott [11] carried out pioneering work on the understanding of self-force and performed approximate calculations for a rigid spherical shell whose centre undergoes arbitrary motion. They obtained the lowest order terms in a power series solution, expanded in powers of the radius *R* of the sphere, its acceleration and derivatives thereof, and combinations of all these. von Laue [12] clarified and simplified the work of earlier authors by expressing it in a manifestly Lorentz-covariant form; see equation (6.4). Dirac [13] adopted a different strategy, starting from a manifestly covariant Lagrangian and reproduced this equation; it is now commonly known as the Lorentz–Dirac or Abraham–Lorentz–Dirac (ALD) equation. To be precise, Dirac's version has a modified inertial term for reasons that we shall discuss below. The ALD equation includes terms of order *R*^{−1} and *R*^{0}, and neglects terms of order *R* and above.

Nodvik [14] brought arguably greater rigour to the derivation using the Lagrangian formalism and presented the power series expansion to *O*(*R*^{2}). Further work has clarified the interpretation of the force and the associated energy-momentum movements in the field and has extended the calculation of the self-force, such that the ‘linear’ contribution can be obtained exactly, in a beautifully neat and well-behaved expression (equation (6.5)). The word ‘linear’ here does not mean, however, that all the contribution at order *R*^{1} is accounted for by this expression, because it neglects terms involving *R* multiplied by nonlinear combinations of the velocity or its derivatives (see [15–18] for details).

Throughout this century-long period, the study of self-force has continued to raise doubts and questions concerning the correct interpretation or application of results. Many of these issues were helpfully clarified by Yaghjian [16] and Rohrlich [19,20]. For example, Yaghjian showed that there is no ‘preacceleration’ in the classical electromagnetic theory of small charged bodies; its supposed existence (discussed by authors too numerous to list) was based on a misunderstanding of the approximations involved in the ALD equation. Numerous authors and textbooks continue to assert that the ALD equation has pathological runaway solutions but this too is an artefact of its incorrect application, as we discuss in §6, and as Gralla *et al.* [4] and Rohrlich [21], among others, have commented. A further, and different, source of confusion is the effect of Poincaré stresses in a body undergoing acceleration.

To have a concrete result with no approximation, for a non-trivial case, remains very helpful both for pedagogy and as a test case for more general but approximate formulae. Most previous work has been devoted to the case of arbitrary motion, endeavouring to extend the set of available approximate results and understand them better. This work takes the complementary path of restricting the motion to a simple case, and solving it exactly. We find the electromagnetic self-force for a spherical charged shell undergoing constant proper acceleration while maintaining constant proper size and shape. To our knowledge, this has not previously been done. The method of calculation involves a useful general observation about the electromagnetic potentials in the case of arbitrary bodies undergoing rigid hyperbolic motion.

## 2. Strategy

The classical electromagnetic contribution to the self-force for an arbitrary distribution of charge, undergoing arbitrary motion, is
2.1where *ρ* is the volume density of charge, the integral is taken over all space (though of course it suffices to restrict it to the region where *ρ*≠0) at some instant of time *t* in a chosen inertial reference frame, and the fields **E**(*t*,*x*,*y*,*z*), **B**(*t*,*x*,*y*,*z*) are those produced at each point by the set of charges in question (owing to their previous behaviour at some set of source events). Note that if *ρ* is everywhere finite, then the fields are also finite and continuous, and furthermore the contribution to the field at a given point (*x*,*y*,*z*) made by the charge within a distance *d* of that point vanishes in the limit (because the quantity of charge scales as *ρd*^{3} but the fields of a point charge scale as 1/*d*^{2}). Hence, no ambiguity arises about whether the integral above correctly excludes the non-physical idea of a source interacting with itself: only pairwise interactions should be included, and only pairwise interactions contribute to the integral.

In the case of a distribution of charge that is rigid, there must be further forces acting (for example, the Poincaré stresses) and these further forces may be deemed ‘internal’ and yet have a non-vanishing sum over the body. Hence, the total self-force includes both the electromagnetic part that is the main subject of this paper and a further part that we shall discuss in §6. The non-vanishing contribution from the ‘internal’ forces does not break Newton's third law but rather is required by it, because there is a continuous transfer of momentum between the shell and the surrounding electromagnetic field, mediated via the electric charge.

In the case of a spherical shell of charge, we treat a shell of finite thickness *w*, so that the volume density of charge is finite, and then explore the limit , such that the surface charge density *σ*=*ρw* remains finite. We shall consider a shell which moves such that there exists a frame in which the whole shell is at rest at some moment, so the **v**∧**B** term vanishes from the force. Let *R* be the inner radius of the shell in such a frame (the shell being spherical in that frame), and assume the charge density *ρ* is uniform between the inner and outer surfaces at *r*=*R* and *r*=*R*+*w*. For a sufficiently thin shell, the field within the material of the shell (where *ρ*≠0) can be approximated by the linear form
2.2where **E**(*r*,*θ*,*ϕ*) is the field at radius *r* from the centre of the sphere, *s*=*r*−*R* is the distance through the shell, and we suppressed explicit indication of the dependence of the fields on polar angles *θ*,*ϕ*. Hence,
2.3where d*S* is an element of surface. Carrying out the integration over *s* gives, in the limit ,
2.4where is the field on the interior surface of the shell and is the field on the exterior surface of the shell. In other words, to calculate the self-force, it suffices to use the average of these two fields. This deals with the discontinuity in **E** in the limit . Now, we know from Gauss’ law what the size and direction of this discontinuity is: its size is *σ*/*ϵ*_{0} and its direction is normal to the surface. For a spherically symmetric case, *σ* is the same everywhere over the shell and we have
2.5As the contribution from is radial and spherically symmetric, its integral gives zero contribution to the self-force. Therefore, we can also write
2.6In other words, one can use either the field on the interior surface or the field on the exterior surface to calculate the self-force. For convenience, we will use the field on the interior surface.

## 3. Simple calculation of the lowest order approximation

Consider a particle undergoing hyperbolic motion in the *x*-direction, so that relative to some frame its position as a function of time is *y*_{s}=*z*_{s}=0 and
3.1where *L*=1/*a*_{0} is a natural distance scale set by the proper acceleration *a*_{0}. The potentials and fields of a point charge *q* undergoing such motion are easily calculated (e.g. [22–24]). Adopting rectangular coordinates and *c*=1, one finds the scalar and vector potentials (Liénard–Wiechart potentials)
3.2where *Q*=*q*/4*πϵ*_{0} and
3.3In the following, we will only need the solution at *t*=0, which is the moment when the particle is at rest in the frame under consideration. At this moment, one finds the magnetic field is **B**=0 and the electric field is (adopting now cylindrical coordinates)
3.4

Now, consider a spherical shell of charge undergoing rigid hyperbolic motion. By *rigid* motion, we mean motion such that, at any moment, there is an instantaneous rest frame in which all parts of the body are at rest, and the physical dimensions of the body in each such frame are constant (i.e. the same in all successive instantaneous rest frames); e.g. [24]. Such rigid motion has the property that, once the initial conditions and the worldline of one particle in the body are given, the worldlines of all other particles are fixed. For the case of constant proper acceleration (i.e. hyperbolic motion) of one part of a body, the resulting constraint takes a particularly simple form: every part of the body undergoes constant proper acceleration, but with a proper acceleration varying with distance from a plane normal to the acceleration. Such a body is at rest relative to the well-known ‘constantly accelerating reference frame’; the special plane is a horizon which can conveniently be placed at *x*=0 by a suitable choice of origin. The worldline of each part of the shell is given again by (3.1), but now we must allow *L* to be chosen for each part such that that part comes to rest at the correct position at *t*=0. This is easily done: one sets *L* equal to the location of the part at *t*=0. For example, for a spherical shell of proper radius *R* centred at (*L*_{0},0,0), one must use , where *θ* is the azimuthal angle relative to the centre of the shell and the *x*-axis.

The observation of the previous paragraph makes it possible to write down an integral expression for the field at the centre of the shell. A moment's thought about the geometry leads one to conclude that the integrand is obtained from equation (3.4) with the replacements , , , leading to
3.5and *E*_{y}=*E*_{z}=0. The integration is straightforward. One obtains
3.6

To find the self-force of such a shell, to the lowest order approximation, is now extremely easy. One simply multiplies this expression by *q*
3.7where to reduce clutter we introduced *e*^{2}≡*q*^{2}/(4*πϵ*_{0}). To see that this expression is accurate to the lowest order, as claimed, argue as follows. The self-force is in fact given by either of the integrals in (2.6), of which we choose the first. The field at the interior surface, **E**_{−}, may be obtained from a Taylor expansion of the field throughout the empty space in the interior of the shell, expanded about the point (*L*_{0},0,0). One finds that **E**_{−} is given by equation (3.6) plus further terms forming a power series in (*x*_{s}−*L*_{0})/*L*_{0}. Using the symmetry of the charge distribution, the linear terms cancel in the calculation of the self-force, and the next contribution is at order (*R*/*L*_{0})^{2} compared with the first. In summary, the field is, to good approximation, uniform throughout the interior (figure 1), and, to find the self-force, one simply multiplies this uniform field by the total charge of the shell. Thus, we obtain equation (3.7). This is the simplest derivation of it of which I am aware.

## 4. The potentials

We now turn to the task of calculating the electric field, and hence the self-force, exactly. We will obtain the field from the potentials via **E**=−∇*ϕ*−∂**A**/∂*t*. In empty space ∇⋅**E**=0, hence
4.1where we used (3.2) in the final step. This result is valid for the potentials of a single constantly accelerating point charge and, because for a distribution of charge the potentials simply add and the equation is linear, it remains valid for an arbitrary distribution of point charges, each undergoing constant proper acceleration in the *x*-direction (not necessarily all having the same acceleration). Evaluating ∂*A*_{x}/∂*t* for a single point charge using (3.2), one finds, at *t*=0,
4.2It follows that the electric field can be obtained from
4.3and equation (4.1) can be written as
4.4This is a linear homogeneous second-order differential equation for the potential in free space. It may be regarded as a replacement for the Laplace equation; it applies in the free space around a point charge undergoing hyperbolic motion in the *x*-direction.

Using the same argument concerning rigid motion as we presented after equation (3.4), we will now show that equation (4.4) has a wider range of validity. Consider a distribution of charge undergoing rigid hyperbolic motion in the *x*-direction. Then in the instantaneous rest frame, the total scalar potential at any field point (*x*,*y*,*z*) is given by
4.5where *ρ* is the charge density and is the potential per unit charge obtained from equation (3.2)
4.6Note that we are evaluating the potential quite correctly using the distribution of charge at its ‘present’ location (*t*=0), not the location it had at the source events, because we can take advantage of the fact that we already know how to calculate what contribution each source event makes to the potential ‘now’ (at *t*=0) for the simple type of motion under consideration.

The important property of equation (4.5) for our purposes is that every source charge presents a contribution that is calculated by substituting the *same* value for *x* into the formula for , and therefore every contribution satisfies (4.4) with a common value of *x*, and therefore (4.4) applies to the net resulting *ϕ* (the one given by (4.5)). Hence, equation (4.4) describes the potential in the free space around, or in cavities within, any distribution of charge that is undergoing rigid hyperbolic motion. (In obtaining it, we have made a gauge choice that makes at positions far from any finite charge contained in a finite region.)

The problem of finding the potential inside an accelerating spherical shell can now be addressed by seeking a potential function *ϕ* that satisfies (4.4) and that agrees with a suitable boundary condition.

### (a) Solution of the differential equation

In order to solve (4.4), introduce another function *V* (*x*,*y*,*z*)≡*ϕ*/*x*. Then one finds that *V* satisfies
4.7If *V* =*V* (*x*) (i.e. a solution with no dependence on *y* and *z*), this leads to a Cauchy-type equation with general solution *V* (*x*)=*a*_{0}+*a*_{1}/*x*^{2}. More generally, the equation is not readily separable, but we can look for solutions of a form suitable to the type of problem we are treating. We would like solutions which are non-singular at *ρ*=0 and which have axial symmetry, so we try the ansatz
4.8where *ρ*=(*y*^{2}+*z*^{2})^{1/2} is the radial coordinate in a cylindrical coordinate system. Substituting this into (4.7) and using ∇^{2}*ρ*^{n}=*n*^{2}*ρ*^{n−2}, one finds
4.9As this is to be satisfied at all *ρ*, we can set the coefficient of *ρ*^{2(N−k−1)} equal to zero
4.10Integrating, we have
4.11which applies for values of *k* between 0 and *N*−1. The coefficient of *ρ*^{2N} in (4.9) gives
4.12This has the general solution *f*_{0}=*a*_{0}+*a*_{1}/*x*^{2}, which can be substituted into the right-hand side of (4.11) in order to find *f*_{1}, and hence *f*_{2} and so on. Table 1 shows the first few solutions found this way.

In the rest of this paper, we will restrict attention to solutions having the form
4.13where the are constants and *k*≥0. This restriction is merely for convenience: one can expand *V* (*x*,*ρ*) for any given charge distribution in terms of any sufficiently complete set of basis solutions, and this is one such set for our problem. This restriction does not give the expected divergence in *V* at *x*=0 with finite *k*, but in the limit it can match *V* at points arbitrarily close to *x*=0. This is sufficient for our purpose because, in the case of a body accelerating with fixed proper dimensions, the whole of the body must be at *x*>0 (otherwise part of it extends over the horizon at *x*=0 and the body must break), so, when calculating the field at the surface or inside the body, we would not require *V* (0,*ρ*).

Substituting (4.13) into (4.11), we obtain
4.14Hence,
4.15and are undetermined constants to be found from boundary conditions. For example, with *N*=2 one finds is undetermined, then and is undetermined. These are in turn used to find and . Finally, is undetermined. For any given value of *N*, we thus find a solution involving a set of *N*+1 undetermined constants, each of which multiplies a polynomial function of *x* and *ρ*. The polynomial function which appears with the factor is the one involving . To be precise, it is . By using the above recurrence relation, the terms in the sum are easily found, and one obtains
4.16where is a binomial coefficient and _{2}*F*_{1} is the hypergeometric function. For , we thus obtain an infinite set of linearly independent solutions of equation (4.7). The first few of these functions are shown in the left-hand column of table 1, in each case multiplied by (*n*+1) to make the coefficients all integer.

The electric field given by these solutions is (using equation (4.3) and *ϕ*=*xV* _{n})
4.17
4.18and
4.19
4.20

## 5. Interior field and self-force of the accelerating spherical shell

We now apply the ideas in §4 to the case of the spherical shell of charge undergoing rigid hyperbolic motion. Without loss of generality, choose distance units such that the centre of the shell comes to rest at *x*=1 (this amounts to setting *L*_{0}=1 in equation (3.7)). Then the surface of the shell is at *ρ*^{2}=*R*^{2}−(*x*−1)^{2}. By substituting this into (4.18), one finds the contribution to the electric field at the (interior surface of the) shell for each of the potential solutions identified in §4. For a uniformly charged spherical shell with unit total charge, the contribution to the self-force is then given by
5.1
5.2For example, the first few values of this expression are given by
5.3

To find the self-force of the shell, it remains to identify which linear combination of these solutions matches the potential, and hence the field, in the interior of a uniformly charged shell undergoing rigid hyperbolic motion. To find this, we first evaluate the potential on the axis of the shell, by performing the integral in (4.5) for *y*=*z*=0
5.4
5.5where and *β*_{±}=(1±*R*)^{2}−*x*^{2}. This function is plotted in figure 2 (note, the function is real because the argument of the log has unit modulus when *s* is real). For 1−*R*<*x*<1+*R* (that is, for points inside the sphere), it takes the form
5.6We will now find the potential throughout the interior of the spherical shell by finding a function that satisfies (4.7) and that matches the known potential and all its derivatives with respect to *x* at the point (*x*,*y*,*z*)=(1,0,0) (the centre of the sphere). This may be compared with matching a boundary condition along an infinite line, because, for an analytic function, the value of the function and all its derivatives at a point suffice to define the function along a line by Taylor expansion. Of course the potential given by the shell actually has a discontinuous change in slope at *x*=1±*R*, but this is irrelevant to the method. By using *ϕ* and all its derivatives at one point, we gain sufficient information to fix the potential throughout the interior of the shell.

We employ the solution
5.7where *V* _{n}(*x*,*ρ*) is given by (4.16), the constants *a*_{n} are to be determined, and the solution becomes exact in the limit . We find the constants by writing down the Taylor expansion of both *V* (*x*,0) and *ϕ*(*x*,0,0) about the point *x*=1, and equating coefficients of powers of (*x*−1). This gives a set of *N*+1 simultaneous equations for the *a*_{n} which can be solved by matrix inversion. For example, for *N*=2 one finds
5.8The self-force is then given by
5.9where the expression becomes exact in the limit and for given finite *N* one obtains an exact expression for the terms up to order *R*^{2N−3} (see appendix A). The above values for *a*_{0},*a*_{1},*a*_{2} when *N*=2 yield, for example,
5.10in which the first two terms are exact. Further details of the Taylor expansions and the solution coefficients are given in appendix A. The end result is that the self-force on the uniformly charged spherical shell undergoing rigid hyperbolic motion is
5.11where I have calculated this result for terms up to order *R*^{101}, and I conjecture its validity at all orders. This is the main result of this paper. At *L*_{0}=1 and *e*^{2}=1, the first few terms in this series are
5.12It is remarkable that the result takes such a simple form. One expects a power series in odd powers of *R*, but it is striking that the coefficients have such a simple expression. The coefficients in the Taylor series for *ϕ*(*x*,0,0) are, by contrast, much more complicated.

The lowest order terms in expression (5.12) for *f*_{self} can be found by hand. The series expansion and matrix inversion method described above is essentially simple if laborious; it can easily be executed in a computer algebra system such as Mathematica. This is how the series up to order *R*^{101} was calculated. The problem of finding the expression for an arbitrary coefficient in the series, and thus confirming the (eminently reasonable) conjecture that the result is valid at all orders, is more difficult and remains open. As a further check, the field inside the shell was calculated by direct numerical integration of the fields of a point charge given in equation (3.4), and compared with the one obtained from equations (4.18) and (4.20), multiplied by the coefficients *a*_{n} and summed. This confirms, to within numerical precision, that one obtains the field correctly at points away from the *x*-axis (as well as on it).

Numerical integration was also used to check the overall result (5.11). The field at the interior surface of the shell was obtained by calculating the field (numerically) at four points near the surface, and then extrapolating to the surface using a cubic fit; this enables the singular behaviour to be avoided. The numerically calculated self-force is compared with prediction (5.11) in figure 3. By fitting a polynomial in *R*^{2} to the numerical calculation of *Rf*_{self}, the first few coefficients in (5.11) were confirmed to good approximation (one part in 10^{9,6,3} for the term of order *R*^{−1,1,3}, respectively). However, the numerical calculation rapidly runs out of precision in obtaining the higher order terms by this ‘brute force’ method, whereas the analytical result is easy to obtain for many more orders.

## 6. Discussion

Equation (5.11) is the result of a fully relativistic calculation and is believed to be exact. It gives the spatial part of the self-four-force in the instantaneous rest frame. The temporal part is zero in that frame. As the resulting four-vector is parallel to the four-acceleration, it may be written as
6.1where *v* is the four-velocity (in an index-free notation) and the dot signifies the derivative with respect to proper time. The equation of motion of the spherical shell of charge is
6.2where F_{other} is the four-force applied to the shell by some other influence, which must, in this case, take such a form that, when combined with F_{self}, it gives to the shell the motion that has been assumed. *m*_{0} is the ‘bare’ rest mass of the shell—that is, the rest mass which enters on the right-hand side of the equation of motion *of the material of the shell itself*. This does not include a contribution from the energy-momentum density of the electromagnetic field sourced by the charge of this shell, which is quite correct. To find out about the dynamics of the field sourced by the shell, one can appeal to the conservation of energy momentum, which is perfectly obeyed in classical electromagnetism. The force *from* the field *on* the charge (i.e. the electromagnetic self-force *f*_{self}) implies the presence of an equal and opposite ‘force on’ (i.e. rate of injection of momentum into) the electromagnetic field. Hence, the field sourced by the spherical shell must be acquiring energy momentum at the rate −F_{self}.

Now, we assumed that the motion of the shell was rigid. This is allowed in relativity but requires that the forces conspire to produce such motion. Therefore, the four-force F_{other} both must oppose the tendency of the electromagnetic force to explode or otherwise deform the proper shape of the shell and must provide the correct amount of force to result in the assumed net acceleration of the shell. It makes sense, therefore, to write
6.3where F_{ext} is provided by an external force-producing entity such as a rod pushing on the sphere, and F_{P} is deemed ‘internal’, that is, it may be regarded as the integral of the Poincaré stresses.

To clarify the separation into F_{ext}, F_{P} and F_{self}, it helps to imagine a deliberately artificial scenario. Consider a non-rigid latex balloon with a uniform coating of charge attached to it by glue, and suppose that inside the balloon there is a programmable device which can push out or pull in the interior surface of the balloon (the latex itself having negligible tension). We plan to push this balloon from the outside using a wooden rod, so as to cause hyperbolic motion of its centre. Before doing so, we first program the device to produce, starting from some preset starting time, whatever forces are calculated to maintain rigid motion of the balloon, given that it is going to be pushed externally by our rod. The internal device cannot, of course, produce any arbitrary combination of forces but must respect Newton's third law. Therefore, we shall have to plan for the external force to have a suitable form in order to obtain a self-consistent solution. Once this is done, we can launch the experiment. Then the three terms F_{ext}, F_{P}, F_{self} are provided by the rod, the internal device and the electromagnetic field sourced by the balloon, respectively.

In this separation of forces, F_{P} may or may not be called a self-force. From the point of view of the surface of the balloon, F_{P} is an ‘external’ force, i.e. one not sourced by the rubber of the balloon's surface, nor by the charge and its associated electromagnetic field. However, from the point of view of an outside observer observing the sphere has a whole, F_{P} is a self-force, because it contributes something in addition to the force F_{ext} that ‘the rest of the world’ is providing.

The assumed motion can also be obtained to good approximation in a less artificial scenario. For a very stiff sphere, for example one made of diamond and given a uniform coating of charge, F_{ext} could, for example, be the force from a uniform external electric field applied in the *x*-direction. Such an applied field will tend to stretch the sphere (as it gives to all parts of the sphere the same acceleration in the instantaneous rest frame, whereas we want the part momentarily at rest at given *x* to have acceleration proportional to 1/*x*). The self-force is somewhat larger at the back of the sphere, and therefore also tends to stretch the sphere. If the object's natural shape were an oblate spheroid, this stretching would result in the assumed spherical shape in the instantaneous rest frame, and then rigid hyperbolic motion will result in a uniformly applied electric field. Another way to realize the conditions approximately is to hold the sphere fixed in a nearly uniform gravitational field.

In the limit of a small sphere, one expects to obtain the ALD equation
6.4where we use the metric signature Tr(*g*)=+2 so . The three terms on the right-hand side are called the inertial, Schott and loss (or radiation) terms, associated, respectively, with the provision of four-momentum to the field bound to the charge, its redistribution within the bound field and the provision of four-momentum to the radiated field. Hyperbolic motion is characterized by , so exhibits a well-known balance between the last two terms. Hence, we expect , which is the first term (the one given by *n*=0) in our series (5.11). The next term in the series is of order and is not accounted for in the approximations leading to equation (6.4).

Another important approximate expression for the self-force of a spherical shell undergoing arbitrary rigid motion (that is, relativistically rigid motion in which the proper acceleration may vary) is [15,16]
6.5where *v*(*τ*) is the four-velocity at proper time *τ* and *τ*_{R}=2*R* is the time taken for light to cross the sphere. This result is more accurate than (6.4) in that it evaluates a Taylor expansion for the field to all orders in the derivatives of *v*, but it still makes a linear approximation in which all nonlinear powers of the acceleration and its derivatives are neglected. In the instantaneous rest frame, the spatial part of this equation gives the three-force
6.6which for hyperbolic motion () leads to
6.7Comparing this with (5.11), we find that for hyperbolic motion equation (6.5) correctly gives the inertial term but it should not be expected to, and does not, correctly give the term of order , which is the leading term after the inertial term is accounted for (in the special case of hyperbolic motion).

*An aside on point-like particles*. For the avoidance of all confusion, it should be emphasized that the ALD equation is a valid approximation of the force to order *O*(*R*^{0}), but nevertheless it should not be used in the limit for a sphere of finite charge and finite observed mass. This is because the ‘observed mass’ includes the inertia of the field, which becomes infinite in the limit . Here, the observed mass *m* is defined as the ratio of external force to acceleration for a body at rest and with negligible second or higher derivatives of velocity. It is given by *m*=(*m*_{0}+*m*_{P})+2*e*^{2}/3*R*, where *m*_{P} is the contribution from Poincaré stresses. *m*_{P} can be negative, but (*m*_{0}+*m*_{P}) cannot (according to standard classical mechanics), and therefore *m* must exceed 2*e*^{2}/3*R*. The ALD equation exhibits pathological behaviour such as preacceleration and runaway solutions, but this is an artefact of its approximate nature and these are not expected in an exact treatment. Neither of these problems arises in the more accurate equation (6.5), as long as *m*>2*e*^{2}/3*R* [1,25]. One cannot by this route obtain an equation of motion for a point-like particle with finite charge and finite observed mass. In classical physics, such an entity is simply impossible (it has infinite field energy and infinitely negative bare mass); in quantum physics it is also impossible, in the sense that a delta-function-like wave function is never achieved in practice. Gralle *et al.* [4] provide an alternative approach, showing that a reduced-order form of the ALD equation is correct to the same order in *R* but avoids the pathological behaviour, and therefore gives a reasonable approximation to the behaviour of small charged bodies.

In an important and extensive paper, Nodvik [14] gave a treatment of self-force in electromagnetism using the calculus of variations to obtain covariant Euler–Lagrange equations for a body undergoing relativistically rigid motion. He then obtained a perturbative expansion for the self-force up to order *R*^{2} in our terminology. His general expression (eqns (7.21)–(7.24) of [14]) allows for rotation and an arbitrary distribution of charge. In the case of a non-rotating spherical shell, it yields
6.8where v^{(n)}=d^{n}v/d*τ*^{n}. Applied to the case of constant proper acceleration and setting *e*^{2}=*a*_{0}=1, this yields
6.9which should be compared with our equation (5.12). The two results are in agreement for the even powers of *R* but not for the odd powers. This is because equation (5.12) gives the contribution purely from the electromagnetic field sourced by the shell, whereas (6.9) is the sum of this and the contribution from Poincaré stresses. This fact is, however, less than clear in Nodvik's presentation, which lumps the self-electromagnetic and the other part together. The same issue was already noted by Dirac [13] and is exhibited, for example, in the lowest order term (the inertial term). As the inertial term can be lumped together with the mass term in the equation of motion and called ‘renormalization’, this difference has no easily observable consequence. However, this should not be taken to mean that there is any doubt about the calculation. The electric field predicted by Maxwell's equations is certainly the one we have calculated, and there is no need to abandon the Lorentz force equation when calculating the force on a continuous distribution of charge; it is the simplest equation consistent with energy-momentum conservation. Therefore, equation (3.7) is also correct. In the method adopted first by Dirac and then by Nodvik and many later workers, the rigidity of the charge distribution is accounted for by including it as a constraint in the variational procedure, while insisting that all equations are manifestly covariant. This means that, when the electromagnetic forces in the physical scenario do not on their own give the assumed motion (such as rigid motion), the application of the constraint automatically gives rise to a further force which, although it may have some other physical origin, such as a strong nuclear force or the programmable device inside our imagined balloon, appears in the equations as a modification of the force called ‘electromagnetic’. Having said that, it is appropriate to remark that what one usually wants to know in practice is the total self-force for a given rigid body, i.e. the sum F_{self}+F_{P} in our notation, and this is what Nodvik has provided.

Finally, in view of the fact that for hyperbolic motion , and using *v*^{2}=−1, we have . By repeatedly differentiating , one finds
6.10Hence, another way to write equation (6.1) is
6.11

## Appendix A. Series solution for the field

We would like to find the derivatives of *ϕ*(*x*,0,0) with respect to *x* at *x*=1. This amounts to finding the Taylor expansion of *ϕ*(*x*,0,0) about *x*=1. To this end, introduce *h*≡*x*−1, then equation (5.6) can be written as
A1where
A2with
A3The first 11 terms in the Taylor expansions of *ϕ*_{−} and *ϕ*_{+} are

To find the self-force, we use form (5.7), choosing a finite value for *N* and finding the coefficients *a*_{n} by matching the above Taylor series. To illustrate, we display the working for the case *N*=3. One finds the set of equations
The solution is
One can now find an approximation to *f*_{self} by using these values in (5.9). One finds
A4It is straightforward to perform a similar calculation at higher values of *N*. One thus obtains both an increasingly accurate estimate of the self-force and also a power series approximation to the electric field inside the spherical shell. The terms in the power series for *V* (*h*,*ρ*) are correct up to order *h*^{k}*ρ*^{p} with *k*+*p*=*N*, and consequently the terms in the power series for the electric field are correct up to order *h*^{k}*ρ*^{p} with *k*+*p*=*N*−1. Therefore, the series for *f*_{self} is guaranteed to be correct up to order *R*^{N−2} (not *N*−1 because each *a*_{n} coefficient involves a 1/*R* term).

Comparing (A4) with expression (5.12), which was obtained using a higher value of *N* so as to obtain a higher order approximation to the potential, we observe that the solution with *N*=3 has correctly reproduced the *O*(1/*R*) term and the *O*(*R*) term as expected, and it has also given the *O*(*R*^{3}) term correctly. Is this just a coincidence? Perhaps not, because a similar observation can be made about the *O*(*R*) term in equation (5.10). By examining the results for values of *N* in the range of 2–20, I found that the method at given *N* yields the self-force correctly up to order *R*^{2N−3}. In other words, it is surprisingly effective, converging more rapidly than was expected. However, I have not been able to discover why. This question might be explored by expanding the potential directly in terms of the *V* _{n} functions, rather than going via the polynomial series. That is one possible direction for future work.

We will finish by illustrating this point in its simplest form, which is found when *N*=2. The case *N*=2, calculated as above by matching the Taylor series on the *x*-axis, leads to the following approximation for the *x*-component of the electric field:
A5whereas the correct power series expansion to this order is
A6The *N*=2 approximation does not give the terms of order *h*^{2}/*R* and *ρ*^{2}/*R* correctly. However, to find the self-force, we evaluate the field at *ρ*^{2}+*h*^{2}=*R*^{2} and then integrate over *h* between −*R* and +*R*. The calculation using the correct expression for *E*_{x} yields, for the coefficient of the term of order *R* in *f*_{self},
and the calculation using the *N*=2 approximation to *E*_{x} yields

- Received July 18, 2013.
- Accepted October 31, 2013.

- © 2013 The Author(s) Published by the Royal Society. All rights reserved.