Physical interpretation of the Schott energy of an accelerating point charge and the question of whether a uniformly accelerating charge radiates

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1 IOP PUBLISHING Eur. J. Phys. 31 (2010) EUROPEAN JOURNAL OF PHYSICS doi: / /31/5/006 Physical interpretation of the Schott energy of an accelerating point charge and the question of whether a uniformly accelerating charge radiates David R Rowland Student Services, The University of Queensland, Brisbane QLD 4072, Australia d.rowland@uq.edu.au Received 11 May 2010, in final form 15 June 2010 Published 9 July 2010 Online at stacks.iop.org/ejp/31/1037 Abstract A core topic in graduate courses in electrodynamics is the description of radiation from an accelerated charge and the associated radiation reaction. However, contemporary papers still express a diversity of views on the question of whether or not a uniformly accelerating charge radiates suggesting that a complete physical understanding of the energy content of the fields surrounding an accelerating charge is still missing. It is argued in this paper that the missing insight is the precise physical meaning of the somewhat mysterious Schott energy which is shown to be simply the difference between the energy in the bound electromagnetic fields of the accelerating charge and the amount of energy in the bound fields of a uniformly moving charge which has the same instantaneous velocity. This difference arises because the bound fields of a charge cannot respond rigidly when the state of motion of a charge is changed by an external force. During uniform acceleration, the rate of change of this difference is just the negative of the rate at which radiation energy is created, and hence the power needed to accelerate a charged particle uniformly is just that which is required to accelerate a neutral particle with the same rest mass even though the charge is radiating. The errors in other analyses are also identified. 1. Introduction A core topic in graduate courses of electromagnetism is the modelling of radiation reaction from an accelerating point or point-like charge like an electron. Fundamental to a clear exposition of the topic is a conceptual understanding of energy and momentum balance for the problem. However, in this regards recent expositions in the literature still give conflicting accounts. For example, when it comes to uniform rectilinear acceleration, Eriksen and Grøn /10/ $30.00 c 2010 IOP Publishing Ltd Printed in the UK & the USA 1037

2 1038 DRRowland [1] state that the radiated energy-momentum as predicted by the relativistic generalization of Larmor s formula comes from the so-called Schott energy-momentum, while Singal [2] has argued that there is no radiation in this case and Heras and O Connell [3] have argued that radiation is only created during the periods of transition to and from motion with a uniform acceleration and motion with a constant velocity. One source of the above confusion comes from the fact that the Abraham Lorentz equation of motion for a charged particle 1 [4] m 0 v = f e + m 0 τ 0 v (1) implies that for constant acceleration, the m 0 τ 0 v term vanishes. Since this term is commonly taken to be the radiation reaction, this result seems to imply that the work done by the external force, f e, during constant acceleration only alters the kinetic energy of the particle and hence cannot be contributing to the creation of radiation. In (1), m 0 is the observed rest mass of the particle, v its three velocity, the over dots indicate time derivatives, and in SI units, τ 0 = q 2 /(6πε 0 m 0 c 3 ) [4], where c is the speed of light, q is the charge on the particle and ε 0 is the permittivity of free space. For an electron, τ s and to convert to Gaussian units, as are often used in the literature, replace ε 0 with 1/(4π). To see that there is a problem with the above argument, take the inner product of (1)with v to find that the rate at which the external force does work on the particle is given by dw e = d ( ) 1 dt dt 2 m 0 v 2 m 0 τ 0 v v, (2) where dw e /dt = f e v. While the first term on the right-hand-side of (2) appears to be the time rate of change of the kinetic energy of the particle as one would expect, the second term is not the Larmor formula for the rate at which energy is radiated by an accelerating charge [9]. Thus, as pointed out in [1], we seem to have a problem with energy balance for arbitrary accelerations, not just a constant acceleration. That this might have been expected comes from the fact that for non-relativistic velocities, if θ is the angle measured from the direction of acceleration, then the energy radiated from an accelerating charge has an approximately sin 2 θ dependence [9]. Hence, the net reaction from this radiated energy must be virtually zero. Consequently, the reaction term in (1) cannot be simply describing the reaction due to radiation. As is well known, the problem with (2) not including the Larmor formula can be fixed by using the differential identity, v v = d( v a)/dt a 2, where a = v, to obtain dw e = d ( ) 1 dt dt 2 m 0 v 2 m 0 τ 0 v a + m 0 τ 0 a 2. (3) In (3), m 0 τ 0 a 2 is Larmor s formula for the rate at which a non-relativistic charged particle radiates energy, and m 0 τ 0 v a U S, (4) the so-called Schott energy (taking into account relativistic corrections, U S = m 0 τ 0 γ 4 v a, where γ = (1 v 2 /c 2 ) 1/2 is the Lorentz factor [1]). Equation (3) invites the interpretation that the work done by the self-force term in (1), m 0 τ 0 v, goes into the creation of irreversibly lost radiation energy together with the Schott energy which can be reversibly lost or gained 1 Note that several authors have argued that to overcome the problems of runaway motion and pre-acceleration inherent in the Abraham Lorentz equation or its relativistic generalization, classical charged bodies must have a certain minimum size which leads to a slightly different equation of motion [5 8]. Such a modification to the Abraham Lorentz equation will not be considered in this paper as it would only lead to small corrections at the expense of making the relevant calculations prohibitively difficult as the energies and momenta in the velocity and acceleration fields of an extended charged body would need to be calculated.

3 Physical interpretation of the Schott energy of an accelerating point charge 1039 by the particle. It will be argued below that this interpretation is basically but not precisely correct. Now going back to the case of constant acceleration, (3) suggests that during constant acceleration, rather than there being no radiation of energy, energy is radiated as usual according to the Larmor formula but this is compensated for by a corresponding decrease in the Schott energy leaving the net change in energy just the change in kinetic energy that would be experienced by a neutral particle with the same rest mass experiencing the same external force. While this seems a straightforward interpretation, care must be taken because formal manipulations do not always give equations with physically clear interpretations (see e.g. [10]), and the appearance of the Schott term has in some classic electrodynamics texts been taken as an indication that the Abraham Lorentz equation is only on average true [4, 11]. Consequently, there is a need to confirm that the above interpretation of (3) is physically meaningful. A key to the required confirmation of the above interpretation of (3) is evidently a clear understanding of what the Schott energy is physically and why it arises. However, in this regard the existing literature is lacking, with the Schott energy having thus far only being vaguely defined in the literature as an acceleration energy which comes from the fields of the particle and which can be considered to be bound to the particle by virtue of it being a state function of the particle s instantaneous velocity and acceleration [1], or as being like the reactive part of the impedance of an antenna [12]. However, neither of these definitions really explain what the Schott energy actually is and why it arises. Furthermore, the Schott energy has the following mysterious properties which need explaining. First, unlike kinetic or radiation energy, it can be negative as well as positive, a fact which causes problems for Yaghjian s [12] claim that it is like the reactive part of the impedance of an antenna because the energy stored in inductors and capacitors can only be non-negative. However, like kinetic energy it is frame-dependent: move to a frame in which the particle is instantaneously at rest, and U S is found to be zero in that frame (this is another reason for concluding that U S is linked to an energy bound to the charge). In addition, no matter how large it is, if a particle stops accelerating it all seems to magically disappear, while if a particle starts accelerating, it can seem to pick up an arbitrarily large amount almost instantaneously if v c. Because of the above-mentioned issues, the goal of this paper is to provide a clear physical interpretation of the Schott energy and the reasons why it arises so as to make a slight variation of the interpretation of (3) given above physically compelling. 2. The critical clue The clue which led to the interpretation of the Schott energy which will be presented in this paper came from the following observation. Consider the case of a rectilinearly moving charge which enters a region in which the external force is in the direction of propagation and increases with position as the particle progresses. In this situation, d a/dt >0, and so the Abraham Lorentz equation, equation (1), predicts that the particle would accelerate faster than would be predicted by the external force alone. This observation led to the question: when could an object accelerate at a rate faster than would be predicted on the basis of its mass and the applied force? Well consider what happens when you lift a long flexible and slender pole or plank of timber off the ground. Initially, while the pole is flexing, you will not be supporting the entire mass of the pole and so the point at which you lift the pole would initially accelerate faster than would be predicted on the basis of your applied force and the total mass of the pole. Thus, the point of contact of a non-rigid extended object can accelerate

4 1040 DRRowland Figure 1. For a particle moving with a constant velocity in the positive x-direction, this figure shows the difference between a Lorentz-contracted sphere (dashed line) centred on the instantaneous current position of the charge, x(t), and the retarded sphere (solid line) with the same radius r = c(t t r ) centred on the position, x(t r ), which is where the particle was at time t r < t. U bc and U bret differ because the energy in region A is included in the calculation of U bret but not U bc, while region B is included in the calculation of U bc but not U bret. The example shown is for β = 0.8. faster than would be predicted by the applied force and the total mass of the accelerated object because some of the mass of the pole is initially left behind. How does the above example apply to a charge particle? Well, classical models of an electron posit that the electrostatic field of the electron makes a contribution to the observed rest mass of the electron [13], and special relativity dictates that if the state of motion of an electron is affected by an external force, knowledge of this change in motion cannot propagate instantaneously into the bound field of the charge. That is, the charge behaves like an extended elastic body rather than a point mass, just like the slender pole mentioned above. If this analogy is valid, then it suggests that the Schott energy arises as a result of more distant parts of the bound field being left behind (or overshooting when deceleration is considered). The next few sub-sections provide a quantitative proof of the general validity of this analogy. 3. Understanding the energy in the fields of an accelerating charge Note that from this point on we will need to move to a fully relativistic analysis. The relativistic generalizations of equations (1) (4) can be found in [1], from which I will just quote the needed results Background: energy in the fields of a point charge moving with a constant velocity In subsequent sub-sections, two different results will be needed. First, if the electromagnetic fields of a point charge moving with a constant velocity v are expressed in terms of the current position of the charge, then it can be shown that the amount of energy in the fields outside a Lorentz contracted sphere centred on the current position of the charge and with a radius of r in the rest frame of the charge (see figure 1)isgivenby[14] U bc (β, r) = γ(1+β 2 /3)U 0 (r). (5) In (5), β = v/c (and so the Lorentz factor γ = (1 β 2 ) 1/2 ), and the subscript b refers to this energy as being bound to the charge, while the subscript c is to indicate that the cut-off radius for the energy integral is centred on the current position of the charge. In addition, U 0 (r) = q 2 /(8πε 0 r) (6) is the energy in the electric field of a stationary spherical shell of radius r and with charge q distributed uniformly over the surface of the shell [13] and the subscript 0 on U 0 (r) isto

5 Physical interpretation of the Schott energy of an accelerating point charge 1041 indicate that this energy is measured in the rest frame of the charge when it is moving with a constant velocity (in Gaussian units, U 0 (r) = q 2 /(2r)). In contrast, if the current fields of a point charge are expressed in terms of the retarded position of the charge, and the energy outside a sphere of radius r centred on the retarded position of the charge (see figure 1) is calculated, then it is found that [2] U bret (β, r) = γu bc (β, r) = ( γ 2 β 2) U 0 (r). (7) The equivalence of the two terms on the right hand side of (7) follows simply from expanding γ in terms of β. Here again the subscript b indicates that bound fields are being referred to, while the additional ret is to indicate that the cut-off radius for the field energy integral is centred on the retarded position of the charge. That U bret is a factor of γ larger than U bc can be understood qualitatively by considering figure 1. Since region A is excluded from the calculation of U bc and region B is excluded from the calculation of U bret, it follows that U bret U bc = U(A) U(B). And since it can be shown that region A is closer to x(t) than region B and the energy density of the fields is higher closer to the charge, it is plausible that U(A) > U(B). In the above expressions for U bc and U bret, the dependence of these quantities on the velocity of the charge and the radius outside which the energy is being calculated have been made explicit for later convenience when considering shells of different radii around a charge moving with a changing velocity The distribution of electromagnetic energy around an accelerating charge Consider now a charge moving along the x-axis with a constant velocity v 1 for times t < t 1 and then accelerating with a constant proper acceleration g along the direction of motion (this special case is chosen because exact algebraic expressions for the fields can be obtained [15]. See the appendix for more mathematical details on the properties of constant proper acceleration, but note that the zero of time has been chosen such that if the hyperbolic motion was extended over an infinite period of time, the charge would have been momentarily at rest at t = 0). The electric field lines outside the charge at some time t > t 1 are shown in figure 2. These field lines are a combination of a Lorentz-contracted Coulomb field centred on the virtual current position of the charge x v (t) (i.e. the position where the charge would have been at time t if it had not started accelerating) outside a sphere with retarded radius r 1 = c(t t 1 ), together with curved field lines inside this sphere. Clearly the field lines are not those of a charge moving at a constant velocity given by the current velocity v(t), nor even those of a charge moving with a constant velocity v(t) together with some radiation fields. Hence, given that the bound fields of a charge are posited to contribute to the observed mass energy of the charge [16], figure 2 shows that when accelerating, the entire mass energy making up m 0 does not all move at the same instantaneous velocity, at least in the sense that the field outside an accelerating charge at time t is not simply the field of a charge moving at the constant velocity v(t) together with a radiation field 2. This observation brings into question 2 Strictly speaking, the bound fields do not have a well-defined velocity since they are not wave-like in nature. However, equations of motion like (1) are really about rates of change of momentum rather than velocity, and the momentum density (and hence momentum) of an electromagnetic field is well defined in a vacuum being just ε 0 E B. And, following a similar process as the one I am about to use for energy, it can be shown that the net momentum in the bound fields differ from those of an identical charge moving with a uniform velocity at the same instantaneous velocity as the accelerating charge. In fact, the difference is just such that for a non-relativistic charge, (1) written as d(m 0 v m 0 τ 0 v)/dt = f ext can be interpreted as saying that the rate of change of the total bound momentum equals the externally applied force (recall that for a non-relativistic particle, the total radiated momentum is essentially zero and so has a zero rate of change).

6 1042 DRRowland Figure 2. Electric field lines for a charged particle moving initially in the negative x-direction with a constant velocity β = when at time t = t 1 it starts to accelerate uniformly in the positive x-direction. The particle s location at t = t 1 is labelled x(t 1 ) in the diagram, its current position by x(t), and its virtual position at the current time by x v (t) (the particle s virtual position is where it would have been if it had not started accelerating at t = t 1 ). The dashed circle in the diagram is the location of the light cone emanating from x(t 1 ) at the current instant. Outside this light cone the particle s fields are simply Lorentz-contracted Coulomb fields focused on x v (t), while inside the light cone the field is that of a uniformly accelerating charge. The instant shown is the instant at which the charge has momentarily come to rest. In the diagram, distances have been normalized to c 2 /g,whereg is the magnitude of the uniform acceleration. the validity of interpreting 1 2 m 0 v 2 in (2)or(γ 1)m 0 c 2 when considering a relativistic particle, as being the instantaneous kinetic energy of an accelerating charge, an idea I will now elaborate on. As is well known, the electromagnetic field of an accelerating charge can be split into a so-called velocity/generalized Coulomb field and an acceleration/radiation field. Since the electromagnetic energy momentum tensor is quadratic in the field, this means that the total energy in the field has terms involving only velocity fields, only acceleration fields and an interaction energy being the product of velocity and acceleration fields [1]. Since only the terms involving only acceleration fields contribute to the energy content a large distance away from a charge and move with the speed of light, they are taken to describe the energy content U r of the radiation fields. The remaining energy is taken to be bound to the charge and will be referred to as U b. (Eriksen and Grøn [1] refer to U b and U r as U I and U II, respectively.) This split into bound and radiation fields also makes physical sense to the extent that Teitelboim (cited in [1, 17]) has shown that there is no interchange of energy between U b and U r, except possibly at the position of the charge (see also [18]). Given the above ideas and terminology, together with the results presented in section 3.1, it is now possible to interpret physically the results of Eriksen and Grøn s analyses in [17]for the case described by figure 2 (note that Eriksen and Grøn use Gaussian units while SI units are used in this paper). As shown in figure 3, space outside the current position of the charge is divided into three regions. In region A, the field is just that of a charge moving with the constant velocity v 1, and since this region is centred on a retarded position of the charge, the total energy in region A is given by U b (A) = U bret (β 1,r 1 ), (8) where r 1 = c(t t 1 ) and β 1 β(t 1 ). Region B lies between two spheres with retarded radii r 2 = c(t t 2 ) and r 1 = c(t t 1 ), with t 1 < t 2 < t. Since the charge has been accelerating between t 1 and t 2, U(B) has contributions

7 Physical interpretation of the Schott energy of an accelerating point charge 1043 Figure 3. Schematic of the regions surrounding an accelerating charge where fields and field energies have been calculated. The innermost ellipse is the cross-section through the Lorentzcontracted sphere (i.e. it is an ellipsoid) centred on the current position x(t) of the charge and with radius r 0 (not shown on this diagram; see figure 4) in the rest frame of the particle which is used to provide a cut-off for the energy integrals to keep them finite. Region C lies between this ellipsoid and a retarded sphere tangential to the ellipsoid. x(t 2 ) is the particle s position at time t = t 2 and r 2 = c(t t 2 ). x(t 1 ) is the particle s position when it started accelerating and r 1 = c(t t 1 ). Thus region A outside of the retarded sphere of radius r 1 is a region where the particle s fields are just Lorentz contracted Coulomb fields, while in region B both generalized Coulomb and radiation fields exist. from both bound and radiation fields. Using (7) to interpret Eriksen and Grøn s results in [17], one finds that U b (B) = U bret (β 2,r 2 ) U bret (β 1,r 1 ), (9) where β 2 β(t 2 ), and t2 U r (B) = U r (t 1,t 2 ) = R L dt = m 0 τ 0 g 2 (t 2 t 1 ). (10) t 1 In (10), R L = m 0 τ 0 A A is the relativistic generalization of the Larmor formula for the rate at which an accelerating charge radiates energy [19] and A is the four-acceleration of the particle. For rectilinear motion, A A = γ 6 a 2, where a is the magnitude of the three acceleration, and for acceleration with a constant proper acceleration of magnitude g, γ 6 a 2 = g 2 (see the appendix). The remarkably simple results of (9) and (10) belie the complexity of the calculations needed to obtain them. As a first step in understanding how the energy in the fields outside an accelerating charge differ from that of a charge moving at a constant velocity, note that (8) and (9) imply that the total amount of bound energy in regions A and B (i.e. external to the sphere with retarded radius r 2 )isgivensimplybyu b (A + B) = U bret (β 2,r 2 ). In contrast, if the charge had have been travelling at the current velocity for its entire history, the amount of energy in regions A and B would have been given by U bret (β(t), r 2 ). Using (7), these two amounts differ by 4 3( γ 2 2 β2 2 γ 2 β 2) U 0 (r 2 ), where γβ γ(t)β(t). From(A.4) in the appendix, for the motion considered here, γ i β i = gt i /c during the period of acceleration. Using this result together with t 2 = t δt, r 2 = cδt and U 0 (r 2 ) = (r 0 /r 2 )U 0 (r 0 ) (which follows from (6)), it follows to first order in δt that when δt/t 1 U b (A + B) U bret (β(t), r 2 ) 8γβr 0 U 0 (r 0 )/(3L) = 2U S, (11) where L = c 2 /g and the last result follows from (A.7). In other words, the energy in the bound fields in regions A and B differ from those of a charged particle moving with the current velocity by twice the Schott energy. For a particle speeding up, this is negative and hence represents a shortfall, while if the particle is slowing down, the difference is positive

8 1044 DRRowland and hence there is too much energy in the bound fields. These results make physical sense for the following reason. Since from (7) it can be seen that the energy in the bound field of a charge moving at a constant velocity is a monotonically increasing function of β, one would expect U b (A + B) U bret (β(t), r 2 ) to be negative when the particle is speeding up as the fields surrounding the charge are those for particles moving slower than the current speed, and conversely positive when it is slowing down as the fields surrounding the charge are then those for particles moving faster than the current speed, as is the case for U S (recall from (4) that U S v a). Also, since U b (A + B) = U bret (β 2, r 2 ) and the acceleration determines (in part) how much β 2 differs from β(t), one would also expect U S to depend on acceleration. A third region, C, is needed because retarded spheres (eikonals or light cones) are not centred on the current position of the charge and so bias the energy calculations as described in section 3.1. Thus, region C is the volume between a Lorentz contracted sphere of radius r 0 centred on the current position of the charge and the spherical eikonal of radius r 2. To calculate the energy in the fields in region C, in appendix A of [17], Eriksen and Grøn consider r 2 to be of order r 0, which is taken to become vanishingly small in the point particle limit, and use this to obtain manageable approximations to the fields. (More specifically, it is required that r/(c 2 /g) 1, where r is r 0 or r 2 and c 2 /g is a characteristic length scale of the problem. For example, c 2 /g delimits the boundary between where the velocity field dominates and where the acceleration field dominates.) Using (5), (7) and the relativistic version of (4) to interpret Eriksen and Grøn s results, one finds that U b (C) U bc (β(t), r 0 ) U bret (β(t), r 2 ) U S (β(t), a(t)). (12) Since in (12), U bc (β(t), r 0 ) U bret (β(t), r 2 ) is how much bound energy would have been in region C if the particle had have been travelling at its current velocity for its entire history, (12) shows that again as expected, the bound energy in region C differs from what it would have been if the particle had have been travelling at its current velocity for its entire history. However, in this case the difference has what seems to be the opposite sign to what one might expect based on the arguments used above for region B. The result is in fact physically plausible for the following reasons. First, note from figure 3 that the region C is a sample of the energy around the charge biased to the region behind rather than in front of the charge (for the example shown). The significance of this observation comes from figure 2 where it is seen that the fields lines bend so as to be convex in the direction of the acceleration. Now recall that the density of field lines is related to the magnitude of the field and hence the energy density in the field. Consequently, in the case of figure 2, the field lines are more dense to the left of the particle than to the right so that the energy density is likewise higher to the left than to the right. As a consequence, as a result of this bending of the field lines, there is more energy in the region C than there would have been if the particle had have been moving at a constant velocity, and vice versa if the particle is slowing down, consistent with the correction term being U S. One might question at this point whether the same argument would apply for region B. The answer is yes it would, but in region B this effect is reduced by the fact that region B is not quite so biased as the sample of the field energy surrounding the particle and since it applies over a much longer period of time, the change in bound energy due to the change in velocity also plays a bigger effect (it does not have a dominant role in region C because the time interval for region C is so small). To finalize the field energy calculations, U r (C) would be needed. However, calculating U r (C) would be a formidable task, and in any event is unnecessary since in the point particle limit, t t 2 can be taken to be much smaller than t 2 t 1, and so the amount of radiation produced between t 2 and t can be neglected in comparison to that produced between t 1 and t 2. Consequently, U r (t 1, t) can be approximated by U r (t 1, t 2 ).

9 Physical interpretation of the Schott energy of an accelerating point charge 1045 Table 1. Summary of results for the bound energy of a charge moving with either a constant velocity or accelerating. In this table, U 0 U 0 (r 0 ) and it has been assumed that the energy stored in the Poincaré stresses [14] is not affected by acceleration (from a classical point of view, Poincaré stresses need to be present or a charged body would fly apart as a result of the mutual electrostatic repulsion of its components). The results for the constant velocity case come from section 7.3 of Møller [14], while the results for the non-zero acceleration case come from combining the constant velocity case results with those of Eriksen and Grøn [17]. Here m bare is the mass the charged particle would have if it was not charged. Constant velocity case Non-zero acceleration case (Bound) field energy γu 0 (1+β 2 /3) γu 0 (1+β 2 /3) m 0 τ 0 γ 4 v a Total mechanical energy including contribution from Poincaré stresses γ(m bare c 2 U 0 β 2 /3) γ(m bare c 2 U 0 β 2 /3) Total bound energy γ(m bare c 2 + U 0 ) ( γm 0 c 2 ) γm bare c 2 + (γ U 0 m 0 τ 0 γ 4 v a) γm 0 c 2 m 0 τ 0 γ 4 v a To calculate the total amount of energy in the bound field of the charge being considered, first note that using the result from (11), (12) can be rewritten as U b (C) U bc (β(t), r 0 ) U b (A + B) + U S (β(t), a(t)), (13) which means that the total bound energy in the field of an accelerating point charge is given by U b (total) = U b (A + B + C) = U bc (β(t), r 0 ) + U S (β(t), a(t)). (14) As expected, this is not simply the amount of energy in the bound fields of a charge moving at a constant velocity equal to the current instantaneous velocity, but amazingly only differs from this by a small amount, namely the Schott energy. In this way, we see that the Schott energy is a correction term which corrects U bc (β(t), r 0 ) for the fact that the particle has not been moving at its current speed for all time. Eriksen and Grøn [20] argue that (13) impliesthat the Schott energy is located in the vicinity of the particle, but (11) and the discussion below it show that in fact the Schott energy is a net effect of the difference between the bound field energy of an accelerating particle and the bound field energy of a particle travelling with the current velocity of the charge. Now that the external field of an accelerating charge has been analysed, using wellknown results and working in analogy to the case for a point charge moving at a constant velocity as shown in table 1, the total bound energy of motion of an accelerating charge can be determined. From these results, the relativistic generalization of the power balance equation (3), namely [1] dw e = d dt dt (γ m 0c 2 + U S ) + R L (t), (15) now has the beautifully simple physical interpretation that the rate at which the external force does work on the charge equals the rate at which the total bound energy of the charge varies plus the rate at which energy is radiated. For the case of uniform rectilinear acceleration, (15) reduces to dw e /dt = d(γm 0 c 2 )/dt. (16) The error in interpreting this result as meaning that when a charge is being accelerated uniformly that all the work done by the external force goes into increasing the kinetic energy

10 1046 DRRowland (a) (b) Figure 4. This figure shows qualitatively how the size of region C, and hence ultimately U S, depends on the velocity of the particle. In both figures, r 0, the effective size of the particle is the same, the difference is that in (a) β = 0.1 while in (b) β = 0.6. of the particle and none goes into radiation is as follows. Because the field of a charge contributes to its mass, momentum and energy of motion, the bound energy of motion (i.e. kinetic energy) is not in fact (γ 1)m 0 c 2, but rather (γ 1)m 0 c 2 m 0 τ 0 γ 4 v a. In this way, (16) can still imply that radiation is being created during uniform acceleration, it is just that it is being created at the same rate at which the total bound energy is decreasing. Another argument against interpreting (16) as meaning no radiation is created during uniform acceleration is as follows. While the above analysis shows that the rates of change of radiated energy and Schott energy cancel for uniform acceleration, from (A.6) it follows that during a finite period of acceleration, at time t the Schott energy is given by U S = m 0 τ 0 g 2 t, while the amount of radiated energy is by (10), U r = m 0 τ 0 g 2 (t t 1 ). Consequently, U S and U r themselves only cancel if t 1 = 0 (i.e. the charge is at rest when it starts its period of acceleration). As a final point for this sub-section, we are now in a position to understand why U S depends on the velocity of the particle as well as its acceleration. The reason is that as shown in figure 4, the size of region C depends on the velocity of the charge, with the size going to zero as the velocity goes to zero. 4. Answers to further questions raised in the literature 4.1. Is the rest mass of a charged particle conserved while it is accelerating? A question that the above analyses raise is whether the fact that the bound energy is equal to γm 0 c 2 m 0 τ 0 γ 4 v a rather than just γm 0 c 2 means that the rest mass of the charge is not conserved during acceleration. Fulton and Rohrlich [21] argue that it does not, pointing out that at each instant of time, the Schott energy is zero relative to the inertial frame in which the particle is momentarily at rest, thus meaning that the rest frame bound energy is always just m 0 c 2 (see table 1). Another way of looking at this is to consider lifting a long flexible pole off the ground. While the mass of the pole does not change during this process, initially one does not have to lift the entire weight of the pole as the pole flexes Is energy only radiated during transitions into and out of uniform acceleration? An argument sometimes presented in the literature, and most recently by Heras and O Connell [3], is that energy is radiated only during the periods of transition into and out of uniform acceleration, not during a period of uniform acceleration. The purpose of this sub-section is to identify the error in this argument.

11 Physical interpretation of the Schott energy of an accelerating point charge 1047 Figure 5. An idealized acceleration programme for a particle undergoing rectilinear acceleration. The particle undergoes uniform acceleration for a period of T seconds, bounded by short periods (i.e. δt T) where the acceleration changes linearly with time so that initially and finally the particle is moving with a constant velocity. For simplicity, only non-relativistic motion in a straight line will be considered in this sub-section, and the particle is taken to have the idealized acceleration program shown in figure 5, from which it follows that g/δt, 0 <t<δt v = 0, δt t T + δt. (17) g/δt, T + δt<t<t +2δt The total amount of work done by the external force over the time period [0, T +2δt] can now be found by integrating (2) with respect to time, which means evaluating T +2δt ( ) 1 T +2δt W e = f e v dt = 0 2 m 0v 2 m 0 τ 0 vv dt. (18) 0 Assuming δt/t 1 so that the velocity of the particle varies negligibly over δt, then if the initial velocity is v i, then the final velocity is v f v i + gt. Then using (17) T +2δt 0 m 0 τ 0 vv dt δt 0 m 0 τ 0 (g/δt)v i dt + T +2δt T +δt m 0 τ 0 (g/δt)(v i + gt ) dt = m 0 τ 0 g 2 T. (19) The final result is just the amount of energy radiated over T predicted by Larmor s formula, but because the integral from δt to T+δt does not contribute to this result, it is sometimes argued that this means that the radiation is happening during the transition periods into and out of uniform acceleration rather than during the period of uniform acceleration itself. For this to be a valid interpretation though, one would have to accept that the amount of energy radiated during the period [0, δt] is given approximately by m 0 τ 0 gv i, a negative amount for the problem under consideration! If radiation is to take on its usual meaning for this problem (i.e. irretrievably lost energy), it must necessarily be non-negative. To find an energy with this property during the initial period [0, δt], integrate the term in vv by parts to find that the work done by the external force in the interval [0, δt] is given by ( ) 1 δt W e = 2 m 0v 2 m 0 τ 0 v v + m 0 τ 0 v 2 dt, (20) 0 which, from table 1, is the change in the bound energy of motion plus the amount of energy radiated according to Larmor s formula (this radiated energy being a positive quantity). Thus, in order for the radiated energy to be non-negative, one must, as argued in this paper, take 1 2 m 0v 2 m 0 τ 0 v v to be the bound energy of motion of a non-relativistic particle rather than just 1 2 m 0v 2. Doing so, one finds for the above problem that m 0 τ 0 v v decreases by approximately

12 1048 DRRowland m 0 τ 0 g 2 T during the period of uniform acceleration, thus explaining where the radiated energy during this period has come from. What the above analysis shows is that in general, if one wants to directly see the rate at which energy is being radiated as a function of time, the non-relativistic power equation needs to be written as (3) rather than what is directly obtained from the equation of motion, namely (2). This conclusion provides a salutary reminder that the correct physical interpretation of the results of formal manipulations of equations of motion is not always as straightforward as one might think What about Einstein s equivalence principle and radiation during uniform acceleration? According to Einstein s equivalence principle (EEP), a uniformly accelerating charge should behave like a charge held in a fixed position in a uniform gravitational field. But since no work is being done on a charge held in a fixed position in a uniform gravitational field, such a charge cannot be radiating relative to that frame of reference, and hence cannot experience a radiation reaction. Hence the force needed to hold the charge at a fixed position in the uniform gravitational field should be simply m 0 g, the same as required to accelerate the charge with a constant proper acceleration of magnitude g in gravity free space as can be shown using the idea of Maxwell Faraday stresses in the electromagnetic field of the charge [23]. The above argument raises the question though: how can it be that a uniformly accelerating charge radiates with respect to an inertial frame of reference but not with respect to a co-moving uniformly accelerating frame of reference? Regarding this question, the following points can be made. First, as shown in figure 6, the hypersurfaces of simultaneity on which an inertial frame versus the Rindler (i.e. uniform acceleration) frame in which the charge is at rest sample the fields of the charge are very different. One consequence of this is that the electric field lines of the charge in the Rindler frame in which it is at rest lie along the geodesics for photons relative to that frame of reference (this can be shown using the equations for the field lines given by Rohrlich [15] and then doing a standard relativistic calculation for the null geodesics of a uniformly accelerating reference frame). This means that relative to the Rindler frame, the photons emitted by the charge are purely longitudinal, not transverse, meaning that they are virtual rather than real (i.e. radiation) photons. 4 That photons can be real or virtual depending on your frame of reference also happens in quantum mechanics, though the other way around: in the well-known Unruh effect, the virtual photons in the quantum vacuum of an inertial frame of reference become real relative to a uniformly accelerating reference frame, consequently producing a thermal heat bath in that frame. Another way of understanding the very different observations of an inertial frame and a Rindler frame is as follows. Note that a Rindler frame is continuously rotating its hyperplane of simultaneity about the spacetime origin of figure 6. It can be shown that at each instant of Rindler time, this rotation is just what is necessary to convert the electric and magnetic fields which are present in the inertial frame of figure 6 into a pure and static electric field. 5 In this way, the slicing of spacetime made by a Rindler frame hides the radiation seen by an inertial frame. 3 See for example the Abraham Minkowski controversy over the amount of momentum carried by an electromagnetic wave in a dielectric medium [22], and the question of the longitudinal momentum density carried by a transverse wave pulse on a taut string [10]. 4 Note that in [23] I failed to appreciate this point and thereby drew the invalid conclusion that because the self-force on a uniformly accelerating charge is just the inertial weight of the bound field, then this means that a uniformly accelerating charge does not radiate. In fact, all these results mean is that a uniformly accelerating charge does not radiate with respect to the Rindler frame in which it is at rest. 5 This can be proven using equation (7.105) for the Faraday bivector of a uniformly accelerating charge from [24] and the idea illustrated in figure 3 of [25].

13 Physical interpretation of the Schott energy of an accelerating point charge 1049 Figure 6. The solid line in this figure is the worldline of a particle undergoing rectilinear uniform acceleration along the x-axis. At the event P along this worldline, the line with the longer dashes indicates a hyperplane of simultaneity for the inertial frame with respect to which the worldline has been drawn. In contrast, the line with the shorter dashes is the hyperplane of simultaneity for the Rindler frame in which the particle is at rest, thus demonstrating the very different parts of the electromagnetic field created by the particle observed by the two reference frames. 5. Conclusions The pedagogical implications of this paper are as follows. First, most if not all treatments of charged particles give the impression that they are rigid point objects. The key observation of this paper is that while this is a reasonable approximation in many instances, when it comes to understanding energy and momentum balance for an accelerating charged particle, it is essential to remember that if the bound field of a charge contributes to its mass, then dynamically the charge is a non-rigid, extended and composite object. As such, for nonrelativistic motion, m 0 v and 1 2 m 0v 2 are not exactly the momentum and bound energy of motion of the composite object, and taking into account the fact that the bound field of a charge does not respond rigidly to changes in the velocity of the charge, one finds that the Schott momentum and energy are corrections to m 0 v and 1 2 m 0v 2 accounting for the non-rigid response of the bound fields (this is where this paper differs from the work of Eriksen and Grøn [1, 17]). Because of these facts, during uniform acceleration when the rate of work done by the external force is d ( 1 2 m 0v 2)/ dt, this cannot be interpreted as meaning no radiation is being created. Instead, it means that in this instance, the rate of change of bound energy plus the rate of increase of radiation energy just happens to equal d ( 1 2 m 0v 2)/ dt (which is in fact needed for Einstein s equivalence principle to hold). A second pedagogical implication is that in the Abraham Lorentz equation, m 0 τ 0 d 2 v/dt 2 should not be referred to as the radiation reaction/damping, but is rather a correction term to the rate of change of the total bound momentum (recall that for a non-relativistic particle, the net radiated momentum is approximately zero and hence so is its rate of change). However, when dotted with v and integrated by parts this term does in fact give the amount of energy radiated. Appendix. Some results for a uniformly accelerating particle This appendix collects together some results for a particle accelerating uniformly (in a relativistic sense) in a straight line needed in this paper and to translate some of the equations in Eriksen and Grøn [17] to the form they take in this paper. Relativistically speaking, uniform acceleration is not a constant three-acceleration, but rather the acceleration that results from a constant external three-force [26]. Writing this constant external force as f e = m 0 g for some constant g, a possible worldline of a particle accelerating uniformly along the x-axis of some inertial frame is x = (L 2 + c 2 t 2 ) 1/2, (A.1)

14 1050 DRRowland where L = c 2 /g. Differentiating (A.1) with respect to t, it follows that the velocity is given by v = dx dt = gt (1+g 2 t 2 /c 2 ) 1/2. (A.2) Note that for simplicity, the zero of time has been chosen to occur at the instant the particle comes momentarily to rest. From (A.2), it follows that γ(t)= (1+g 2 t 2 /c 2 ) 1/2, γv = gt, (A.3) (A.4) and a = dv/dt = g/γ 3. (A.5) From these equations, it further follows that during uniform acceleration, the Schott energy is given by U S = m 0 τ 0 γ 4 va = m 0 τ 0 g 2 t. (A.6) It is also useful to note that since m 0 τ 0 = (4/3)r 0 U 0 (r 0 )/c 3, for a uniformly accelerating particle U S = 4γβr 0U 0 (r 0 ). (A.7) 3L To calculate (17), we need to evaluate γ2 2β2 2 γ 2 β 2.Using(A.4), we find γ2 2 β2 2 γ 2 β 2 = (g 2 /c 2 ) ( t2 2 t 2) = (g 2 /c 2 )( 2tδt + δt 2 ), (A.8) where t 2 = t δt has been used to obtain the last result. If δt/t 1, then (A.8) gives ( ) gt gδt γ2 2 β2 2 γ 2 β 2 2 c c = 2γβr 2 L, (A.9) using δt = r 2 /c and L = c 2 /g. It then follows that for δt/t 1, 4( γ β2 2 γ 2 β 2) U 0 (r 2 ) 8 3 γβr 2 r 0 U 0 (r 0 ) = 2U S. (A.10) L r 2 Note from (12) that r 0 U 0 (r 0 ) is independent of r 0,soU S does not change if the limit r 0 0 is taken. In contrast, if in (A.8) t = 0, γ2 2 β2 2 γ 2 β 2 = (gδt/c) 2, (A.11) and since around t = 0 the particle is stationary to first order in δt (since x 1 2 gδt2 ), this means that r 2 r 0 and δt r 0 /c in this case. Using these results in (A.11), it follows that at t = 0 4 γ 2 3( 2 β2 2 γ 2 β 2) U 0 (r 2 ) 4 ( gr0 ) 2 U0 (r 0 ). (A.12) 3 c Now using gr 0 /c 2 gδt/c δβ, the right-hand-side of (A.12) is approximately 4δβr 0 U 0 (r 0 ), (A.13) 3L which is effectively zero as required (U S = 0 when β = 0) if the speed of the particle varies negligibly in the time it takes light to travel across the effective radius of the particle (one can also argue that unlike (A.10), (A.13) 0asr 0 0 which is the point particle limit).

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