Electrodynamics of Radiating Charges

Abstract

The theory of electrodynamics of radiating charges is reviewed with special emphasis on the role of the Schott energy for the conservation of energy for a charge and its electromagnetic field. It is made clear that the existence of radiation from a charge is not invariant against a transformation between two reference frames that has an accelerated motion relative to each other. The questions whether the existence of radiation from a uniformly accelerated charge with vanishing radiation reaction force is in conflict with the principle of equivalence and whether a freely falling charge radiates are reviewed. It is shown that the resolution of an electromagnetic “perpetuum mobile paradox” associated with a charge moving geodetically along a circular path in the Schwarzschild spacetime requires the so-called tail terms in the equation of motion of a charged particle.

1. Introduction

The nonrelativistic version of the equation of motion of a radiating charged particle was discussed already more than a hundred years ago by Lorentz [1], 𝑚₀𝐚=𝐟_ext+𝑚₀𝜏₀𝑑𝐚𝑑𝑇,𝜏₀=𝑞²6𝜋𝜀₀𝑚₀𝑐³, (1.1) where 𝐚 is the ordinary (Newtonian) acceleration of the particle. The time 𝜏₀ is of the same order of magnitude as the time taken by light to move a distance equal to the classical electron radius, that is, 𝜏₀≈ 10⁻²³ seconds. The general solution of the equation is 𝐚(𝑇)=𝑒^𝑇/𝜏0𝐚(0)−1𝑚𝜏₀^𝑇₀𝑒^{−𝑇′/𝜏0}𝐟_ext𝑇^′𝑑𝑇^′. (1.2) Hence, the charge performs a runaway motion, that is, it accelerates away even when 𝐟_ext=0 unless one chooses the initial condition 𝑚𝜏₀𝐚(0)=^∞₀𝑒^{−𝑇′/𝜏0}𝐟_ext𝑇^′𝑑𝑇^′. (1.3) By combining (1.1) and (1.3), one obtains 𝑚𝐚(𝑇)=^∞₀𝑒^−𝑠𝐟_ext𝑇+𝜏₀𝑠𝑑𝑠. (1.4) This equation shows that the acceleration of the charge at a point of time is determined by the future force. Hence, when runaway motion is removed preacceleration appears.

The relativistic generalization of the equation was originally found by Abraham in 1905 [2]. A new deduction of the Lorentz covariant equation of motion was given by Dirac in 1938 [3]. This equation is therefore called the Lorentz-Abraham-Dirac equation, or for short, the LAD equation. In the 4-vector notation invented by Minkowski in 1908, and referring to an inertial frame in flat spacetime, the equation of motion of a radiating charged particle takes the form 𝐹^𝜇_ext+𝐹^𝜇_𝐴=𝑚₀𝑈̇^𝜇, (1.5) where 𝑈^𝜇=𝑑𝑋^𝜇/𝑑𝜏 is the 4-velocity of the particle and the dot denotes the ordinary differentiation with respect to the proper time of the particle. Here, 𝐹^𝜇_𝐴≡𝑚₀𝜏₀𝐴̇^𝜇−𝑔²𝑈^𝜇,𝑔²=𝐴_𝛼𝐴^𝛼, (1.6) is called the Abraham 4-force, 𝑔 is the proper acceleration of the charged particle with respect to an inertial frame, and 𝐴^𝜇𝐞_𝜇=𝑑𝑈𝑑𝜏^𝜇𝐞_𝜇 (1.7) is the 4-acceleration of the particle.

Let the components of the 4-acceleration be (𝐴⁰,𝐀). Fritz Rohrlich called the spatial component of the Abraham 4-force, 𝐟_𝐴, for the field reaction force, and separated it in two forces, 𝐟_𝐴=𝐟_𝑆+𝐟_𝑅, where 𝐟_𝑆 is the Schott force (called the acceleration reaction force by Rohrlich) and 𝐟_𝑅 the radiation reaction force. The Schott force is given by 𝐟_𝑆=𝑚₀𝜏₀𝑑𝐀,𝑑𝑇 (1.8) where 𝑇 is the inertial laboratory time, and the radiation reaction force is 𝐟_𝑅=−𝑚₀𝜏₀𝑔²𝐯, (1.9) where 𝐯 is the ordinary velocity of the particle.

The field reaction force is 𝐟_𝐴=𝑚₀𝜏₀𝑑𝐀𝑑𝑇−𝑚₀𝜏₀𝑔²𝐯. (1.10) For a freely moving particle both the Schott force and the field reaction force vanish.

Gal’tsov and Spirin [4] have reviewed and compared two different approaches to radiation reaction in the classical electrodynamics of point charges: a local calculation of the self-force, using the equation of motion and a global calculation consisting in integration of the electromagnetic energy-momentum flux through a hypersurface encircling the worldline. With reference to Dirac [3] and Teitelboim [5], they interpreted the Schott force physically in the following way: the Schott force is the finite part of the derivative of the momentum of the electromagnetic field, which is bound to the charge.

In an inertial reference frame the Abraham 4-force may also be written as𝐹^𝜇_𝐴=𝑚₀𝜏₀𝛾(𝐯⋅𝐠̇,𝐠̇), (1.11) where the vector 𝐠 is the proper acceleration of the charged particle with respect to an inertial frame. The Abraham 4-force is written in terms of the field reaction force as 𝐹^𝜇_𝐴=𝛾𝐯⋅𝐟_𝐴,𝐟_𝐴. (1.12) Hence, the field reaction force can be written as 𝐟_𝐴=𝑚₀𝜏₀𝐠̇. (1.13) According to Larmor’s formula, the energy radiated by the particle per unit time is 𝑃_𝐿=𝑚₀𝜏₀𝑔². (1.14) It should be noted that this version of Larmor’s formula is only valid with respect to an inertial frame meaning that it is only valid in a frame where there is no acceleration of gravity.

In flat spacetime uniformly accelerated motion is defined by 𝐠̇=0. Hence, no field reaction force acts upon a uniformly accelerated charge. However, if one investigates the electromagnetic field of such a charge, one finds that a uniformly accelerated charge emits radiation in accordance with Larmor’s formula.

The 4-acceleration is defined as the covariant directional derivative of the 4-velocity along the four-velocity. Since free particles move along geodesic curves that are defined by the condition that their tangent vectors, that is, the 4-velocity, are connected by parallel transport, it follows that in curved as well as in flat spacetime a freely falling charge has vanishing 4-acceleration. Hence, in an inertial frame a freely moving charge will not radiate. However, it will be shown in Section 4 that this is not so in a non-inertial frame where the acceleration of gravity does not vanish. Then, a generalization of (1.14) will be needed.

2. Localization and Physical Interpretation of the Schott Energy

In a recent article on relativistic particle motion and radiation reaction in electrodynamics Hammond [6] has discussed the question whether energy is conserved for uniformly accelerated motion of a radiating charge. He wrote that the Schott energy (defined in (2.5) below) destroys our concept of what conservation of energy should be. Hammond pointed out an important problem that is rather disturbing as long as the Schott energy only appears as a term in the energy conservation equation without any physical interpretation telling us what sort of energy it is. In order to remedy this defect in classical electrodynamics Eriksen and Grøn [7–10] and also Rowland [11] worked out a physical interpretation of the Schott energy in terms of the energy of the electromagnetic field produced by an accelerating charge. The results of this research will be summarized in this section.

From the equation of motion (1.5) we get the energy equation 𝐯⋅𝐅_ext=𝛾⁻¹𝑚₀𝑈̇⁰−𝐹⁰_𝐴=𝑚₀𝛾³𝐯⋅𝐚−𝐯⋅𝐟_𝐴=𝑑𝐸_𝑘𝑑𝑇−𝐯⋅𝐟_𝐴, (2.1) where 𝐚 is the ordinary acceleration and 𝐸_𝑘=(𝛾−1)𝑚₀𝑐² is the kinetic energy of the particle. In an instantaneous inertial rest frame of the charge the radiation reaction force 𝐟_𝑅 vanishes. The reason is that in this frame the charge radiates isotropically. Hence, in this frame the field reaction force 𝐟_𝐴 reduces to the Schott force 𝐟_𝑆, which here takes the form 𝐟_𝑆=𝑚₀𝜏₀𝑑𝐠𝑑𝑇,𝐠=𝑑𝐯.𝑑𝑇 (2.2) For particles moving with a velocity 𝑣≪𝑐 we can use this expression for the Schott force, and the work done by this force is ^𝑇₀𝐟_𝑆⋅𝐯𝑑𝑇=𝑚₀𝜏₀^𝑇₀𝑑𝐠𝑑𝑇⋅𝐯𝑑𝑇=𝑚₀𝜏₀[𝐠⋅𝐯]^𝑇₀−𝑚₀𝜏₀^𝑇₀𝑔²𝑑𝑇. (2.3) The first term vanishes if the acceleration or the velocity vanishes at the boundary of the integration region or for periodic motion, integrating over a whole number of periods. Then, the work performed by the Schott force accounts for the radiated energy as given by the last term.

The power due to the field reaction force is 𝐯⋅𝐟_𝐴=𝑚₀𝜏₀𝑑𝛾𝑑𝑇⁴𝐯⋅𝐚−𝑃_𝐿=−𝑑𝐸_𝑆−𝑑𝑇𝑑𝐸_𝑅,𝑑𝑇 (2.4) where 𝐸_𝑅 is the energy of the radiation field and 𝐸_𝑆 is the Schott energy defined by 𝐸_𝑆≡−𝑚₀𝜏₀𝛾⁴𝐯⋅𝐚=−𝑚₀𝜏₀𝐴⁰. (2.5) The Schott energy was called “acceleration energy” by Schott [12]. It is negative if the acceleration is in the same direction as the velocity and positive if it is in the opposite direction. By means of (2.4) the energy equation (2.1) may now be written as 𝑑𝑊_ext𝑑𝑇=𝐯⋅𝐅_ext=𝑑𝑅𝑑𝑇^𝐾+𝐸_𝑆+𝐸_𝑅. (2.6) In the case of uniformly accelerated motion, 𝐠̇=0, the field reaction force, 𝐟_𝐴, vanishes. From (2.4) and (2.6) it follows that in this case 𝐯⋅𝐅_ext=𝑑𝐸_𝐾.𝑑𝑇 (2.7) Hence, in this case all of the work performed by the external force goes to increase the kinetic energy of the charged particle. From (2.6) it is seen that then all of the radiated energy comes from the Schott energy. But what is the Schott energy?

In the article “The equations of motion of classical charges” Rohrlich wrote [13]: “If the Schott energy is expressed by the electromagnetic field, it would describe an energy content of the near field of the charged particle which can be changed reversibly. In periodic motion energy is borrowed, returned, and stored in the near-field during each period. Since the time of energy measurement is usually large compared to such a period only the average energy is of interest and that average of the Schott energy rate vanishes. Uniformly accelerated motion permits one to borrow energy from the near-field for large macroscopic time-intervals, and no averaging can be done because at no two points during the motion is the acceleration four-vector the same. Nobody has so far shown in detail just how the Schott energy occurs in the near-field, how it is stored, borrowed, and so forth.”

This challenge was taken up by Eriksen and Grøn, who gave an answer in a series of articles [7–10]. Pearle [14] has summarized works of Fulton and Rohrlich [15] and Teitelboim [5] on this problem and writes: “The physical meaning of the Schott term has been puzzled over for a long time. Its zero component represents a power which adds “Schott acceleration energy” to the electron and its associated electromagnetic field. The work done by an external force not only goes into electromagnetic radiation and into increasing the electron’s kinetic energy, but it causes an increase in the “Schott acceleration energy” as well. This change can be ascribed to a change in the “bound” electromagnetic energy in the electron’s induction field, just as the last term of (2.6) can be ascribed to a change in the “free” electromagnetic energy in the electron’s radiation field. What meaning should be given to the Schott term? Teitelboim has argued convincingly that when an electron accelerates, its near-field is modified so that a correct integration of the electromagnetic four-momentum of the electron includes not only the Coulomb 4-momentum(𝑞²/8𝜋𝜀₀𝑟)𝑈^𝜇, but an extra four-momentum of the bound electromagnetic field.”

It remains to obtain a precise localization of the Schott field energy. Eriksen and Grøn [10] obtained the following result. The Schott energy is inside a spherical light front touching the front end of a moving Lorentz contracted charged particle. At the point of time 𝑇 the radius of the eikonal consisting of light emitted at the point of time 𝑇_𝑄2 that represents the boundary of the distribution of the Schott energy is (Figure 1) 𝑇−𝑇_𝑄2=𝑟₀1+𝑣/𝑐.1−𝑣/𝑐 (2.8) Here, 𝑟₀=𝜀 represents the proper radius of the particle and 𝑣 is the absolute value of its velocity. The horizontal extension of the hatched region is 𝐿=2𝑐𝑇−𝑇_𝑄2−2𝑟₀1−𝑣²𝑐²=2𝑟₀𝑣𝑐(1+𝑣/𝑐)().1−𝑣/𝑐 (2.9) Note that 𝐿→0 when 𝑣→0. Unless the velocity of the charge is close to that of light, the Schott energy is localized just outside the surface of the charged particle. This is illustrated in Figure 2.

Figure 1 

The figure shows a Lorentz contracted charged particle with proper radius 𝜀 moving to the right with velocity 𝑣. The field is observed at a point of time 𝑇, and at this moment the centre of the particle is at the position 𝑋(𝑇). The circle is a field front produced at the retarded point of time 𝑇𝑄2 when the centre of the particle was at the position 𝑋(𝑇𝑄2). The field front is chosen such that it just touches the front of the particle. The Schott energy is localized in the shaded region between the field front and the ellipsoid representing the surface of the particle. In the figure the velocity is chosen to be 𝑣=0.6.

Figure 2 

The figure shows how the size of the region 𝐶, as given by (2.9), depends on the velocity of the particle. In both parts of the figure the radius 𝑟0 of the particle is the same. The difference is that in (a) 𝛽=0.1 while in (b) 𝛽=0.6.

We can now see, from a field point of view, why the Schott energy 𝐸_𝑆 depends on the velocity of the particle as well as its acceleration and vanishes in the rest frame of the charge. The reason is that the size of the region 𝐶 containing the Schott energy depends on the velocity of the charge, with the size going to zero as the velocity goes to zero.

With Eriksen and Grøn’s results as a point of departure Rowland [11] has worked out a new interpretation of the Schott energy. He considered a charged particle moving initially in the negative 𝑥-direction with a constant velocity 𝛽=𝑣/𝑐=−0.745. At time 𝑡=𝑡₁ it starts to accelerate uniformly in the positive 𝑥-direction. The particle’s location at 𝑡=𝑡₁ is labeled 𝑥(𝑡₁) in Figure 3, its current position 𝑥(𝑡), and its virtual position at the current time 𝑥_𝑣(𝑡) (the particle’s virtual position is where it would have been if it had not started accelerating at 𝑡=𝑡₁). The dashed circle in the figure is the location of the light cone emanating from 𝑥(𝑡₁) at the current instant. Outside this light cone the particle’s field is simply a Lorentz contracted Coulomb field focused on 𝑥_𝑣(𝑡), while inside the light cone the field is that of a uniformly accelerating charge with circular field lines.

Figure 3 

The figure shows the electrical field of a charged particle moving initially in the negative 𝑥-direction with a constant velocity 𝛽=−0.745. At time 𝑡=𝑡1 it starts to accelerate uniformly in the positive 𝑥-direction. The particle’s location at 𝑡=𝑡1 is labeled 𝑥(𝑡1), its current position 𝑥(𝑡), and its virtual position at the current time 𝑥𝑣(𝑡). The dashed circle in the diagram is the location of the light cone emanating from 𝑥(𝑡1) at the current instant. Outside this light cone the particle’s field is simply a Lorentz contracted Coulomb field focused on 𝑥𝑣(𝑡), while inside the light cone the field is that of a uniformly accelerating charge with circular field lines.

Consider for a moment a charged particle that moves with constant velocity in the positive x-direction. Figure 4 shows the difference between a Lorentz contracted sphere (dashed line) centered at the instantaneous current (𝐶) position of the charge, 𝑥(𝑡), and the retarded (ret) sphere (solid line) with the same radius 𝑟=𝑐(𝑡−𝑡_𝑟) centered at the position, 𝑥(𝑡_𝑟), of the particle at time 𝑡_𝑟<𝑡. The bound (b) field energies 𝐸_𝑏cur and 𝐸_𝑏ret outside the current and retarded surfaces differ because the energy in region 𝐴 is included in the calculation of 𝐸_𝑏ret but not 𝐸_𝑏cur, while region 𝐵 is included in the calculation of 𝐸_𝑏cur but not 𝐸_𝑏ret. The example shown is for 𝛽=0.8.

Figure 4 

The figure shows the difference between a Lorentz contracted sphere (dashed line) centered at the instantaneous current (cur) position of the charge, 𝑥(𝑡), and the retarded (ret) sphere (solid line) with the same radius 𝑟=𝑐(𝑡−𝑡𝑟) centered at the position, 𝑥(𝑡𝑟), of the particle at time 𝑡𝑟<𝑡.

The energy in the electric field outside a spherical shell of radius 𝑟 and with charge 𝑞 moving with constant velocity in the laboratory, as measured in the rest frame of the shell, is 𝐸₀(𝑟)=𝑞²8𝜋𝜀₀𝑟. (2.10) The bound energy in the electric field outside a Lorentz contracted spherical shell, centered at the current position of the charge, is𝐸_𝑏cur𝛽𝛽,𝑟=𝛾1+3𝐸₀(𝑟). (2.11) The bound energy in the electric field outside a spherical shell, centered at the retarded position of the charge, is 𝐸_𝑏ret𝛽,𝑟=𝛾𝐸_𝑏cur=4𝛽,𝑟1+3𝛾²𝛽²𝐸₀(𝑟). (2.12) The space outside the current position of the charge is divided into three regions (Figure 5).

Figure 5 

The regions surrounding an accelerating charge where field energies are calculated. Here 𝑥(𝑡1) is the particle position when it started accelerating, and 𝑟1=𝑐(𝑡−𝑡1). Thus there is just a Lorentz contracted Coulomb field in the region 𝐴, while there exist both generalized Coulomb and radiation fields in region 𝐵. The innermost ellipse is the cross-section through the Lorentz contracted sphere (i.e., it is an ellipsoid) centered at the current position 𝑥(𝑡) of the charge and withradius 𝑟0 (not shown on this diagram) in the rest frame of the particle. Hence the charge has a finite radius that is necessary to provide a cut-off for the energy integrals to keep them finite. Region 𝐶 where Eriksen and Grøn localized the Schott energy, lies between this ellipsoid and a retarded sphere tangential to the ellipsoid. Furthermore, 𝑥(𝑡2) is the particle position at time 𝑡=𝑡2 and 𝑟2=𝑐(𝑡−𝑡2). The relationship between 𝑡 and 𝑡2 is given in (2.8).

In region 𝐴 the field is just that of a charge moving with the constant velocity 𝑣₁, and since this region is centered at a retarded position of the charge, the total energy in region 𝐴 is given by (2.12) 𝐸_𝑏(𝐴)=41+3𝛾²𝛽²𝑞²8𝜋𝜀₀𝑟. (2.13) Region 𝐵 lies between two spheres with retarded radii 𝑟₂=𝑐(𝑡−𝑡₂) and 𝑟₁=𝑐(𝑡−𝑡₁), with 𝑡₁<𝑡₂<𝑡. Since the charge has been accelerating between 𝑡₁ and 𝑡₂, 𝐸(𝐵) has contributions 𝐸_𝑏 and 𝐸 from, respectively, bound and radiation fields. Using (2.12) to interpret Eriksen and Grøn’s results, one finds that 𝐸_𝑏(𝐵)=𝐸_𝑏ret𝛽₂,𝑟₂−𝐸_𝑏ret𝛽₁,𝑟₁, (2.14) where 𝛽₂=𝛽(𝑡₂), and 𝐸_𝑟(𝐵)=𝐸_𝑟𝑡₁,𝑡₂=^𝑡2_𝑡1𝑃_𝐿𝑑𝑡=𝑚₀𝜏₀𝑔²𝑡₂−𝑡₁. (2.15) Here, 𝑃_𝐿 is the radiated effect as given by Larmor’s formula (1.14). The bound energy in the region 𝐴+𝐵 outside the sphere with retarded radius 𝑟₂ is 𝐸_𝑏(𝐴+𝐵)=𝐸_𝑏ret𝛽₂,𝑟₂. (2.16) If the charge had been moving with its current velocity, that is, its velocity at the point of time 𝑡, for its entire history, the energy in the region 𝐴+𝐵 would have been 𝐸_𝑏ret(𝛽(𝑡),𝑟₂). Rowland has shown that to the first order in 𝛿𝑡=𝑡−𝑡₂𝐸_𝑏(𝐴+𝐵)−𝐸_𝑏ret𝛽(𝑡),𝑟₂≈2𝐸_𝑆. (2.17) In other words the bound field energy in 𝐴+𝐵 differs from that of a charged particle moving with the current velocity by twice the Schott energy. The acceleration determines in part how much 𝛽₂ differs from 𝛽(𝑡). It is therefore not surprising that 𝐸_𝑆 depends upon the acceleration.

The region 𝐶 is the volume between a Lorentz contracted sphere of radius 𝑟₀ with center at the current position of the charge and a spherical light front of radius 𝑟₂. Using (2.10), (2.11), and (2.6) to interpret Eriksen and Grøn’s results, one finds that the energy of the fields in the region 𝐶 is 𝐸_𝑏(𝐶)≈𝐸_𝑏cur𝛽(𝑡),𝑟₀−𝐸_𝑏ret𝛽(𝑡),𝑟₂−𝐸_𝑆𝛽(𝑡),𝑎(𝑡). (2.18) The first term at the right- hand side is the bound energy in the electric field outside a Lorentz contracted spherical shell, centered at the current position of the charge as given in (2.6). The second term is the bound energy in the electric field outside a spherical shell, centered at the retarded position of the charge. The difference, Δ𝐸_𝑏cur=𝐸_𝑏cur(𝛽(𝑡),𝑟₀)−𝐸_𝑏ret(𝛽(𝑡),𝑟₂), is the energy that would have been in the region 𝐶 if the charged particle had moved with constant velocity for its entire history. For the accelerated particle the field lines are curved as shown in the Figure 6, and there is more energy in the region 𝐶 than if the particle had moved with constant velocity. Equation (2.18) can be written as 𝐸_𝑆=Δ𝐸_𝑏cur−𝐸_𝑏(𝐶), (2.19) which leads to the following interpretation: the Schott energy is the difference between the bound field energy of an accelerated charged particle if the particle had moved with its current velocity for its entire history and the actual bound field energy of the particle. With this understanding one may say that the Schott energy is that part of the bound field energy of an accelerated charged particle that is due to its acceleration, and it is localized close to the particle.

Figure 6 

We have here plotted the rapidity 𝛼=arctanh𝛽 and its rate of change as functions of proper time. Here 𝛼(−∞)=−0.8 and we have chosen 𝜏2=14𝜏0.

We can now provide a description of how the LAD equation implies energy-momentum conservation even during the strange runaway motion. This understanding comes from the detailed investigations of energy-momentum relationships of an accelerated charge and its electromagnetic field that has been reviewed above. It is important to recognize that an accelerated charge is not an isolated system. It is the charge and its electromagnetic field that are the isolated system. During runaway motion there is an increasingly negative Schott energy and an increasing Schott momentum directed oppositely to the motion of the charge. It follows from (2.6) with 𝑊_ext=0 that the energy and momentum of the charge and its field are conserved during runaway motion. The bound field close to the charge where the Schott energy is acts upon the charge, accelerates it, increases its kinetic energy and also is transformed into radiation energy, while the charge acts back on the field in accordance with Newton’s third law. These are internal forces in the isolated charge-field system. The internal forces involving the Schott energy and its transformation to radiation energy then cause a redistribution of energy and momentum between the particle, its near field, and the radiation it emits.

3. Motion of a Charge Moving into a Constant Electric Field Which Stops the Charge and Accelerates It Back

We will now consider a situation analyzed by Eriksen and Grøn [16–18] that makes clear the essential role played by the Schott energy for energy conservation. The particle enters a region H with a constant electric field at 𝜏=𝜏₁ giving the particle a constant proper acceleration in the opposite direction of its velocity and leaves the electrical field at 𝜏=𝜏₂. Assuming 𝜏₀≪𝜏₂−𝜏₁, that is, that the time 𝜏₀ is much less than the time the charge is inside the region 𝐻, the solution of the LAD equation is 𝛼̇=𝑔𝑒^{(𝜏−𝜏1)/𝜏0},𝛼=𝛼₀+𝑔𝜏₀𝑒^{(𝜏−𝜏1)/𝜏0},𝜏≤𝜏₁, (3.1) where 𝛼₀<0 is the limiting initial value of the rapidity for 𝜏→−∞. There is preacceleration both before the particle enters the electric field and before it leaves it (Figure 6).

The values of the kinetic energy of the particle and the Schott energy, respectively, for 𝜏=−∞, are 𝐸_𝐾=𝑚₀cosh𝛼₀−1,𝐸_𝑆=0. (3.2) At the moment 𝜏=𝜏₁ when the particle enters 𝐻, the values of these energies are 𝐸_𝐾=𝑚₀cosh𝛼₁−1,𝐸_𝑆2=−3𝑄²𝑔sinh𝛼₁. (3.3) To second order in 𝑔𝜏₀, which we assume is much less than the velocity of light, 𝑐=1, the changes in the kinetic energy and the Schott energy from 𝜏=−∞ to 𝜏=𝜏₁ are Δ𝐸_𝐾=𝛾₀𝑚₀𝑣₀𝑔𝜏₀+12𝛾₀𝑚₀𝑔²𝜏²₀,Δ𝐸_𝑆=−𝛾₀𝑚₀𝑣₀𝑔𝜏₀−𝛾₀𝑚₀𝑔²𝜏²₀. (3.4) From the energy equation (2.6), which for 𝜏<𝜏₁ takes the form Δ𝐸_𝑅+Δ𝐸_𝐾+Δ𝐸_𝑆=0, it follows that the radiated energy during this period is Δ𝐸_𝑅=12𝛾₀𝑚₀𝑔²𝜏²₀. (3.5)From (3.3)–(3.5) it follows that the particle during the preacceleration gets an increase of the Schott energy which is nearly equal to the loss of kinetic energy of the particle. Only a minor part of the particle loss in kinetic energy (second order in 𝑔𝜏₀) is radiated away.

The change of velocity of the particle, tanh𝛼₁−tanh𝛼₀, during the preacceleration may be expressed as 𝑣₁−𝑣₀=sinh𝑔𝜏₀𝛾₁𝛾₀, (3.6) which shows that 𝑣₁→𝑣₀ when 𝛾₀→∞.

We shall now consider the energy budget during the time when the charge is within 𝐻, that is, for 𝜏₁<𝜏<𝜏₂. From (2.6) and (3.1) we get an energy equation 𝑊̇_ext=𝐸̇_𝑅+𝐸̇_𝐾+𝐸̇_𝑆 with the following rates of change per unit proper time: 𝑊̇_ext=𝑚₀𝑔sinh𝛼,𝐸̇_𝐾=𝑚₀𝐸̇𝛼̇sinh𝛼,_𝑆=𝑚₀2𝑔−𝛼̇sinh𝛼−3𝑄²𝛼̇²cosh𝛼,𝐸̇_𝑅2=−3𝑄²𝛼̇²cosh𝛼. (3.7) The equations shall now be interpreted for the case that 𝛼̇=𝑔, that is, when the motion of the particle is approximately hyperbolic. According to (3.1) this is the case when 𝜏≪𝜏₂−𝜏₀, that is, when the particle is not too near its exit from 𝐻. In this region 𝑊̇_ext=𝐸̇_𝐾 so that there is no other effect of the external force than a change in the kinetic energy. Thus, the sum of the kinetic energy of the charge and its potential energy in the field of force accelerating it in 𝐻 is approximately constant, and the radiated energy is taken from the Schott energy, which according to (3.7) decreases uniformly with time, 𝑑𝐸_𝑆=𝐸̇𝑑𝑇𝛾^𝑆2=−3𝑄²𝑔². (3.8) When the particle arrives at the point where it turns back, there is no Schott energy left, and at this moment the energy that has been radiated by the particle is equal to its loss of kinetic energy during the preacceleration. The energy has been radiated as a pulse with energy Δ𝐸_𝑅 given in (3.5), during the preacceleration, and then with a constant effect (2/3)𝑄²𝑔² during the hyperbolic motion. The situation changes when the particle approaches the position where it leaves 𝐻, that is, when 𝜏≈𝜏₂−𝜏₀. Then, the particle experiences a new nonnegligible preacceleration, which reduces the acceleration from ≈𝑔to 0, and the emitted power is reduced from ≈(2/3)𝑄²𝑔² to 0. The velocity still increases during this period, but less than in the case of hyperbolic motion. The Schott energy, which until now (in 𝐻) has decreased at a constant rate, increases from the negative value −(2/3)𝑄²𝑔𝛾𝑣 to zero. All the energies 𝐸_𝑅,𝐸_𝐾,and𝐸_𝑆 increase during this preacceleration. The energy is provided by the work of the external force 𝐹_ext=𝑚₀𝑔, or in other words from the loss of potential energy of the particle in the field of this force. In the region where the motion can be considered as hyperbolic, 𝛼̇=𝑔=constant, and the reaction force 𝑚₀𝜏₀𝛼̈ vanishes. Here, 𝐹_ext=𝑚₀𝑔 is the only force acting upon the particle, and 𝐸_𝐾+𝐸_𝑃=constant. This is no longer the case when the particle approaches the exit of 𝐻, where the preacceleration makes 𝛼̈≠0.

In order to make a complete energy budget in the region 𝐻, we must know the proper time 𝜏₂ when the particle leaves 𝐻. The position 𝑋(𝜏) of the particle at a point of time 𝜏 is given by 𝑋(𝜏)−𝑋₁=^𝜏_𝜏1𝛾𝑣𝑑𝜏=^𝜏_𝜏2sinh𝛼𝑑𝜏, (3.9) where 𝛼 is given by (3.1). The point of time 𝜏₂ when the particle leaves 𝐻 is found from the equation 𝑋(𝜏₂)=𝑋₁. Solving the integral (to the second order in 𝑔𝜏₀) the equation reads 𝑔^𝜏2_𝜏1sinh𝛼𝑑𝜏=1−𝑔²𝜏²₀cosh𝛼₁+𝑔𝜏₂−𝜏₁−cosh𝛼₁, (3.10) where we have utilized 𝜏₀≪𝜏₂−𝜏₁. We get the following solution to the second order in 𝑔𝜏₀: 𝜏₂−𝜏₁1=−𝑔2𝛼₁+𝑔²𝜏²₀coth𝛼₁, (3.11) where 𝛼₁=𝛼₀+𝑔𝜏₀ is the rapidity of the particle at the moment it enters 𝐻. The term −2𝛼₁/𝑔, which is dominating, is the proper time that the particle would have spent inside 𝐻 if the motion had been hyperbolic. Then 𝛼̇=𝑔, so the travelling proper time would be Δ𝜏=Δ𝛼/𝑔, where Δ𝛼=−2𝛼₁ is the increase of 𝛼 during the motion in 𝐻. Equation (3.11) tells that 𝜏₂−𝜏₁ is a little larger than this value.

Inserting 𝜏₂ from (3.11) into the the expression for 𝛼 in equation (3.1), we get 𝛼𝜏₂=−𝛼₁−𝑔𝜏₀−𝑔²𝜏²₀coth𝛼₁, (3.12) which gives cosh𝛼𝜏₂=31+2𝑔²𝜏²₀cosh𝛼₁+𝑔𝜏₀sinh𝛼₁. (3.13) From this we find the following (negative) changes of the kinetic energy of the charge and its Schott energy during the period, 𝜏₁<𝜏<𝜏₂, when the charge moves in 𝐻: Δ𝐸_𝐾=𝑚₀𝑔𝜏₀𝛾₁𝑣₁+32𝑚₀𝑔²𝜏²₀𝛾₁,Δ𝐸_𝑆=𝑚₀𝑔𝜏₀𝛾₁𝑣₁. (3.14) Since the total work performed by the external force 𝐹_ext upon the charge during its motion in 𝐻 vanishes, the energy equation (2.6) gives for the energy radiated by the charge during this motion Δ𝐸_𝑅=−Δ𝐸_𝐾−Δ𝐸_𝑆=−𝑚₀𝑔𝜏₀𝛾₁𝑣₁−32𝑚₀𝑔²𝜏²₀𝛾₁. (3.15) The dominating term, −2𝑚₀𝑔𝜏₀𝛾₁𝑣₁, may be interpreted as the energy radiated by a particle with exact hyperbolic motion. This is seen as follows. Using 𝑔=𝛼̇=𝛾𝑑𝛼𝑑𝑇=cosh𝛼𝑑𝛼=𝑑()𝑑𝑇sinh𝛼,𝑑𝑇 (3.16) for hyperbolic motion, leads to Δ(sinh𝛼)=𝑔Δ𝑇. (3.17) Thus, the time that the charge stays inside 𝐻 is 2Δ𝑇=−𝑔sinh𝛼₁. (3.18) The dominating term in (3.15) may be written as 𝑚₀𝑔²𝜏₀−2sinh𝛼₁𝑔=𝑚₀𝑔²𝜏₀2Δ𝑇=3𝑄²𝑔²Δ𝑇 (3.19) in agreement with Larmor’s formula.

According to (3.14) the Schott energy and the kinetic energy decrease by about the same amount, which means that the Schott energy and the kinetic energy give approximately the same contribution to the radiated energy.

To get the complete energy budget from 𝜏=−∞ to 𝜏=∞ we utilize 𝑊_ext=0 and 𝐸_𝑆(−∞)=𝐸_𝑆(∞)=0. Then, according to (3.15), Δ𝐸_𝑅=Δ𝐸_𝐾=0, where Δ𝐸_𝑅 is the sum of the expressions in (3.4). Expressing the relationships in terms of the velocity 𝑣₀ (which is negative) at 𝜏=−∞, we find that the radiated energy is Δ𝐸_𝑅=Δ𝐸_𝐾=𝐸_𝐾(−∞)−𝐸_𝐾(∞)=−2𝑚₀𝑔𝜏₀𝛾₀𝑣₀−3𝑚₀𝑔²𝜏²₀𝛾₀. (3.20) The kinetic energy, the mechanical energy (i.e., the sum of the kinetic and potential energy), the Schott energy, and the radiated energy as functions of proper time are shown in Figures 7 and 8.

Figure 7 

Kinetic energy and Schott energy in units of 𝑚0 as functions of proper time for the motion given in (3.1).

Figure 8 

Mechanical energy, that is, 𝐸𝐾+𝐸𝑃, Schott energy, and radiated energy as functions of proper time. Here 𝐸𝑃 is the potential energy in the force field 𝑚0𝑔 with 𝐸𝑃=0 for 𝑋>𝑋1, and 𝐸𝑃=𝑚0𝑔(𝑋1−𝑋). Note that 𝐸𝐾+𝐸𝑃+𝐸𝑆+𝐸𝑅=constant. All energies are in units of 𝑚0.

Let us summarize what happens to the particle and its energy from 𝜏=−∞ to 𝜏=∞. The charge comes from an infinitely far region with constant velocity. It moves towards a region 𝐻 with, say, a constant electrical field antiparallel to its direction of motion. Approaching 𝐻 it gets an increasing preacceleration, which causes the kinetic energy of the particle to decrease. A Schott energy of about the same magnitude appears. Also a small amount of energy is radiated away by the particle. In the region 𝐻 the particle moves approximately hyperbolically until it experiences a new preacceleration before it leaves 𝐻. During the hyperbolic part of the motion the external work performed by the field force upon the particle is used only to change the kinetic energy of the particle. The particle radiates at a constant rate, and the radiated energy comes from the Schott energy, which decreases steadily during this part of the motion. Before the particle leaves 𝐻 the preacceleration decreases the acceleration towards zero. The particle still radiates although the Schott energy now increases. What happens all together while the particle is in 𝐻 is that the kinetic energy and the Schott energy decrease by about the same amount, giving about the same contribution to the radiated energy. When the particle has left 𝐻 and disappears towards an infinite remote region, the Schott energy has vanished again. The particle has lost kinetic energy, and this loss of energy is equal to the energy that the particle has radiated.

Pauri and Vallisneri [19] have commented this situation in the following way:

“We can imagine that the energy flux radiated by the charge during the uniformly accelerated motion is being borrowed from the divergent energy of the electromagnetic field near the charge, which effectively acts as an infinite reservoir. While draining energy, the field becomes more and more different from the pure velocity field of an inertial charge; when hyperbolic motion finally ends, the extended force must provide all the energy that is necessary to re-establish the original structure of the field.” They did not mention the Schott energy, but its existence is implicit in what they wrote.

4. The Principle of Equivalence and Noninvariance of Electromagnetic Radiation

The foundation of the principle of equivalence is that at a certain point of spacetime every free particle instantaneously at rest falls with the same acceleration independent of its composition. A consequence of this is that if a proton and a neutron are falling from the same point in spacetime, they will fall together with the same acceleration. However, the proton will emit electromagnetic radiation and the neutron not. So where does the radiated energy come from?

Think of the following two situations.(1)Consider a freely falling charge, for example, in a uniformly accelerated reference frame (UAF). It emits radiation in accordance with Larmor’s formula. However, as observed by a comoving observer it is at rest in an inertial frame. Hence, this observer would say that the charge does not radiate.(2)Consider now a charge with uniform acceleration in the inertial frame. This charge, too, emits radiation in accordance with Larmor’s formula. But it is at rest in UAF.

If the detection of radiation depended only on the state of motion of the charge, an observer in UAF would detect radiation from a charge at rest in UAF. However, in this situation there is nothing that can provide the radiation energy since the situation is static, so the assumption that whether a charge radiates or not depends only upon its state of motion, cannot be correct.

Both situations (1) and (2) mean that the existence of radiation cannot be invariant against a transformation between an inertial and noninertial reference frame.

In order to have a discussion of the question whether the existence of radiation from a charged particle is invariant against a transformation between an inertial- and an accelerated reference frame we need a precise definition of electromagnetic radiation. This will be given below following Rohrlich [20].

4.1. The Definition of Electromagnetic Radiation

The rate of radiation energy emission will be defined as a Lorentz invariant, but not generally invariant, quantity. Hence, in this section all components of tensor quantities will refer to an inertial reference frame (and we use units so that 𝑐=1). The component formulae will be generalized to expressions valid also with respect to a uniformly accelerated reference frame in Section 4.4.

Given a point charge 𝑞 following a trajectory 𝑋^𝜇(𝜏), the electromagnetic field produced by the charge, 𝐹^𝜇𝜈, is measured at a point 𝑃 with coordinates 𝑋^𝜇_𝑃. The point 𝑃 is connected to an emission point 𝑄 by a null vector 𝑅^𝜇=𝑋^𝜇_𝑃−𝑋^𝜇, that is, 𝑅_𝜇𝑅^𝜇=0. The spatial distance between 𝑃 and 𝑄 in the inertial system in which the charge is instantaneously at rest is 𝜌=−𝑈_𝜇𝑅^𝜇>0, (4.1) where 𝑈^𝜇 represents the 4-velocity of the charge at the retarded point 𝑄. We now define a spacelike vector 𝑁^𝜇=1𝜌𝑅^𝜇−𝑈^𝜇, (4.2) obeying 𝑁_𝜇𝑁^𝜇=1,𝑁_𝜇𝑈^𝜇=0. (4.3) The vector 𝑅^𝜇 can therefore be written as 𝑅^𝜇=𝜌𝑁^𝜇+𝑈^𝜇. (4.4) In this notation the Liénard-Wiechert potentials are 𝐿^𝜇𝑈=𝑒^𝜇𝜌𝑞,𝑒≡4𝜋𝜀_0.. (4.5) The retarded electromagnetic field is given by 𝐹^𝜇𝜈=−2𝑒𝜌²𝑁^[𝜇𝑈^𝜈]−2𝑒𝜌𝑈^[𝜇𝐴^𝜈]+𝑁^[𝜇𝑈^𝜈]𝐴_𝑁+𝐴^𝜈,𝐴_𝑁=𝐴_𝜆𝑁^𝜆. (4.6) Here, the antisymmetrization bracket around the upper indices is defined by 𝑎^[𝜇𝑏^𝜈]=(1/2)(𝑎^𝜇𝑏^𝜈−𝑎^𝜈𝑏^𝜇). The first term is the generalized Coulomb field, and the second term, which vanishes if and only if the components of the four-acceleration with respect to an inertial frame vanish, is the radiation field, by definition.

The electromagnetic energy tensor is 𝑇_𝜇𝜈=𝜀₀𝐹^𝜆_𝜇𝐹_𝜆𝜈−14𝜂_𝜇𝜈𝐹_𝛼𝛽𝐹^𝛼𝛽, (4.7) where the components of the Minkowski metric tensor have been used since all quantities are decomposed in an inertial frame in this section. Inserting (4.6) into this expression one finds the energy-momentum tensor at a point 𝑃 due to the field emission at the retarded point 𝑄 on the world line of a point charge q, 𝑇^𝜇𝜈=𝜀₀𝑒²𝜌⁴𝑁^𝜇𝑁^𝜈−𝑈^𝜇𝑈^𝜈−12𝜂^𝜇𝜈+2𝜀₀𝑒²𝜌³𝐴_𝑁𝑉^𝜇𝑉^𝜈−𝑉^(𝜇𝑈^𝜈)𝐴_𝑁+𝐴^𝜈+𝜀₀𝑒²𝜌²𝑉^𝜇𝑉^𝜈𝐴²_𝑁−𝑔². (4.8) Here, 𝑉^𝜇=𝑁^𝜇+𝑈^𝜇, and the symmetrization bracket around the upper indices is defined by 𝑎^(𝜇𝑏^𝜈)=(1/2)(𝑎^𝜇𝑏^𝜈+𝑎^𝜈𝑏^𝜇).

Let Σ represent a spherical surface in the instantaneous inertial rest frame of the charge at the point 𝑄. We shall calculate the rate of change of electromagnetic field energy inside this surface in the limit of a very large radius so that the surface is in the wave zone of the charge. Energy-momentum conservation can be expressed in the following way: the rate of change of electromagnetic energy-momentum inside this surface is equal to the flux of electromagnetic energy-momentum through the surface, 𝑑𝑃^𝜇𝑑𝜏=−lim_Σ→∞𝑑𝑑𝜏_Σ𝑇^𝜇𝜈𝑑³𝜎_𝜈=−lim_Σ→∞𝑑𝑑𝜏_Σ𝑇^𝜇𝜈𝑁_𝜈𝑑𝜏𝜌²𝑑Ω. (4.9) Inserting expression (4.8) gives 𝑑𝑃^𝜇𝑑𝜏=𝜀₀𝑒²lim_Σ→∞Σ𝑉^𝜇𝑔²−𝐴²_𝑁𝑑Ω=8𝜋𝜀₀3𝑒²𝑔²𝑈^𝜇. (4.10) This 4-vector is the total rate of energy and momentum emission in form of radiation. The Lorentz invariant emitted power is 𝑃_𝐿=−𝑈_𝜇𝑑𝑃^𝜇=𝑑𝜏8𝜋𝜀₀3𝑒²𝑔² (4.11) in agreement with Larmor’s formula (1.14).

Rohrlich [20] then showed that the radius of the spherical surface Σ can be chosen to be small. The sphere need not be in the wave zone of the charge. This was demonstrated in the following way. The rate of energy and momentum which crosses the surface Σ in the direction 𝑁^𝜇 per unit solid angle is 𝑑𝑃^𝜇𝜀𝑑𝜏𝑑Ω=−⁰𝑒²2𝑁^𝜇𝜌²+𝜀₀𝑒²𝜌𝐴^𝜇−𝐴_𝑁𝑁^𝜇+𝜀₀𝑒²𝑉^𝜇𝑔²−𝐴²_𝑁. (4.12) This quantity is a 4-vector. The first two terms are both spacelike vectors and may be interpreted as the Coulomb 4-momentum and the cross-term between Coulomb and radiation fields. The last term is a null vector and describes pure radiation. It is independent of 𝜌. Since the two first terms are orthogonal to the velocity vector, we get −𝑈^𝜇𝑑𝑃^𝜇𝑑𝜏𝑑Ω=𝜀₀𝑒²𝑔²−𝐴²_𝑁. (4.13) This is the invariant formulation of Poynting’s formula for the radiation energy rate per unit solid angle. Because this expression is independent of 𝜌, it follows that the radiated power through the surface Σ is given by 𝑃_𝐿=𝑑𝑑𝜏_Σ𝑇^𝜇𝜈𝑑³𝜎_𝜈=8𝜋𝜀₀3𝑒²𝑔², (4.14) that is, we do not need to take the limit of an infinitely large radius of the surface Σ. This result permits one to establish a criterion for testing whether a charge is emitting radiation at a given instant, by measuring the fields only and without having to do so at a distance large compared to the emitted wave length.

The criterion for radiation is as follows: given the world line of a charge and an arbitrary instant 𝜏₀ on it. Consider a sphere Σ of arbitrary radius 𝑟 in the instantaneous inertial rest system 𝑆₀ of the charge at the proper time 𝜏₀ with center at the charge at that instant. Measure the electromagnetic fields 𝐹^𝜇𝜈 on Σ at the time 𝜏₀+𝑟 and evaluate the integral 𝑃_𝐿𝜏₀=𝑇^0𝑘𝑁_𝑘𝑟²𝑑Ω=_Σ𝐒⋅𝐍𝑑²𝜎, (4.15) where 𝐍 is the Poynting vector. The value of this integral is the Lorentz invariant rate of radiation energy 𝑃_𝐿 at time 𝜏₀ and vanishes if and only if the charge did not radiate at that instant.

Ginzburg [21] has pointed out that according to this definition a uniformly accelerated charge radiates although there is no wave zone in this case, and it is not suitable, then, to speak about the appearance of photons.

It should be noted that an accelerated observer may very well measure a different rate of radiation from that given in (4.15) or even no radiation at all. The concept of radiation emitted from a charged particle as measured by an accelerated observer will be described in Section 4.4.

4.2. On the Concept “Gravitational Field”

The principle of equivalence is usually stated as follows: the physical effects of a homogeneous gravitational field due to a mass distribution are equivalent to the physical effects of the artificial gravitational field in an accelerated reference frame. However, Hammond [6] writes that in the general theory of relativity the curvature of spacetime replaces the Newtonian concept of a gravitation field. For a uniformly accelerated frame (in fact for any accelerated frame in Minkowski spacetime) the curvature tensor vanishes: there is no gravitational field.

However, in applications of the principle of equivalence it is necessary to distinguish between the tidal and the nontidal components of a gravitational field. This distinction was made clear by Grøn and Vøyenli [22, 23], and a related discussion of the concept “gravitational field” has been given by Brown [24].

The distinction between a tidal and a nontidal gravitational field is based on the geodesic equation and the equation of geodesic deviation. Consider two nearby points 𝑃₀ and 𝑃 in spacetime and two geodesics, one passing through 𝑃₀ and one through 𝑃. Let 𝐧 be the distance vector between 𝑃₀ and 𝑃. The geodesics are assumed to be parallel at 𝑃₀ and 𝑃, so that (𝑑𝐧/𝑑𝜏)_𝑃0=0. Using (53) of [25] we find that a Taylor expansion about the point 𝑃₀ gives the following formula for the acceleration of a free particle at 𝑃: 𝑑²𝑥^𝑖𝑑𝜏²_𝑃=−Γ^𝑖_𝛼𝛽𝑢^𝛼𝑢^𝛽_𝑃0−Γ^𝑖_{𝛼𝛽,𝛾}𝑢^𝛼𝑢^𝛽_𝑃0𝑛^𝛾. (4.16) The first term at the right hand side represents the acceleration of a free particle at 𝑃₀ and contains, for example, the centrifugal acceleration and the Coriolis acceleration in a rotating reference frame.

We define the gravitational field strength at the point 𝑃, 𝐠, as the acceleration of a free particle instantaneously at rest. Then, the spatial components of the four-velocity vanish. Using the proper time of the particle as time coordinate gives 𝑢⁰=1, and (4.16) simplifies to 𝑔^𝑖=−Γ^𝑖_00𝑃0−Γ^𝑖_00,𝛾𝑃0𝑛^𝛾. (4.17) Grøn and Vøyenli [22] have shown that in a stationary metric this equation can be written as 𝑔^𝑖=−Γ^𝑖₀₀+Γ^𝑘₀₀Γ^𝑖_𝑗𝑘−Γ^𝜎_0𝑗Γ^𝑖_𝜎0𝑛^𝑗−𝑅^𝑖_0𝑗0𝑛^𝑗, (4.18) where the Christoffel symbols and the components of the Riemann curvature tensor are evaluated at the point 𝑃₀. The first term at the right hand side of this equation represents the acceleration of gravity at the point 𝑃₀, that is, it represents the uniform part of the gravitational field. The second term represents the nonuniform part of the gravitational field, which is also present in a noninertial reference frame in flat spacetime, for example, the non-uniformity of the centrifugal field in a rotating reference frame. The last term represents the tidal effects, which in the general theory are proportional to the spacetime curvature.

This suggests the following separation of a gravitational field into a nontidal part and a tidal part 𝑔^𝑖=𝑔^𝑖_NT+𝑔^𝑖_T, (4.19) where the non-tidal part is given by 𝑔^𝑖_NT=−Γ^𝑖₀₀+Γ^𝑘₀₀Γ^𝑖_𝑗𝑘−Γ^𝜎_0𝑗Γ^𝑖_𝜎0𝑛^𝑗 (4.20) and the tidal part by 𝑔^𝑖_T=−𝑅^𝑖_0𝑗0𝑛^𝑗. (4.21) The non-tidal gravitational field can be transformed away by going into a local inertial frame. The tidal gravitational field cannot be transformed away. The mathematical expression of these properties is that the non-tidal gravitational field is given by Christoffel symbols, and they are not tensor components. All of them can be transformed away. But the tidal gravitational field is given in terms of the Riemann curvature tensor of spacetime, which cannot be transformed away.

A gravitational field caused by a mass distribution may be called a permanent gravitational field. In general such a field has both a tidal and a non-tidal component. However, the gravitational fields experienced in rotating or accelerated reference frames in flat spacetime are purely non-tidal.

With this background, making it clear that there exists a gravitational field in a uniformly accelerated reference frame in flat spacetime, a brief review of earlier works on the compatibility of the equivalence principle with the invariance or noninvariance of radiation produced by an accelerated charge will now be presented.

4.3. Previous Discussions on Radiation by an Accelerated Charge and the Principle of Equivalence

The present discussion will be based on the following formulation of the principle of equivalence. According to the principle of equivalence the physical effects of the non-tidal gravitational field in a noninertial frame of reference are equivalent to the effects of the non-tidal component of a permanent gravitational field caused by a mass distribution. The adjective “non-tidal” means that the principle of equivalence does only have a local validity. It is valid in a region of spacetime sufficiently small that tidal effects of a permanent gravitational field cannot be measured with the equipment of the laboratory. Hence, according to the principle of equivalence in such a laboratory one cannot perform any experiment making it possible to decide, for example, whether one is in an accelerated laboratory in space or in a laboratory at rest on the surface of the Earth.

In 1955 Bondi and Gold [26] discussed radiation from a uniformly accelerated charge in relation to the principle of equivalence. They showed that a uniformly accelerating charge radiates and then wrote: “The principle of equivalence states that it is impossible to distinguish between the action on a particle of matter of a constant acceleration or of static support in a gravitational field. This might be thought to raise a paradox when a charged particle, statically supported in a gravitational field, is considered, for it might be thought that a radiation field is required to assure that no distinction can be made between the cases of gravitation and acceleration—But there can be no static radiation field, and as the whole system is static the electromagnetic field cannot depend upon time.” They resolved this apparent contradiction by arguing that hyperbolic motion requires a homogeneous gravitational field of infinite extension and that such a field does not exist in nature.

In 1960 Fulton and Rohrlich [15] took up the discussion whether the existence of radiation from a uniformly accelerated charge is in conflict with the principle of equivalence. First they performed a calculation based on Maxwell’s equations showing that a uniformly accelerated charge radiates. Then they used the LAD-equation to show that in spite of the vanishing radiation reaction upon a uniformly accelerating charge this is not in conflict with energy conservation. Finally they discussed the principle of equivalence and wrote: “A particle which is falling freely in a homogeneous gravitational field should appear to an observer who is falling with it, like a particle at rest in an inertial frame. When the particle is charged, the observer can establish the presence of a gravitational field by looking for radiation. If he observes radiation from the charge, he knows that he and the charge are falling in a gravitational field; if he observes no radiation, he knows that he and the particle are in a force free region of space.”

Fulton and Rohrlich solved the problem in a similar way as Bondi and Gold, noting that radiation can only be determined far from the emitting charge and that the principle of equivalence has only a local validity, and hence the problem disappears.

The problem discussed by Bondi and Gold and by Fulton and Rohrlich and the way they solved it indicate that at this time it was an underlying assumption that the existence of radiation from an accelerated charge is invariant against a transformation between an accelerated and an inertial reference frame, and hence that an observer in the accelerated frame for whom the charge is permanently at rest will measure radiation from the charge.

Rohrlich [27] presented a discussion of the problem in 1963 containing a breakthrough in the understanding of the invariance properties of electromagnetic radiation. He calculated the electromagnetic field of a charge at rest in a uniformly accelerated reference frame (UAF) and found a purely electric static field. An observer at rest in AF experiences a static homogeneous gravitational field (SHGF). Rohrlich thus concluded that a charge at rest in a SHGF does not radiate. As observed from an inertial frame this charge is accelerated and radiates in agreement with Larmor’s formula. Hence, the next conclusion is that, the existence of radiation from an accelerated charge is not invariant against a transformation from an inertial frame to the accelerated permanent rest frame of the charge.

Rohrlich then referred to Rosen’s [28] calculation of the electromagnetic field of a freely falling charge in a SHGF, which showed that the freely falling charge emits radiation. As observed from a comoving observer this charge is permanently at rest in an inertial frame and obviously does not radiate. So the next conclusion is that the existence of radiation from a freely falling charge is not invariant against a transformation from the accelerated frame with a SHGF to the inertial rest frame of the charge. Rohrlich noted that since radiation is not a generally invariant concept the question whether the charge really radiates is meaningless unless it is referred to a particular coordinate system. He also emphasized that in this connection a definition of radiation that does not depend on taking the wave-zone limit is essential [20] and concluded: “These considerations are important in that they confirm the validity of the principle of equivalence also for charged test particles.”

Similar results were obtained by Kovetz and Tauber [29] in 1969. But instead of defining a SHGF as that in a uniformly accelerated reference frame in flat spacetime, they defined it by taking a local limit of the Schwarzschild spacetime. Their results agree with those of Rohrlich: a charge falling freely in a SHGF radiates as measured by a supported observer but not as measured by a comoving freely falling observer, and a supported charge in a SHGF radiates as measured by a freely falling observer but not as measured by a supported one.

In the same year Ginzburg [21] presented a thorough discussion of the relationship of the non-invariance of radiation and the principle of equivalence. He arrived at the same results as Rohrlich and Kovetz and Tauber, and he also discussed the physical meaning of the non-invariance of radiation from the point of view of the photon picture of radiation. Similar results were later deduced by direct calculation, essentially using Rohrlich’s definition of radiation, by Piazzese and Rizzi [30], by Ginzburg and Ya Eroshenko [31], and by Shariati and Khorrami [32]. Matsas [33] obtained related results in an analysis based upon the photon picture of radiation.

By studying the electromagnetic field associated with a uniformly accelerated charge Boulware [34] made in 1980 an important discovery that he summarized in the following way: “The equivalence principle paradox that the co-accelerating observer measures no radiation while a freely falling observer measures the standard radiation of an accelerated charge is resolved by noting that all the radiation goes into the region of space time inaccessible to the co-accelerating observer.” The same result was later deduced in a slightly simpler way by de Almeida and Saa [35].

There have been some later works where these results have not been accepted [36–39]. They seem to be based upon a different conception of radiation associated with the wave zone and a possible existence of photons, leading to a generally invariant character of radiation. However, the main stream definition of radiation is that of Rohrlich [20], making radiated effect Lorentz invariant, but not generally invariant, which is the definition adopted in here. On the other hand Singal [36] argued that neither a freely falling charge nor a charge at rest in a SHGF is observed to radiate irrespective of the state of motion of the observer, and Fabbri [38] concluded similarly for a freely falling charge. These results were countered by Parrott [40]. Furthermore, Logunov et al. [37] claimed: “A charge at rest in an inertial system cannot radiate, and this assertion does not depend on the system of coordinates in which the charge is considered. It is for this reason that the charge emits no radiation in the accelerated system either.” Concerning this Ginzburg et al. [41] wrote: “A charge may be considered to emit radiation if the flow of the Poynting vector across the surface surrounding the charge differs from zero. Such a definition, which is reasonable and justified, implies that a uniformly accelerated charge may radiate although there is no wave zone in this case, and one cannot speak about the appearance of photons. Logunov and co-workers understand by radiation only the field in the wave zone.” Finally Lyle [39] has written a book on this problem, but he too seems implicitly to have defined radiation in the same way as Logunov and coworkers and hence is outside the main stream physics on these matters.

4.4. Mathematical Expression of the Noninvariance of Radiation

According to Larmor’s formula (1.14), the effect radiated from an accelerated charge is proportional to the scalar product of the 4-acceleration of the charge with itself. This is a scalar and should be invariant against arbitrary coordinate transformations. However, the components of the 4-acceleration written with capital letters are assumed to refer to an inertial frame. Hence, when one is going to write down a formula for the effect radiated by an accelerated charge, which is to be valid also in an accelerated frame, Larmor’s formula has to be generalized. Such generalizations have been given by Kretzschmar and Fugmann [42, 43] and by Hirayama [44, 45].

Following Hirayama we shall write down Larmor’s formula in a uniformly accelerated reference frame. Using the Rindler coordinates the line element in the uniformly accelerated reference frame takes the form 𝑑𝑠²=−𝑔²𝑥²𝑑𝑡²+𝑑𝑥²+𝑑𝑦²+𝑑𝑧². (4.22) Here 𝑔 is the acceleration of gravity in the uniformly accelerated reference frame defined in the same way as the surface gravity of the Kerr-Newman spacetime, namely as 𝑥̈/𝑢^𝑡, where 𝑥̈ is the acceleration of a free particle instantaneously at rest as given by the geodesic equation, 𝑥̈=−Γ^𝑥_𝑡𝑡𝑡²̇. Calculating the Christoffel symbol from the metric (4.22) gives Γ^𝑥_𝑡𝑡=𝑔²𝑥. Using that the four-velocity of a particle instantaneously at rest is 𝑢^𝑡=𝑡̇=(−𝑔_𝑡𝑡)^−1/2=1/𝑔𝑥, we obtain 𝑥̈/𝑢^𝑡=(−𝑔_𝑡𝑡)^−1/2Γ^𝑥_𝑡𝑡=𝑔.

The 4-velocity and 4-acceleration of a charged particle moving along the 𝑥-axis in the Rindler frame are 𝑣^𝜇=𝑑𝑥^𝜇()𝑔𝑑𝜏=𝛾1,𝑣,0,0,𝛾=²𝑥²−𝑣^2−1/2,𝑎^𝜇=𝑑𝑣^𝜇𝑑𝜏+Γ^𝜇_𝛼𝛽𝑣^𝛼𝑣^𝛽=𝛾⁴𝑎+𝑔²𝑥−2𝑣²𝑥𝑣,𝑔²𝑥,0,0, (4.23) where 𝑣=𝑑𝑥/𝑑𝑡 and 𝑎=𝑑𝑣/𝑑𝑡. Inserting 𝑣=𝑎=0, we find that the 4-velocity and 4-acceleration of the reference particles in the Rindler frame are 𝑢^𝜇=1𝑔𝑥,0,0,0,𝑔^𝜇=10,𝑥,0,0. (4.24) Calculating the acceleration scalar of the reference particles, we find that it is equal to the acceleration of gravity in the Rindler frame, (𝑔_𝜇𝑔^𝜇)^1/2=𝑔. The projection tensor on the plane orthogonal to the world line of the charged particle has components ℎ^𝜇_𝜈=𝛿^𝜇_𝜈+𝑣_𝜈𝑣^𝜇. (4.25) Hirayama [44] defines an acceleration 4-vector 𝛼^𝜇=ℎ^𝜇_𝜈𝑎^𝜈−𝑔^𝜈−𝑔𝑢^𝜈. (4.26) At a moment, 𝜏=𝜏₀, when the particle is at rest, we have that 𝑣^𝜇=𝑢^𝜇. Hence, at this moment the Hirayama acceleration vector reduces to 𝛼^𝜇𝜏₀=𝑎^𝜇𝜏₀−𝑔^𝜇. (4.27) This shows that 𝛼^𝜇 represents the acceleration of a particle instantaneously at rest relative to an observer permanently at rest in the Rindler frame. It will be called the Hirayama 4-acceleration.

The generalized formula for the radiated effect valid in uniformly accelerated frames as well as in inertial ones is 𝑃_𝑅𝑈=𝑚₀𝜏₀1+𝑔𝑥𝑐²²𝛼^𝜇𝛼_𝜇. (4.28) This is not an ad hoc definition but based upon a thorough analysis given in [46] of the near field and far field produced by an accelerated charge as described from a uniformly accelerated reference frame.

4.5. The Freely Falling Charge

For a charge that falls freely from a position 𝑥₀ in a uniformly accelerated reference frame one finds for the position and the velocity 1+𝑔𝑥𝑐²=1+𝑔𝑥/𝑐²cosh𝑔𝑡/𝑐,𝑣=−𝑔1+𝑔𝑥𝑐²²sinh𝑔𝑡/𝑐cosh².𝑔𝑡/𝑐 (4.29) The components of the Hirayama 4-acceleration are 𝛼^𝑡=𝑒^{𝑔𝑡/𝑐}sinh𝑔𝑡/𝑐cosh²𝑔𝑡/𝑐𝑔1+𝑔𝑥₀/𝑐²²,𝛼^𝑥𝑒=−^{𝑔𝑡/𝑐}cosh²𝑔𝑡/𝑐1+𝑔𝑥₀/𝑐². (4.30) Hence, the acceleration scalar of the charges particle is 𝛼^𝜇𝛼_𝜇=𝑒^{2𝑔𝑡/𝑐}cosh𝑔𝑡/𝑐1+𝑔𝑥₀/𝑐²². (4.31) Inserting this into the generalized formula for the radiated effect in the uniformly accelerated reference frame (4.24) gives 𝑃_𝑅𝑈=𝑚₀𝜏₀𝑔²𝑒^{2𝑔𝑡/𝑐}. (4.32) The radiated energy is 𝐸_𝑅=^𝑡₀𝑃_𝑅𝑈𝑚𝑑𝑡=⁰𝜏₀2𝑔𝑒^{2𝑔𝑡/𝑐}−1. (4.33) The Schott energy is given by 𝐸_𝑆=−𝑚₀𝜏₀𝑣𝛼^𝑥. (4.34) Inserting the expressions for 𝑣 and 𝛼^𝑥 from (4.29) and (4.30), respectively, gives 𝐸_𝑆𝑚=−⁰𝜏₀2𝑔𝑒^{2𝑔𝑡/𝑐}−1=−𝐸_𝑅. (4.35) Hence, for a freely falling charged particle all of the radiated energy comes from the Schott energy.

5. An Electromagnetic Perpetuum Mobile?

Consider a charged particle moving freely in a circular path around the Earth. It has vanishing 4-acceleration. Hence the Abraham 4-vector vanishes, too. This means that it moves along a closed geodesic curve just like a neutral particle. It radiates in accordance with the Larmor formula. This is like a perfect perpetuum mobile [47]. One seems to get radiation energy for nothing from a system working cyclically.

The solution to this problem is that the LAD-equation is valid only in flat spacetime. The LAD-equation was generalized to an equation of motion for radiating charges in curved space by DeWitt and Brehme [48] and by Hobbs [49]. There is a large amount of recent literature on the equation of motion of a charged particle in curved spacetime. An eminent review with a large number of references has been given by Poisson et al. [50].

The generalized equation was applied to a particle moving freely in curved space by B. S. DeWitt and C. M. DeWitt [51]. They found that in the weak field and slow motion approximation the force upon the charge due to the curvature of space can be separated in a conservative part, 𝐟_C, and a nonconservative part, 𝐟_NC. Hence the equation of motion of a radiating charge moving freely in curved space takes the form 𝑚₀𝐫̈=𝐟_ext+𝐟_C+𝐟_NC. (5.1) For a particle moving slowly far outside the Schwarzschild radius of a mass 𝑚₀ B. S. DeWitt and C. M. DeWitt found the expressions 𝐟_C=32𝑚₀𝜏₀𝐺𝑀𝐫/𝑟⁴,𝐟_NC=−𝑚₀𝜏₀𝐫̇𝐺𝑀𝑟³𝐫̇−3𝐫⋅𝑟⁵𝐫. (5.2)

B. S. DeWitt and C. M. DeWitt write that 𝐟_C arises from the fact that the total mass of the particle is not concentrated at a point but is partly distributed as electric field energy in the space around the particle. The non-conservative force 𝐟_NC arises from a back-scatter process in which the Coulomb field of the particle, as it sweeps over the “bumps” in spacetime, receives “jolts” that are propagated back to the particle. This force gives rise to radiation damping even for a charge moving freely in curved spacetime. Hence, the radiated effect comes from the work performed by this force.

Therefore, the radiated effect should be equal to the work performed by this force per unit time, 𝑃=𝐟_NC⋅𝐫̇. (5.3) The motion of the particle is given by 𝐫=𝑟𝑒_𝑟,𝐯=𝐫̇=𝑟𝜔𝐞_𝜃,𝐚=𝐯̇=−𝑟𝜔²𝐞_𝑟,𝐚̇=−𝑟𝜔³𝐞_𝜃, (5.4) where 𝑎=|𝐚|=𝐺𝑀𝑟⁻². Hence, according to Larmor’s formula the radiated effect is 𝑚𝑃=⁰𝜏₀𝐺²𝑀²𝑟⁴. (5.5) With circular motion the velocity is orthogonal to the radius vector, 𝐫⋅𝐫̇=0. Then, (5.3) reduces to 𝐟_NC=𝑚₀𝜏₀𝐫̇𝐺𝑀𝑟³. (5.6) The work performed by unit time by this force is 𝑃_NC=𝐟_NC⋅𝐫̇=−𝑚₀𝜏₀̇𝐫̇𝐺𝑀𝐫⋅𝑟³𝑚=−⁰𝜏₀𝐺²𝑀²𝑟⁴, (5.7) which is minus the radiated effect given in (5.5). This means that due to the non-conservative part of the force due to the curvature of spacetime the charged particle loses energy at the same rate as it radiates energy, and hence energy is conserved.

6. Conclusion

The Schott energy, which is proportional to the product of the velocity and acceleration of the charge, is essential for conservation of energy of a radiating charge. It provides the energy radiated by a uniformly accelerated charge. The Schott energy is increasingly negative if the acceleration has the same direction as the velocity.

The Schott energy can be interpreted as the difference between the bound field energy the charge would have had if it had moved with its current velocity for its entire history and its actual bound field energy.

The existence of electromagnetic radiation is not invariant against a transformation involving acceleration. A freely falling charge radiates as observed by a stationary observer, but not by an inertial observer comoving with the charge.

The usual radiation reaction force vanishes for a charge moving freely in curved spacetime. However, it is replaced by a force due to the curvature of space, which accounts for the energy radiated by the particle.

Acknowledgments

This paper is based on a presentation given at the 2nd GIF Workshop February 19–23, 2012 at the Jerusalem College of Technology. The author would like to thank the organizers of the workshop, Yaakov Friedman and Tzwi Scarr, for friendly hospitality and the invitation to contribute. The author also appreciates comments from a referee that contributed in improving the paper.