Physica A: Statistical Mechanics and its Applications
Volume 320, 15 March 2003, Pages 329-356

Superluminal radiation by uniformly moving charges

Roman TomaschitzCorresponding Author Contact Information, E-mail The Corresponding Author

Department of Physics, Hiroshima University, 1-3-1 Kagami-yama, Higashi-Hiroshima 739-8526, Japan

Received 26 June 2002. 
Available online 16 November 2002.


The emission of superluminal quanta (tachyons) by freely propagating particles is scrutinized. Estimates are derived for spontaneous superluminal radiation from electrons moving close to the speed of the Galaxy in the microwave background. This is the threshold velocity for tachyon radiation to occur, a lower bound. Quantitative estimates are also given for the opposite limit, tachyon radiation emitted by ultra-relativistic electrons in linear colliders and supernova shock waves. The superluminal energy flux is studied and the spectral energy density of the radiation is derived, classically as well as in second quantization. There is a transversal bosonic and a longitudinal fermionic component of the radiation. We calculate the power radiated, its angular dependence, the mean energy of the radiated quanta, absorption and emission rates, as well as tachyonic number counts. We explain how the symmetry of the Einstein A-coefficients connects to time-symmetric wave propagation and to the Wheeler–Feynman absorber theory. A relation between the tachyon mass and the velocity of the Local Group of galaxies is suggested.

Author Keywords: Superluminal Liénard–Wiechert potentials; Quantum tachyons; Spontaneous tachyon radiation; Detailed balancing; Longitudinal radiation; Aether

PACS classification codes: 05.30.Ch; 42.25.Bs; 11.10.Lm; 98.70.Vc

Article Outline

1. Introduction
2. Superluminal radiation fields, their energy, and the power radiated
3. Does a uniformly moving charge radiate?
4. Quantization of the superluminal spectral densities and the radiant power
5. Spontaneous emission and absorption outside the lightcone: Einstein coefficients for free charges
6. Conclusion

1. Introduction

We will explore the spontaneous emission of tachyons by uniformly moving sources. In a relativistic setting such as electrodynamics, freely moving charges do not radiate and radiating particles slow down by radiation losses. (We will consider point charges without an internal structure.) Some explanations as to the context are therefore in order.

When considering superluminal signals, we have to give up relativity or causality, as Lorentz boosts can change the time order of spacelike connections [1, 2, 3, 4 and 5]. We will maintain causality, and model superluminal signals in an absolute spacetime as defined by the expanding galaxy grid, the rest frame of the microwave background. We may try a wave theory or a particle picture as the starting point. The latter has been studied for quite some time but did not result in viable interactions of tachyons with matter [6, 7, 8 and 9]. So we suggest to model tachyons as wave fields with negative mass-square, coupled by minimal substitution to subluminal particles.

Whatever the specifics of the superluminal wave equation, there is only one Green function supported outside the light cone; it is time symmetric, half-retarded, half-advanced. To achieve fully retarded wave propagation, an absorber is needed, capable of turning advanced modes into retarded ones [10, 11, 12, 13, 14 and 15]. A causal theory of superluminal signals requires an absolute space, quite independently of the actual mechanism of signal transfer. On this basis we can identify space itself as the absorber medium, the ether, the medium of wave propagation [16].

Having settled for a wave theory, we have to define the interaction of the superluminal modes with matter. This is the crucial point; after all, what else can one expect from a theory of tachyons other than suggestions as to where to search for them? We will maintain the best established interaction mechanism, minimal substitution, by treating tachyons as a sort of photons with negative mass-square, a real Proca field minimally coupled to subluminal particles [17 and 18]. Although great care is taken to maintain the analogy to electrodynamics, there are some basic differences. There is no gauge freedom but there is longitudinal radiation, even more pronounced than the transversal counterpart, due to the mass term in the wave equation. More importantly, this is not only a theory of superluminal wave motion, but also a theory of the absolute cosmic spacetime, this cannot be disentangled. The universal frame of reference is generated by the galaxy grid; it is the rest frame of the ether, the absorber medium, as well as the rest frame of the cosmic background radiations [19 and 20]. Uniform motion and rest are distinguishable states, and in this context we will show that freely moving charges can radiate superluminal quanta. They even do so without slowing down, as the radiated energy is drained from the absorber, from the oscillators of the ether. Superluminal radiation by inertial charges is but a manifestation of the absolute nature of space.

In Section 2 we will derive the superluminal power radiated by a classical point charge in arbitrary motion. We will discuss transversal and longitudinal radiation, its angular dependence, time symmetry outside the lightcone, the absorber field, retardation, and tachyonic Liénard–Wiechert potentials [21 and 22]. In Section 3, we specialize to uniformly moving charges and calculate the transversal and longitudinal spectral densities. In Section 4 these densities are quantized, and we discuss their asymptotic limits with respect to the speed of the radiating charge. We find a threshold velocity, a lower bound on the speed of the source, for tachyon radiation to occur. This is a pure quantum effect absent in the classical theory. This threshold happens to numerically coincide with the speed of the Galaxy in the microwave background, which suggests a connection between the tachyon mass and the velocity of the Local Group of galaxies in the ether,

Here, vLG/c≈2.10×10−3 is inferred from the temperature dipole anisotropy of the microwave background [23], and the electron–tachyon mass ratio Image is derived from Lamb shifts in hydrogenic ions [18]. At the end of Section 4, we derive estimates for superluminal radiation (spectral range, power, tachyonic mean energy and number counts, spectral maxima) by electrons in linear colliders and supernova shocks; in this way illustrating the three asymptotic regimes, that is, the ultra-relativistic limit, the non-relativistic limit, and the extreme non-relativistic limit close to the threshold velocity (1.1). In Section 5, we calculate the tachyonic emission rates for freely moving electrons in second quantization, in particular the Einstein A-coefficients for spontaneous emission [24]. The symmetry of the A-coefficients is linked to the spontaneous absorption of absorber quanta balancing the spontaneous tachyon emission. In Section 6, the conclusion, we further discuss radiation by inertial charges, the underlying spacetime concept, the ether, the absorber theory, and compare with the relativistic spacetime view.

2. Superluminal radiation fields, their energy, and the power radiated

The Proca equation [17] with negative mass-square, Fμνmt2Aμ=c−1jμ, can equivalently be written as (□+mt2)Aμ=−c−1jμ, subject to the Lorentz condition Aμ=0. The sign conventions for tachyon mass and field tensor are mt>0 and Fμν=Aν,μAμ,ν, for metric and d'Alembertian, ημν=diag(−c2,1,1,1) and □=ημνμν, respectively. The tachyon mass mt has the dimension of an inverse length, being a shortcut for mtc/planck constant over two pi. We find Image , estimated from Lamb shifts in hydrogenic systems [18]. The Lagrangian and the energy–momentum tensor of the free Proca field read

and the above field equations follow from L=LP+c−1Aαjα. The tachyonic E and B fields are related to the vector potential by

Ei=c−1Fi0=c−1(backward differenceA0−∂A/t), Fij=var epsilonijkBk ,Bk=(1/2)var epsilonkijFij=rotA, Aα=(A0,A) ,
so that FαβFαβ=−2(E2B2). The field equations decompose into

divB=0, rotE+c−1B/∂t=0 ,divE=ρ−c−1mt2A0, rotBc−1E/∂t=c−1j+mt2A ,
where we identified jμ=(ρ,j). The Lorentz condition c−2A0/∂t=divA apparently follows from the field equations and current conservation, ∂ρ/∂t+divj=0. The vector potential is completely determined by the current and the E and B fields, there is no gauge freedom due to the tachyon mass.

We represent the spatial component of the vector potential as

, Image , and the same relations hold for the time component, the charge and current densities, and the E and B fields. We consider tachyonic charges, by definition subluminal, located in the vicinity of the coordinate origin. The charges should be confined to a bounded region, so that we can use their asymptotic fields when calculating the energy flux radiated through a large sphere centered at the origin. In the subsequent example of uniformly moving charges, cf. Section 3, we will show how to circumvent this restraint by time averaging. The asymptotic radiation fields can be decomposed into transversally and longitudinally polarized components Image . To this end, we define and Image , with n=x/r, and find


This is completely general, there are no specific assumptions on the current, other than being localized in the vicinity of the coordinate origin, a bounded domain, that is. A discussion of superluminal Green functions and the derivation of (2.4) is given in [16]. The only classical Green function outside the lightcone is time-symmetric, half-retarded, half-advanced. Its convolution with the current results in a time-symmetric vector field Image , where Image stands for Image or Image , and the advanced field Image is likewise given by (2.4) with the substitution k(ω)→−k(ω). An absorber medium, the ether, is needed to cancel the advanced component of Image and to supply the missing half of the retarded field [12]. The oscillators of the ether [16 and 20] generate the absorber field, Image , which, when added to Image , results in the fully retarded Image in (2.4). In short, the retarded potential is a superposition of the time-symmetric field of the radiating particle and the absorber field. This is a crucial difference to electromagnetic radiation based on a retarded Green function. There is no radiation damping resulting from spontaneous tachyon radiation, since the energy balance for the time-symmetric field is zero; every outgoing mode has an incoming counterpart. The radiated energy stems from the absorber field, from the oscillators of the ether. The Lorentz force of the absorber field may be compared to inertia, and the derivation of the absorber field from the oscillators of the ether reminds us of the Mach principle, the attempt to extract the inertial force from the galaxy background. In both cases, the result is known beforehand, whatever the derivation.

The Fourier transforms of the field strengths and the time component of the 4-potential are readily calculated by making use of ((2.2) and (2.4)) and the Lorentz condition, Image . The polarized components read in leading order

The real-time field strengths ET,L(x,t) and BT,L(x,t) relate to these Fourier transforms as defined after (2.3), and so does the zero component of the 4-potential, A0T,L(x,t).

To illustrate the meaning of the integral transform Image defined in (2.5), we consider a subluminal particle x0(t), Image, arbitrarily moving in the vicinity of the coordinate origin. The particle carries a tachyonic charge q, resulting in the current density

We use the shortcuts vT(x,t)colon, equalsvn(n·v) and vL(x,t)colon, equalsn(n·v), and write (2.5) as

The asymptotic Liénard–Wiechert potentials and the corresponding field strengths are given by ((2.4) and (2.6)) with this Image inserted.

We turn to the energy density and the flux vector, which can be read off from ((2.1) and (2.2)),

T00=(1/2)(E2+B2)−(mt2/2)(c−2A02+A2) ,T0m=cE×B+mt2A0A .
Thus we find the transversal and longitudinal densities and the corresponding energy flux as

ρET(x,t)not, vert, similar(1/2)(ET2+BT2mt2AT2), STnot, vert, similarcET×BT ,

with the asymptotic fields ((2.4) and (2.6)) inserted. We have identified (ρET,ST) with T0μ, and (ρEL,SL) stands for −T0μ, so that the time-averaged densities are positive definite in either case. The averaging is readily carried out by means of the Fourier modes listed in ((2.4) and (2.6)). We find for the respective products of the transversal modes

The superscript T always stands for ‘transversal’ and is not to be confused with the time variable. In the integrand, we have already put ω=ω′ at several places, to save notation. The integral transform Image of the current can be singular, cf. (2.7) and Section 3, and therefore, we refrain from this identification in Image . A limit representation of the Dirac function,

will be used to avoid ill-defined squares of δ functions. According to (2.10), the time-averaged transversal flux and the energy density can be written as

where we insert the Fourier representations (2.12) to obtain

and analogously for Image . The longitudinal averages, cf. ((2.4) and (2.6)),

are substituted into

cf. (2.11), and we arrive at

The radiant power is obtained by integrating the flux through a sphere of radius r→∞,

P=PT+PL, PT,Lcolon, equalsr2n·left angle bracketST,Lright-pointing angle bracket dΩ ,
with the solid angle element dΩ=sin θ dθ dphi. Here, we use the asymptotic Pointing vectors ((2.15) and (2.18)), with the transforms Image of the current as defined in (2.5) or (2.8). This is applicable to any type of particle motion.

In Section 4, we will replace the classical current in the above formulas by current matrices, appealing to the correspondence principle. To this end, we assume the classical current to consist of a single Fourier mode ωmn:

so that Image , with an arbitrary Image . (The subscript mn is chosen for future reference.) We define the truncated Fourier transform

Such truncations result in smooth limit representations of the δ function, cf. (2.13), which admit unambiguous squares. The dω and dω′ integrations in ((2.15) and (2.18)) get trivial for large T, if we use Image in (2.5) with the current (2.21) inserted. We thus find the radiant powers, cf. (2.19):


We have defined here, cf. (2.5),

with Image and Image , where ncolon, equalsx/r. The longitudinal current transform Image in (2.24) depends on the tachyonic charge density only. To see this, we use the identity

valid up to a divergence; this is a consequence of current conservation as stated after (2.20). Hence,

Formulas ((2.22) and (2.23)) for the radiant power are exact; there is no multipole expansion involved. (We will return to them in 4 and 5, when quantizing.) The same holds for the power derived in (2.19) (with the asymptotic flux vectors ((2.15) and (2.18)) substituted), which is completely general, applying to any conserved current. In the next section we will work out the simplest example, radiation by uniformly moving charges.

3. Does a uniformly moving charge radiate?

We turn to the conceptually most interesting case, superluminal radiation emitted by uniformly moving charges. We derive here the classical theory, the first and second quantization will be carried out in the subsequent sections. We consider a tachyonic charge q, moving along the z-axis, z=vt, 0less-than-or-equals, slantv<c, so that ne3=cos θ, n=x/r. The integral transform (2.8) of the transversal and longitudinal current projections is easily calculated:

where k(ω) is negative for negative ω, cf. (2.4), and δ(1)(ω;T) is defined in (2.13). We have restricted the trajectory to a finite time interval [−T/2,T/2], so that the asymptotic formulas ((2.4) and (2.6)) apply, also compare (2.21). The time-averaged transversal Pointing vector is readily assembled, cf. (2.15):

By making use of (2.13) and

we may write this as

The argument of the δ function in (3.3) can only get zero for cos θ>0, therefore the Heaviside function Θ(cos θ). In (3.2), the limit T→∞ can be performed without compromising the asymptotics in (2.4). In this limit, the singular accelerations inflicted by the artificial, but technically convenient discontinuous truncation in (3.1) do not show in the time averages. We thus find the transversally radiated power, cf. ((2.19) and (3.4)):

The spectral energy density is identified by a variable change according to (3.3):

with ωmaxcolon, equalsmtvγ as the highest frequency radiated. The tachyon mass mt is a shortcut for mtc/planck constant over two pi and γ is the subluminal Lorentz factor (1−v2/c2)−1/2, so that ωmax is just an mt/m fraction of the electron energy. Another way to obtain the spectral density is to insert ((3.2) and (3.1)) into (2.19), and to perform the dω′ integration as above, followed by the angular integration:

This derivation is simpler, but conceals the angular dependence, explicit in (3.4).

The longitudinal flux is calculated via (2.18)

which can be evaluated in the same way as (3.2), resulting in

We thus find the longitudinal power

which in turn leads to the spectral density

with ωmax defined after (3.6). Alternatively, we may interchange the dω and the angular integrations as done in (3.7).

which coincides with (3.11). Flux vector and energy density relate in the usual way, Image , with vgr=c2/vph, and vph=v cos θ. There is no backward radiation, that is, for cos θless-than-or-equals, slant0. In the limit θ→π/2, the emitted tachyons approach infinite speed and zero energy. Radiation angle and frequency relate via ω=k(ω)v cos θ. To restore the units, we have to substitute mtmtc/planck constant over two pi in the above formulas. A detailed discussion of the spectral densities and powers will be given in the next section, after quantization. The classical formulas derived here are only valid if v/cmuch greater-thanmt/m. The Planck constant does not show in this constraint; however, the tachyon mass already enters in the classical field equations by the combination mtc/planck constant over two pi, cf. the beginning of Section 2.

4. Quantization of the superluminal spectral densities and the radiant power

We will investigate how far quantization modifies the classical picture given in Section 3, tachyon radiation by a structureless particle in uniform motion. To derive the quantized version of the spectral densities ((3.6) and (3.11)), we replace the classical current by the current matrix of a subluminal quantum particle carrying tachyonic charge as outlined at the end of Section 2. In doing so, we assume the correspondence principle; in Section 5, we will demonstrate that the spectral densities and powers calculated in this way can be recovered from the spontaneous emission rates in second quantization. We will not consider spin or antiparticles, and content ourselves with positive frequency solutions of the Klein–Gordon equation. The inclusion of spin is interesting if the electron orbits in a magnetic field, resulting in tachyonic cyclotron and synchrotron radiation, but there are otherwise no conceptual changes, the current being replaced by the matrix elements of the spinor current followed by polarization averages.

We start with the Klein–Gordon equation of a subluminal particle, c−2ψ,tt−Δψ+m2ψ=0, where m is a shortcut for mc/planck constant over two pi. We define the 4-current functionals

ρ(ψ,phi)colon, equalsiq(phi*ψ,t−ψphi,t*), j(ψ,phi)colon, equals−iqc2(phi*backward differenceψ−ψbackward differencephi*) ,
and note the continuity equation ρ,t+divj=0, where ψ and phi are arbitrary solutions of the wave equation. We use the separation ansatz ψi=ui exp(−iωit), ωi>0, and define the shortcuts ρmncolon, equalsρ(ψmn) and jmncolon, equalsjmn), as well as Image and Image , with ωmncolon, equalsωm−ωn. We hence find the time-separated wave equation Δui=(m2c−2ωi2)ui, as well as the Hermitian current matrices

We consider periodic boundary conditions on a box of size L and conveniently normalized eigenfunctions:

with ki=2πni/L and niset membership, variantZ3. The frequencies depend on the wave vectors via the subluminal dispersion relation ki2i2/c2m2. The current matrices Image and Image in (4.2) are composed with the ui in (4.3), and we substitute them into ((2.24) and (2.26)) (where all spatial integrations extend over the box size):



Kmncolon, equalskmknkmn)n, ncolon, equalsx/r.
Here, δ(1)(k;L) is the three-dimensional analog to the truncated integral representation of the δ function in (2.13); the limit procedure outlined there has likewise an obvious 3-d generalization by factorization, which we use when squaring these Image in the integrands of the classical powers ((2.22) and (2.23)):


The solid angle integration refers to the unit vector n and is easily done by means of the substitution Image . Hence,


The total power radiated is obtained by summing over the final states and performing the continuum limit:

We introduce polar coordinates for kn, with km as polar axis, and integrate dPtotT,L over the angular variables. This is easily done by means of the δ functions in ((4.8) and (4.9)), if we replace d3kn with Image . We thus obtain



Dmncolon, equals4km2kn2−(km2+kn2k2mn))2 ,
where Θ is the Heaviside function. The tachyonic wave vector relates to the subluminal frequencies by Image , with ωmnm−ωn. The dispersion relation for the subluminal charge is Image , and the same for km and ωm. The initial state is denoted by a subscript m, the final state by n, so that for emission ωmn>0. This designation of ‘initial’ and ‘final’ is arbitrary, just for the purpose of defining the radiation modes. By making use of the dispersion relations, we write Dmn as a function of ωmn:

with Image . There are two zeros, Dmnmn±)=0, where

Emission means ωmn>0, thus we can ignore the negative root and we will write ωmaxm) for ωmn+. It is easy to see that ωmax is positive only if ωm0, cf. (4.14), and ωmax0)=0. Clearly, ωmgreater-or-equal, slantedmc from the outset. It is likewise evident that Dmnmn)>0 for 0<ωmnmax and negative for larger frequencies. If ωm0, then Θ(Dmnmn)) in ((4.11) and (4.12)) vanishes for all ωmn>0, and hence ωm0 is a necessary condition for the emission of superluminal quanta. The spectral range is 0<ωmnmax, defined by Θ(Dmn)=1.

The total power radiated is Image , cf. ((4.11) and (4.12)). To obtain the frequency distributions, we introduce ωmn as integration variable. Using the dispersion relation for kn, we find ωnmn=−c2kn dkn and Image . Finally, ωmaxm)less-than-or-equals, slantωmmc for ωm0, which is easily seen from (4.15). (There is a double zero at Image .) Thus we can replace the upper integration boundary by ωmax and drop Θ(Dmn) in ((4.11) and (4.12)). We write in the following ω for ωmn, and define the densities pT,L(ω) dωcolon, equals−dPtotT,Lmn). We thus find, via the sub- and superluminal dispersion relations as stated after (4.13), the transversally and longitudinally radiated powers, the number counts, and the respective spectral functions:




where Image . The upper edge ωmax of the spectral range is positive only if ωm0. Spontaneous emission can only occur if the subluminal source surpasses a finite threshold energy ω0. This is unparalleled in the classical radiation theory, cf. Section 3.

At the upper edge of the spectrum, we have pTmax)=0, cf. ((4.14) and (4.15)), but the longitudinal density pL(ω) is still positive at ωmax. It may even happen that the integration in (4.16) is cut off before the maximum of pL(ω) is reached, so that pL(ω) is increasing throughout the spectral range, cf. the discussion following (4.26). The tachyonic mean energy is planck constant over two piωavT,Lcolon, equalsPtotT,L/NtotT,L, the emission rates NtotT,L (tachyons per unit time) are defined in (4.16). To get the dimensions right in ((4.17), (4.18) and (4.19)), we still have to rescale the masses, m(t)m(t)c/planck constant over two pi. The integrals in (4.16) are elementary, and we find the total transversally emitted power and the transversal count rate as


Here, ωmax is the break frequency defined in (4.17), the power scale is set by

and 0<arctan<π/2. The longitudinal power and count rate are

with ω0 defined in (4.17).

For the rest of this section, we will study asymptotic limits of the spectral densities, powers and number counts derived above. There are three asymptotic regimes giving a comprehensive picture of the radiation. To see this, we introduce the shortcut Image and write ωm=mcγ, with the subluminal γ=(1−v2/c2)−1/2. Since ωm0, we have apparently α<1 or

which is equivalent to ωmax>0. The velocity v refers to the subluminal particle. Condition (4.23) defines the threshold velocity vmin for tachyon radiation. To figure out the asymptotic regimes of the spectral functions with regard to v, we parametrize ωmax and ωm with α:

It is evident that ωmaxmmt/m. If α2much less-thanmt2/m2much less-than1, which defines the extreme non-relativistic regime, we find ωmaxmcα2. In the non-relativistic limit, mt2/m2much less-thanα2much less-than1, we find ωmaxmtcα. In the ultra-relativistic regime, with α2≈1 (and mt2/m2much less-than1), we find ωmaxmtc(1−α2)−1/2. This is to be compared to ωmmc in the two non-relativistic regimes, and to ωmmc(1−α2)−1/2 in the ultra-relativistic limit. (All these estimates are meant as leading orders in asymptotic double series expansions.) The extreme non-relativistic limit only applies to a very narrow velocity range, to velocities close to the threshold vmin, which is evident from γ=(1−α2)−1/2γmin.

We can now compare the foregoing to the classical radiation theory of Section 3. The upper edge of the spectrum, ωmax in (4.24), coincides with its classical counterpart defined after (3.6) in the limit mt2/m2much less-thanα2. This is the condition for the classical theory to apply. In this limit we can apparently identify α≈v/c, and it is also evident that ωmmuch greater-thanωmax (where ωmax is the highest frequency radiated, and ωm is the energy of the source). In the transversal spectral density (4.18), we may therefore replace ω0 by mc and drop the subsequent term ωmω, in this way recovering the classical density (3.6). The same reasoning applies to the longitudinal density pL(ω), which coincides with the classical formula (3.11) if we drop the ωmω and ω2/4 terms in (4.19). The powers ((4.20) and (4.22)) are turned into the classical ones in ((3.5) and (3.10)), by discarding all terms explicitly depending on the mt/m ratio.

The peak frequency of the transversal spectral density pT(ω), cf. (4.18), is a zero of

where yTcolon, equalsωpeakTm. We calculate this maximum in the three asymptotic regimes enumerated above. If α2much less-thanmt2/m2much less-than1, we can ignore the fourth order in (4.25) as well as the second, and find ωpeakTmcα2/2, just in the middle of the spectral range. If mt2/m2much less-thanα2much less-than1, we find, by dropping the fourth and first-order terms, Image , which is likewise located almost in the center of the frequency range. Finally, in the ultra-relativistic regime, α2≈1, we may again drop the first- and fourth-order terms in (4.25), so that the density is peaked at ωpeakTmtc.

The maximum of the longitudinal spectral function (4.19) is found by solving

with yLcolon, equalsωpeakLm, and we may always assume mtcmmuch less-than1. There are two positive solutions; the first, yL≈2, lies outside the integration range in (4.16), the relevant one is ωpeakLmtc, stemming from the second and zeroth orders of (4.26). Also this peak lies beyond ωmax if α2less-than-or-equals, slant1/2, since ωmaxmtc for α2≈1/2. Hence, if α2less-than-or-equals, slant1/2, then pL(ω) is increasing throughout the spectral range; if α2greater-or-equal, slanted1/2, it admits a maximum at ωpeakLmtc like the transversal density.

We turn to the asymptotic limits of the radiant powers and the count rates in ((4.20) and (4.22)). In the extreme non-relativistic regime, α2much less-thanmt2/m2much less-than1,

In the non-relativistic limit, mt2/m2much less-thanα2much less-than1,

The non-relativistic mean frequencies ωavT,L are close to the transversal spectral peak ωpeakT. In the extreme relativistic regime, α2≈1, we find

and the same formulas for the longitudinal radiation with the −1 after the log-terms dropped. The parameter α defining the three asymptotic regimes has been introduced after (4.22); it relates to the subluminal velocity of the source and the tachyon mass by

We still have to rescale the masses, m(t)m(t)c/planck constant over two pi, in all formulas of this section, and we define the tachyonic fine structure constant as αqcolon, equalsq2/(4πplanck constant over two pic), which is not to be confused with the expansion parameter α. We illustrate the quantities listed in ((4.27), (4.28) and (4.29)) with a freely moving electron as source. The electron–tachyon mass ratio is mt/m≈1/238, resulting in a tachyonic Compton wavelength of planck constant over two pi/(mtc)≈0.92 Å, and the quotient of tachyonic and electric fine structure constants reads αqe≈1.4×10−11, inferred from Lamb shifts in hydrogenic systems [18]. We will use αq≈1.0×10−13 and mt≈2.15 keV/c2. The quantities in ((4.27), (4.28) and (4.29)) can easily be assembled with these ratios and mtc2/planck constant over two pi≈3.27×1018 s−1.

As an example for the extreme non-relativistic limit (4.27), we assume the electron at vLG/c≈2.10×10−3, which is the velocity of the Galaxy in the microwave background, inferred from the dipole anisotropy of the background temperature, cf. the review article of Smoot and Scott in [23]. This speed coincides with the threshold velocity vmin in (4.23), recovered by putting α=0 in (4.30). This suggests that the velocity of the Local Group in the ether is linked to the tachyon mass as stated in (1.1). I do not have a real explanation for that, perhaps it is just a coincidence, but it is most intriguing indeed that vLG is the speed at which free electrons cease to emit tachyons, cease to drain energy from the ether.

In the extreme non-relativistic regime, α2much less-thanmt2/m2 (that is 0<αmuch less-than10−3), the electronic speed (parametrized as in (4.30)) is virtually independent of α, and very nearly coincides with the threshold velocity. We find with the above constants, cf. (4.27):

PtotT (eVs−1)≈3.1×1015α6, NtotT (s−1)≈1.8×1010α4,planck constant over two piωavT (eV)≈1.7×105α2 ,PtotL (eVs−1)≈9.4×1015α4, NtotL (s−1)≈3.7×1010α2,planck constant over two piωavL (eV)≈2.55×105α2 ,planck constant over two piωmax (eV)≈5.1×105α2, planck constant over two piωpeakT (eV)≈2.55×105α2,
and ωpeakL≈ωmax. The speed of the radiated quanta at the peak frequencies is readily found by the Einstein relation, planck constant over two piωpeakT,L=mtc2(vtach2/c2−1)−1/2. For α→0, the peak frequencies converge to zero, so that vtach/cmtc2/(planck constant over two piωpeakT,L) can attain virtually any value, with modest energy transfer, however. This applies to electrons freely propagating very close to the speed of the Local Group, vLG≈627 km/s, below this threshold no tachyons can be emitted in uniform motion.

The normal non-relativistic regime as defined by (4.28) is covered by 10−3much less-thanαmuch less-than1; in this case we may identify α≈v/c, cf. (4.30), and find

PtotT (keVs−1)≈3.5×105α3, NtotT (s−1)≈3.3×105α2,planck constant over two piωavT,L (keV)≈1.1α ,PtotL (keVs−1)≈3.5×105α, NtotL (s−1)≈3.3×105 ,planck constant over two piωmax (keV)≈2.15α, planck constant over two piωpeakT (keV)≈1.2α, ωpeakL≈ωmax .
In contrast to the extreme non-relativistic limit, these quantities scale with the particle speed, and the energy scale of this radiation is, by at least a factor of 103, larger.

Next-generation linear colliders will yield electrons with E≈0.5 TeV or γ≈9.785105 and our first ultra-relativistic example. In (4.29) we put 1−α2≈γ−2, and find

PtotT (GeVs−1)≈9.35, NtotT,L (s−1)≈5.1×105, planck constant over two piωavT (keV)≈18.2 ,PtotL (GeVs−1)≈9.7, planck constant over two piωavL (keV)≈18.9 ,planck constant over two piωmax (GeV)≈2.1, planck constant over two piωpeakT,L (keV)≈2.15 .
The spectral range is much larger than in the previous examples, and the longitudinal spectral density has a genuine maximum coinciding with the transversal peak frequency; in contrast to the non-relativistic limits, where the longitudinal density is truncated at the break frequency ωmax before the peak is reached. The spectral peaks are not very pronounced, they deviate from the mean frequencies by one order of magnitude. A further increase of the Lorentz factor does not substantially change the radiant powers and the number counts in (4.29), although it strongly affects the shape of the spectral densities, since ωmaxpeakT,Lnot, vert, similarγ. For instance, electrons shock-accelerated to E≈200 TeV (γ≈3.91×108) in supernova remnants [25, 26 and 27] radiate

PtotT (GeVs−1)≈13.6, NtotT,L (s−1)≈5.1×105, planck constant over two piωavT (keV)≈26.4 ,PtotL (GeVs−1)≈13.9, planck constant over two piωavL (keV)≈27.1 , planck constant over two piωmax (TeV)≈0.84 ,
with planck constant over two piωpeakT,L as in (4.33). At this point, one could be tempted to define a radiation lifetime, something like E/P. However, tachyon radiation is generated by a time-symmetric Green function and an absorber field. Contrary to electromagnetic radiation, there is no deceleration due to radiation loss, the energy spontaneously radiated is contained in the absorber field, supplied by the oscillators of the absorber. In the next section, we will demonstrate that the classical time symmetry (discussed after (2.5)) has its quantal counterpart in the symmetry of the Einstein A-coefficients, the spontaneous emission being balanced by spontaneous absorption.

5. Spontaneous emission and absorption outside the lightcone: Einstein coefficients for free charges

We will study induced and spontaneous radiation in second quantization. A non-relativistic example to that effect, tachyonic transitions between bound states in a Coulomb potential, has already been given in [24]. Here, we consider tachyon radiation by freely propagating electrons. In this case, the Einstein coefficients can be calculated without multipole approximations. The B-coefficients reflect the symmetry of the induced radiation, however, the A-coefficients are symmetric as well. In electrodynamics, there is no time-symmetric counterpart to spontaneous emission, but outside the lightcone there is spontaneous absorption, the radiated energy being recovered from the absorber medium. The Green function is time symmetric, and so is spontaneous radiation. The spontaneous absorption corresponds to the advanced component of the classical radiation field, cf. Section 2. The quantum statistics of the free tachyon field was studied in [20], we repeat some formulas needed to compile the matrix elements of the Hamiltonian. Then we calculate and balance the emission rates for uniformly moving charges. Finally we show that the spectral densities ((4.18) and (4.19)) derived by means of the correspondence principle survive the second quantization. In this paper, the Fourier transforms of the dielectric and magnetic permeabilities of the ether are put equal to one, Image , that is, we assume a negligible refractivity and absorptivity, cf. [20]. Otherwise we would have to specify more parameters, apart from the tachyon mass and the tachyonic fine structure constant.

We start with the plane wave decomposition of the spatial component of the vector potential

with kcolon, equalsn/L. The summation is over integer lattice points n in R3, corresponding to periodic boundary conditions in (2.3), so that the L−3/2exp(ikx) are orthonormal and complete in a box of size L. The var epsilonk,1 and var epsilonk,2 are arbitrary real unit vectors (linear polarization vectors) orthogonal to var epsilonk,3colon, equalsk0=k/|k|, so that the var epsilonk constitute an orthonormal triad for every n. The amplitudes â(k,λ) are arbitrary complex numbers.

The Fourier coefficients Â0(k) of the time component A0(x,t) of the 4-potential are defined as in (5.1), and the same holds for the field strengths,

For A to be a solution of (2.3) (with ρ=0, j=0), the dispersion relation,

k22/c2+mt2 ,
has to be satisfied, which we henceforth assume; ω and kcolon, equals|k| are positive.

We split the potential and the field strengths into transversal and longitudinal components:

The amplitudes â(k,λ), λ=1,2,3, can be arbitrarily prescribed. Time averages of products such as Image , over a period of 2π/ω, are readily calculated according to We find the spatially integrated and time-averaged energy T00 and the flux vector T0m in (2.9) as, cf. (5.4),

The interference term of the longitudinal and transversal modes vanishes in the averaging procedure. The sign change of the longitudinal components of energy and flux, anticipated in (2.11), will be effected by Fermi statistics. By comparing the individual modes in these series, we find

The group velocity vgr follows from the dispersion relation (5.3), dω/dk=c2k/ω, cf. the discussion after (3.12). We introduce rescaled Fourier coefficients ak in the preceding time averages,

â(k,λ)=:2−1/2cplanck constant over two pi1/2ω−1/2ak, â(k,3)=:2−1/2planck constant over two pi1/2ω1/2mt−1ak,3 ,
where λ=1,2, so that the field energy and the flux get amendable to statistical interpretation


These time averages are the starting point for quantization. We sketch here only very shortly the overall reasoning, for details see [20]. The Fourier coefficients ak are interpreted as operators, and the complex conjugates ak* as their adjoints ak+. We use commutation relations, [ak, ak′,λ′+]=δkkδλλ′, for the transversal modes λ=1,2, which admit the occupation number representation

Anticommutators, [ak,3,ak′,3+]+kk, are employed for the longitudinal modes, to turn the longitudinal energy (5.9) into a positive definite operator. These Fermi operators admit the representation

where the occupation numbers are now restricted to zero and one. The time-averaged transversal Hamilton operator for the free tachyon field and the transversal flux operator are thus given in (5.8), with the Fourier amplitudes akak* replaced by the operator product ak+ak. The energy and flux operators of the longitudinal radiation are obtained by the substitution ak,3ak,3*→−ak,3+ak,3 in (5.9). The partition function is easily assembled, the lattice sums being replaced by the continuum limit [28 and 29], and we find the spectral densities of the transversal and longitudinal radiations as


We turn to the interaction with subluminal matter. As in Section 4, we consider a spinless quantum particle, a Klein–Gordon field coupled to the tachyonic vector potential by minimal substitution. We write the Lagrangian of the coupled system as L=LP+Lψ, with the Lagrangian LP of the free Proca field as in (2.1), and

Lψcolon, equalsc−2tAψ∂tA*ψ*backward differenceAψ backward differenceA*ψ*−(mc/planck constant over two pi)2ψψ* ,∂tAcolon, equalst−i(q/planck constant over two pic)A0, backward differenceAcolon, equalsbackward difference−i(q/planck constant over two pic)A .
The energy density of the free matter field reads

Hψfreecolon, equalsc−2ψ,tψ,t*+backward differenceψbackward differenceψ*+(mc/planck constant over two pi)2ψψ*=c−2,tψ,t*−ψ*ψ,tt) ,
the second equality is valid up to a divergence, and we used the free field equation as stated before (4.1). The 4-current, the time separation, and the spectral resolution are given in ((4.1), (4.2) and (4.3)). We expand the free Klein–Gordon field, Image , with arbitrary complex amplitudes bn, normalized eigenfunctions un, cf. (4.3), and positive frequencies ωn. We thus find the energy of the free field, E=∫Hψfreed3x=∑nplanck constant over two piωnbnbn*, via the orthonormality (4.3). In the 4-current (4.1), we at first put phi=ψ and then expand the wave field, so that

with Image and Image defined in ((4.2) and (4.3)). The interaction Hamiltonian can be read off from the Lagrangian (5.1),

up to terms of O(q2). Hence, by means of (4.1),

Here we substitute the Fourier expansions (5.15) as well as those of the tachyon field defined by ((5.1), (5.4) and (5.7)). Finally, we replace the bmbn* in (5.15) by operator products bn+bm, and the tachyonic field amplitudes ak(*) by operators ak(+) as done after (5.11) for the free field. The subluminal spinless Klein–Gordon field is quantized in Bose statistics, [bm,bn+]=δmn, so that the representation (5.10) is applicable, and the (anti)commutator brackets and representations for the tachyonic operators ak(+) are stated in ((5.10) and (5.11)).

First we study interaction with transversal tachyons. We consider a fixed linear polarization λ (that is, no summation over λ in the Fourier series). The transversal component of the interaction Hamiltonian (5.17) reads HintTcolon, equalsplanck constant over two pi−1c−3ATj(ψ), where we substitute the Fourier decompositions ((5.1), (5.4) and (5.7)), and (5.15),

The amplitudes have been replaced by operators bi(+) and ak(+) as indicated after (5.17). The transversal ak,λ=1,2(+) satisfy Bose statistics. We compile the matrix elements of (5.18) with an initial state m and a final state n representing a single subluminal particle and the absorption or emission of a tachyon of polarization λ,

The nk are tachyonic occupation numbers for a state of polarization λ. At this point, k is a discrete lattice vector, cf. (5.1). The left angle bracketTabs,emTright-pointing angle bracket just differ by a sign change of the wave vector in the exponential. (The upper sign always refers to absorption.) The preceding formulas are standard time-dependent perturbation theory with a periodic potential [30]; the nk-dependent factors stem from the bosonic representation (5.10). The tachyonic wave vector k relates to the tachyonic frequency ωk by the dispersion relation (5.3); k and ωk are positive, and the ωmncolon, equalsωm−ωn refer to energy levels of the free wave equation, cf. (4.3). The initial state will be denoted by a subscript m and the final state by n, so that a positive ωmn stands for emission.

We turn to the longitudinal component of the interaction (5.17), HintL=HintL(1)+HintL(2), where HintL(1)=−planck constant over two pi−1c−3ALj(ψ) and HintL(2)=−planck constant over two pi−1c−3A0ρ(ψ), with the Fourier series for AL and A0 defined in ((5.1) and (5.4)) and (5.7). We find, analogously to (5.18),


We have here restored the units, mtmtc/planck constant over two pi. The longitudinal operators ak,3(+) anticommute, the representation (5.11) applies, and we assemble the matrix elements of the longitudinal interaction operator as

Here, nk is an occupation number in Fermi statistics, zero or one, and (−)n<m denotes the sign factor occurring in the fermionic representation (5.11); k0=k/k is the tachyonic unit wave vector. The generalization of the matrix elements ((5.19) and (5.21)) to a refractive and absorptive spacetime can be found in [20]. Finally we return to ((4.1), (4.2) and (4.3)), and inspect the integral ∫(umΔun*un*Δum)e±ikx d3x, once by applying the Gauss theorem, and once by using the Klein–Gordon equation. In this way we derive Image , valid under the integral sign, cf. (2.25). Thus, we can express the longitudinal T-matrix by the charge density alone:

where we used energy conservation, ωk=minus-or-plus signωmn in (5.21), as well as the tachyonic dispersion relation (5.3) (with mtmtc/planck constant over two pi.)

Once the matrix elements are known, the transition rate for transversally induced absorption and emission in a given polarization λ is obtained by a standard procedure [30],

valid for large times t, with the smooth Dirac limit function δ(1) as defined in (2.13). We have here replaced the box-summation by the continuum limit, L3(2π)−3∫dk, and the occupation numbers by their averages left angle bracketnkright-pointing angle bracket=(eβplanck constant over two piωk−1)−1, cf. (5.12). dΩ=sin θ dθ dphi, the solid angle element of the tachyonic wave vector, and k(ω) is given in (5.3). The same formula also applies to spontaneous radiation, wemT,sp, but with the left angle bracketnkright-pointing angle bracket-factor dropped, since wemT,sp stems from the +1 under the root in (5.19). In the limit t→∞, the dω-integration in (5.23) gets trivial, and by substituting (5.19) we find


dwemT,spnot, vert, similar(eβplanck constant over two piω−1)dwemT,ind=:AmnT(k,λ) dΩ ,
where ω (and k(ω)) is taken at |ωmn|. The upper sign refers to absorption, and m to the initial state. The transversal tachyonic spectral density ρT is defined in (5.12). (The spectral densities in (5.12) refer to the tachyonic heat bath triggering the induced radiation.) The total emission rate is dwemT=dwemT,ind+dwemT,sp. In equilibrium, induced emission and absorption compensate each other, due to the detailed balancing symmetry BmnT(k,λ)=BnmT(−k,λ), which follows from the hermiticity of the current matrices (4.2). The spontaneous emission of transversal tachyons is temperature independent, unaffected by the tachyonic heat bath, in contrast to the longitudinal emission discussed below. The unpolarized transversal radiation rates are obtained by replacing Image in (5.24) by the transversal current, Image , where k0colon, equalsk/k and

cf. ((4.2) and (4.3)).

The spontaneous emission rate (5.25) is symmetric, AmnT(k,λ)=AnmT(−k,λ), reflecting the time symmetry of the classical radiation field, cf. Section 2. (The radiation discussed in the previous sections is all spontaneous.) The retarded field, which we have quantized, results from the absorber field complementing the time-symmetric field of the particle, as pointed out after (2.5). The net energy balance of the time-symmetric field is zero, as the spontaneous emission of a tachyon is accompanied by the absorption of an absorber quantum. This restores the initial state of the source in the reverse transition. Spontaneous absorption stands as the quantal analog to the advanced modes of the time-symmetric classical wave field. Induced transitions are not affected by the absorber field, and in equilibrium induced emission and absorption cancel each other, due to the mentioned symmetry of the B-coefficients. In the energy balance for the equilibrium distribution ρT(ω) in (5.12), the different Boltzmann weights are accounted for by the A-coefficients,

and the occupation numbers relate by Nm/Nn=exp(−βplanck constant over two piωmn), quite independent of the statistics.

We turn to longitudinal radiation. The induced absorption/emission rate for longitudinal tachyons is composed analogously to the transversal rates (5.23),

with left angle bracketTabs/emLright-pointing angle bracket in (5.22). The fermionic occupation numbers are replaced in the continuum limit by the averages left angle bracketnkright-pointing angle bracket=(eβplanck constant over two piωk+1)−1, cf. (5.12). Hence,

where m denotes the initial state, both for absorption and emission, and ω=|ωmn|. The longitudinal spontaneous emission is identified as follows. The nk in (5.21) can only take the values zero and one, so that the factor 1−nk does not change if squared. Thus the total emission rate is dwemL=dwem,T=0L,sp−dwemL,ind, with dwemL,ind as defined by (5.29) and

dwem,T=0L,spcolon, equals(eβplanck constant over two piω+1) dwemL,ind .
This is the spontaneous transition rate in the zero temperature limit, obtained from (5.28) with the nk-factors dropped. At finite temperature, the spontaneous emission is dwemL,sp=dwem,T=0L,sp−2dwemL,ind, so that the total emission dwemL=dwemL,ind+dwemL,sp. Hence,

dwemL,spnot, vert, similar(eβplanck constant over two piω−1) dwemL,ind=tanh(βplanck constant over two piω/2) dwem,T=0L,sp=:AmnL(k) dΩ ,
which reduces in the absence of a tachyonic heat bath to dwem,T=0L,sp The basic symmetries BmnL(k)=BnmL(−k) and AmnL(k)=AnmL(−k) also extend to longitudinal radiation, so that the induced transitions cancel each other, and a spontaneous transition is instantaneously restored by an absorber quantum. The longitudinal spontaneous emission (5.31) is temperature dependent and vanishes in the high temperature limit. At finite temperature, the equilibrium condition, cf. (5.27),

BmnL(kL(ω)+AmnL(k)=BnmL(−kL(ω) exp(βplanck constant over two piωmn) ,
requires the longitudinal density ρL(ω) in (5.12).

I take this opportunity to correct a mistake in the dipole approximation of the longitudinal transition probability calculated in [20]. The squared ratio planck constant over two piωji/(mtc2) in (5.17), (5.23) and (5.27) of [20] should be inverted. At 2.2 MeV, the ratio of the longitudinal and transversal dipole transition rates reads ωLT≈3.8×10−7, from which we conclude that the longitudinal background radiation has reached equilibrium within 1018 s. This is to be compared with a cosmic age of H0−1≈14 Gyr≈4.4×1020 s. The reasoning behind this is explained in [20].

At zero temperature, the power spontaneously radiated by a freely propagating charge was calculated in Section 4 by means of the correspondence principle, which amounts to identify in ((2.20), (2.21), (2.22), (2.23), (2.24), (2.25) and (2.26)) Image and Image with the hermitian current matrices Image and Image in (5.26). The powers ((4.20) and (4.22)) can be recovered from the emission rates dwemT,sp and dwem,T=0L,sp in ((5.24), (5.25) and (5.29)), and (5.30). The angular-integrated power radiated at ω=ωmn is apparently

We consider unpolarized transversal radiation, which means to replace Image in dwemT,sp by the transversal current Image . If we substitute the current (5.26) into the powers ((2.22) and (2.23)), we obtain (5.33); thus the spectral densities ((4.18) and (4.19)) also hold in second quantization.

6. Conclusion

The absorber theory [12] was motivated by Dirac's covariant version of radiation damping [15], where the absorber field, half-retarded minus half-advanced, enters as Lorentz force. In the non-relativistic derivation of Abraham and Lorentz [14] it does so as well, of course, but in a less explicit way. In any case, this field is not perceived as stemming from an absorber medium, but rather as generated by the charge itself. In Dirac's theory, the absorber field does not show as radiation field in the equations of motion, but is exclusively applied along the trajectory of the charge, defining the damping force. Here we have elaborated on superluminal radiation fields at large distance from the source, the opposite limit. The asymptotic fields are quite sufficient to calculate the spectral densities and the radiant power, classically as well as in second quantization. It is not advisable to rely on the short distance behavior of Green functions; the self-energy problem indicates that the Maxwell theory may just be the asymptotic limit of a non-linear Born–Infeld type of electrodynamics [31]. If so, one cannot use the linearized theory in the vicinity of the radiating sources. The same holds for the Proca field.

Wheeler and Feynman designed the absorber theory for electrodynamics, and they interpreted the half-retarded minus half-advanced Liénard–Wiechert potential, cf. Section 2, as generated by an absorber medium, which they proposed to be the collection of electric charges in the universe [12]. They used this potential in an action-at-a-distance electrodynamics [10, 11 and 13], in an attempt to solve the radiation damping problem. In the Maxwell theory, we do not consider an absorber medium because there is a retarded Green function. Outside the lightcone, however, retardation can only be achieved by an absorber field, as the Green function supported there is time symmetric. A causal theory of superluminal signals needs an absolute spacetime, since Lorentz boosts do not preserve the time order in spacelike connections. Once the absolute nature of space is acknowledged, it is only a small step to identify space itself as the absorber medium, the ether, whose microscopic oscillators generate the absorber field [16 and 20].

I conclude by comparing the absolute spacetime underlying superluminal radiation to the relativistic spacetime view. Radiation by inertial charges may be unimaginable in relativity theory, but in the absolute cosmic spacetime this is easy to comprehend, since accelerated and inertial frames are treated on the same basis. There is a universal reference frame, the rest frame of the ether, generated by the comoving galaxy grid and manifested by the microwave background and other background radiations. The spectral density of the radiation is determined by the velocity of the uniformly moving charge. This is not a relative velocity, it stands for the absolute motion of the charge in the ether. Relative velocities only affect the appearance of the radiation in moving frames. In the rest frames of inertial observers, the radiation field may appear advanced, the transversal and longitudinal modes may appear tangled, or they may not appear at all, as it happens in the rest frame of the radiating charge [24], but all this is a consequence of the observer's individual motion. Whatever the appearance of the superluminal radiation field in a moving frame, the observer can infer the radiation in the rest frame of the ether (such as the power, the spectral densities and the frequencies radiated) by measuring the absolute velocity of the charge in the microwave background.

More generally, the relativity principle asserts that the laws of nature are the same in all inertial frames, in particular, uniform motion and rest are not distinguishable in this respect. In the absolute cosmic spacetime, the laws of nature are inherent in the rest frame of the ether, and their appearance in inertial frames is determined by the observer's state of motion. This is in sharp contrast to relativity theory, where the laws of nature are thought of as attached to individual and equivalent inertial frames. The absolute spacetime concept is centered at the state of rest, tantamount to the universal reference frame generated by the galaxy grid. Particles move in the ether, subjected to the flow of cosmic time as defined by the galactic recession, without resort to the inertial frames and proper times of individual observers. This is again in strong contrast to relativity theory, where inertial frames are the substitute for the universal rest frame. In the absolute cosmic spacetime, the crucial distinction is not between inertial and accelerated frames, but simply between motion and rest, and therefore it is not surprising that uniformly moving charges radiate.


The author acknowledges the support of the Japan Society for the Promotion of Science. The hospitality and stimulating atmosphere of the Centre for Nonlinear Dynamics, Bharathidasan University, Trichy, the Institute of Mathematical Sciences, Madras, and the Tata Institute of Fundamental Research, Bombay, are likewise gratefully acknowledged. I would like to thank Nandor Balazs and George Contopoulos for exciting discussions.


1. G. Feinberg Phys. Rev. 159 (1967), p. 1089. 

2. G. Feinberg Sci. Am. 222 2 (1970), p. 69.

3. R. Newton Science 167 (1970), p. 1569. 

4. R. Tomaschitz Chaos Solitons Fractals 7 (1996), p. 753. 

5. R. Tomaschitz Chaos Solitons Fractals 8 (1997), p. 761. 

6. Th. Des Coudres Arch. Néerland. Sci. (II) 5 (1900), p. 652.

7. S. Tanaka Prog. Theor. Phys. 24 (1960), p. 171. 

8. Ya.P. Terletsky Sov. Phys. Dokl. 5 (1961), p. 782.

9. G. Feinberg Phys. Rev. D 17 (1978), p. 1651. 

10. A.D. Fokker Physica 9 (1929), p. 33.

11. A.D. Fokker Physica 12 (1932), p. 145.

12. J.A. Wheeler and R.P. Feynman Rev. Mod. Phys. 17 (1945), p. 157. 

13. J.A. Wheeler and R.P. Feynman Rev. Mod. Phys. 21 (1949), p. 425.

14. H.A. Lorentz Theory of Electrons, Dover, New York (1952).

15. P.A.M. Dirac Proc. Roy. Soc. A 167 (1937), p. 148.

16. R. Tomaschitz Class. Quant. Grav. 18 (2001), p. 4395. 

17. A. Proca J. Phys. 7 (1936), p. 347. 

18. R. Tomaschitz Eur. Phys. J. B 17 (2000), p. 523. 

19. R. Tomaschitz Celest. Mech. Dyn. Astron. 77 (2000), p. 107.

20. R. Tomaschitz Physica A 307 (2002), p. 375. 

21. E. Wiechert Arch. Néerland. Sci. (II) 5 (1900), p. 549.

22. A. Sommerfeld Proc. K. Ned. Akad. Wet. (Sec. Sci.) 7 (1904), p. 346.

23. K. Hagiwara et al.Phys. Rev. D 66 (2002), p. 010001. 

24. R. Tomaschitz Physica A 293 (2001), p. 247. 

25. K. Koyama et al.Nature 378 (1995), p. 255. 

26. O.C. de Jager et al.Astrophys. J. 457 (1996), p. 253. 

27. S.P. Reynolds and J.W. Keohane Astrophys. J. 525 (1999), p. 368. 

28. B. Mashhoon Phys. Rev. D 8 (1973), p. 4297. 

29. S. Singh and R.K. Pathria Phys. Rev. A 30 (1984), p. 442.

30. L.D. Landau and E.M. Lifshitz Quantum Mechanics: Non-relativistic Theory, Pergamon, London (1991).

31. M. Born and L. Infeld Proc. Roy. Soc. A 144 (1934), p. 425. 

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