Equilibrium for Classical Zero-Point Radiation: Detailed Balance Under Scattering by a Classical Charged Harmonic Oscillator

TL;DR

This paper demonstrates that relativistic classical scattering of zero-point radiation by a charged harmonic oscillator maintains detailed balance across all harmonics, supporting the classical derivation of the Planck spectrum.

Contribution

It provides the first example of detailed balance of zero-point radiation under relativistic scattering, extending previous nonrelativistic results to all harmonics.

Findings

Detailed balance holds at fundamental and first harmonic frequencies.

Zero-point radiation spectrum is linear in frequency.

Balance likely extends to all harmonics in relativistic scattering.

Abstract

It has been shown repeatedly over a period of 50 years that the use of relativistic classical physics and the inclusion of classical electromagnetic zero-point radiation leads to the Planck blackbody spectrum for classical radiation equilibrium. However, none of this work involves scattering calculations. In contrast to this work, currently accepted physical theory connects classical physics to only the Rayleigh-Jeans spectrum. Indeed, in the past, it has been shown that a nonlinear classical oscillator (which is necessarily a nonrelativistic scattering system) achieves equilibrium only for the Rayleigh-Jeans spectrum where the random radiation present at the frequency of the second harmonic of the oscillator motion has the same energy per normal mode as the radiation present at the fundamental frequency. Here we continue work emphasizing the importance of relativistic versus nonrelativistic analysis. We consider the scattering of random classical radiation by a charged harmonic oscillator of small but non-zero oscillatory amplitude (which can be considered as a relativistic scattering system) and show that detailed radiation balance holds not only at the fundamental frequency of the oscillator but through the first harmonic corresponding to quadrupole scattering, provided that the radiation energy per normal mode at the first harmonic is double the radiation energy per normal mode at the fundamental frequency. This condition corresponds exactly to the zero-point radiation spectrum which is linear in frequency. It is suggested that for this relativistic scattering system, the detailed balance for zero-point radiation holds not only for the fundamental and first harmonic but extends to all harmonics. Here we have the first example of the detailed balance of zero-point radiation under relativistic scattering.

Equations

$$\mathbf{E}(\mathbf{r},t)=\sum _{\mathbf{k}}\sum _{\lambda =1}^2\widehat{ϵ}(\mathbf{k},\lambda )\mathfrak{h}(\omega )\cos [\mathbf{k}\cdot \mathbf{r}-\omega t+\theta (\mathbf{k},\lambda )]$$

$$\mathbf{B}(\mathbf{r},t)=\sum _{\mathbf{k}}\sum _{\lambda =1}^2\hat{k}\times \widehat{ϵ}(\mathbf{k},\lambda )\mathfrak{h}(\omega )\cos [\mathbf{k}\cdot \mathbf{r}-\omega t+\theta (\mathbf{k},\lambda )]$$

$$\mathfrak{h}(\omega )={\left(\frac{8\pi \mathcal{E}(\omega )}{a^3}\right)}^{1/2}$$

$$\mathbf{E}(\mathbf{r},t)=\sum _{\mu ,\mathbf{k},\lambda }\widehat{ϵ}\frac{\mathfrak{h}}{2}FRA$$

$$\mathbf{B}(\mathbf{r},t)=\sum _{\mu ,\mathbf{k},\lambda }\hat{k}\times \widehat{ϵ}\frac{\mathfrak{h}}{2}FRA$$

$$\sum _{\mu }f(\mu )=f(-1)+f(+1)$$

$$F=\exp [i\mu \mathbf{k}\cdot \mathbf{r}],$$

$$R=\exp [-i\mu \omega t],$$

$$A=\exp [i\mu \theta (\mathbf{k},\lambda )].$$

$$m\ddot{x}=-m{\omega }_0^{2}x+m\Gamma \stackrel{...}{x}+e{E}_x(0,t)$$

$$x(t)=\frac{e}{2m}\sum _{\mu ,\mathbf{k},\lambda }{ϵ}_x\mathfrak{h}\frac{RA}{C}$$

$$C=-{\left(\mu \omega \right)}^2+{\omega }_0^2-i\Gamma {\left(\mu \omega \right)}^3.$$

$$⟨x^2⟩$$

$$={\left(\frac{e}{2m}\right)}^2\sum _{{\mu }_1,{\mathbf{k}}_1{,\lambda }_1}\sum _{{\mu }_2,{\mathbf{k}}_2{,\lambda }_2}{ϵ}_{x1}{ϵ}_{x2}{\mathfrak{h}}_1{\mathfrak{h}}_2\frac{R_1{R}_2}{C_1{C}_2}⟨A_1{A}_2⟩$$

$$⟨A_1{A}_2⟩$$

$$={\delta }_{{\mu }_1,-{\mu }_2}{\delta }_{{\mathbf{k}}_1,{\mathbf{k}}_2}{\delta }_{{\lambda }_1,{\lambda }_2}={\delta }_{1(-2)}$$

$$⟨x^2⟩$$

$$=2{\left(\frac{e}{2m}\right)}^2\sum _{{\mathbf{k}}_1{,\lambda }_1}{ϵ}_{x1}^2{\mathfrak{h}}_1^2\frac{1}{C_1{C}_{(-1)}}$$

$$k_x=l\frac{2\pi }{a},k_y=m\frac{2\pi }{a},k_z=n\frac{2\pi }{a},$$

$$\sum _{\lambda }{ϵ}_x^2=\frac{k^2-k_x^2}{k^2},$$

$$\sum _{\mathbf{k},\lambda }{ϵ}_x^2=\sum _{l,m,n}\sum _{\lambda }{ϵ}_x^2\approx {\left(\frac{a}{2\pi }\right)}^3\int d^{3}k\sum _{\lambda }{ϵ}_x^2={\left(\frac{a}{2\pi }\right)}^3\int d^{3}k\frac{(k^2-k_x^2)}{k^2}.$$

$$\int d\Omega (1-{\cos }^2\theta )={\int }_0^{2\pi }d\varphi {\int }_0^{\pi }d\theta \sin \theta (1-{\cos }^2\theta )=8\pi /3.$$

$$⟨x^2⟩$$

$$\approx \frac{4}{3\pi }{\left(\frac{e}{m}\right)}^2\frac{{\omega }_0^2\mathcal{E}({\omega }_0)}{c^3}{\int }_0^{\infty }d\omega \frac{1}{(2{\omega }_0{)}^2(-\omega +{\omega }_0{)}^2+(\Gamma {\omega }^3{)}^2}$$

$$=\frac{4}{3\pi }{\left(\frac{e}{m}\right)}^2\frac{{\omega }_0^2\mathcal{E}({\omega }_0)}{c^3}\frac{\pi }{2{\omega }_0^4}\left(\frac{3m{c}^3}{2e^2}\right)=\frac{\mathcal{E}({\omega }_0)}{m{\omega }_0^2}$$

$${\int }_0^{\infty }d\omega \frac{1}{(2{\omega }_0{)}^2(-\omega +{\omega }_0{)}^2+(\Gamma {\omega }^3{)}^2}$$

$$=\frac{\pi }{2{\omega }_0\Gamma {\omega }_0^3}=\frac{\pi }{2{\omega }_0^4}\left(\frac{3m{c}^3}{2e^2}\right)$$

$${\int }_{-\infty }^{\infty }\frac{dx}{a^2{x}^2+b^2}=\frac{\pi }{ab}.$$

$$⟨x^2⟩=\frac{\mathcal{E}({\omega }_0)}{m{\omega }_0^2}.$$

$$E_x(\hat{i}x,t)\approx E_x(0,t)+x[{\partial }_{x^{\prime }}{E}_x(\hat{i}{x}^{\prime },t){]}_{x^{\prime }=0}+...,$$

$$\ddot{x}=-{\omega }_0^{2}x+(e/m)\left\{E_x(0,t)+x[{\partial }_{x^{\prime }}{E}_x(\hat{i}{x}^{\prime },t){]}_{x^{\prime }=0}+...\right\}.$$

$$x=x_1+x_2+x_3+...$$

$${\ddot{x}}_1=-{\omega }_0^2{x}_1+(e/m)E_x(0,t),$$

Full text

I Introduction

I.0.1 Aspects Missing from the Classical Physics of 1900

One suspects that the history of physics would be quite different if the physicists at the turn of the 20th century had not been unaware of two crucial ideas of classical physics: 1) the existence of classical electromagnetic zero-point radiation, and 2) the importance of special relativity.  Had the earlier physicists included these aspects, they would have extended the explanatory power of classical physics to include blackbody radiation, the decrease of specific heats at low temperature, and the behavior of van der Waals forces.  As it happened, these aspects were first described in connection with quantum ideas, and the use of quantum physics has been extended until it is the dominant physical theory of the present day.  Only during the past half-century has a small group of physicists gone back to classical physics to note the extensions of that theory made possible by the inclusion of classical zero-point radiation and relativity.M1 B1975a delaPena1996 Cole2003a H-B2015

The problem of the equilibrium spectrum of random radiation, the blackbody radiation problem, provided one of the dilemmas for physics at the turn of the 20th century.  The work on blackbody radiation within classical physics has been reviewed recently.B2018c  Although the work involves a variety of points of view, there is no treatment of the equilibrium spectrum of radiation under *scattering *by a relativistic classical system.  Here we provide the first relativistic scattering calculation and show that classical zero-point radiation is indeed an equilibrium spectrum.

Relativity imposes strong restrictions on systems.  Two relativistic mechanical systems are 1) relativistic point charges in classical electrodynamics, and 2) small harmonic oscillator systems within classical electromagnetism.  The first system (that of relativistic point charges in classical electrodynamics) is a familiar relativistic system; the second system involving harmonic oscillators is not obviously relativistic.  However, one can imagine a harmonic oscillator system of small amplitude as arising from the classical electrodynamic system where two identical charged particles $q$ are held at some fixed distance apart, and a third charged particle $e$ of the same sign is placed between the two charges $q$ and is allowed to oscillate along the line connecting the two identical charges $q$.  In the approximation of small amplitude of oscillation, the electrostatic potential experienced by the third particle $e$ becomes a harmonic potential, and the relativistic and nonrelativistic particle motions for the particle $e$ agree with each other since higher powers of the particle velocity can be ignored.  Within this simple harmonic motion of small amplitude (ignoring terms in $v/c$), there is no role for the speed of light in vacuum $c$.

I.0.2 Radiation-Spectrum Stability

If this harmonic-oscillator system involving the oscillating charge $e$ is bathed in random classical electromagnetic radiation, then (using only the dipole approximation for radiation) the harmonic oscillator system comes to equilibrium at an oscillator energy equal to the energy of the radiation normal modes of the same frequency as the oscillator frequency.  This energy-balance result connecting a point harmonic-oscillator system with random radiation has been known since Planck’s work at the end of the 19th century.Planck  What does not seem to be recognized is that this mechanical system, when treated beyond the dipole approximation, involves detailed radiation balance at the radiation harmonics of the fundamental oscillator frequency provided that the random radiation spectrum corresponds to that of classical electromagnetic zero-point radiation.

Most physicists are satisfied to repeat the erroneous textbook claimerr that classical physics leads inevitably to the Rayleigh-Jeans spectrum for radiation equilibrium.  The truth is far more nuanced.B2018c  It is indeed true that if one use a nonrelativistic classical theory such as (nonrelativistic) classical statistical mechanicsER2p12 or considers scattering by a nonrelativistic nonlinear dipole oscillator treated in the dipole radiation approximation,B1976a then one arrives at the Rayleigh-Jeans spectrum.  However, if one uses relativistic classical physics and includes classical electromagnetic zero-point radiation, then one arrives at the Planck spectrum for classical radiation equilibrium.B2018c

In work carried out more than forty years ago, it was shownB1976a that the addition of a nonlinear term to a harmonic oscillator led to a nonrelativistic, nonlinear mechanical oscillator which scattered electromagnetic radiation (treated in the radiation-dipole approximation) toward the Rayleigh-Jeans spectrum.  Thus the Rayleigh-Jeans spectrum was stable under scattering by this nonrelativistic nonlinear system, but any other spectrum of random classical radiation (including Lorentz-invariant zero-point radiation) was changed by the scattering of this system, and the radiation spectrum was pushed toward the Rayleigh-Jeans spectrum.  The crucial aspect is that the scattering system was a nonrelativistic classical system.vanVleck

In the calculation in the present article, we show that a simple harmonic oscillator treated with its radiation multipole moments (not just the electric dipole moment) leaves the spectrum of classical electromagnetic zero-point radiation invariant.  Our calculation goes through only the quadrupole order, but there are good reasons to expect the validity of the scattering results to hold for all the multipole moments.  Here the crucial aspect is that harmonic oscillator motion in the small-amplitude regime is the same in both relativistic and nonrelativistic physics, whereas the treatment of electromagnetic multipole radiation emission and absorption is fully within the relativistic regime.  The spectrum of classical electromagnetic zero-point radiation is invariant under Lorentz transformation, and we should expect that the spectrum will be preserved only by a relativistic classical scattering system.  Here we give the first example of a relativistic scattering calculation; indeed it involves equilibrium at the zero-point radiation spectrum.

I.0.3 Outline of the Article

We start out by reviewing the treatment of random radiation within classical physics.  We follow this with a review of the exact steady-state behavior of a point harmonic oscillator when located in a bath of random radiation.  Then we repeat the calculation in a form involving energy balance for energy absorption and emission, since this is the form which will be used for the quadrupole terms appearing for non-zero amplitude.  After demonstrating that the average oscillator amplitude can be obtained correctly from the energy balance at the fundamental frequency of the oscillator, we turn to energy balance at the first harmonic of the oscillator frequency when the oscillator amplitude is non-zero.  We first calculate the energy absorbed by the oscillator from the random radiation spectrum during a short time interval $\tau,$ and then we obtain the radiation emission.  We find that energy balance at the first harmonic requires that the radiation per normal mode at the first harmonic should be double the energy per normal mode at the fundamental.  This is precisely the statement that the spectrum of random radiation must correspond to the zero-point radiation spectrum which is linear in frequency.  Our calculation demonstrates explicitly that classical scattering by this relativistic system does not lead to the Rayleigh-Jeans spectrum but rather to Lorentz-invariant zero-point radiation.  We then explain why we expect the detailed balance to extend to all the radiation harmonics of the harmonic-oscillator system.  Finally, we comment upon our current understanding of blackbody radiation within classical physics.

II Set-Up for the Detailed-Balance Calculation

II.1 Random Classical Electromagnetic Radiation

We consider a large volume $V$ containing source-free isotropic random radiation.  The random radiation present in the enclosure of volume $V$ can be given as a sum over plane waves with periodic boundary conditionsB2016b

[TABLE]

[TABLE]

where the wave vector $\mathbf{k}$ takes the values $\mathbf{k=}\widehat{\mathbf{x}}l2\pi/a+\widehat{y}m2\pi/a+\widehat{z}n2\pi/a$ for $l,~{}m,~{}n$ running over all positive and negative integers, the constant $a$ is the length of a side of the box for periodic boundary conditions with volume $V=a^{3},$ the random phases $\theta(\mathbf{k},\lambda)$ are distributed independently and uniformlyRice over the interval $(0,2\pi],$ and the amplitude is given by

[TABLE]

where $\mathcal{E}(\omega)$ is the energy per radiation normal mode of frequency $\omega=ck=c|\mathbf{k}|.$ Because our calculations will seem complicated, we will introduce compact notations.note  The random electric and magnetic fields in Eqs. (1) and (2) will be written compactly as

[TABLE]

[TABLE]

where in addition to the sums over $\mathbf{k}$ and $\lambda,$ we include the sum over $\mu=\pm 1$,

[TABLE]

for any function $f(\mu).$  Here the abbreviated notation writes $\widehat{\epsilon}$ for $\widehat{\epsilon}(\mathbf{k},\lambda),$ $\mathfrak{h}$ for $\mathfrak{h}(\omega),$ and

[TABLE]

[TABLE]

and

[TABLE]

II.2 Dipole Oscillator in a Harmonic Oscillator Potential

The charged particle of mass $m$ and charge $e$, constrained to move along the $x$-axis in a harmonic potential, has the equation of motion (for small amplitude)

[TABLE]

where the energy emission and absorption has been evaluated in the point dipole radiation limit, and where $\Gamma=2e^{2}/(3mc^{3}).$  Here the radiation damping term $m\Gamma\dddot{x}$ includes only the electric dipole radiation emission, (and not the damping associated with the electric quadrupole or higher multipole moments), and the driving radiation field $\mathbf{E}(\mathbf{r},t)$ is evaluated at the center of the oscillator, $\mathbf{r}=0$.  This linear equation has been solved many times before.B2016b In our compact notation of Eq. (4), the (real) steady-state solution takes the form

[TABLE]

where the symbol $C$ stands for

[TABLE]

The factor $F$ (appearing in Eq. (7)) does not appear in Eq. (11) because the driving electric field is evaluated at $\mathbf{r}=0.$

The average value of $x^{2}$ can be evaluated as

[TABLE]

Now averaging over the random phases  $\theta(\mathbf{k,}\lambda)$, we have

[TABLE]

where the last expression $\delta_{1(-2)}$ involves a short-hand notation.  Now summing over the Kronecker delta terms arising in Eq. (14), equation (13) becomes

[TABLE]

since $\widehat{\epsilon}(\mathbf{k},\lambda)$ and $\mathfrak{h}(\omega)$ are independent of $\mu,$ while $R_{1}R_{(-1)}=1,$ and $C_{1}C_{(-1)}=(-\omega^{2}-\omega_{0}^{2})^{2}+(\Gamma\omega^{3})^{2}.$

Next we assume that the distance $a$ used for the periodic boundary conditions on the radiation is very large so that the sum can be approximated as an integral,.  Therefore we can write

[TABLE]

and

[TABLE]

and so approximate the sums as

[TABLE]

Choosing $x$ as the polar axis, the needed angular integration over $\mathbf{k}$ involves

[TABLE]

Then using Eqs. (3), (12), (18),and (19), we have

[TABLE]

where we have assumed that the integrand is sharply peaked at $\omega_{0}$ and therefore have substituted $\omega_{0}$ for every appearance of $\omega,$ except where the combination $\omega-\omega_{0}$ occurs.  Also, we have evaluated the integral as

[TABLE]

using

[TABLE]

Thus we find

[TABLE]

We notice that the average amplitude of the motion decreases as the mass $m$ increases for fixed $\omega_{0}.$

III General Scattering Calculation

The steady-state solution in Eq. (11) is exact for the linear equation given in Eq. (10).  However, equation (10) is only an approximation to the actual physical situation because it ignores the non-zero amplitude of the oscillator motion.  Often one speaks of a “point” dipole oscillator.  The condition for the “relativistic” compatibility of the motion is that $|\omega_{0}x|<<c$ so that the charged particle speed is small compared to the speed of light.  However, even if this conditon is satisfied, equation (10) still ignores the radiation emission and absorption associated with the non-zero excursion of the charged particle. The solution in Eq. (11) holds for a point dipole oscillator whose excursion is so small that the dipole approximations may be applied consistently and radiation at all the higher harmonics can be ignored.  We wish to show that this same expression given in Eq. (11) holds even when we consider the radiation emission and absorption associated with a finite non-zero excursion of the oscillator where the radiation at the harmonics is treated in detail.

In order to proceed with the more complicated analysis, we first simplify.  We drop the radiation damping term (to be treated later) in Eq. (10), and consider only the driving by the random radiation where we now include the first correction in our approximation, so that

[TABLE]

giving an equation of motion for the oscillator

[TABLE]

We will use this equation to calculate the energy absorbed by the oscillator from the radiation during a short time interval $\tau$, and then will treat separately the energy emitted by the oscillator into the radiation field.

We write a series expansion for the oscillator displacement

[TABLE]

where the subscript refers to the number of factors of $\mathfrak{h}$ which appear in the steady-state expression.  Substituting this expansion into Eq. (24), we can separate the terms depending on the number of powers of $\mathfrak{h}$.  In first order in $\mathfrak{h}$, we have

[TABLE]

whereas in second order, we find

[TABLE]

III.1 Energy Balance Calculation for the Dipole Terms

Equation (26) corresponds to the same approximations as in Eq. (10) except that the radiation damping term has been omitted.  Thus equation (26) has no energy-loss mechanism while it is driven by the forcing term.  The solution of the differential equation (26) will involve the sum $x_{1}=x_{1c}+x_{1p}$ of a particular solution $x_{1p}$ of the full equation (26) and a complementary solution $x_{1c}$ of the homogeneous equation, chosen so as to meet the initial conditions for the displacement and velocity of the oscillator at time $t=0.$  We are interested in energy balance, and so we will calculate the average amount of energy delivered to the oscillator by the forcing electromagnetic field during a short time interval $\tau,$ and will then balance this energy against the energy radiated by the oscillator during this same short time interval $\tau.$  Both the amount of energy absorbed and the amount of energy emitted by the oscillator may depend upon the displacement and velocity of the oscillator at time $t=0.$  Since we have the solution for the oscillator motion in the dipole approximation, we will take the initial conditions from the full solution given in Eq. (10).  Of course, since we have the full solution for the dipole approximation, we could use this solution directly when calculating the energy absorption and emission in the dipole case.  However, since in calculating the quadrupole energy absorption and emission, we will not have the exact solution, it seems wise to show the consistency of the basic procedure even though this is redundant in the dipole case.

Provided $\omega\neq\omega_{0,}$ a particular solution of Eq. (26) corresponds to the steady-state solution obtained from the driving field in Eq. (1),

[TABLE]

where

[TABLE]

The summand in Eq. (28) diverges at $\omega=\omega_{0}.$ The complementary solution to Eq. (26) is a solution to the homogeneous equation, so that

[TABLE]

where $R_{0}=\exp[-i\mu_{0}\omega_{0}t].$  The constants $X$ and $Y$ are determined such that at time $t=0,$ there is agreement with Eq. (11), so

[TABLE]

and

[TABLE]

However, the terms $x(0)$ and $\dot{x}(0)$ from Eq. (11) are finite at $\omega=\omega_{0}$ and will not contribute to resonant energy pick-up from the driving electric field.  Therefore these terms can be omitted.  Thus we need only $x_{1c}(0)\approx-x_{1p}(0)$ and $\dot{x}_{1c}(0)\approx-\dot{x}_{1p}(0)$ from Eqs. (28) and (30), giving

[TABLE]

and

[TABLE]

Then the complementary solution in Eq. (30) can be written as

[TABLE]

The energy absorbed from the random radiation in the $x_{1}$ approximation for the oscillator displacement is

[TABLE]

We first consider the last term involving $\dot{x}_{1p}$ in Eq. (28) and $E_{x}(0,t)~{}$in Eq. (4).   Taking the average over the random phases, we are dealing with

[TABLE]

where $\left\langle A_{1}A_{2}\right\rangle$ is given in Eq. (14).  Now summing over the second set of indices, we have

[TABLE]

since $\epsilon_{x1}^{2}$ and $\mathfrak{h}_{1}^{2}$are independent of $\mu_{1}$ while the expression in Eq. (38) contains exactly one factor of $\mu_{1}=\pm 1.$

The average over the random phase for the term involving $\dot{x}_{c}$ in Eq. (36) involves Eqs. (35) and (4), giving

[TABLE]

where we have averaged over the random phases as in Eq. (14) and then summed over $\mu_{2},$ $\mathbf{k}_{2},$ and $\lambda_{2}.$  Now we need the integral

[TABLE]

Then combining Eqs. (40) and (41), we have

[TABLE]

Only if $\mu_{0}=\mu_{1}$ will we have a resonant denominator involving $\omega-\omega_{0}$.  Therefore retaining only the resonant terms, we write

[TABLE]

where we have used

[TABLE]

Now we need to use Eqs. (18) and (19) so as to convert from sums to integrals.  Then we have

[TABLE]

where we have usedGR432

[TABLE]

The average energy $\left\langle W_{\mathbf{p}}\right\rangle$ radiated by an electric dipole during a time $\tau$ is given by Larmor’s formula

[TABLE]

Comparing the energy pick-up in Eq. (46) and loss in Eq. (48), we have

[TABLE]

or

[TABLE]

exactly as found above in Eq. (22).

III.2 Energy Balance for the Quadrupole Terms

Now we treat the energy balance involving the quadrupole moment when the oscillator behavior is given by the same random motion as was determined using the dipole approximation for radiation.  We now consider the terms in Eq. (24) which involve two factors of $\mathfrak{h,}$ arising from Eq. (11) and Eq. (1),

[TABLE]

Once again, we can solve this differential equation in terms of the sum of a particular solution $x_{2p}$ and a complementary solution $x_{2c}$ of the homogeneous equation.   From the time dependence in $R_{1}$ and $R_{2},$ we can obtain the particular solution $x_{2p}$ as

[TABLE]

where

[TABLE]

The complementary solution involves $x_{2c}(t)=\mathcal{X}\cos\omega_{0}t+\mathcal{Y}\sin\omega_{0}t,$ when $x_{2}(0)=x_{2c}(0)+x_{2p}(0)$ and $\dot{x}_{2}(0)=\dot{x}_{2c}(0)+\dot{x}_{2p}(0).$  Now we are taking the oscillator behavior as unchanged from the radiation dipole behavior so that $x_{2}(0)=0$ and $\dot{x}_{2}(0)=0,$ so therefore $x_{2c}(0)=-x_{2p}$ and $\dot{x}_{2c}(0)=-\dot{x}_{2p}.$  Then noting

[TABLE]

we find

[TABLE]

and

[TABLE]

so that

[TABLE]

and

[TABLE]

We want to find the average power absorbed by the oscillator in connection with $x_{2}$ during the short time interval $\tau,$

[TABLE]

where we have dropped the term $\left\langle\int_{0}^{\tau}dt\dot{x}_{2}eE_{x}(0,t)\right\rangle$ because it involves an odd number of random phases $A$ and hence vanishes.  Now the term $\left\langle\int_{0}^{\tau}dt(\dot{x}_{2p})ex[\partial_{x}E_{x}(\widehat{i}x,t)]_{x=0}\right\rangle$  in Eq.(59) also vanishes because it involves four sums over $\mu$ but an odd number of factors of $\mu$: one factor of $\mu$ appearing in Eq. (52), one factor of $\mu$ appearing from the time derivative, and a third factor of $\mu$ appearing from the spatial derivative of the electric field.  The calculation is analogous to that for the term $\left\langle\int_{0}^{\tau}dt\dot{x}_{1p}eE_{x}(0,t)\right\rangle$ which vanished in Eq. (38).

This leaves

[TABLE]

where

[TABLE]

Introducing Eqs. (58) and (61) into Eq. (60), we obtain an expression with an odd number of sums over $\mu$ and an odd number of factors of $\mu,$

[TABLE]

Now averaging over the random phases, we have

[TABLE]

Resonant behavior will occur only when the random phase $\theta_{1}$ is matched with $\theta_{3}$ and the phase $\theta_{2}$ is matched with $\theta_{4},$ so that the only term which is resonant and hence of interest is the center term $\delta_{1(-3)}\delta_{2(-4)}$ in Eq. (63).  Taking only this center term and summing over the indices labeled $3$ and $4$, while noting  that $\epsilon_{x}$ and $\mathfrak{h}$ do not depend on $\mu,$ while $\mu_{2}\mu_{(-2)}=-1,$ we have

[TABLE]

Now because the denominator involves $C_{1}C_{(-1)}=[-(\mu_{1}\omega_{1})^{2}+\omega_{0}^{2}-i\Gamma(\mu_{1}\omega_{1})^{3}][-(-\mu_{1}\omega_{1})^{2}+\omega_{0}^{2}-i\Gamma(-\mu_{1}\omega_{1})^{3}]=(-\omega_{1}^{2}+\omega_{0}^{2})^{2}+(\Gamma\omega_{1}^{3})^{2},$ while the constant $\Gamma$ is assumed small, the function in the sum over $\mathbf{k}_{1}$ is sharply peaked at $\omega_{1}=\omega_{0}.$  Then we approximate the sum over $\mathbf{k}_{1}$ by an integral and set $\omega_{1}=\omega_{0}$ everywhere except where the combination $\omega_{1}-\omega_{0}$ appears.  We extend the lower limit on the integral to $-\infty,$ and so evaluate the sum using Eqs. (18) and (19) as

[TABLE]

Now the time integral needed in Eq. (64) is

[TABLE]

while the constant $D_{1+2}$ is

[TABLE]

We will have resonance near $\omega_{2}\approx 2\omega_{0}$ only if $\mu_{1}=-\mu_{0}=-\mu_{2}.$  Thus performing the sums over $\mu_{1}$ and $\mu_{2},$ and retaining only the resonant terms, equation (64) becomes

[TABLE]

Next, summing over $\mu_{0},$ we reduce the expression to

[TABLE]

since

[TABLE]

At this point, we convert the sum to an integral using Eq. (18) and note the angular integration in $\mathbf{k}$

[TABLE]

to obtain

[TABLE]

The expression involves resonance at $\omega_{2}=2\omega_{0}.$  Thus we approximate all the terms in $\omega_{2}$ as $2\omega_{0}$ except those which involve the resonant combination $\omega_{2}-2\omega_{0}$, and we extend the lower limit of integration to $-\infty.$  The needed integral is given in Eq. (47) so that

[TABLE]

III.3 Radiation Emitted by the Quadrupole Moment

In order to obtain the radiation emitted by the quadrupole moment, we consider the radiation emitted by a point charge moving along the $x$-axis, $\mathbf{r}_{e}(t)=\widehat{i}x_{e}(t).$  The current density is

[TABLE]

The vector potential due to this current density is

[TABLE]

The vector potential in Eq. (75) starts with the dipole term and then includes the quadrupole term.  For particle motion along a straight line, there is no magnetic dipole term.  Now we are interested in only the radiation from the moving charge $e$, and so we go to the radiation zone.  Furthermore, we are concerned with only the quadrupole contribution, which in the radiation zone becomes

[TABLE]

from the derivative $dr/dx=x/r.$  Now we need the magnetic field $\mathbf{B}_{eQ}=\nabla\times\mathbf{A}_{eQ}.$  Then in the radiation zone, we have

[TABLE]

Now treating the $x$-axis as the polar axis, the radiation power emitted per unit solid angle is $dP/d\Omega=[c/(4\pi)]r^{2}B_{eQ}^{2},$ corresponding to

[TABLE]

To obtain the total power radiated, we integrate over all solid angles as

[TABLE]

Then the total power radiated by the quadrupole term is

[TABLE]

Inserting the expression for $x_{e}(t)$ given in Eq. (11), we have

[TABLE]

Upon squaring the square bracket in Eq. (81), we will be dealing with three terms.  The first term from inside the square bracket (corresponding to the square of the first line in Eq. (81)) is

[TABLE]

When we average over the random phases as in Eq. (63), this term becomes

[TABLE]

since

[TABLE]

Now for small charge $e,$ the function $(C_{1}C_{(-1)})^{-1}$ is sharply peaked at $\omega_{1}=\omega_{0}.$  The term involving $(-i\mu_{1}\omega_{1})^{3}(-i\mu_{2}\omega_{2})^{3}$ is odd in both $\mu_{1}$ and $\mu_{2}$ and therefore will cancel completely.  We evaluate the remaining term in the usual way from Eqs. (15)-(22) as

[TABLE]

The cross term in the square bracket in Eq. (81) is

[TABLE]

When we average over the random phases as in Eq. (63), this term becomes

[TABLE]

Only the last term in Eq. (91) is even both in the number of sums over $\mu$ and in the number of factors of $\mu,$ and so does not vanish.  We evaluate this term in the usual way as

[TABLE]

The third term in the square bracket of Eq. (81) (corresponding to the square of the second line of the equation) is

[TABLE]

When we average over the random phases as in Eq. (63), this term becomes

[TABLE]

Only the last term in Eq. (94) has an even number of terms both in the sums over $\mu$ and in the number of factors of both $\mu_{1}$ and $\mu_{2}$ and so is nonvanishing.  We evaluate this term in the usual way as

[TABLE]

Summing the terms in Eqs. (87), (92), and (95), equation (81) becomes

[TABLE]

Equating the average energy absorbed at frequency $2\omega_{0}$ in Eq. (73) with the average energy emitted at $2\omega_{0}$ from Eq. (96) during the short time interval $\tau$, we have

[TABLE]

Energy balance requires

[TABLE]

This condition states that the spectrum is linear in frequency, which is precisely the character of the Lorentz-invariant spectrum of zero-point radiation.

IV Comments on the Physical Situation

IV.1 Higher Multipoles and Radiation

The mechanical harmonic-oscillator scattering system which we have considered may be regarded as relativistic provided that $\omega_{0}^{2}\left\langle x^{2}\right\rangle<<c^{2}.$  The usual point-dipole approximation for both the mechanical motion and the interaction with radiation corresponds to the limit $\left\langle x^{2}\right\rangle\rightarrow 0$ while $e^{2}\rightarrow\infty$ in such a way that the dipole moment squared $p^{2}$ is finite $p^{2}=e^{2}\left\langle x^{2}\right\rangle=const.$  In the limit $\left\langle x^{2}\right\rangle\rightarrow 0,$ the mechanical system clearly satisfies the limit of nonrelativistic speed, and all the radiation emission and absorption takes place at the fundamental frequency; there is no radiation interaction at the harmonics since all the higher multipole moments above the dipole moment vanish.  In the analysis of the present article, we avoid the limit $\left\langle x^{2}\right\rangle\rightarrow 0.$  The amplitude of oscillator motion is required to be so small that the speed of the particle is nonrelativistic, but the quadrupole moment is non-vanishing.  The quadrupole moment involves two factors of length (not just the one factor of length needed for the dipole moment) and so vanishes in the usual point-dipole limit.  Indeed, all the multipole moments above the dipole moment have additional factors of length and so require a non-zero amplitude of motion in order to remain non-zero.

IV.2 Role of the Constant c

The mechanical motion of the oscillator involves no factors of the speed of light $c$ so long as the speed $v$ of the oscillator is small $v<<c$.  On the other hand, the radiation energy emitted and the radiation energy absorbed at each harmonic involve the same number of factors of $c$, so that the condition of radiation balance, harmonic-by-harmonic, gives no role for the ratio $v/c.$  Thus the radiation at the fundamental frequency involves factors of $c^{-3}$ for both the emitted radiation and the absorbed radiation, as seen in Eq. (49), while the radiation at the first harmonic involves balancing factors of $c^{-5},$ as seen in Eq. (97). Accordingly, as far as an analysis harmonic-by-harmonic is involved, there is no connection between the oscillator speed $v$ and the radiation speed $c.$ It is only when we sum the series for the radiation emission or absorption that we discover that there is a singularity associated with the radiation interaction when the particle speed $v$ approaches the speed of light $c.$Burko

IV.3 Adiabatic Invariance

In the past, it has been shown that the adiabatic invariance of the point harmonic oscillator fits with the adiabatic invariance of classical electromagnetic zero-point radiation.B1978c  Thus as the frequency of the oscillator is changed adiabatically, the harmonic oscillator remains in radiation balance with the zero-point radiation.  In the earlier work, it was emphasized that the adiabatic invariance in the presence of radiation depended crucially upon the absence of any interaction with radiation harmonics. Based upon the present work, we see that this adiabatic invariance extends beyond the point oscillator out to an oscillator of finite non-zero amplitude.  A small oscillator of non-zero excursion has detailed balance with zero-point radiation at higher harmonics.  Since the zero-point radiation is invariant under a $\sigma_{ltU^{-1}}$-scale transformationB1989a , the adiabatic invariance under a change of oscillator frequency will continue for a harmonic oscillator of small but non-zero amplitude.

IV.4 Limitations on the Approximation

The charged particle in a harmonic-oscillator potential which is used in the calculation of this article has a distinct limitation.  As described here, the system involves acceleration-based radiation emission, but does not allow any velocity-dependent damping proportional to the random radiation which is present.  Thus our description does not allow the treatment of random radiation involving velocity-dependent damping.  Now thermal radiation has a preferred inertial frame, and any particle moving relative to this preferred inertial frame will experience a velocity-dependent damping proportional to the thermal radiation which is present.B1969b  On the other hand, zero-point radiation is Lorentz invariant and so involves no velocity-dependent damping.  The mathematical description used in the present article is accurate for zero-point radiation only and does not extend to thermal radiation at non-zero temperature.

V Discussion of the Connection Between Blackbody Radiation and

Relativity

V.1 Relativistic Invariance is Strongly Restrictive

The basic physical ideas involved in the present calculation are not well known and deserve a broader audience.  Many physicists are unaware of the restrictive nature of the requirement that a system should be relativistic.  The first three conservation laws of dynamics, associated with symmetries under space translations, time translations, and rotations, involve conservation of linear momentum, energy, and angular momentum; these conservation laws appear in both nonrelativistic and relativistic theories.  However, the fourth conservation law associated  with Galilean symmetry or relativistic symmetry is quite different between nonrelativistic and relativistic systems.B2009a  The conservation law for Galilean invariance of nonrelativistic dynamics essentially repeats the information of the law of conservation of linear momentum and so gives no restrictions on allowed nonrelativistic systems.  On the other hand, the conservation law associated with Lorentz invariance has profound limitations on which systems are relativistic. For example, the no-interaction theorem of Currie, Jordan, and SudarshanCurrie states that any relativistic interaction between particles beyond point interactions requires the introduction of a field theory, and relativistic field theories are restricted still further.

V.2 Simplest Equilibrium Radiation Spectra

Within classical physics, there are two spectra for electromagnetic radiation which take particularly simple forms.  One of these is the Rayleigh-Jeans spectrum which associates the same energy (taken as $k_{B}T)$ with every radiation normal mode.  This spectrum involves one parameter, the energy $k_{B}T$ per normal mode, and does not distinguish any length or any frequency.  This spectrum is associated with nonrelativistic physics and in particular with the equipartition theorem of nonrelativistic classical statistical mechanics.

The second simple radiation spectrum is that of classical electromagnetic zero-point radiation which has an energy linear in the frequency (energy taken as $\hbar\omega/2)$ with every radiation normal mode.  Again, this spectrum involves one parameter $\hbar$ with units corresponding to an angular momentum or to an energy $\times$ time, and does not distinguish any length or frequency.  The spectrum is associated with relativity; zero-point radiation is the unique (up to a multiplicative constant) Lorentz-invariant spectrum of random classical radiation, and it takes the same form in every inertial frame.

It is striking that the Planck blackbody radiation spectrum including zero-point radiation is the smoothest possible interpolation between the Rayleigh-Jeans spectrum at low frequencies and the zero-point radiation spectrum at high frequencies.B2003c

V.3 Scatterers for Classical Radiation Equilibrium

Electromagnetic radiation can not bring itself to equilibrium.  Rather, there must be some mechanical scattering system, some “black” particle (a particle which scatters radiation toward the equilibrium spectrum), which enforces the radiation equilibrium.  Since radiation equilibrium is determined by a mechanical system, we certainly expect that the nature of the scattering system will influence the radiation equilibrium spectrum.  Indeed for classical mechanical systems, nonrelativistic scatterers leave the Rayleigh-Jeans spectrum unchanged, while relativistic scatterers leave the zero-point spectrum unchanged.

The simplest scattering system which allows a transition corresponding to that found for the Planck spectrum (including zero-point radiation) between the Rayleigh-Jeans spectrum at low frequency and the zero-point spectrum at high frequency is that of a charged particle of charge $e$ and mass $m$ in a Coulomb potential where the ratio $mc^{2}/k_{B}T$ provides the mechanical transition parameter matching the radiation transition parameter $\hbar\omega/k_{B}T.$  We notice that for the Coulomb potential, high mechanical mass $m$ is associated with high frequency $\omega$.  The Coulomb potential is one of the few mechanical systems where large mass $m$ is associated with high frequency, and small mass $m$ is associated with low frequency.

V.4 Harmonic Oscillator Scatterers

Although hydrogen-like scatterers involving a Coulomb potential are the relativistic systems which are expected to scatter classical electromagnetic radiation toward the Planck spectrum with zero-point radiation, it seems exceedingly difficult to work with the Coulomb potential as a scatterer.  On the other hand, it is vastly easier to treat a charged harmonic oscillator as a scattering system for electromagnetic radiation.  Indeed, at the end of the 19th century, Planck calculated the behavior of a charged harmonic oscillator taken in the point-size limit when bathed in random classical electromagnetic radiation.  Although Planck had initially hoped that the oscillators would serve as “black” particles and determine the spectrum of electromagnetic radiation, it became clear that small (point-limit) harmonic oscillator systems simply acquired an energy which matched the energy of the radiation normal modes at the oscillator frequency.  A point harmonic dipole oscillator did not determine the equilibrium radiation spectrum.  Acting as a scatterer, the oscillator may change the angular distribution of the radiation, but the point oscillator did not change the frequency spectrum of the random electromagnetic radiation.  Planck subsequently turned to statistical mechanics for the harmonic oscillator in an attempt to understand equilibrium for the electromagnetic radiation.  And the application of the nonrelativistic equipartition theorem to an oscillator scatterer is still used in physics textbooks as a way of obtaining the Rayleigh-Jeans radiation spectrum.err

V.5 Extensions for Harmonic Oscillator Scatterers

We wish to avoid the use of statistical mechanics in our exploration of radiation equilibrium, but we would like to use the calculational simplicity of harmonic oscillator systems.  Now point oscillator systems are unsatisfactory because they interact with random radiation at a single frequency.  However, there are two natural extensions of the point-limit harmonic oscillator system which will bring the system into contact with the full radiation spectrum.  One involves the introduction of a small nonlinear term in the harmonic oscillator potential so as to introduce higher harmonics in the oscillator mechanical motion.  The second possible modification is the consideration of all the radiation harmonics associated with a charged particle in a purely harmonic potential but with an amplitude of finite, non-zero excursion.

V.5.1 Nonlinear-Oscillator Scatterer

The radiation scattering due to an oscillator with a small nonlinear term was considered in 1976.B1976a The introduction of a small nonlinear term in the harmonic-oscillator potential leads to mechanical motion which involves harmonics of the basic oscillator motion so that the particle displacement becomes $x(t)=a_{1}\cos(\omega_{0}t+\phi_{1})+a_{2}\cos(2\omega_{0}t+\phi_{2})+...$.  The ratio $a_{n}/a_{1}$ of the amplitudes $a_{n}$ compared to the initial harmonic-oscillator amplitude $a_{1}$ depends upon the arbitrary strength of the nonlinear term in the potential.  Although radiation emission and absorption are still treated in the dipole approximation, the presence of the harmonics in the mechanical motion brings the oscillator into contact with not only the radiation at the fundamental oscillator frequency $\omega_{0}$ but also the radiation at the multiples $n\omega_{0}$ of the fundamental frequency $\omega_{0}.$  Although the ratios $a_{n}/a_{1}$ may be arbitrary, it turns out that the radiation spectrum which has the same energy per normal mode at every frequency (the Rayleigh-Jeans spectrum) remains unchanged by scattering from this system.  Indeed, there have been several calculation, going back to van Vleck’s work of 1924 showing that nonrelativistic nonlinear mechanical scattering systems treated in the dipole limit for their radiation interaction leave the Rayleigh-Jeans spectrum invariant.vanVleck

V.5.2 Finite-Amplitude Harmonic Oscillator

The second possible extension of the small harmonic oscillator scatterer is what is treated in the calculations of the present article.  We consider not a change in the harmonic oscillator mechanical potential but rather a calculation of the radiation emitted and absorbed at the radiation harmonics of the fundamental oscillator frequency.  The oscillator potential remains unchanged as a harmonic oscillator potential and the free oscillator motion $x(t)=a_{1}\cos(\omega_{0}t+\phi_{1})$ remains unchanged.  However, relativistic classical electrodynamics involves radiation at all the harmonics for any finite non-zero amplitude of oscillation.  Thus the relativistic aspects enter not through the mechanical oscillator motion but through the radiation theory.  Our analysis calculates the radiation energy balance for the second harmonic corresponding to quadrupole radiation and shows that the spectrum which remains unchanged is the zero-point radiation spectrum.  This is the first classical scattering calculation showing explicitly that a relativistic scattering system indeed leaves the relativistically-invariant zero-point radiation spectrum unchanged.

VI Acknowledgement

The present calculation was prompted by the work of Professor Daniel C. Cole and of Dr. Wayne Cheng-Wei Huang and Professor Herman Batelaan showing the influence of radiation harmonics back on mechanical systems’ motions at the fundamental frequency.  I wish to thank Professor Michael C. Boyer for helpful suggestions regarding the presentation of the calculational results.