Larmor Temperature, Casimir Dynamics, and Planck’s Law

Abstract

Classical radiation from a single relativistically accelerating electron is investigated where the temperature characterizing the system highlights the dependence on acceleration. In the context of the dynamic Casimir effect with Planck-distributed photons and thermal black hole evaporation, we demonstrate analytic consistency between the ideas of constant acceleration and equilibrium thermal radiation. For ultra-relativistic speeds, we demonstrate a long-lasting constant peel acceleration and constant power emission, which is consistent with the idea of balanced equilibrium of Planck-distributed particle radiation.

1. Introduction

It is fascinating that black holes (BH), with surface gravity, ${\kappa }_{\mathrm{BH}}=c^4/4GM$, have ‘quantum’ (with the reduced Planck constant, ℏ) temperature [1],

$$T_{\mathrm{BH}}=\frac{\hslash {\kappa }_{\mathrm{BH}}}{2\pi c{k}_B},$$

(1)

because, in part, the radiated particles in equilibrium are frequency-distributed with a Planck factor, and the power emitted scales according to $P\sim T^2$, substantiating black holes as one-dimensional information channels [2]. Here, G is the Newtonian constant of gravitation, M is the BH mass, c is the speed of light, and $k_B$ is the Bolzmann constant.

In this paper, we help to make the case for a classical analog to Equation (1). We present new details supporting the idea of a moving point charge radiation effect, quite similar in form to Equation (1) yet fully classical in origin. The thermal radiation originates from a single accelerating electron. For clarity, we provide overlap with [3], but the novel results here focus on temperature and the analytic expressions of time dependence. Our results are concerned with the equilibrium period of an electron’s radiation: when the power emitted is uniform and classical thermodynamics applies, the emission has a temperature proportional to the peel acceleration, $\kappa$, of the electron; the latter term is defined and explained in Section 3.3. One finds the ‘classical’ (no ℏ) temperature [4],

$$T_{\mathrm{electron}}=\frac{{\mu }_0{e}^2\kappa }{2\pi k_B},$$

(2)

which is commensurate with constant power emission [3]. This temperature is in the Stoney scale [5]; see [6]. Here, e denotes the electron’s charge and ${\mu }_0$ is the vacuum magnetic permeability. Interestingly, this occurs during Planck-distributed radiation from an analog moving mirror (dynamical Casimir effect [7,8,9]) accelerated along the same specific trajectory (provided in Ref. [3]). A horizontal leveling of the power is visually observed at extremely ultra-relativistic final speeds of the electron. A notion of temperature is congruent with the power P, emitted by the electron scaling according to $P\sim T^2$ (this quantity is the ${\overline{P}}_c$ defined in Equation (18)), revealing similar Bekenstein one-dimensional behavior and power–temperature scaling [2]. In this paper, we investigate the arguments supporting this conjecture and the classical temperature (2), as well as the analogy to the quantum temperature (1).

1.1. Analog Bridge

First, let us consider the straightforward action correspondence between Equations (1) and (2),

$$\hslash \to \frac{e^2}{{ϵ}_{0}c}={\mu }_{0}c{e}^2,$$

(3)

where the ℏ value is,

$$\hslash =1.054\times {10}^{-34}\mathrm{J}\mathrm{s},$$

(4)

and the smaller action (or angular momentum) classical quantity is

$${\mu }_{0}c{e}^2=9.671\times {10}^{-36}\mathrm{J}\mathrm{s}.$$

(5)

Notice that

$$\frac{\hslash }{{\mu }_{0}c{e}^2}\approx 10.91.$$

(6)

For a given acceleration scale, $\kappa$, the classical temperature (2) is nominally about a magnitude order smaller than the quantum temperature (1). The analog ‘substitution’ (3) can help one bridge the analog connection between the elementary particle and black hole via a substitution $\hslash \to {\mu }_{0}c{e}^2$ in Equation (1) gives Equation (2). We justify and generalize this in what follows.

1.2. Temperature Definition

Temperature is a collective property and is almost always defined with an assemblage of particles. The helpfullness of thermodynamics is particularly salient in a regime with a large number of particles (in this case, the large amount of radiated particles are infinite soft thermal photons).

We emphasize that what is meant by ‘the temperature of electron radiation’ in Equation (2) is a temperature extracted by averaging the photon energy radiated over many realizations of the same decay experiment with a single asymptotically ultra-relativistic electron. Only in this context does it make sense to consider a single electron radiating photons with a defined temperature.

Here, the frequency distribution is also analogous to the moving mirror particle production, which is Planck-distributed; see Section 4 below. The connection to black hole temperature is limited in the sense that an explicit Planck distribution has not been derived for classical electron radiation, unlike the moving mirror Planck distribution, which is a result of the Bogolubov beta coefficients originating from the quantum fields in a curved space approach, see, e.g., [8,9,10,11]. However, the connection is explicitly tethered by the power–temperature scaling of the (1+1)-dimensional Stefan–Boltzmann law; see Section 5 below. Importantly, this notion of electron radiation temperature is dynamically convenient because it signals a corresponding period of uniform peel acceleration (which is defined in Section 3.3).

1.3. Extension Bridge

The bridge (3) is not limited to Equations (1) and (2). It proves helpful as an action correspondence in general (see Refs. [12,13,14,15]) between the quantum moving mirror model (denoted below by the q subscript) and the classical moving point charge model (denoted by the c subscript). This is observed, respectively, in the power (see, e.g., [16,17]),

$$P_q=\frac{\hslash {\alpha }^2}{6\pi c^2},P_c=\frac{{\mu }_0{e}^2{\alpha }^2}{6\pi c},$$

(7)

where $\alpha$ is the proper acceleration of the mirror or electron, and self-force [18,19],

$$F_q=\frac{\hslash {\alpha }^{\prime }\left(\tau \right)}{6\pi c^2},F_c=\frac{{\mu }_0{e}^2{\alpha }^{\prime }\left(\tau \right)}{6\pi c},$$

(8)

with the prime indicating the derivative with respect to proper time, $\tau$, (not coordinate time, t) (see, e.g., [18,20]), for any limited horizonless trajectory whose acceleration is asymptotically zero (asymptotic inertia).

Moreover, the bridge also occurs specifically between the spectral radiance of a particular moving mirror model and lowest-order inner bremsstrahlung (IB) during beta decay [3], contained therein. More generally, if one examines the infrared limit, this is observed in the frequency independence of the spectral energy per unit bandwidth (see, e.g., [16,21]),

$$I_q=\frac{\hslash }{2{\pi }^2}\left(\frac{\eta }{s}-1\right),I_c=\frac{{\mu }_{0}c{e}^2}{2{\pi }^2}\left(\frac{\eta }{s}-1\right),$$

(9)

where $s=tanh\eta$ is the final speed of the mirror or electron as a fraction of c, and $\eta$ is the final rapidity.

In what follows, we check the consistency of our claims by analyzing the correspondence from different sides. In Section 2, we consider the radiation from an accelerated electron more closely, discussing the relevant scales of the problem. In Section 3, we show that such an electron is characterized by constant-in-time characteristic quantities, thus supporting the thermal regime. Section 4 presents the Planck spectrum for a moving mirror model and its connection to the moving point charge and black holes, also supporting thermality and the analog bridge of Equations (1) and (2). In Section 5, we provide several derivations of the Stefan–Boltzmann law in the relevant contexts; the results serve as an independent confirmation of the quadratic dependence in temperature. Section 6 gives the conclusions.

2. Energy Radiated by an Electron

2.1. Total Energy Emitted

To obtain the energy per unit bandwidth from Equation (9), one associates the ultra-violet (UV) scale, ${\omega }_{\max }$, of the system with the acceleration scale, $\kappa$:

$${\omega }_{\max }=\frac{\pi \kappa }{12c},$$

(10)

such that, using the first equation of Equation (9), the quantum spectral energy per unit bandwidth,

$$I_q=\frac{\mathrm{d}{E}_q}{\mathrm{d}\omega }\to E_q=\frac{\hslash \kappa }{24\pi c}\left(\frac{\eta }{s}-1\right),$$

(11)

or with the second equation of Equation (9), the classical spectral energy per bandwidth,

$$I_c=\frac{\mathrm{d}{E}_c}{\mathrm{d}\omega }\to E_c=\frac{{\mu }_0{e}^2\kappa }{24\pi }\left(\frac{\eta }{s}-1\right).$$

(12)

With the clarity of the International System of Units (SI), this demonstrates that the two different models have analogous energy emission scaling. Notice that energy (12) can be expressed as

$$E_c=\frac{{\mu }_0{e}^2\kappa }{48\pi }\left[\frac{1}{s}ln\left(\frac{1+s}{1-s}\right)-2\right],$$

(13)

where, again, as in Equation (9), s is the final constant speed of the electron as a fraction of the speed of light. As shown in Section 3 below, for a thermal plateau, ultra-relativistic speeds are required, $s\sim 1$ (although this is not required in order to obtain the Planck-distributed photon thermal spectrum). Only the classical version Equation (12) [22] or lowest-order IB energy [23] have been directly observed in experiments; see, e.g., [24].

2.2. UV Cutoff and Temperature

For some orientations, consider now re-expressing the temperature (2) of the electron radiation in terms of the maximum appreciable energy emitted, $\hslash {\omega }_{\max }=\hslash \pi \kappa /12c$ (see Equation (10)),

$$T=\frac{6}{{\pi }^2}\frac{{\mu }_{0}c{e}^2}{\hslash }\frac{E_{\gamma }}{k_B},$$

(14)

where energy range of the detected photons is UV-limited by $E_{\gamma }=\hslash {\omega }_{\max }$ and can be expressed from Equation (14) as

$$E_{\gamma }=\frac{{\pi }^2}{6}\frac{\hslash }{{\mu }_{0}c{e}^2}{k}_{B}T\approx 18k_{B}T.$$

(15)

This provides some perspective on the dependence of the system on the cutoff when in thermal equilibrium. Consistency of the temperature formulas (2) and (14) is confirmed below; see Section 2.3, Section 3, Section 4, and Section 5 below.

Including these UV limits, experimental evidence of lowest-order IB energy emitted during beta decay confirms the consistency of the theoretically derived frequency independence of the spectral energy per unit bandwidth (9); see, e.g., [24]. In what follows, we support the physical notion of temperature in this context by providing corroborative analytic results confirming the mathematical validity of Equation (2).

2.3. Scale Dependence

The analog between black hole temperature and electron radiation temperature, introduced in Section 1, has limitations. Black hole temperature, $T=\hslash \kappa /2\pi c{k}_B$, varies dependent on the surface gravity, ${\kappa }_{\mathrm{BH}}=c^4/4GM$, of the black hole, while electron radiation temperature, $T={\mu }_0{e}^2\kappa /2\pi k_B$, varies on the acceleration scale, $\kappa =12c{\omega }_{\max }/\pi$, inherently a function of the UV scale of the system, ${\omega }_{\max }$. Hence, because the charge of every electron is the same, the fine structure does not change in this context, and the temperature of the electron’s acceleration radiation is UV-dependent, so the two expressions differ with respect to both intuition and scale. In this context, it is convenient to consider the universality of the soft factor [25] and the thermal character of the infinite zero-energy photons emitted in this regime. Indeed, the thermality here is connected to every scattering process in the deep infrared, at least in the instantaneous collision reference frame [26]. Thus, there is an argument for the relevance of Equation (2) beyond the bremsstrahlung context.

To this end, we point out that Equation (2) is relevant for Feddeev–Kulish dressed states, where equivalent particle count and energy results [27] suggest one can can derive a ‘cloud temperature’. Analog systems with corresponding results are also subject to thermal character. For instance, ‘mirror temperature’ is an appropriate assignment in the context of the dynamical Casimir effect [28], as we demonstrate straightforwardly with the spectral computation (see Equation (26) below). Moreover, since the internal structure of the source cannot be discerned by long wavelengths, these results can necessarily be extended in analog to curved spacetime final states [29], where ‘black hole temperature’ leading to a leftover remnant becomes a helpful characterization of the system. We leave these extensions for future investigations.

3. Thermal Plateaus

In a system with well-defined thermality, one naturally expects to observe an equilibrium, which implies that characteristic quantities describing this system remain constant in time (up to small fluctuations). For example, a black body immersed in a heat bath radiates the same amount of energy each second, i.e., the radiation power remains constant.

Since we talk about temperature and thermality for the electron/mirror setup, it is desirable to see that this system is indeed in a regime where its characteristic quantities remain constant; in this Section, we explore this in detail. We find that indeed there is a time window where the characteristic quantities remain constant and exhibit plateaus. In the ultrarelativistic limit, $s\to 1$, of high final speeds, these plateaus become wide, thus validating the regime of thermality.

3.1. Constant Power Emission

As expected for thermal equilibrium, a stable emission period of constant power is measured by a far-away observer. This is best represented as the change in energy with respect to retarded time, $u=t-r/c$ (r is the distance to the origin), and written as Larmor power, $\overline{P}=\mathrm{d}E/\mathrm{d}u,$ such that $\overline{P}=P\mathrm{d}t/\mathrm{d}u=P/(1-\beta )$, where $P={\mu }_0{e}^2{\alpha }^2/6\pi c$, see Equation (7). Here, $\beta$ is the velocity normalized by c.

The main example here is the trajectory directly related to the lowest-order inner bremsstrahlung in the radiative beta decay [3],

$$\mathit{r}\left(t\right)=\frac{sc}{\kappa }W\left(e^{\kappa t/c}\right)\widehat{\mathit{r}},$$

(16)

Here, $\widehat{\mathit{r}}$ is the unit vector in the $\mathit{r}$ direction, while W is the Lambert product logarithm defined as a solution to equation, $w{e}^w=x$, such that $w=W\left(x\right)$ and $W\left(0\right)=0$. The Larmor power can be computed analytically. Its expression, formulated in terms of u, reads:

$$\overline{P}\left(u\right)=\frac{{\mu }_0{e}^2{\kappa }^2{s}^2{W}^2(W+1-s)}{6\pi c{(W+1)}^4{((1+s)W+1-s)}^3},\mathrm{where}W\equiv W\left[e^{\kappa u/c}(1-s)\right].$$

(17)

Equation (17) has a plateau when the final speed of the electron is near the causal limit, $s\to 1$. Consider analytically two separate limits of high speeds and late times, which reveal, using Equation (2),

$${\overline{P}}_c\equiv \underset{u\to \infty }{lim}\underset{s\to 1}{lim}\overline{P}\left(u\right)=\frac{{\mu }_0{e}^2{\kappa }^2}{48\pi c}=\frac{\pi }{12}\frac{k_B^2}{{\mu }_{0}c{e}^2}{T}^2.$$

(18)

Figure 1 shows $\overline{P}\left(u\right)$ at high final asymptotic speeds, $s\sim 1$, and illustrates the constant power plateau indicative of thermal emission.

Keep in mind that we are working with classical (3+1)-dimensional radiation of an electron. Therefore, we notice that Equation (18) is a (1+1)-dimensional classical power–temperature relation, with scaling identical to the standard quantum (1+1)-dimensional Stefan–Boltzmann law [30], which describes (3+1)-dimensional black hole power radiance (see, e.g., [2]),

$$P_q=\frac{\pi k_B^2}{12\hslash }{T}^2.$$

(19)

In the same way that a single-spatial-dimensional Planck distribution yields Equation (19), an analog Planck distribution, J (without ℏ), or spectral energy density in angular frequency space,

$$J\left(\omega \right)=\frac{1}{2\pi }\frac{{\mu }_{0}c{e}^2\omega }{e^{{\mu }_{0}c{e}^2\omega /k_{B}T}-1},$$

(20)

where Equation (3), $\hslash \to {\mu }_{0}c{e}^2$, can be applied (as an example), integrated over angular frequency,

$${\int }_0^{\infty }J\left(\omega \right)\mathrm{d}\omega =\frac{\pi }{12}\frac{k_B^2}{{\mu }_{0}c{e}^2}{T}^2,$$

(21)

which results in Equation (18). It is natural to suppose a distribution similar to $J\left(\omega \right)$ (20) might be responsible for Equation (18); see, e.g., [4]. Such a distribution could lend support for the action correspondence (3), but also corroborate the temperature (2)). It appears that such a distribution would only characterize the radiation during a long-lived constant power emission phase at sufficiently high speeds, $s\sim 1$. Nevertheless, independent of any $J\left(\omega \right)$ supposition and the difficulties commensurate with such speculation, the power emission (17) possesses a plateau consistent with Equation (18).

3.2. Constant Radiation Reaction

Having seen the power plateau in $\overline{P}\left(u\right)$ originating from $P={\mu }_0{e}^2{\alpha }^2/6\pi c$, let us now turn to the self-force, $F={\mu }_0{e}^2{\alpha }^{\prime }\left(\tau \right)/6\pi c$, and the associated power, which we call ‘Feynman power’ [31], $\overline{F}\left(u\right)=F\mathrm{d}r/\mathrm{d}u=F\beta /(1-\beta )$, as a function of u,

$$\overline{F}\left(u\right)=\frac{{\mu }_0{e}^2{\kappa }^{2}sW(s-W-1)\left(2(s+1)W^2+s+W-1\right)}{6\pi c{(W+1)}^4{((s+1)W-s+1)}^3},W\equiv W\left[e^{\kappa u/c}(1-s)\right].$$

(22)

Taking the same two separate consecutive limits of high speeds and late times, as in Equation (18), reveals

$$\underset{u\to \infty }{lim}\underset{s\to 1}{lim}\overline{F}\left(u\right)=-\frac{{\mu }_0{e}^2{\kappa }^2}{48\pi c}=-\frac{\pi }{12}\frac{k_B^2}{{\mu }_{0}c{e}^2}{T}^2.$$

(23)

Figure 2 shows the period of constant Feynman power. This plot, similar to Figure 1 of the Larmor power, $\overline{P}$, also exhibits a constant period during which the electron emits particles in thermal equilibrium. Equation (23) substantiates Equation (2).

3.3. Constant Peel Acceleration

Direct corroboration of an extended period of thermal equilibrium is given by the object $\overline{\kappa }\left(u\right)=v^{″}\left(u\right)/v^{\prime }\left(u\right)$, where $v=t+r/c$ is the advanced coordinate and the prime denotes a partial derivative with respect to the independent variable, the retarded time, u, in this case. This quantity is called the ‘peeling function’ and has been used in the relativity literature; see, e.g., [32,33]. Following precedent, we call it the ‘peel acceleration’ or ‘peel’ for short.

The peel acceleration typically accompanies thermal particle radiation. For instance, it has been used as a measure of what Carlitz–Willey [34] called ‘local acceleration’. The result for IB is [3]

$$\overline{\kappa }\left(u\right)=\frac{2\kappa sW}{{(W+1)}^2(1+(s+1)W-s)},W\equiv W\left[e^{\kappa u/c}(1-s)\right].$$

(24)

In the limit of high speeds and late times, one finds:

$$\underset{u\to \infty }{lim}\underset{s\to 1}{lim}\overline{\kappa }\left(u\right)=\kappa .$$

(25)

The peel acceleration, $\overline{\kappa }\left(u\right)$, is related to the Lorentz-invariant proper acceleration, $\alpha$, via the relation, $\overline{\kappa }=2\alpha e^{\eta }$, or via the first derivative of the rapidity with respect to retarded time, $\overline{\kappa }\left(u\right)=2{\eta }^{\prime }\left(u\right)$.

Figure 3 shows the peel acceleration. A quasi-constant peel acceleration is in agreement with the equilibrium of a thermal distribution and constant power emission; however, it is important to underscore the fact that a constant peel acceleration does not describe uniform proper acceleration of the electron.

4. Planck Spectrum

4.1. Moving Mirror Model

In the moving mirror model (see, e.g., [9,10]), the Bogolubov beta coefficients corroborate radiative equilibrium via an explicit Planck distribution. For IB during beta decay, the Planck distribution is explicitly manifest in Equation (26). Accelerating boundaries radiate soft particles whose long wavelengths lack the capability to probe the internal structure of the source [25]. In the spirit of the analogy, the moving mirror spectrum, with the peel acceleration, $\kappa$, supports the appropriate notion of temperature for the soft spectrum of the electron’s IB. Combining the results for each side of the mirror [28] by adding the squares of the Bogolubov beta coefficients, the overall spectrum reads [3]:

$$|{\beta }_{\omega {\omega }^{\prime }}{|}^2=\frac{2c{s}^2\omega {\omega }^{\prime }}{\pi \kappa (\omega +{\omega }^{\prime })}\frac{a^{-2}+b^{-2}}{e^{2\pi c(\omega +{\omega }^{\prime })/\kappa }-1}.$$

(26)

Here, $a=\omega (1+s)+{\omega }^{\prime }(1-s)$, and $b=\omega (1-s)+{\omega }^{\prime }(1+s)$, where $\omega$ is the out-frequency mode and ${\omega }^{\prime }$ is the in-frequency mode of the massless scalar field [10]; Davies-Fulling notation [9] is used here. See Figure 4 for an illustration of the symmetry between the frequency modes of the beta modulus (26).

For a consistency check, using the retarded time clock of the observer, the following integrations hold:

$$E_c=-{\int }_{-\infty }^{\infty }\overline{F}\left(u\right)\mathrm{d}u={\int }_{-\infty }^{\infty }\overline{P}\left(u\right)\mathrm{d}u,$$

(27)

along with

$$E_q={\int }_0^{\infty }{\int }_0^{\infty }\hslash \omega {|{\beta }_{\omega {\omega }^{\prime }}|}^{2}d\omega d{\omega }^{\prime },$$

(28)

or

$$E_q={\int }_0^{\infty }{\int }_0^{\infty }\hslash {\omega }^{\prime }{|{\beta }_{\omega {\omega }^{\prime }}|}^{2}d\omega d{\omega }^{\prime },$$

(29)

demonstrating consistency with the conservation of energy. Importantly, this also demonstrates the consistency of the analogy between quantum mirrors and classical electrons.

The Planck spectrum of Equation (26) is robust to both high-frequency approximations and low-frequency approximations. This is made particularly explicit by considering high final speeds, $s\approx 1$, then using either frequency approximation. Consider the high frequency, ${\omega }^{\prime }\gg \omega$ approximation [1]. To leading order, one retrieves

$$|{\beta }_{\omega {\omega }^{\prime }}{|}^2=\frac{c}{2\pi \kappa \omega }\frac{1}{e^{2\pi c{\omega }^{\prime }/\kappa }-1}.$$

(30)

Likewise, considering high final speeds and the low-frequency, ${\omega }^{\prime }\ll \omega$, approximation switches the prime on the $\omega$’s, leading to (see, e.g., [35])

$$|{\beta }_{\omega {\omega }^{\prime }}{|}^2=\frac{c}{2\pi \kappa {\omega }^{\prime }}\frac{1}{e^{2\pi c\omega /\kappa }-1},$$

(31)

demonstrating Planck factor validity to either frequency approximation. The spectrum plot of the moving mirror radiation (Figure 4) illustrates the explicit Planck factor, which demonstrates the particles are distributed with a temperature given in Equation (2): $N\left(\omega \right)=\int \mathrm{d}{\omega }^{\prime }{|{\beta }_{\omega {\omega }^{\prime }}|}^2$.

4.2. Relation to Electrons and Black Holes

The moving mirror model, or dynamical Casimir effect (DCE), is closely related to electron radiation and black holes. Having the radiation spectrum of the mirror, it is possible to obtain the radiation spectra for these related systems. Let us explain the details.

The connection between DCE and point charge radiation has been suggested long ago (by Unruh and Wald [19] and by Ford and Vilenkin [18]), and has been developing since; see [12,13,14,15,16,36]. Eventually, this led to the realization that there is an exact functional identity between the radiation spectra in these models [4,37]. In papers [4,37], the corresponding transformation recipe was derived and checked; it was established that an electron corresponding to the mirror equation (26) radiates with the spectrum,

$$I\left(\omega \right)=\frac{{\mu }_{0}c{e}^2}{2{\pi }^2}\left(\frac{{tanh}^{-1}s/c}{s/c}-1\right)\frac{2\pi c\omega /\kappa }{e^{2\pi c\omega /\kappa }-1}.$$

(32)

One can immediately see the aforementioned Planck form of the spectrum with the same temperature as the mirror.

Thermal emission is not so surprising considering the Larmor power plateau (Figure 1), Feynman power plateau (Figure 2), and acceleration plateau (Figure 3). It is also in agreement with the close analogy for quantum and classical quantities of powers [16,17] and self-forces [18,20] between mirrors and electrons.

Black hole evaporation [1], and, in particular, the collapse of a null shell in the s-wave approximation, can also be described as a DCE [29,38]. This black hole–moving mirror correspondence has been successfully applied, for example, to such important spacetimes as Schwarzschild [39,40], Reissner–Nordström [41], and Kerr [42] metrics. In the triarchy ‘moving mirrors–electrons–black holes’, the quantum-classical temperature relation between Equation (1) and Equation (2) has been found, supporting the analog bridge of Section 1.1. It is an interesting question about which geometry corresponds to mirror Equation (26); we leave this for a future investigation.

5. Stefan–Boltzmann Law

It is natural to consider how the classical power scales according to the (1+1)-dimensional Stefan–Boltzmann law [30],

$$P\sim T^2,$$

(33)

rather than the (3+1)-dimensional Stefan–Boltzmann law,

$$P\sim A{T}^4,$$

(34)

which governs the power radiated from a black body in terms of its temperature. A first heuristic answer is the classical electron is a point particle with no area. We note that in flat spacetime, Equation (34) is the relevant contrasting expression for the energy transmission of a single photon polarization out of a closed hot black body surface with temperature T and area A into 3-dimensional space.

Ultimately, a better understanding may be related to black hole radiance. The scaling could occur for the same reason that black holes are one-dimensional information channels [2], whose power also scales according to $P\sim T^2$. In the context of Equation (2), the electron’s constant power peaks at exactly Equation (18), which is the analog of the known all-time constant equilibrium emission of the quantum stress tensor for the eternal thermal Carlitz–Willey moving mirror [11] and the late-time Schwarzschild mirror [39]. The 1+1 spacetimes corresponding to these mirrors exhibit horizons and have been considered as analogous to black holes; see, e.g., [29].

A complete investigation concerning the entropy and information flow related to the quadratic temperature dependence of the electron’s power emission is a worthwhile study but is outside the scope of this study. Nevertheless, in Section 5.1–Section 5.4, we make some necessary preliminary progress regarding exploring this Stefan–Boltzmann law in the context of its origin from electromagnetic spectral analysis, statistically maximized entropy, and classical thermodynamics.

5.1. Classical Stefan–Boltzmann

Using the aforementioned Stoney scale [5], the classical temperature of radiation from an electron is, regarding Equation (2),

$$T=\frac{{\mu }_0{e}^2\kappa }{2\pi k_B}.$$

(35)

Contrast this with the Kelvin scale and the temperature resembles the quantum Davies–Fulling–Unruh effect,

$$T=\frac{\hslash \kappa }{2\pi c{k}_B},$$

(36)

except, here, $\kappa$ is the peel (not uniform proper acceleration). The Davies–Fulling–Unruh expression is well-understood as a quantum effect and the proposed temperature of radiation emitted by an electron in the literature, e.g., [43,44,45,46].

However, the classical reasoning for Equation (35) is two-fold: dynamics and spectral analysis. Dynamically, one can compute the power [3] and find it agrees with the Stefan–Boltzmann law, $P\sim T^2$, at the plateau for high speeds, $s\approx 1$. The spectral analysis done in Ref. [3] confirms the Planck distribution, using the spectral distribution, $\mathrm{d}I\left(\omega \right)/\mathrm{d}\Omega =\mathrm{d}E/\mathrm{d}\omega \mathrm{d}\Omega$ [4],

$$I\left(\omega \right)=\frac{{\mu }_{0}c{e}^2}{2{\pi }^2}\left(\frac{\eta }{s}-1\right)\frac{M}{e^M-1},$$

(37)

where the dimensionless M is an analog to $\hslash \omega /\left(k_{B}T\right)$:

$$M\equiv \frac{{\mu }_{0}c{e}^2}{k_{B}T}\omega ,$$

(38)

with temperature (35). Moreover, as we see in the next sub-section; the characteristic frequency of the photons confirms the Stefan–Boltzmann law using basic classical electromagnetic spectral analysis.

5.2. Stefan–Boltzmann from Spectra

Let us assume thermal emission is described by a heuristic and characteristic frequency of the radiation when the electron is ultra-relativistic. Then, this frequency is

$$\frac{P}{k_{B}T}={\int }_0^{\infty }\mathcal{I}\left(f\right)\mathrm{d}f,$$

(39)

where the left-hand side is the ratio of the thermal power divided by the average energy in equilibrium, $k_{B}T$, as given by the equipartition theorem for the canonical ensemble. Here,

$$\mathcal{I}\left(f\right)=\frac{I\left(f\right)}{I_{\mathrm{infra}}}=\frac{M}{e^M-1}$$

(40)

is the dimensionless spectrum, as a function of frequency, $f=\omega /\left(2\pi \right)$, so that $M=\frac{{\mu }_{0}c{e}^2}{k_{B}T}2\pi f$. Here, $I_{\mathrm{infra}}$ is the infrared limit of the spectrum; see Equation (58) in Ref. [4] and Equation (9) above. Integrating Equation (39) over f provides the required result for the power,

$$P=\frac{\pi }{12}\frac{k_B^2}{{\mu }_{0}c{e}^2}{T}^2,$$

(41)

which is same $T^2$ temperature scaling as the (1+1)-dimensional Stefan–Boltzmann law [30] describing black hole radiance [47] and electron radiance [3], as derived straight from the dynamics of the trajectory using the proper acceleration via the Larmor power.

While the quadratic scaling of temperature in Equation (41) describes thermal noise power transfer in one-dimensional optical systems [30],

$$P=\frac{\pi }{6}\frac{k_B^2}{\hslash }{T}^2,$$

(42)

the most known case of one-dimensional thermal radiation is Johnson noise or Nyquist noise of electrical circuits [48],

$$P=\frac{\pi }{12}\frac{k_B^2}{\hslash }{T}^2,$$

(43)

which is also proportional to temperature squared, yet with an emissivity of $ϵ=1/2$. This lower emissivity arises from the fact that photons in electrical networks are polarised, and thus the resistors act as gray bodies rather than black bodies.

5.3. Stefan–Boltzmann from Entropy

Consider the classical accelerating electron in thermal equilibrium with its environment. By the second law of thermodynamics, the probability distribution, $p\left(n\right)$, must be such as to maximize the system entropy. Following Oliver [48], we determine $p\left(n\right)$, where n is an integer. Here, n is the number of photons emitted by the ensemble system. We start with the definition of Gibbs entropy for the electron,

$$S_e=-k_B\sum _{n=0}^{\infty }p\left(n\right)lnp\left(n\right),$$

(44)

with constraints of unitarity and averaging

$$\sum _{n=0}^{\infty }p\left(n\right)=1,\sum _{n=0}^{\infty }np\left(n\right)=\overline{n}.$$

(45)

The first constraint demands n must be some integer. The second constraint provides the average number of photons present where $\overline{n}$ need not be an integer.

The above summations will not vary as the distribution is varied as long as the entropy is maximized. A linear sum of all three,

$$\sum p\left(n\right)lnp\left(n\right)+\mathcal{A}\sum np\left(n\right)+\mathcal{B}\sum p\left(n\right),$$

(46)

where $\mathcal{A}$ and $\mathcal{B}$ are constants, have then zero variation,

$$\sum \left[lnp\left(n\right)+1+\mathcal{A}n+\mathcal{B}\right]\delta p\left(n\right)=0,$$

(47)

for small perturbations, $\delta p\left(n\right)$, of $p\left(n\right)$. This is satisfied if

$$lnp\left(n\right)+1+\mathcal{A}n+\mathcal{B}=0,$$

(48)

which provides a probability distribution,

$$p\left(n\right)=e^{-1-\mathcal{B}}{e}^{-\mathcal{A}n}.$$

(49)

Using the averaging and unitarity constrain, one obtains:

$$p\left(n\right)=\left(1-e^{-\mathcal{A}}\right)e^{-n\mathcal{A}},\overline{n}=\frac{1}{e^{\mathcal{A}}-1}.$$

(50)

Using the distribution $p\left(n\right)$ in the entropy, one then finds:

$$\frac{S_e}{k_B}=\frac{ln\left(e^{\mathcal{A}}-1\right)}{e^{\mathcal{A}}-1}-\frac{ln\left(-e^{-\mathcal{A}}+1\right)}{-e^{-\mathcal{A}}+1}.$$

(51)

The electron can be imagined to absorb an average non-integer classical energy (no ℏ),

$$\overline{W}\left(\omega \right)=\overline{n}{\mu }_{0}c{e}^2\omega ,$$

(52)

in analog to $\overline{n}\hslash \omega$, suitable for the Stoney scale (nominally, ${\mu }_{0}c{e}^2$ is, as we have seen, more than ten times smaller than ℏ). For a gray body with absorptivity a, average absorbed radiation is $\overline{n}\hslash \omega ·a$. In our setup, $a\sim {\mu }_{0}c{e}^2/\hslash$; see Section 5.5.

The average energy (52) comes from the surrounding outside thermal environment, producing an entropy change,

$$S_o=-\frac{\overline{n}{\mu }_{0}c{e}^2\omega }{T}=k_B\frac{1}{e^{\mathcal{A}}-1}M,$$

(53)

in the rest of the system ‘outside’ the electron. The dimensionless M is the Stoney temperature scale version of $\hslash \omega /\left(k_{B}T\right)$:

$$M\equiv \frac{{\mu }_{0}c{e}^2}{k_{B}T}\omega .$$

(54)

The total change, $\Delta S=S_e+S_o$, in entropy will progress until, in equilibrium, $\Delta S$ is maximized. Taking a derivative of $\Delta S$ with respect to $\mathcal{A}$ provides

$$\frac{1}{4}{\csc \mathrm{h}}^2\left(\frac{\mathcal{A}}{2}\right)(M-\mathcal{A})=0.$$

(55)

This is true if $\mathcal{A}=M$. The probability distribution and $\overline{n}$ is then written as

$$p\left(n\right)=\left(1-e^{-M}\right)e^{-nM},\overline{n}=\frac{1}{e^M-1}.$$

(56)

The average classical energy absorbed by the electron is, therefore

$$\overline{W}\left(\omega \right)=\overline{n}\mu c{e}^2\omega =\frac{1}{e^M-1}M{k}_{B}T.$$

(57)

The total thermal power is found by

$$P={\int }_0^{\infty }\overline{W}\left(\omega \right)\frac{\mathrm{d}\omega }{2\pi }=\frac{{\left(k_{B}T\right)}^2}{{\mu }_{0}c{e}^2}\frac{1}{2\pi }{\int }_0^{\infty }\frac{M}{e^M-1}\mathrm{d}M,$$

(58)

whose integral is ${\pi }^2/6$, so that

$$P=\frac{\pi }{12}\frac{k_B^2}{{\mu }_{0}c{e}^2}{T}^2,$$

(59)

which scales as the (1+1)-dimensional Stefan–Boltzmann law. This is classical thermal noise from a single accelerating electron [4].

5.4. Stefan–Boltzmann from Thermodynamics

The Stefan–Boltzmann law can be derived from thermodynamics alone in two steps (see the original paper [49]). In this derivation, we do not assume any particular form of the spectral frequency distribution.

5.4.1. Maxwell Relations

First, consider the Maxwell relations for the entropy. Let U be the radiation energy, and then $U=\rho \left(T\right)V$, where $\rho \left(T\right)$ is the energy density (we suppose that it depends only on the temperature T). Then, one has:

$$\mathrm{d}U=T\mathrm{d}S-p\mathrm{d}V,$$

(60)

from which it follows that

$$\mathrm{d}S=\frac{1}{T}(\mathrm{d}U+p\mathrm{d}V)=\frac{1}{T}\left(V\rho T\mathrm{d}T+(\rho +p)\mathrm{d}V\right).$$

(61)

From this, one can read off the first derivatives:

$$\begin{array}{cc}\hfill {\left(\frac{\partial S}{\partial T}\right)}_V& =\frac{V}{T}\frac{\mathrm{d}\rho }{\mathrm{d}T},\hfill \\ \hfill {\left(\frac{\partial S}{\partial T}\right)}_T& =\frac{\rho +p}{T}.\hfill \end{array}$$

(62)

Computing the second derivative, ${\partial }^{2}S/\partial T\partial V$, in two different ways, one obtains:

$${\left(\frac{\partial p}{\partial T}\right)}_V=\frac{\rho +p}{T}.$$

(63)

To finish the derivation, one needs an equation of state. Let us derive it.

5.4.2. Equation of State in 3+1 Dimensions

We start with the (3+1)-dimensional case. Consider the radiation inside a perfectly reflecting box (the perfect reflectivity assumption is actually not mandatory, but it simplifies the derivations.); see Figure 5. When radiation waves hit the box wall, the waves are reflected and therefore transfer some of the momentum to the wall. Take some area element, $\mathrm{d}A$, of the wall, and let the x axis be perpendicular to the wall in this vicinity. Let $\theta$ be the angle between the x axis and the wavevector of an incoming electromagnetic wave. Total momentum of the radiation coming from the solid angle, $\mathrm{d}\Omega$, is given by the energy divided by c:

$$|\overrightarrow{\mathtt{P}}|=\frac{1}{c}·\rho ·c\mathrm{d}tcos\theta ·\mathrm{d}A·\frac{\mathrm{d}\Omega }{4\pi }=\rho cos\theta \frac{\mathrm{d}\Omega }{4\pi }\mathrm{d}A\mathrm{d}t.$$

(64)

The momentum transfer is twice the x-projection:

$$\Delta {\mathtt{P}}_x=2|\overrightarrow{\mathtt{P}}|cos\theta =2\rho {cos}^2\theta \frac{\mathrm{d}\Omega }{4\pi }\mathrm{d}A\mathrm{d}t.$$

(65)

Dividing the momentum Equation (65) by $\mathrm{d}t$, one obtains the force, and then, dividing by $\mathrm{d}A$, the pressure. Integration over the solid angle, $\mathrm{d}\Omega$, one side of the wall finally provides

$$p=\frac{2\rho }{4\pi }\underset{0}{\overset{2\pi }{\int }}\mathrm{d}\varphi \underset{0}{\overset{1}{\int }}{cos}^2\theta \mathrm{d}cos\theta =\frac{1}{3}\rho .$$

(66)

Plugging Equation (66) into Equation (63) yields an ordinary differential equation, whose solution,

$$\rho =C·T^4,$$

(67)

is determined only up to an arbitrary constant, C. Note that, in general, this constant must be positive, but, otherwise, it is not restricted by the above derivation.

5.4.3. Other Dimensions

What happens in lower dimensions?

Consider a lower dimensional system embedded in 3-dimensional space. This means that one can still consider electromagnetic waves even for a one-dimensional system. Then, the only point to modify in the derivation in Section 5.4.2 is the solid angle part.

When the space is two-dimensional, there is no solid angle, and $\theta$ represents the polar angle. The integration in this case provides the result $p=\rho /2$. In a one-dimensional system, there are no angles at all, and one obtains: $p=\rho$.

From this, one can derive the corresponding Stefan–Boltzmann laws. We summarize the results in

Table 1. Stefan–Boltzmann law in various dimensions. See text for details.

Dimension	Equation of State	Stefan–Boltzmann Law
1	$p=\rho$$\phantom{/2}$	$P\sim T^2$
2	$p=\rho /2$	$P\sim T^3$
3	$p=\rho /3$	$P\sim T^4$

. For a body immersed in an equilibrium radiation heat bath, the radiated energy per second is proportional to the energy density of the ambient radiation (the proportionality coefficient depends on the surface area and absorptivity).

5.4.4. Lessons

Using only thermodynamics, the Stefan–Boltzmann law is determined by the dimensionality of the space, up to a coefficient. The coefficient is undetermined in this derivation: it depends on the physical system under consideration and is not fixed. In particular, Equation (41) is acceptable because of the scaling.

It is quite instructive to find that Equation (41) represents the Stefan–Boltzmann law in one-dimensional space; indeed what one would expect from a moving mirror in one spatial and one temporal dimension.

5.5. Electron as a Gray Body

In general, the Stefan–Boltzmann law relates the total power to the temperature. In 1+1 dimensions [30],

$$P_{\mathrm{black}\mathrm{body}}=\frac{\pi k_B^2}{6\hslash }A{T}^2;$$

(68)

see, e.g., the discussion below Equation (10) in Ref. [30]. Here, A is the body’s surface area: in 1+1 dimensions, $A=1$ (one side) or $A=2$ (two sides). In the moving mirror setup, it is natural to take $A=1$ as the observer is most often on one (the right) side of the mirror.

We can compare Equation (68) to the power in our setup. Regarding the quantum radiation power in Equation (19),

$$P_q=\frac{\pi k_B^2}{12\hslash }{T}^2,$$

(69)

one can see that Equation (69) corresponds to a gray body with absorptivity, $a=1/2$ (see, e.g., Section 3 in Ref. [30]).

The classical radiation power in Equation (18), when re-expressed in terms of the quantum temperature scale (see Equation (3)), becomes

$${\overline{P}}_e=\frac{{\mu }_0{e}^2{\kappa }^2}{48\pi c}=\frac{{\mu }_{0}c{e}^2}{\hslash }\frac{\pi }{12}\frac{k_B^2}{\hslash }{T}^2,$$

(70)

which is also proportional to the square of the temperature, $T^2$. One can see that Equations (68) and (70) are not quite the same; they are off by a factor (we take $A=1$, which is natural from the mirror’s perspective):

$${\overline{P}}_e=\frac{{\mu }_{0}c{e}^2}{2\hslash }{P}_{\mathrm{black}\mathrm{body}},\frac{{\mu }_{0}c{e}^2}{2\hslash }\approx 0.0458.$$

(71)

The existence of a coefficient tells us that the electron may be considered a (1+1)-dimensional quantum-radiating gray body that absorbs only about 4.58% of the incoming radiation.

The physical meaning of this absorptivity is intriguing and deserves further investigation, which we leave for future studies.

6. Conclusions

In this paper, we have helped to develop the analogy between the dynamical Casimir, black hole, and electron radiation temperature. We have found periods of constant power and radiation reaction, indicative of thermal equilibrium. Indeed, by analogy with the dynamical Casimir effect, we have demonstrated thermality, in part, by the symmetry between frequency modes in the analog spectrum for the radiation of an accelerated electron, which, at ultra-relativistic speeds, manifests explicit uniform plateau radiation emission commensurate with the spectral Planck distribution. The constant temperature is consistent with the constant periods of power, self-force, and peel acceleration.