Quantum Power Distribution of Relativistic Acceleration Radiation: Classical Electrodynamic Analogies with Perfectly Reflecting Moving Mirrors

Abstract

We find the quantum power emitted and distribution in 3 + 1-dimensions of relativistic acceleration radiation using a single perfectly reflecting mirror via Lorentz invariance, demonstrating close analogies to point charge radiation in classical electrodynamics.

1. Introduction

There are deep connections between point charge radiation in classical electrodynamics and quantum vacuum acceleration radiation from the perfectly reflecting point mirrors of DeWitt–Davies–Fulling [1,2,3]. For instance in 1982 Ford and Vilenkin [4] demonstrated that the force on a 1 + 1 moving mirror has the same covariant expression as the Lorentz–Abraham–Dirac (LAD) radiation reaction force of an arbitrary moving point charge in 3 + 1 classical electrodynamics. The connection extends to scalar source changes demonstrated, for instance, in 1992 by Higuchi, Matsas and Sudarsky [5] through the discovery that photon emission from a uniformly accelerated classical charge in the Minkowski vacuum corresponds to emission of a zero-energy Rindler photon into an Unruh thermal bath. In 1994, Hai [6] found the quantum energy flux integrated along a large sphere in the asymptotic future as the Larmor formula for the power radiated by a moving scalar charge with respect to an inertial observer. In 2002, Ritus [7,8,9,10] found symmetries linking creation of pairs of massless bosons or fermions by an accelerated mirror in 1 + 1 space and the emission of single photons or scalar quanta by electric or scalar charges in 3 + 1 space.

The above studies are just some of the fascinating connections, nowhere near exhaustive, so far discovered. Recent studies ([11,12,13]) have also strengthened the connection between quantum acceleration radiation and classical radiation of point charges, namely the works by Landulfo, Fulling and Matsas [12], who found that zero-energy Rindler modes are not mathematical artifacts but are critical to understanding the radiation in both the classical and quantum realm, confirming that Larmor radiation emitted by a charge can be seen as a consequence of the Unruh thermal bath. Moreover, Cozzella, Fulling, Landulfo, and Matsas [13] concluded that uniformly accelerated pointlike structureless sources emit only zero-energy Rindler particles using Unruh–deWitt detectors [14]. The analogies hold true with respect to the uniformly accelerated moving mirror, which emits zero energy flux but produces non-zero particle counts as computed from the beta Bogolubov coefficients (e.g., [15]).

Investigations are underway aimed at extending the 1 + 1 dimensional moving mirror model to 3 + 1 dimensions, and understanding the production of scalar particles in the relativistic regime [16]. These efforts are carried out with the goal of direct detection of relativistic moving mirror radiation [17,18,19], and they complement the growing accumulation of observations of the dynamical Casimir effect (see references therein [20]). Extension of 1 + 1 dimensions to 3 + 1 dimensions, particularly in connection to electrodynamics [21], should be done with caution because, for instance, there is a genuine difference between a two-dimensional world and an effective two-dimensional world arising when a charged particle is placed on a straight line in ambient space. In the former case, the vector potential A behaves in space as r, and hence the charged particle does not radiate at all, whereas in the latter case, A exhibits Coulomb-like behavior, $1/r$, so that usually Larmor radiation results [22].

Experimental verification will be facilitated by knowing the distribution of detected radiation. Recent studies have only just solved for the five classes of uniformly accelerated trajectory distribution [23] in classical electrodynamics (effective Unruh-like temperatures have also been calculated [24]). Unfortunately, mathematically, no settled upon or convincing [25] covariant expression has yet been derived to express the power distribution in a frame-independent formulation [26]. In order to know the distribution of quantum power from a relativistic moving mirror we are also forced to abandon cherished covariant language, while at the same time maintaining the principle of Lorentz invariance.

In this work we compute the quantum power Larmor formula for relativistic moving mirror radiation, the explicit non-covariant quantum power distribution, and apply the results to the simple case of abrupt mirror creation, violent acceleration, and near-instantaneous final constant velocity state of motion for the spectrum of radiation of the mirror. Along the way, we highlight the direct analogies to classical radiation from a moving point charge in electrodynamics.

Our paper is organized as follows: in Section 2, we obtain a total power definition for quantum radiation emitted by a single relativistic moving mirror in 1 + 1-dimensions. We highlight that it has identical form to the relativistic generalization of the Larmor formula for power emitted by an accelerated point charge. In Section 3, we derive the angular distribution in time of the quantum radiation of the mirror in 3 + 1 dimensions using Lorentz invariance. Here we utilize the ansatz that proper acceleration magnitude is a Lorentz scalar independent of direction or dimension. We determine that the quantum power distribution has the same form as in classical electrodynamics. In Section 4, we derive the radiation integral, demonstrating that the approach in electrodynamics equally applies to the mirror. Finally, in Section 5, we specialize our results to an abruptly created, rapidly accelerated, constant velocity moving mirror connecting the frequency-independent spectrum with that of beta decay. We discuss and conclude in Section 6 and Section 7, respectively. We use natural units throughout, $\hslash =c=1$. For conversion from the SI electrodynamics analog, one requires: $q\to 1$, ${\mu }_0\to 1$, ${ϵ}_0\to 1$. For the commonly used Gaussian units one converts by $4\pi {ϵ}_0\to 1$ and ${\mu }_0\to 4\pi$.

2. Relativistic Quantum Larmor Formula

The energy flux, $\mathcal{F}$, radiated by the mirror moving along a trajectory $p\left(u\right)$ in null coordinates where $u=t-x$ and p is the advanced time, $v=t+x$, is derived from the Davies–Fulling quantum stress tensor [2],

$${\mathcal{F}}^R\left(u\right)=-\frac{1}{24\pi }\{p\left(u\right),u\},$$

(1)

where the total energy emitted to the right of the mirror is (e.g., Walker [27]):

$$E^R={\int }_{-\infty }^{\infty }{\mathcal{F}}^R\mathrm{d}u,$$

(2)

defining the Schwarzian derivative by (e.g., Fabbri-Navarro-Salas [28]):

$$\begin{array}{cc}\hfill \left\{p\left(u\right),u\right\}& =\frac{p^{‴}}{p^{\prime }}-\frac{3}{2}{\left(\frac{p^{″}}{p^{\prime }}\right)}^2.\hfill \end{array}$$

(3)

This energy Equation (2) can be expressed in lab coordinates as (see Equation (2.34) of Good, Anderson, and Evans [29]):

$$E^R=\frac{1}{12\pi }{\int }_{-\infty }^{\infty }{\alpha }^2(1+\dot{x})\mathrm{d}t,$$

(4)

where $\alpha$ is the proper acceleration of the mirror. The energy radiated to the left, $E^L$, is found by the same expression but with a parity flip, $\dot{x}\to -\dot{x}$, so that the total radiated energy is:

$$E=E^R+E^L=\frac{1}{6\pi }{\int }_{-\infty }^{\infty }{\alpha }^2\mathrm{d}t.$$

(5)

This allows us to identify and define a relativistic quantum power for the moving mirror, $P\equiv \mathrm{d}E/\mathrm{d}t$,

$$E={\int }_{-\infty }^{\infty }\frac{dE}{dt}\mathrm{d}t\equiv {\int }_{-\infty }^{\infty }P\mathrm{d}t,$$

(6)

which gives, from Equation (5), a familiar acceleration scaling:

$$P_{\mathrm{mirror}}=\frac{{\alpha }^2}{6\pi }.$$

(7)

The quantum vacuum scalar radiation power, Equation (7), (in SI units $P_{\mathrm{mirror}}=\hslash {\alpha }^2/6\pi c^2$) emitted by the mirror enjoys the same scaling as the relativistic Larmor formula,

$$P_{\mathrm{electron}}=\frac{2}{3}\frac{q^2{\alpha }^2}{4\pi {ϵ}_0{c}^3}=\frac{q^2{\alpha }^2}{6\pi {ϵ}_0{c}^3}\to \frac{q^2{\alpha }^2}{6\pi },$$

(8)

in classical electrodynamics, where we start in SI units and “→” implies conversion to natural units where ${ϵ}_0=c=1$, and $\alpha$ is the magnitude of the proper acceleration of the moving point charge (e.g., electron).

3. Angular Distribution in Time

The relativistic quantum generalization of Larmor’s power formula, Equation (7), is a Lorentz invariant scalar,

$$P=P^{*}=\frac{{\alpha }^2}{6\pi },$$

(9)

where $P^{*}$ is the instant rest frame power and P is the lab frame power. Here we make a critical assumption about the $1+1$-dimensional result (Equation (9)): the universality of the Lorentz invariant scalar in any frame suggests it is independent of dimension. It is just a number after all, with no associated direction (see Appendix A for elucidation). We take this as an ansatz and find it possible to proceed. In this case, a simple definition for the angular distribution of the power Equation (9) of a moving mirror in $3+1$ dimensions can be written as (we imagine the boundary condition is still defined as a perfectly reflecting point, not a two-dimensional surface, but see the end of this section for more discussion):

$$P=\int \frac{d{P}^{*}}{d{\Omega }^{*}}d{\Omega }^{*}=\int \frac{d{P}^{*}}{d{\Omega }^{*}}\sin {\Theta }^{*}d{\Theta }^{*}d{\varphi }^{*}.$$

(10)

Here the * designates the instantaneous rest frame, and the angle ${\Theta }^{*}$ is between the acceleration 3-vector ${\mathbf{a}}^{*}$ and the direction to the observer in the * frame. What we have done is assumed that the power formula derived in $1+1$ dimensions (Equation (9)) holds true in $3+1$ dimensions (this was found 20 years ago by Bekenstein for power emitted by black holes in the context of information transmission [30]). Likewise we will assume the power is distributed in $3+1$ dimensions by using the $3+1$ dimensional Lorentz covariant definition of proper acceleration. Therefore:

$$\frac{d{P}^{*}}{d{\Omega }^{*}}=\frac{{|{\mathbf{a}}^{*}\times {\widehat{\mathbf{n}}}^{*}|}^2}{16{\pi }^2}=\frac{{\alpha }^2{\sin }^2{\Theta }^{*}}{16{\pi }^2},$$

(11)

where the instantaneous rest frame unit vector, ${\widehat{\mathbf{n}}}^{*}$, is:

$${\widehat{\mathbf{n}}}^{*}=\sin {\theta }^{*}\cos {\varphi }^{*}{\widehat{x}}^{*}+\sin {\theta }^{*}\sin {\varphi }^{*}{\widehat{y}}^{*}+\cos {\theta }^{*}{\widehat{z}}^{*},$$

(12)

and ${\mathbf{a}}^{*}$ is the proper acceleration 3-vector in the instant rest frame, where ${\alpha }^2\equiv -a_{\mu }{a}^{\mu }$. The acceleration 4-vector in the instant rest frame has components $(0,{\mathbf{a}}^{*})$, i.e., $\alpha$ is the invariant Lorentz scalar, the acceleration felt by the moving mirror itself—its “property” [31]. The numerical $16{\pi }^2$ factor in Equation (11) originates from:

$${\int }_0^{2\pi }{\int }_0^{\pi }\frac{{\sin }^2{\Theta }^{*}}{16{\pi }^2}\sin {\Theta }^{*}d{\Theta }^{*}d{\varphi }^{*}=\frac{1}{16{\pi }^2}\frac{8\pi }{3}=\frac{1}{6\pi }.$$

(13)

See Figure 1 for an illustration of the usual angle convention we have adopted. We now wish to find the power distribution starting from Equation (11). We will find the dimensional allocation without appeal to fields or potentials, using only the principle of Lorentz invariance.

First, let us transform the solid angle to the lab frame. Under Lorentz transform the solid angle is (e.g., [25]):

$$d{\Omega }^{*}=\frac{d\Omega }{{\gamma }^2{(1-\beta \cos \theta )}^2},$$

(14)

and the energy density is $d{U}^{*}=\gamma (1-\beta \cos \theta )dU$, while the square of the acceleration scalar is written,

$${\alpha }^2={\gamma }^6(a^2-{(\beta \times a)}^2).$$

(15)

Hence, we write, using $dt=\gamma d{t}^{*}$, the distribution from the source, Equation (11), in the lab frame coordinates

$$\frac{dP}{d\Omega }=\frac{1}{{\gamma }^4{(1-\beta \cos \theta )}^3}\frac{d{P}^{*}}{d{\Omega }^{*}},$$

(16)

obtaining:

$$\frac{dP}{d\Omega }=\frac{1}{16{\pi }^2}\frac{{\gamma }^2[a^2-{(\mathbf{\beta }\times \dot{\mathbf{\beta }})}^2]}{{(1-\beta \cos \theta )}^3}{\sin }^2{\Theta }^{*}.$$

(17)

Since the distribution is needed in the lab frame, the rest frame angle ${\Theta }^{*}$ should be described in terms of lab frame angles $\theta$, $\varphi$, and $\chi$ (see Appendix C). To put Equation (17) in a more explicitly useful (and recognizable) form without reference to the angle ${\Theta }^{*}$, we would like to show that:

$$g^2{\gamma }^2[a^2-{(\mathbf{v}\times \mathbf{a})}^2]{\sin }^2{\Theta }^{*}={[\widehat{\mathbf{n}}\times ((\widehat{\mathbf{n}}-\mathbf{v})\times \mathbf{a})]}^2,$$

(18)

or more concisely, that:

$$|{\mathbf{a}}^{*}\times {\widehat{\mathbf{n}}}^{*}{|}^2=\frac{{\gamma }^4}{g^2}{[\widehat{\mathbf{n}}\times ((\widehat{\mathbf{n}}-\mathbf{v})\times \mathbf{a})]}^2.$$

(19)

To do this, we express the instantaneous rest frame 3-vector acceleration, ${\mathbf{a}}^{*}$, in terms of the lab frame 3-vector acceleration, a, via the appropriate usual Lorentz acceleration transformation (e.g., [32]):

$${\mathbf{a}}^{*}=\frac{\mathbf{a}}{{\gamma }^2{(1-v^2)}^2}-\frac{(\mathbf{a}·\mathbf{v})\mathbf{v}(\gamma -1)}{v^2{\gamma }^2{(1-v^2)}^3}+\frac{(\mathbf{a}·\mathbf{v})\mathbf{v}}{{\gamma }^2{(1-v^2)}^3}.$$

(20)

We can transform instant rest angles $({\theta }^{*},{\varphi }^{*})$ to lab angles $(\theta ,\varphi )$ by:

$${\sin }^2{\theta }^{*}=\frac{{\sin }^2\theta }{{\gamma }^2{g}^2},{\cos }^2{\theta }^{*}=\frac{{(\cos \theta -\beta )}^2}{g^2},$$

(21)

where g is the Doppler factor, $g\equiv 1-\beta \cos \theta$ and ${\varphi }^{*}=\varphi$. This gives:

$$|{\mathbf{a}}^{*}\times {\widehat{\mathbf{n}}}^{*}{|}^2={|\widehat{\mathbf{n}}(\widehat{\mathbf{n}}·\mathbf{a})-\mathbf{a}-\mathbf{v}(\widehat{\mathbf{n}}·\mathbf{a})+\mathbf{a}(\widehat{\mathbf{n}}·\mathbf{v})|}^2\frac{{\gamma }^4}{g^2}.$$

(22)

From $\mathbf{A}\times (\mathbf{B}\times \mathbf{C})=\mathbf{B}(\mathbf{A}·\mathbf{C})-\mathbf{C}(\mathbf{A}·\mathbf{B})$, the right-hand side of Equation (22) is equal to the right-hand side of Equation (19). Therefore, our power distribution in the lab frame, Equation (17), is (see Appendix D for a demonstration of well-known limits):

$$\frac{dP}{d\Omega }=\frac{1}{16{\pi }^2}\frac{{[\widehat{\mathbf{n}}\times ((\widehat{\mathbf{n}}-\mathit{\beta })\times \dot{\mathit{\beta }})]}^2}{{(1-\beta \cos \theta )}^5}.$$

(23)

This form for the quantum power distribution of scalar radiation from a relativistic moving mirror is that of the widely used textbook result for the classical power distribution of an arbitrarily moving electric point charge (e.g., [33,34,35]). Note that this derivation does not rely upon fields or potentials and depends solely on the validity of Lorentz invariance of proper acceleration and its vector decomposition in 3 + 1 dimensions. Although the derivation method of Equation (23) was for quantum radiation from a moving mirror, it is also valid for classical radiation from a moving electron.

The exact equation of motion of the system in 3 + 1 dimensions has not been defined. The derivation does not rely upon the fields directly, but the system obeys the homogeneous Klein–Gordon equation everywhere except at the mirror and some boundary conditions at the mirror. The main result of this paper is to demonstrate that the constraints of Lorentz invariance and dependence only on acceleration completely determine the radiation distribution, given the single integrated power quantity Equation (9). Barring the details of the (1 + 1)-dimensional derivation of Equation (9), the (3 + 1)-dimensional result is independent of the precise boundary condition at the mirror (or whatever is located at the singular point).

4. Angular Distribution in Frequency

In this section we confirm a procedure to obtain the widely-used classical radiation integral of electrodynamics but for the case of quantum scalar radiation from a moving mirror. This is needed to find the angular distribution in frequency. We demonstrate that the derivation and assumptions applied in the context of classical electromagnetic fields is applicable to the quantum scalar field.

Converting to the frequency domain requires considering the power distribution in the time domain as an energy density distribution in both time and angle, and defining angular distribution in frequency,

$$\frac{dP\left(t\right)}{d\Omega }=\frac{d^{2}U}{dtd\Omega },\frac{dI\left(\omega \right)}{d\Omega }=\frac{d^{2}U}{d\omega d\Omega }.$$

(24)

where we write:

$$\frac{dU}{d\Omega }=\underset{-\infty }{\overset{+\infty }{\int }}dt\frac{dP\left(t\right)}{d\Omega }=\underset{0}{\overset{\infty }{\int }}d\omega \frac{dI\left(\omega \right)}{d\Omega }.$$

(25)

Parseval’s theorem can be used to deduce that:

$$\frac{dU}{d\Omega }=\underset{-\infty }{\overset{+\infty }{\int }}dt\frac{dP\left(t\right)}{d\Omega }=\frac{1}{2\pi }\underset{-\infty }{\overset{+\infty }{\int }}d\omega \frac{dP\left(\omega \right)}{d\Omega }.$$

(26)

We apply a reality condition to the integrand such that $P\left(t\right)=P^{*}\left(t\right)$ and $P\left(\omega \right)=P^{*}(-\omega )$, which gives an even integral,

$$\frac{dU}{d\Omega }=\frac{1}{\pi }\underset{0}{\overset{+\infty }{\int }}d\omega \frac{dP\left(\omega \right)}{d\Omega }.$$

(27)

The angular distribution in frequency can be found by a derivative,

$$\frac{d}{d\omega }\frac{dU}{d\Omega }=\frac{d}{d\omega }\frac{1}{\pi }\int d\omega \frac{dP\left(\omega \right)}{d\Omega },$$

(28)

which amounts to, using the notation of Equation (24):

$$\frac{dI\left(\omega \right)}{d\Omega }=\frac{d^{2}U}{d\omega d\Omega }=\frac{1}{\pi }\frac{dP\left(\omega \right)}{d\Omega }.$$

(29)

We now have picked up a $\pi$ and a Fourier transform. We plug in Equation (23), which is the time-domain Equation (24), into:

$$\frac{dP\left(\omega \right)}{d\Omega }=\underset{-\infty }{\overset{+\infty }{\int }}dt\frac{dP\left(t\right)}{d\Omega }{e}^{i\varphi },$$

(30)

where $\varphi =\omega (t_r-\widehat{\mathit{n}}·{\mathit{r}}_0\left(t_r\right))$, using the radiation zone approximation. This assumes the observation point is very far from the regions of space where the acceleration is non-zero: ${\widehat{\mathit{n}}}_r\approx \widehat{\mathit{r}}$ is constant. The mirror always moves on some arbitrary trajectory ${\mathit{r}}_0\left(t\right)$ with velocity $\mathit{v}\left(t\right)=\dot{{\mathit{r}}_0}\left(t\right)$. Then, using Equation (29), as well as expressing the Fourier transform over retarded time, $dt=gd{t}_r$, gives the widely-used analog formula (see, e.g., Jackson [33] or Zangwill [34]):

$$\frac{dI\left(\omega \right)}{d\Omega }=\frac{1}{16{\pi }^3}{\left|\underset{-\infty }{\overset{\infty }{\int }}d{t}_r\frac{[\widehat{\mathit{n}}\times (\widehat{\mathit{n}}-\mathit{\beta })\times \dot{\mathit{\beta }}]}{{(1-\beta \cos \theta )}^2}{e}^{i\varphi }\right|}^2.$$

(31)

An important virtue of the form of Equation (31) is that the integrand is zero when the mirror acceleration is zero, which will always be the case for collision scattering-type and open-orbit situations where the mirror is subjected to a force for a finite amount of time. In the next section we will look specifically at such a situation in the case of abrupt creation of the mirror itself.

5. Specialized Case: Abrupt Mirror Creation

The sudden creation of a fast moving mirror can be viewed, for our purposes, as the violent acceleration of the mirror initially at rest to some final velocity within a very short time interval, or, alternatively, as the sudden switching on of the boundary describing the moving mirror in the same short time interval.

Consider the initial situations, where the the mirror is either static or non-existent: there is no radiation. Now consider the more subtle final situation, where a Lorentz boost to the frame of the moving mirror renders the final constant velocity state of motion as trivially non-radiating as that of a static or non-existent mirror. Thus, it is the intermediate situation, where the radiation is sourced from the non-trivial physics during the short time period, in which there is violent acceleration or sudden creation.

Our goal is to calculate the spectrum of this perfectly reflecting boundary. While our calculation can be interpreted both as an accelerated boundary with violent acceleration to constant velocity, and the sudden creation of the mirror, these systems are not the same! As pointed out in the previous sections, the specification of the particular equation of motion and boundary condition in 3 + 1 dimensions is not given. Examples can be found in previous work, such as that by Brown and Louko, who described the case of smooth creation of a two-sided Dirichlet mirror in (1 + 1)-dimensional flat space-time that generates a flux of real quanta [19]. Also relevant is the creation of boundaries in 3 + 1 dimensions and the resulting particle production; that induced by the creation that changes the gravitational field itself as worked out for cosmic strings by Parker [36] (unlike a point Dirichlet mirror, it is known that cosmic strings have effects on the quantum fields around them in 3 + 1 dimensions). For our particular spectral result, the systems cannot be distinguished from each other. That is to say, while sudden creation and violent acceleration are not the same physical processes, the nature of our approximation gives a regime of validity where the result can be interpreted as either sudden creation or violent acceleration. The spectrum will critically depend on a short time period approximation of Equation (31), where all that matters is before $t<0$ there is no radiation contribution to the integral and after $t>0$ the mirror is moving at constant velocity relative to a faraway observer. It is this limiting short-period form that can be straightforwardly integrated.

To compute the spectrum, we first start with the use of a perfect differential identity (see Appendix B for a proof):

$$\frac{\widehat{\mathit{n}}\times (\widehat{\mathit{n}}-\mathit{\beta })\times \dot{\mathit{\beta }}}{{(1-\beta \cos \theta )}^2}=\frac{d}{d{t}_r}\left[\frac{\widehat{\mathit{n}}\times (\widehat{\mathit{n}}\times \mathit{\beta })}{1-\beta \cos \theta }\right],$$

(32)

where the derivatives are evaluated at retarded time. We apply this identity to perform an integration by parts on Equation (31),

$$\begin{array}{ccc}\hfill \frac{dI\left(\omega \right)}{d\Omega }& =& \frac{1}{16{\pi }^3}\left|{\left.\frac{\widehat{\mathit{n}}\times (\widehat{\mathit{n}}\times \mathit{\beta })}{1-\beta \cos \theta }{e}^{i\varphi }\right|}_0^{+\infty }\right.-{\left.i\omega \underset{0}{\overset{\infty }{\int }}d{t}_r\widehat{\mathit{n}}\times (\widehat{\mathit{n}}\times \mathit{\beta })e^{i\varphi }\right|}^2,\hfill \end{array}$$

(33)

where the boundary terms vanish. Again, we have defined $\varphi =\omega t_r-\mathit{k}·{\mathit{r}}_0\left(t_r\right)$, where $k=\omega$. Using $\widehat{\mathit{n}}\approx \widehat{\mathit{r}}$ as a constant vector means it comes outside the integral, and since $|\widehat{\mathit{r}}\times (\widehat{\mathit{r}}\times \mathit{\beta }){|}^2={|\widehat{\mathit{r}}\times \mathit{\beta }|}^2$ for any vector $\mathit{\beta }$, we obtain the angular spectrum of radiated energy as:

$$\frac{dI\left(\omega \right)}{d\Omega }=\frac{{\omega }^2}{16{\pi }^3}{\left|\widehat{\mathit{r}}\times \underset{0}{\overset{\infty }{\int }}dt\mathit{\beta }\left(t\right)\exp [-i(\mathit{k}·{\mathit{r}}_0\left(t\right)-\omega t)]\right|}^2.$$

(34)

The integral is zero from $-\infty$ to 0 but non-zero from 0 to $+\infty$. The non-zero contribution comes because $\mathit{\beta }\left(t\right)=\mathit{\beta }$ for $t>0$ with trajectory function ${\mathit{r}}_0\left(t\right)=\mathit{\beta }t$. Using $\mathit{k}=\omega \widehat{\mathit{r}}$,

$$\frac{dI\left(\omega \right)}{d\Omega }=\frac{{\omega }^2}{16{\pi }^3}{\left|\widehat{\mathit{r}}\times \mathit{\beta }\right|}^2{\left|\underset{0}{\overset{\infty }{\int }}dt\exp [-i\omega (\widehat{\mathit{r}}·\mathit{\beta }-1)t]\right|}^2.$$

(35)

Notice the pre-factor ${\omega }^2$ frequency dependence, which will ultimately cancel out after appropriate integration. The integral diverges at late times (upper limit), so we use a convergence regulator $e^{-ϵt}$ and set $ϵ\to 0$ after integration. Using $\widehat{\mathit{r}}·\mathit{\beta }=\beta \cos \theta$, where $\theta$ is the angle between $\mathit{\beta }$ and the observation point, the integral is:

$$\underset{0}{\overset{\infty }{\int }}dt\exp [-i\omega (\beta \cos \theta -1)t-ϵt]=\frac{i}{iϵ+\omega (1-\beta \cos \theta )},$$

(36)

where we can now set $ϵ\to 0$, and write the square of the integral as:

$${\left|\underset{0}{\overset{\infty }{\int }}dt\exp [-i\omega (\widehat{\mathit{r}}·\mathit{\beta }-1)t]\right|}^2=\frac{1}{{\omega }^2{(1-\beta \cos \theta )}^2},$$

(37)

demonstrating that frequency dependence cancels out exactly from Equation (35). Using $\widehat{\mathit{r}}\times \mathit{\beta }=\beta \sin \theta$, the result is:

$$\frac{dI\left(\omega \right)}{d\Omega }=\frac{1}{16{\pi }^3}\frac{{\beta }^2{\sin }^2\theta }{{(1-\beta \cos \theta )}^2}.$$

(38)

This is angular distribution of energy radiated per unit frequency in the frequency domain for the abrupt creation or violent acceleration of a moving mirror in 3 + 1 dimensions. We can see that in this application, the angular distribution of energy radiated per unit frequency (and the total energy radiated per unit frequency in the next subsection) are independent of frequency.

Total Energy & Particles

The total energy radiated per unit frequency is found by the integral

$$I\left(\omega \right)=\frac{1}{16{\pi }^3}\int d\Omega \frac{{\beta }^2{\sin }^2\theta }{{(1-\beta \cos \theta )}^2}$$

(39)

Using $d\Omega =\sin \theta d\theta \mathrm{d}\varphi$, the numerator scales by ${\sin }^3\theta$ and integrating $\theta$ from $(0,\pi )$ and $\varphi$ from $(0,2\pi )$, we obtain: a simple expression in terms of the final rapidity, $\eta ={\tanh }^{-1}\beta$, and final velocity, $\beta$, of the mirror,

$$I\left(\omega \right)=\frac{1}{2{\pi }^2}\left(\frac{\eta }{\beta }-1\right).$$

(40)

For non-relativistic speeds $\beta \ll 1$, the radiated intensity is negligible and scales as $I\left(\omega \right)={\beta }^2/(6{\pi }^2)$. For ultra-relativistic speeds, $I\left(\omega \right)=\eta /(2{\pi }^2)$. See Figure 2 for a plot of Equation (40).

Ultimately, the spectrum does not depend on the frequency $\omega$ because the mirror is made to be in instantaneous motion at $t=0$ with velocity $\mathit{v}$. A more physical picture will have the velocity approached in some very short time interval $\Delta t$. In this case the spectrum will die off and be negligible when $\omega \gg 1/\Delta t$. The number of scalars per unit energy range is given by:

$$N\left(\omega \right)=\frac{1}{2{\pi }^2\omega }\left(\frac{\eta }{\beta }-1\right).$$

(41)

Their total energy radiated has a maximum frequency dependence:

$$E_{\mathrm{rad}}=\underset{0}{\overset{{\omega }_{\max }}{\int }}\omega N\left(\omega \right)d\omega =\frac{1}{2{\pi }^2}\left(\frac{\eta }{\beta }-1\right){\omega }_{\max }.$$

(42)

We reiterate that the mirror is assumed to be created with constant velocity such that the acceleration period occurs within a very short time interval. The spectrum function is negligibly small when the frequency is much larger than the inverse time interval of acceleration. The intensity distribution, Equation (40), has the same form as a typical inner bremsstrahlung spectrum [33] in classical electrodynamics. In the case of beta decay [33,34], e.g., the electron plays the role of the mirror.

6. Discussion

First, we should comment on the regime of validity of $P={\alpha }^2/\left(6\pi \right)$ and the assumptions that were required to obtain it. Equation (4) has undergone an integration by parts (for details see Equation (2.34) of [29]), which is only valid if globally the acceleration of the mirror is asymptotically zero in both the past and future. Moreover, the boundary terms only disappear if the mirror is sub-light speed asymptotically, which is an even stronger constraint.

This seemingly inconsequential subtlety almost certainly has much to do with the longstanding debate [37,38] over whether a uniformly accelerated point charge radiates (consider the clear distinctions needed between the electromagnetic power received by a set of far-off observers and the instantaneous mechanical power loss of the charge in [39]); in this case we comment that the power formula does not apply to global uniform acceleration, as that would violate the previously mentioned assumptions giving non-vanishing boundary terms. That is, a globally uniformly accelerated mirror does not radiate [15] energy but does radiate particles (as is well known [40,41,42]). This is also in line with the fact that uniformly accelerated point-like structure-less sources emit only zero-energy Rindler particles [13].

Second, we comment on the assumption that the Lorentz power scalar holds in higher dimensions. Conformal invariance breaks down in higher dimensions [18] but our derivation, underscoring the Larmor formula as a Lorentz scalar independent of direction, does not ostensibly require conformal invariance. The moving mirror in 1 + 1 dimensions permits exact solutions to the field equation because of conformal invariance but our result does not appeal to an exact explicit solution to the wave equation of motion. It only appeals to the dynamics of the mirror as computed through the general renormalized quantum stress tensor as given by the Schwarzian derivative. In general contexts, the quantum stress tensor, as opposed to particle production, is much easier to obtain. This is true in higher dimensions where conformal invariance no longer holds (see [18] and references therein, e.g., Refs [15,16,17,18,19]).

Third, we comment on the general applicability of the power distribution of Equation (23). This formula applies to very general trajectories (albeit globally asymptotically sub-light speed and time-like; although this does not stop one from being able to locally compute the power distributions of the five classes of uniform acceleration, which for an electric charge will be the same form as for uniformly accelerated moving mirrors [23]. It is unclear whether the effective temperatures will also carry-over [24]). For instance, a mirror moving on a circular arc near the speed of light will emit a synchrotron-type of radiation in the form of a narrow and intense beam directed tangent to the arc, implying that a fixed observer will see a brief flash or pulse of radiation every time the mirror moves directly toward them.

This power distribution, Equation (23), in the context of vacuum acceleration radiation, warrants study because of its importance in connection to synchrotron radiation. Disentanglement in contexts where it is relevant will be essential in confirming the source’s origin. While astrophysical synchrotron radiation is a powerful indicator of the presence of magnetic fields and particle acceleration mechanisms near pulsars and black holes, the possibility that similar radiation could point to quantum amplification of vacuum fluctuations due to an accelerated boundary condition is of interest to a wide range of physicists concerned with relativistic quantum fields and information.

7. Conclusions

We have investigated the acceleration radiation emitted by a single relativistic perfect point mirror in 3 + 1 dimensions and its analogies with a moving point charge in electrodynamics. Namely,

• We found the quantum power formula for moving mirrors and identified it with Larmor’s form.
• The quantum Larmor formula is not applicable for eternal uniform acceleration.
• We generalized the $1+1$-dimensional moving mirror model to 3 + 1 dimensions in the context of distributed power. This was done with the ansatz that the scalar power in $1+1$ dimensions (which is proportional to the proper acceleration magnitude squared) is also the scalar power in 3 + 1 dimensions, i.e., the scalar will be invariant under Lorentz transformations in 3 + 1 dimensions and obeys Larmor’s form. The only covariant choice of definition for scalar magnitude of proper acceleration in 3 + 1 dimensions is that constructed by the usual Lorentz 4-vector acceleration.
• Consequently, we derived the power distribution and found its form is in analog to the well-known power distribution in electrodynamics (e.g., the apportioning responsible for synchrotron radiation.). This was done by Lorentz invariant vector decomposition of relativistic acceleration. The allotment is a result of the motion of the mirror only and the derivation did not rely on the use of fields or potentials, i.e., there are no requisite Lienard–Wiechert potentials or electric and magnetic field counterparts for the quantum scalar radiation.
• We derived the spectrum of a moving mirror abruptly created and violently accelerated to a constant velocity. In analog to beta decay in classical electrodynamics, we found the appearance of the moving mirror plays the role of the moving electron.

Knowing the radiated distribution of power for the moving mirror is a necessary first step toward precise orientation for accurate detection. The work here determines that the distribution for quantum scalar radiation from a moving mirror is the same form as that of accelerated electron radiation in classical electrodynamics. The close analogy suggests direction for future work, including the application and interpretation of moving mirror trajectories in analog to the exactly known spectra associated with the moving point charge, including, for example, and not limited to, the well-known bending magnet trajectory, the undulator trajectory, and the collinear acceleration burst trajectory.