Light and Airy: A Simple Solution for Relativistic Quantum Acceleration Radiation

Abstract

We study the quantum radiation of particle production by vacuum from an ultra-relativistic moving mirror (dynamical Casimir effect) solution that allows (possibly for the first time) analytically calculable time evolution of particle creation and an Airy particle spectral distribution. The reality of the beta Bogoliubov coefficients is responsible for the simplicity, and the mirror is asymptotically inertial at the speed of light, with finite energy production. We also discuss general relations regarding negative energy flux, the transformation to the 1-D Schrödinger equation, and the incompleteness of entanglement entropy.

1. Introduction

The dynamical Casimir effect (DCE) [1,2,3,4] is a celebrated and multidisciplinary phenomena that plays an important role in many areas of physics ranging from quantum fields, atomic physics, condensed matter, and applications in nanotechnology all the way to astrophysics, cosmology and gravitation. The overarching reach of the DCE results from the fact that it is sourced, like the Unruh effect [5], by the amplified zero-point fluctuations of quantized fields, inherent to physical systems. Notable theoretical studies [6,7,8] have helped lead to experiments (the first being [9]) which have been successful at verifying the existence of the DCE (see a pedagogical overview here: [10]). The quantum acceleration radiation of the DCE is well-connected to the Hawking effect [11], potentially bringing experimental data to bear on the quantum relationship between gravity and acceleration.

Studying acceleration radiation with finite energy production is physically well-motivated. In the case of black hole evaporation, for example, this is a conspicuous sign that the evolution has finished, energetic radiation has stopped, and conservation of energy is upheld. The canonical moving mirror model of DeWitt–Davies–Fulling [2,3,4], for a single perfectly reflecting boundary point in flat (1+1)-D spacetime, has solutions demonstrating in a simple way total finite energy production (e.g., the four decade old solution of Walker-Davies which first derived a finite amount of energy creation [12]). Recently, several finite energy mirror solutions have been found that demonstrate close connections to strong gravitational systems. These gravity analog models are called accelerated boundary correspondences (ABCs). The infinite energy ABC solutions correspond to the most well-known spacetimes, e.g., Schwarzschild [13], Reissner-Nordström (RN) [14], Kerr [15], and de Sitter [16]. The finite energy ABC solutions closely characterize interesting well-known curved spacetime end-states, including extremal black holes (asymptotic uniformly accelerated mirrors [15,17,18,19,20]), black hole remnants (asymptotic constant-velocity mirrors [21,22,23,24,25,26]) and complete black hole evaporation (asymptotic zero-velocity mirrors [12,27,28,29,30,31,32]).

Despite this progress, it has been very hard to find a mirror solution whose particle spectrum is simple. Only two known solutions have analytic forms, one whose spectrum is an infinite sum of terms [26] and another which is so lengthy as to be prohibitively cumbersome [27,28]. Consequently, analytic time evolution is impossible to find for the above spectra. Further investigation of the particle production at any given moment is hobbled because one must instead resort to numerical analysis and finite sized frequency-time bins utilizing the discrete nature of orthonormal wave packets [30].

Motivated by simplicity, we take a step back and consider that any Bogoliubov transformation can be broken down into two types: (1) the trivial unitary transformation with $\beta$ Bogoliubov coefficient zero, $\beta =0$, indicating no particle production and (2) squeezing transformations where the $\beta \ne 0$ is given by a transformation matrix that is diagonal [33] (see the Bloch-Messiah decomposition or the theory of singular values). The simplest examples of the non-trivial transformations are those where the Bogoliubov coefficients are real-valued. We therefore look for some mirror motion (i.e., ABC) that should lead to a real non-zero beta Bogoliubov coefficient for particle creation, and anticipate corresponding simplicity in the resulting spectrum.

We take the simplest possible choice for global mirror motion with characteristics leading to the desired reality of the Bogoliubov coefficient, and indeed find a simple solution for the particle production spectrum. Remarkably, a transformation to the time domain on this spectrum analytically gives the particle production at any given moment.

Our paper is organized as follows: in Section 2 we give a very brief motivation of the connection between the reality of the beta Bogoliubov coefficient and the mirror trajectory properties. We analyze this accelerated trajectory in Section 3, computing the key relativistic dynamical properties such as rapidity, speed, and acceleration. In Section 4, we derive the energy radiated, by analysis of the quantum stress tensor, and in Section 5 we derive the particle spectrum, finding a unique Airy-Ai form for the radiation and confirming consistency with the stress tensor results. Finally, in Section 6 we compute the time evolution of particle creation analytically. Appendix A and Appendix B discuss some general properties leading to necessary negative energy flux, and connecting to the 1-D Schrödinger equation, respectively. Appendix C is a note on the connection between rapidity and entanglement entropy. Throughout we use natural units, $\hslash =c=1$.

2. Reality, Acceleration, and Inertia

The beta Bogoliubov coefficient controls quantum particle production and to compute it, we need the trajectory of the mirror. Mirror motions can be written in configuration space, $(x,t)$, but also in light-cone (null) coordinates $(u,v)$. Here retarded time is $u=t-x$ and advanced time is $v=t+x$, while the moving mirror trajectory $f\left(v\right)$ gives retarded time location, where u is switched out for f since u is the independent variable and f is the trajectory function. The beta Bogoliubov coefficient is [34]

$${\beta }_{\omega {\omega }^{\prime }}=\frac{-1}{4\pi \sqrt{\omega {\omega }^{\prime }}}{\int }_{-\infty }^{+\infty }dvs.e^{-i{\omega }^{\prime }vs.-i\omega f\left(v\right)}\left(\omega f^{\prime }\left(v\right)-{\omega }^{\prime }\right),$$

(1)

where $\omega$ and ${\omega }^{\prime }$ are the frequencies of the outgoing and incoming modes, respectively, [35].

To maintain finite energy and the simplicity of no information loss, there must not be a horizon at finite time, and the acceleration must vanish at infinity (i.e., the mirror motion must be asymptotically inertial). Under these conditions we can carry out an integration by parts to give

$${\beta }_{\omega {\omega }^{\prime }}=\frac{1}{2\pi }\sqrt{\frac{{\omega }^{\prime }}{\omega }}{\int }_{-\infty }^{+\infty }dv{e}^{-i{\omega }^{\prime }v-i\omega f\left(v\right)}.$$

(2)

To guarantee a real-valued beta Bogoliubov coefficient, the mirror trajectory $f\left(v\right)$ must be an odd function so that the exponential over the symmetric interval turns into a cosine of the argument, i.e., a real valued function. The simplest odd function that accelerates in the required manner is $f\left(v\right)\sim vs.+v^3$. We will find this results in not only interesting dynamics, but analytic calculation of particle production spectrum and time evolution.

3. Trajectory Motion

As motivated in the previous section, we expect the accelerated mirror trajectory

$$f\left(v\right)=vs.+{\kappa }^2\frac{v^3}{3},$$

(3)

to have interesting physical properties. Here $\kappa$ is a quantity related to the acceleration (and the surface gravity in the black hole case).

We can also write the trajectory in spacetime coordinates,

$$t=-x+\frac{1}{\kappa }{(-6\kappa x)}^{1/3},$$

(4)

taking the real cube root, or

$$x=-t-\frac{1}{2\kappa }\left[A_{+}^{2/3}{A}_{-}^{1/3}+A_{+}^{1/3}{A}_{-}^{2/3}\right],$$

(5)

where

$$A_{\pm }=3\kappa t\pm \sqrt{9{\kappa }^2{t}^2+8}.$$

(6)

Note at late times $x\to -t+\mathcal{O}\left(t^{1/3}\right)$. These forms make it obvious that asymptotically the mirror travels at the speed of light.

A spacetime plot with time on the vertical axis is given of the trajectory in Figure 1. A conformal diagram is plotted in Figure 2. We next investigate the dynamics of the trajectory Equation (3).

We compute the rapidity $\eta \left(v\right)$ by $2\eta \left(v\right)\equiv -ln{f}^{\prime }\left(v\right)$ where the prime is a derivative with respect to the argument,

$$\eta \left(v\right)=-\frac{1}{2}ln\left({\kappa }^2{v}^2+1\right).$$

(7)

From the rapidity we may easily compute the velocity $V\equiv tanh\eta$, plugging in Equation (7),

$$V\left(v\right)=-tanh\left[\frac{1}{2}ln\left({\kappa }^2{v}^2+1\right)\right]=\frac{-{\kappa }^2{v}^2}{2+{\kappa }^2{v}^2},$$

(8)

and the proper acceleration, which follows from $\alpha \left(v\right)\equiv e^{\eta \left(v\right)}{\eta }^{\prime }\left(v\right)$,

$$\alpha \left(v\right)=-\frac{{\kappa }^{2}v}{{\left({\kappa }^2{v}^2+1\right)}^{3/2}}.$$

(9)

At $x=t=0=v$, the velocity and acceleration are zero. At asymptotic infinity, the velocity is the speed of light and the acceleration goes to zero. The magnitude of the velocity, Equation (8), along with the proper acceleration, Equation (9), are plotted in Figure 3. Often, an asymptotically initial zero velocity state is more common. Trajectories are normally either globally defined or piece-wise defined where the accelerated piece is glued to the static piece at the origin. Thus, the asymptotic light-speed property distinguishes the Airy trajectory Equation (3). However, what does it mean to start inertial at the speed of light? What exactly is the initial state of the system? Recall that the ‘mirror’ in the moving mirror model is a massless boundary condition imposed on the fields such that the modes are always zero at it. To start inertial is to start time-like, physically well-associated with familiar world-lines (which are by definition, time-like). The modes in the initial vacuum state are the usual plane wave form as is familiar from ordinary quantum field theory in Minkowski space. The only difference here is that the initial asymptotic inertial state of motion saturates the speed limit and pushes the model to its extreme, but causal effects on the field are still in the past light cone of the ‘reflection’ events that disturb the quantum field and amplify zero-point fluctuations. Cause and effect are still contiguously mediated across the flat spacetime.

4. Energy Flux and Total Energy

4.1. Energy Flux

The quantum stress tensor reveals the energy flux emitted by the moving mirror. Typically, one will see [3]

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

(10)

where the energy flux, $F\left(u\right)$, is a function of light-cone coordinate retarded time $u=t-x$ [4,34] and the brackets define the Schwarzian derivative. The trajectory in light-cone coordinates of the mirror is $p\left(u\right)$ which is the advanced time position “v” as a function of retarded time u. However, since we want advanced time v as the independent variable, we write the radiated energy flux using $f\left(v\right)$ [16,21],

$$F\left(v\right)=\frac{1}{24\pi }\{f\left(v\right),v\}{f}^{\prime }{\left(v\right)}^{-2},$$

(11)

where the Schwarzian brackets are defined as usual,

$$\{f\left(v\right),v\}\equiv \frac{f^{‴}}{f^{\prime }}-\frac{3}{2}{\left(\frac{f^{″}}{f^{\prime }}\right)}^2.$$

(12)

For $f\left(v\right)$ given by Equation (3), this yields

$$F\left(v\right)=\frac{{\kappa }^2}{12\pi }\frac{1-2{\kappa }^2{v}^2}{{\left({\kappa }^2{v}^2+1\right)}^4}.$$

(13)

It is clear that asymptotically $F\left(v\right)\to 0$ for both $v\to \pm \infty$. Figure 4 shows the energy flux as a function of advanced time v.

4.2. Total Energy

The total energy measured by a far away observer at ${\mathcal{I}}_R^{+}$ is [36]

$$E={\int }_{-\infty }^{\infty }F\left(u\right)du,$$

(14)

where integration occurs over retarded time (it takes the energy time to reach ${\mathcal{I}}_R^{+}$). Since we are using advanced time v, we write this with $du=\frac{df}{dv}dv$ to get the Jacobian correct,

$$E={\int }_{-\infty }^{+\infty }F\left(v\right)f^{\prime }\left(v\right)dv.$$

(15)

Plugging in Equations (3) and (13) into Equation (15), with Jacobian $du/dv={\kappa }^2{v}^2+1$, the simple result is

$$E=\frac{\kappa }{96},$$

(16)

which is finite and positive.

Physically, the finite value tells us the evaporation process stops, similar to the ABC’s of extremal black holes (asymptotic uniformly accelerated mirrors), black hole remnants (non-horizon sub-light-speed asymptotic coasting mirrors), and complete black hole evaporation (asymptotic static moving mirrors). The fact that the total energy is positive is consistent with the quantum interest conjecture [37] as derived from quantum inequalities [38].

4.3. Negative Energy Flux

As seen from Figure 4, there are regions of negative energy flux (NEF). This is required by the unitarity sum rule (see Appendix A and Appendix B and, e.g., [27]). These regions extend for $\left|v\right|>1/\left(\kappa \sqrt{2}\right)$. The total negative energy is, by symmetry,

$$E_{NEF}=2{\int }_{v=+\frac{1}{\kappa \sqrt{2}}}^{v=+\infty }F\left(v\right)f^{\prime }\left(v\right)dv,$$

(17)

which gives an analytic result

$$E_{NEF}=\frac{\kappa \left(-10\sqrt{2}+3\pi -6{cot}^{-1}\sqrt{2}\right)}{288\pi }=-0.00930\kappa$$

(18)

As a ratio, the emission of NEF to positive energy flux (PEF) is

$$\frac{|E_{NEF}|}{E_{PEF}}\approx 47.1\%.$$

(19)

Note one cannot judge by eye this ratio in Figure 4 due to the redshift Jacobian $f^{\prime }\left(v\right)$ in Equation (17).

The NEF is a result of asymptotic inertia. We point out that the asymptotic inertia, while ensuring the total energy is finite, does not ensure information preservation [39]. Only the absence of a horizon guarantees that. In the Airy case, even though there is no horizon, and thus no information loss, the entanglement entropy diverges. This is because the rapidity diverges, as seen through the relationship $S=-\eta /6$. This indicates that entanglement entropy is not a comprehensive measure of the unitary, finite energy, information preserving dynamics, due to the inertial light speed asymptote (see Appendix C for more detail).

5. Particle Spectrum

The particle spectrum can be obtained from the beta Bogoliubov coefficient, given by Equation (2) in Section 2. For the particular trajectory Equation (3), as promised the Bogoliubov coefficient is real,

$${\beta }_{\omega {\omega }^{\prime }}=\frac{-1}{{\left(\omega {\kappa }^2\right)}^{1/3}}\sqrt{\frac{{\omega }^{\prime }}{\omega }}\mathrm{Ai}\left(\frac{\omega +{\omega }^{\prime }}{{\left(\omega {\kappa }^2\right)}^{1/3}}\right),$$

(20)

which is highly unusual. This corresponds to the Bogoliubov transformation being a pure boost without rotation, i.e., there is no phase on the beta coefficient, giving us a natural choice for both field modes and coefficients (and potentially an action integral whose real part defines the vacuum–vacuum amplitude [40]).

To obtain the particle spectrum, we take the modulus square, $N_{\omega {\omega }^{\prime }}\equiv {\left|{\beta }_{\omega {\omega }^{\prime }}\right|}^2$, which gives

$$N_{\omega {\omega }^{\prime }}=\frac{{\omega }^{\prime }}{{\kappa }^{4/3}{\omega }^{5/3}}{\mathrm{Ai}}^2\left(\frac{\omega +{\omega }^{\prime }}{{\kappa }^{2/3}{\omega }^{1/3}}\right).$$

(21)

The Airy-Ai function is perhaps most well-known as the solution to the time-independent Schrödinger equation for a particle confined within a triangular potential well and for a particle in a one-dimensional constant force field. The spectrum Equation (21), $|{\beta }_{\omega {\omega }^{\prime }}{|}^2$, is explicitly non-thermal and plotted as a contour plot in Figure 5.

This demonstrates a new spectrum of radiation emanating from a moving mirror trajectory. Equation (21) can be compared to the late time (equilibrium after formation) spectra of non-extremal black holes (e.g., Schwarzschild, RN, Kerr),

$$N_{\omega {\omega }^{\prime }}=\frac{1}{2\pi \kappa {\omega }^{\prime }}\frac{1}{e^{2\pi \omega /\kappa }-1},$$

(22)

and extremal black holes (e.g., extremal RN, extremal Kerr, extremal Kerr-Newman),

$$N_{\omega {\omega }^{\prime }}=\frac{e^{-\pi \omega c/\mathcal{A}}}{{\pi }^2{\mathcal{A}}^2}{\left|K_{1+i\omega c/\mathcal{A}}\left(\frac{2}{\mathcal{A}}\sqrt{\omega {\omega }^{\prime }}\right)\right|}^2.$$

(23)

(For extremal Kerr, $c=\sqrt{2}$; for extremal RN, $c=2$; for extremal Kerr-Newman, $c=\mathcal{A}/\kappa$.) Here $\kappa$ is the surface gravity, i.e., $\kappa =1/\left(4M\right)$ in the case of a Schwarzschild black hole, or outer horizon surface gravity for the RN and Kerr non-extremal black holes. In addition, $\mathcal{A}$ is the extremal parameter, or the asymptotic uniform acceleration [20] in the case of the mirror system, while $K_{\nu }$ is the modified Bessel function of the second kind with order $\nu$.

Furthermore, it is remarkable that the spectrum

$$N_{\omega }={\int }_0^{\infty }{N}_{\omega {\omega }^{\prime }}d{\omega }^{\prime },$$

(24)

is analytic,

$$N_{\omega }=\frac{2\sqrt{\overline{\omega }}}{3\kappa }{\mathrm{Ai}}^2\left(\overline{\omega }\right)-\frac{\mathrm{Ai}\left(\overline{\omega }\right){\mathrm{Ai}}^{\prime }\left(\overline{\omega }\right)}{3\kappa {\overline{\omega }}^{3/2}}-\frac{2{\mathrm{Ai}}^{\prime 2}\left(\overline{\omega }\right)}{3\kappa \sqrt{\overline{\omega }}},$$

(25)

where $\overline{\omega }\equiv {(\omega /\kappa )}^{2/3}$. This analytic $N\left(\omega \right)$ spectrum is plotted in Figure 6 for all $\kappa$.

The Airy functions can be reformulated into Bessel functions using the identities

$$\begin{array}{ccc}\hfill \mathrm{Ai}\left(x\right)& =& \sqrt{\frac{x}{3{\pi }^2}}{K}_{1/3}\left(\frac{2}{3}{x}^{3/2}\right)\hfill \end{array}$$

(26)

$$\begin{array}{ccc}\hfill {\mathrm{Ai}}^{\prime }\left(x\right)& =& \frac{-x}{\sqrt{3{\pi }^2}}{K}_{2/3}\left(\frac{2}{3}{x}^{3/2}\right).\hfill \end{array}$$

(27)

This turns Equation (21) into

$$N_{\omega {\omega }^{\prime }}=\frac{1}{3{\pi }^2{\kappa }^2}\frac{q^{\prime }(q+q^{\prime })}{q^2}{K}_{1/3}^2\left(\frac{2{(q+q^{\prime })}^{3/2}}{3q^{1/2}}\right),$$

(28)

which has similarities to the extremal black hole expression. Here $q=\omega /\kappa$, $q^{\prime }={\omega }^{\prime }/\kappa$. For the particle spectrum we get

$$\begin{array}{ccc}\hfill 9{\pi }^2\kappa N_{\omega }& =& 2q{K}_{1/3}^2(2q/3)+K_{1/3}(2q/3)K_{2/3}(2q/3)\hfill \\ & & -2q{K}_{2/3}^2(2q/3).\hfill \end{array}$$

(29)

In the small and large $\omega$ limits the leading order terms are, respectively,

$$\begin{array}{ccc}\hfill N_{\omega }& \to & \frac{1}{6\sqrt{3}\pi \omega },\omega \to 0,\hfill \end{array}$$

(30)

$$\begin{array}{ccc}& \to & \frac{\kappa }{16\pi {\omega }^2}{e}^{-4\omega /\left(3\kappa \right)},\omega \to \infty .\hfill \end{array}$$

(31)

The $1/\omega$ in the small frequency limit (note this is independent of $\kappa$) demonstrates the infrared divergence leading to an infinite total particle count commonly associated with constant-velocity moving mirror solutions [21,22,23,24,25,26], that are not asymptotically static (asymptotic zero-velocity [12,27,28,29,30,31,32]).

To check that the energy is indeed carried away by the particles, we look for consistency between Equation (21) and the total energy, Equation (16), found from the stress tensor. This is done by quantum summing,

$$E={\int }_0^{\infty }{\int }_0^{\infty }\omega N_{\omega {\omega }^{\prime }}d\omega d{\omega }^{\prime },$$

(32)

that is, associating a quantum of energy $\omega$ with the particle distribution and integrating over all the frequencies. The result is pleasingly analytic:

$$E=\frac{\kappa }{96}.$$

(33)

Since this is also the result of Equation (16), the beta spectrum Equation (21), or Equation (29), is consistent with the quantum stress tensor, Equation (13).

The time dependence of particle creation can be computed via wavepacket analysis treated in Hawking [11], and explicitly numerically computed in [27,28]. Wave packet localization, particularly via orthonormal and complete sets in the moving mirror model, was first carried out in detail in [41]. For completeness, we utilize the same code to illustrate particle creation in time and present the results in Figure 7. The rate of emission of particles is finite only in a given time and frequency interval which can be seen by these complete orthonormal family of wave packets constructed from the beta Bogoliubov coefficients, following Hawking’s notation,

$${\beta }_{jn{\omega }^{\prime }}=\frac{1}{\sqrt{ϵ}}{\int }_{jϵ}^{(j+1)ϵ}d\omega e^{2\pi i\omega n/ϵ}{\beta }_{\omega {\omega }^{\prime }},$$

(34)

where $j\ge 0$ and n are integers. These packets are built at future right null infinity, ${\mathcal{I}}_R^{+}$, and peak at delayed exterior time, $u=2\pi n/ϵ$, with width $2\pi /ϵ$. Therefore the vertical axis in Figure 7 has a discrete and intuitive physical interpretation, giving the counts of a particle detector sensitive to only frequencies within $ϵ$ of ${\omega }_j=jϵ$, for a time $2\pi /ϵ$ at $u=2\pi n/ϵ$. Late times correspond to large quantum number n (for the mirror Equation (5), late times have $u\approx 2t[1+\mathcal{O}{\left(\kappa t\right)}^{-2/3}]$). For excellent time resolution, only one frequency bin is needed, where the particles pile up, $j=0$, and a relatively large value of $ϵ$ resolves the count in time. The text of Fabbri-Navarro-Salas [42] also describes the details needed to reconstruct Figure 7 by first packetizing the beta coefficient as done in Equation (34) and then secondly numerically integrating over ${\omega }^{\prime }$ from 0 to ∞, and third, computing the results, $N_{jn}$,

$$N_{jn}={\int }_0^{+\infty }d{\omega }^{\prime }{\left|{\beta }_{jn{\omega }^{\prime }}\right|}^2,$$

(35)

for each individual time bin, n, for a set frequency bin, j (in our fine-grained time resolution case, $j=0$). While this numerical approach evolves the particle count in time, it is not particularly stream-lined, fast, nor arbitrarily accurate. In Section 6, we will find an analytic approach to the evolution process, resolving these issues.

6. Analytic Time Evolution

The spectrum, Equation (25), is simple enough that analytical time evolution without discrete wave packetization is possible – possibly uniquely in the literature. Typically we would like to employ a Fourier transform converting from frequency to time. Since this does not work out in a straightforward manner, we consider that the Fourier transform of a radially symmetric function in the plane can be expressed as a Hankel transform. The Hankel transform, $N_u=H\left(N_{\omega }\right)/2$—where by time symmetry we have divided the spectrum by 2 so that retarded time u ranges from $-\infty$ to $+\infty$—is analytically tractable for the spectrum Equation (25):

$$\begin{array}{ccc}\hfill \frac{384}{\kappa }{N}_u& =& 5{}_3{F}_2\left(\frac{7}{6},\frac{3}{2},\frac{11}{6};1,2;-\frac{9}{16}{u}^2{\kappa }^2\right)\hfill \\ \hfill & +& 4{}_3{F}_2\left(\frac{1}{2},\frac{5}{6},\frac{7}{6};1,1;-\frac{9}{16}{u}^2{\kappa }^2\right)\hfill \\ & -& 7{}_3{F}_2\left(\frac{5}{6},\frac{3}{2},\frac{13}{6};1,2;-\frac{9}{16}{u}^2{\kappa }^2\right).\hfill \end{array}$$

(36)

The particle spectrum dies off at large times as $u^{-1}$, so the total number indeed diverges.

Turning to the energy, a consistency check can be done by Hankel transforming the quantum of energy $\omega N_{\omega }$, and integrating over all time. The result for the transform, $E_u=H\left(\omega N_{\omega }\right)/2$, is

$$E_u=\frac{sinh\theta }{3\sqrt{3}\pi \kappa u^3}-\frac{cosh\theta }{3\sqrt{3}\pi u^2\sqrt{9{\kappa }^2{u}^2+16}},$$

(37)

where $\theta \equiv \frac{1}{3}{sinh}^{-1}\left(\frac{3\kappa u}{4}\right)$. Equation (37) dies off as $u^{-8/3}$ for large times, so the total energy is finite. The result for the total energy by integrating over all time is also analytic,

$$E={\int }_{-\infty }^{+\infty }{E}_{u}du=\frac{\kappa }{96},$$

(38)

which agrees with the total energy as derived by the stress tensor, Equation (16), and the total energy as derived by integration of the particle spectrum with respect to frequency, Equation (33). As far as we know, this is the first solution for analytic time evolution of particle production from the quantum vacuum. Notice there is no need to resort to wavepacket discreteness as the creation is continuous. Nor have we made any analytic approximations. A plot of the evolution is given in Figure 8.

7. Conclusions

An interesting connection between the reality of the beta Bogoliubov coefficient, asymptotic inertia and finite energy, and mirror motion near the speed of light leads to particle radiation by quantum vacuum that is analytic in the energy flux, simple in the particle spectrum—an Airy function—and, remarkably, analytic expression of the time evolution of particle creation.

We evaluate the simplest allowed accelerated mirror with the needed conditions and derive all these physical quantities. The Airy mirror is asymptotically inertial, coasting at the speed of light; the total energy radiated is finite and simply $\kappa /96$ despite a soft particle divergence; the beta Bogoliubov coefficient is given by a real Airy-Ai function; the particle creation time evolution is analytic and exact.

The mirror has no horizon, and so there is no information loss. The finite energy corresponds to the black hole analog case where evaporation ceases, related to extremal black holes, remnants, or complete evaporation. The asymptotic inertia is responsible for finite energy, but inertial motions that asymptotically approach the speed of light do not preserve the interpretation of entanglement entropy derived from the rapidity as an adequate measure of unitarity (see Appendix C).

The radiated flux exhibits regions of negative energy flux (NEF); these are required by unitarity for the conditions present, and we expand on this “necessity of negativity” in the Appendices, showing it follows directly from the asymptotically inertial nature. The lack of a horizon ensures information conservation although information loss does occur from an inertial horizon [39]. We further connect the NEF to the 1-D Schrödinger equation and interpretation of the rapidity as a Lorentz transformation and wavefunction in a potential well defined by the acceleration properties.

While obtaining a real, and simple, Bogoliubov coefficient is a significant advance, we further derive an analytic particle spectrum (integrating over the beta coefficient squared), time evolution (through a Hankel transform), and energy (further integrating over the spectrum times frequency). An exact analytic time evolution solution for particle production from the quantum vacuum may be unique in the literature. No discrete wave packetization is required (although we also show those results, consistent with the analytic one).

The techniques of accelerating boundary correspondences (ABC) and moving mirrors continue to deliver intriguing insights into connections between acceleration (or surface gravity), particle creation, and information. Furthermore, these lead to interesting directions for research in the properties of black holes (for which they serve as analogs) and quantum information, entanglement, and gravity.