Schwarzschild-Planck moving mirror at general coupling $g/\kappa$: closed-form particle spectrum, non-thermal interpolation, and wave-packet resolution

The Schwarzschild-Planck moving mirror, a pointlike reflecting boundary in 1+1 dimensional flat spacetime following the trajectory $gv=-\sinh(2\kappa x)$, is the benchmark exactly solvable model of finite, approximately thermal particle creation (the dynamical Casimir effect in moving-mirror form). Its per-frequency spectrum $N_\omega$ is known exactly through the Bogolubov coefficient, a modified Bessel function of imaginary order, but in closed analytic form only in the ultra-relativistic limit $g\gg\kappa$, where at temperature $T=\kappa/2\pi$ it approaches the Planck form, though not uniformly in frequency; the total particle number is finite and the spectrum infrared-finite at every finite $g/\kappa$. The problem asks for the quantum particle-creation content of this mirror at general coupling $g/\kappa$: an explicit closed-form expression for the spectrum $N_\omega(\omega;g,\kappa)$, a quantitative characterization of the non-thermal regime $g\sim\kappa$, where the energy flux is known to be non-thermal, and wave-packet-resolved (time-resolved) particle counts outside the ultra-relativistic regime, where only the $g\gg\kappa$ plateau is known. No explicit closed form for $N_\omega$ at general $g/\kappa$ has been published to date.

Background

A moving mirror is an idealized model of the dynamical Casimir effect (DCE), the creation of real quanta from the quantum vacuum by a time-dependent boundary condition: in a flat spacetime with one time and one space dimension, a perfectly reflecting point boundary whose position is a prescribed trajectory $x(t)$ excites a massless scalar field. Particle creation is quantified by the Bogolubov coefficients $\beta_{\omega\omega'}$, which relate the positive-frequency mode functions of the remote past to those of the remote future; the spectrum of created particles is $N_\omega=\int_0^\infty|\beta_{\omega\omega'}|^2d\omega'$, and the total number $N=\int_0^\infty N_\omega d\omega$ is well defined only when the mirror is asymptotically static (at rest) at both early and late times, so that in- and out-mode bases exist.

The Schwarzschild-Planck trajectory is, in null (light-cone) coordinates $u=t-x$ and $v=t+x$,

$$gv=-\sinh(2\kappa x),\qquad\text{equivalently}\qquad t(x)=-x-\frac{\sinh(2\kappa x)}{g}.$$

Here $\kappa$ is an acceleration scale playing the role of a black-hole surface gravity, setting the characteristic temperature $T=\kappa/2\pi$, and $g$ is a second, independent scale with dimensions of inverse length: the length $1/g$ acts as a Planck-length cutoff relative to the horizon scale $1/\kappa$, and the same worldline arises as the moving-mirror dual of a Schwarzschild geometry modified at the Planck length (Good and Linder, New J. Phys. 23, 043007 (2021)). The trajectory is a deformation of the black mirror, also called the Schwarzschild mirror, $\kappa v=-e^{2\kappa x}$ — introduced by Good, Anderson, and Evans (Phys. Rev. D 94, 065010 (2016)) as the moving-mirror dual of a Schwarzschild black hole formed by null collapse, and so named because its energy flux is identical to that of a Schwarzschild black hole. Like the classic eternally thermal Carlitz-Willey mirror, it radiates thermally forever, but its asymptotically null motion produces an infinite total particle number. The deformation cuts the mirror off so that it is asymptotically static at both ends; this cutoff makes the total radiated particle number finite, and it also removes the infrared (low-frequency) divergence of the thermal spectrum, so there is no soft-particle divergence. In the regime $g\gg\kappa$ the mirror is ultra-relativistic (its speed is close to that of light over most of its history) and realizes finite thermal particle creation of Casimir light: a finite number of quanta with an approximately Planck spectrum, to leading order for large $g\gg\kappa$ of the form $N_\omega=\bar\Gamma_\omega\,(e^{\omega/T}-1)^{-1}$, a Planck factor multiplied by a frequency-dependent prefactor $\bar\Gamma_\omega$.

The exactly known results for this mirror (Good, Linder, and Wilczek, Phys. Rev. D 101, 025012 (2020), the companion full-length paper) are: the closed-form Bogolubov coefficient

$$|\beta_{\omega\omega'}|^2=\frac{\omega\omega'\,e^{-\pi\omega/\kappa}}{\pi^2\kappa^2\omega_p^2}\left|K_{i\omega/\kappa}\!\left(\frac{\omega_p}{g}\right)\right|^2,\qquad\omega_p=\omega+\omega',$$

where $K_{i\omega/\kappa}$ is a modified Bessel function of purely imaginary order; the energy flux as a function of retarded time, an analytic expression that includes an interval of negative energy flux (quantum field theory in this setting permits intervals of negative flux; for $g\gg\kappa$ the energy of the negative-flux epoch is $E_{\rm NEF}\approx-0.017\kappa$); a large-$g/\kappa$ analytic asymptotic form for $N_{\omega\omega'}$; numerical spectra $N_\omega$ for several values of $g$ showing a non-uniform approach to the Planck form, always with an exact zero at $\omega=0$; and a time-resolved analysis using the wave packets of Hawking, Davies, and Fulling, packets centered on frequencies near $j\varepsilon$ at retarded times $u=2\pi n/\varepsilon$ of width $2\pi/\varepsilon$, whose counts $N_{jn}$ were computed at $g=10^{10}\kappa$ and form a flattened plateau around $n=0$.

A classical analog has also been studied (Ievlev, Good, and Linder, Ann. Phys. 461, 169593 (2024)): a classical point charge on the same worldline has an exactly known spectral distribution, which at emission angles satisfying $\kappa_\theta/g\ll 1$ — with $\kappa_\theta=2\kappa/(1+\cos\theta)$ and $\theta$ the emission angle relative to the direction of motion — averages to a Planck-like spectrum, while a narrow forward region $(\pi-\theta)\lesssim\kappa/g$ is non-thermal; the classical counterpart of the $g/\kappa$ interpolation is thus available in closed form. What is not known is the quantum particle-creation content at general coupling $g/\kappa$: the explicit closed form of $N_\omega(\omega;g,\kappa)$ at general $g/\kappa$, the quantitative quantum characterization of the regime $g\sim\kappa$ where the energy flux is known to be non-thermal, and wave-packet-resolved counts outside the ultra-relativistic regime. Natural units $\hbar=c=1$ are used throughout.

Problem Statement

For the Schwarzschild-Planck moving mirror $gv=-\sinh(2\kappa x)$ in 1+1 dimensions, with $\kappa$ the temperature scale and $g$ the independent cutoff scale, determine the quantum particle-creation content at general coupling $g/\kappa$:

(i) an explicit closed-form expression for the per-frequency spectrum $N_\omega(\omega;g,\kappa)$ in terms of known special functions, valid at general $g/\kappa$ beyond the known large-$g/\kappa$ asymptotic form and showing how the spectrum departs from the Planck form as $g/\kappa$ decreases;

(ii) the quantitative quantum characterization of the regime $g\sim\kappa$, where the energy flux is known to be non-thermal: the departure of the exact spectrum from the Planck form, expressed as the explicit prefactor $R_\omega\equiv N_\omega\,(e^{\omega/T}-1)$ (with $T=\kappa/2\pi$), together with the total particle number $N=\int_0^\infty N_\omega\,d\omega$ and the energy flux, each as an explicit function of $g/\kappa$; and the explicit $g/\kappa$-scaling by which $R_\omega$, $N$, and the energy flux leave their $g\gg\kappa$ thermal-regime values, which furnishes a measurable criterion for whether finite particle creation and thermality coexist or decouple there;

(iii) the wave-packet-resolved (time-resolved) particle counts $N_{jn}$ in the standard Hawking-Davies-Fulling construction outside the ultra-relativistic regime $g\gg\kappa$, where only the flattened plateau is known, including how the packetized description approaches the continuous-frequency one as packet width varies, and how the negative-energy-flux epoch manifests in it.

These three deliverables are jointly required and share one objective: the particle-creation content of the benchmark finite-thermal mirror at general $g/\kappa$, of which the published thermal result is the $g\gg\kappa$ limit. No restriction on $g/\kappa$ or method is intended; natural units $\hbar=c=1$ are used, and $\kappa=1$ may be set where convenient.

References

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