work we have provided complete numerical solutions of the relational equations
of motion for each of the cosmologies described above. These numerical
solutions show the asymptotic potential-free behaviour near the big bang
corresponding to great circles which continue smoothly through the shape space
equator. In the case of the Bianchi spacetimes, these great circles represent
Kasner solutions. As a result of the analytical treatment, we see that the
Mixmaster behaviour of Bianchi IX cosmologies must be resolved to quiescence
by the introduction of a suitable matter field (in this case a scalar field)
in order for the equations of motion to satisfy the Picard-Lindelöf theorem
near the initial singularity. In other words, Bianchi IX quiescence must be
achieved in order for the dynamical system to have a unique, smooth
continuation through the big bang. In the numerical solution examples given
for quiescent Bianchi IX in section 4.4, we see explicitly the final Kasner
epochs and Taub transitions before the system passes through the initial
singularity.

At present it remains unclear how one might develop a quantisation regime for
shape dynamics, but certainly a proper understanding of the classical theory
is required first. In particular, it is encouraging that the initial
singularity is able to be resolved at the classical level within the Shape
Dynamics framework. One might also hope that an investigation of geodesics on
the contact manifold sheds light on possible alterations to the singularity
theorems within the context of shape dynamics.

##  Appendix A Bianchi Numerical Solution Hamiltonians

Figure 37: Numerical solution Hamiltonian for the Bianchi I + free field (red,
dashed) and harmonic potential (blue) models. The discretised equations of
motion for the free-field model become numerically faster than the harmonic
potential model thus the numerical solution is valid over a smaller time
domain.

(a) Bianchi IX numerical solution Hamiltonian for the free-field (red, dashed)
and harmonic potential models.

(b) Plot of the ratio $\mathcal{H}/\mathcal{H}_{1}$ for the Bianchi IX
numerical solution Hamiltonians.

Figure 38:

In figure 38(a) we plot the numerical solution Hamiltonians for Bianchi IX,
the numerical solution Hamiltonians are larger in magnitude than the previous
cases we have looked at in this paper, however the Hamiltonian itself is still
much smaller than the individual terms that it is comprised of. For example
consider the first term in the Hamiltonian 198

$\mathcal{H}_{1}=\frac{1}{2}p_{\beta}^{2}\cos^{2}\beta$ (201)

The relative size of $\mathcal{H}/\mathcal{H}_{1}$ is typically small except
where $\mathcal{H}_{1}\rightarrow 0$, as demonstrated in figure 38(b)

##  Appendix B Gnomonic Projection Example

Consider the motion of a free-particle in two dimensions, in the standard
symplectic mechanics context. The Lagrangian for such a system is simply

$\mathcal{L}=\frac{1}{2}\left(\dot{x}^{2}+\dot{y}^{2}\right)$ (202)

and the solutions, which are straight lines on the $\mathbb{R}^{2}$ plane are
of course obtained trivially. The solutions are characterised by two constant
momenta $(p_{x},p_{y})$ each associated with one of the coordinates on the
plane

$x(t)=x_{0}+p_{x}t,\quad y(t)=y_{0}+p_{y}t$ (203)

In this appendix we show explicitly how, after making a gnomonic projection to
shape space, the solutions of the equations of motion are great circles on
$S^{2}$. Recall that the gnomonic projection is a compactification onto the
surface of a sphere, in this case a 2-sphere. This projection takes a unit
sphere tangent to the plane at the origin and draws a straight line from a
point $P(x,y)$ on the plane to the centre of the tangent sphere. $P(x,y)$ is
then identified with the coordinates on the 2-sphere $(\alpha,\beta)$ where
the line intersects the sphere. The shape space coordinates are the angle
subtended from the pole $\beta$ and the azimuthal angle $\alpha$. Explicitly
the projection map is written as

$\begin{pmatrix}x\\\ y\end{pmatrix}=|\tan\beta|\begin{pmatrix}\cos\alpha\\\
\sin\alpha\end{pmatrix}$ (204)

In these shape space coordinates the Lagrangian in

$\mathcal{L}=\frac{1}{2}\left(\dot{\beta}^{2}\sec^{4}\beta+\dot{\alpha}^{2}\tan^{2}\beta\right)$
(205)

From which one may find the canonical momenta

$p_{\alpha}=\dot{\alpha}\tan^{2}\beta,\quad
p_{\beta}=\dot{\beta}\sec^{4}\beta$ (206)

and obtain the Hamiltonian through a Legendre transformation

$\mathcal{H}=\frac{1}{2}p_{\beta}^{2}\cos^{4}\beta+\frac{p_{\alpha}^{2}}{2\tan^{2}\beta}$
(207)

One may now simply use the symplectic Hamiltons equations of motion [23]

$\dot{q}_{i}=\frac{\partial\mathcal{H}}{\partial
p^{i}},\quad\dot{p}^{i}=-\frac{\partial\mathcal{H}}{\partial q_{i}}$ (208)

to obtain

$\dot{\alpha}=\frac{p_{\alpha}}{\tan^{2}\beta},\quad\dot{p}_{\alpha}=0$ (209)
$\dot{\beta}=p_{\beta}\cos^{4}\beta,\quad\dot{p}_{\beta}=2p_{\beta}^{2}\cos^{3}\beta\sin\beta+p_{\alpha}^{2}\frac{\cos\beta}{\sin^{3}\beta}$
(210)

In symplectic mechanics, the Hamiltonian is a conserved, constant quantity and
therefore one may determine that

$\frac{d\alpha}{d\beta}=\frac{\dot{\alpha}}{\dot{\beta}}=\frac{p_{\alpha}}{\sin^{2}\beta\sqrt{2\mathcal{H}-p_{\alpha}^{2}\cot^{2}\beta}}$
(211)

Equation B has the solution

$\cos(\alpha-\alpha_{0})=\frac{p_{\alpha}}{\sqrt{2\mathcal{H}}\tan\beta}$
(212)

which is the equation of a great circle on the 2-sphere, whose orientation
relative to the $\mathbb{R}^{2}$ is controlled by
$p_{\alpha}/\sqrt{2\mathcal{H}}$.

## References

  * [1] S. W. Hawking and R. Penrose, “The Singularities of Gravitational Collapse and Cosmology,” Proceedings of the Royal Society of London Series A, vol. 314, pp. 529–548, Jan. 1970.
  * [2] C. Clarke and A. Królak, “Conditions for the occurence of strong curvature singularities,” Journal of Geometry and Physics, vol. 2, no. 2, pp. 127–143, 1985.
  * [3] S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Space-Time.  Cambridge Monographs on Mathematical Physics, Cambridge University Press, 2 2023.
  * [4] A. R. Liddle, An introduction to modern cosmology, Third Edition.  2015\.
  * [5] G. Horowitz, “Singularity resolution in string theory,” in APS April Meeting Abstracts, APS Meeting Abstracts, p. B7.003, Apr. 2008.
  * [6] M. Natsuume, “The singularity problem in string theory,” 2001.
  * [7] G. W. Gibbons, G. T. Horowitz, and P. K. Townsend, “Higher-dimensional resolution of dilatonic black-hole singularities,” Classical and Quantum Gravity, vol. 12, p. 297–317, Feb. 1995.
  * [8] B.-F. Li and P. Singh, “Loop Quantum Cosmology: Physics of Singularity Resolution and its Implications,” 4 2023.
  * [9] R. Gambini, J. Olmedo, and J. Pullin, “Quantum black holes in Loop Quantum Gravity,” Class. Quant. Grav., vol. 31, p. 095009, 2014.
  * [10] R. Gambini, J. Olmedo, and J. Pullin, Quantum Geometry and Black Holes.  2023\.
  * [11] A. Ashtekar, T. Pawlowski, and P. Singh, “Quantum nature of the big bang,” Phys. Rev. Lett., vol. 96, p. 141301, 2006.
  * [12] M. Bojowald, “Absence of singularity in loop quantum cosmology,” Phys. Rev. Lett., vol. 86, pp. 5227–5230, 2001.
  * [13] M. Bojowald, “Loop quantum cosmology,” Living Rev. Rel., vol. 8, p. 11, 2005.
  * [14] F. Mercati, “A Shape Dynamics Tutorial,” 8 2014.
  * [15] J. Barbour, “Shape Dynamics: An Introduction,” in Quantum Field Theory and Gravity: Conceptual and Mathematical Advances in the Search for a Unified Framework, pp. 257–297, 2012.
  * [16] J. B. Barbour and B. Bertotti, “Mach’s Principle and the Structure of Dynamical Theories,” Proc. Roy. Soc. Lond. A, vol. 382, pp. 295–306, 1982.
  * [17] D. Sloan, “Dynamical similarity,” Physical Review D, vol. 97, June 2018.
  * [18] T. A. Koslowski, F. Mercati, and D. Sloan, “Through the big bang: Continuing Einstein’s equations beyond a cosmological singularity,” Phys. Lett. B, vol. 778, pp. 339–343, 2018.
  * [19] D. Sloan, “Scalar Fields and the FLRW Singularity,” Class. Quant. Grav., vol. 36, no. 23, p. 235004, 2019.
  * [20] R. Arnowitt, S. Deser, and C. W. Misner, “Dynamical structure and definition of energy in general relativity,” Phys. Rev., vol. 116, pp. 1322–1330, Dec 1959.
  * [21] R. Jha, “Introduction to Hamiltonian formulation of general relativity and homogeneous cosmologies,” SciPost Phys. Lect. Notes, vol. 73, p. 1, 2023.
  * [22] J. M. Isidro and P. d. Córdoba, “On the contact geometry and the poisson geometry of the ideal gas,” Entropy, vol. 20, p. 247, 2018.
  * [23] M. de León and M. Lainz, “A review on contact Hamiltonian and Lagrangian systems,” 11 2020.
  * [24] V. I. Arnold, Mathematical methods of classical mechanics.  1978\.
  * [25] A. Bravetti, H. Cruz, and D. Tapias, “Contact Hamiltonian mechanics,” Annals of Physics, vol. 376, pp. 17–39, Jan. 2017.
  * [26] A. Bravetti, M. de León, J. C. Marrero, and E. Padrón, “Invariant measures for contact hamiltonian systems: symplectic sandwiches with contact bread,” Journal of Physics A: Mathematical and Theoretical, vol. 53, no. 45, p. 455205, 2020.
  * [27] A. C. Silva, “Lectures on symplectic geometry,” Lecture Notes in Mathematics, vol. 1764, 2001.
  * [28] A. Bravetti, C. Jackman, and D. Sloan, “Scaling symmetries, contact reduction and poincaré’s dream,” Journal of Physics A: Mathematical and Theoretical, vol. 56, no. 43, p. 435203, 2023.
  * [29] D. Sloan, “Scale symmetry and friction,” 2021.
  * [30] L. D. Landau and E. M. Lifschits, The Classical Theory of Fields, vol. Volume 2 of Course of Theoretical Physics.  Oxford: Pergamon Press, 1975.
  * [31] G. MONTANI, M. V. BATTISTI, R. BENINI, and G. IMPONENTE, “Classical and quantum features of the mixmaster singularity,” International Journal of Modern Physics A, vol. 23, p. 2353–2503, July 2008.
  * [32] A. Corichi and D. Núñez, “Introduction to the ADM formalism,” Rev. Mex. Fis., vol. 37, pp. 720–747, 1991.
  * [33] C. W. Misner, “Mixmaster universe,” Phys. Rev. Lett., vol. 22, pp. 1071–1074, May 1969.
  * [34] A. Ashtekar, A. Henderson, and D. Sloan, “A Hamiltonian Formulation of the BKL Conjecture,” Phys. Rev. D, vol. 83, p. 084024, 2011.
  * [35] Y. Akrami et al., “Planck 2018 results. X. Constraints on inflation,” Astron. Astrophys., vol. 641, p. A10, 2020.
  * [36] L. Boubekeur and D. H. Lyth, “Hilltop inflation,” Journal of Cosmology and Astroparticle Physics, vol. 2005, p. 010, jul 2005.
  * [37] J. Hoffmann and D. Sloan, “Squared quartic hilltop inflation,” Phys. Rev. D, vol. 104, no. 12, p. 123542, 2021.
  * [38] K. Dimopoulos, “An analytic treatment of quartic hilltop inflation,” Physics Letters B, vol. 809, p. 135688, 2020.
  * [39] R. Kallosh and A. Linde, “On hilltop and brane inflation after planck,” Journal of Cosmology and Astroparticle Physics, vol. 2019, p. 030, sep 2019.
  * [40] A. D. Linde, “Inflationary Cosmology,” Lect. Notes Phys., vol. 738, pp. 1–54, 2008.
  * [41] A. R. Liddle and D. H. Lyth, Cosmological Inflation and Large-Scale Structure.  Cambridge University Press, 2000.
  * [42] V. Ardourel and J. Jebeile, “Numerical instability and dynamical systems,” European Journal for Philosophy of Science, vol. 11, pp. 1–21, May 2021.
  * [43] R. Steinbauer, “The singularity theorems of General Relativity and their low regularity extensions,” 6 2022.
  * [44] A. D. Rendall, “The Nature of spacetime singularities,” 100 Years Of Relativity : space-time structure: Einstein and beyond, pp. 76–92, 2005.
  * [45] T. Thiemann, “Modern canonical quantum general relativity,” 2001.
  * [46] A. Ashtekar, T. Pawlowski, and P. Singh, “Quantum Nature of the Big Bang: An Analytical and Numerical Investigation. I.,” Phys. Rev. D, vol. 73, p. 124038, 2006.
  * [47] A. Ashtekar, T. Pawlowski, P. Singh, and K. Vandersloot, “Loop quantum cosmology of $k=1$ frw models,” Phys. Rev. D, vol. 75, p. 024035, Jan 2007.
  * [48] Łukasz Szulc, W. Kamiński, and J. Lewandowski, “Closed friedmann–robertson–walker model in loop quantum cosmology,” Classical and Quantum Gravity, vol. 24, p. 2621, apr 2007.
  * [49] A. Ashtekar and A. Barrau, “Loop quantum cosmology: from pre-inflationary dynamics to observations,” Classical and Quantum Gravity, vol. 32, p. 234001, nov 2015.
  * [50] H. Gomes and T. Koslowski, “Frequently asked questions about shape dynamics,” Foundations of Physics, vol. 43, p. 1428–1458, Nov. 2013.
  * [51] F. Mercati and D. Sloan, “Traversing through a black hole singularity,” Phys. Rev. D, vol. 106, no. 4, p. 044015, 2022.
  * [52] E. Anderson, “The problem of time in quantum gravity,” 2012.
  * [53] J. Barbour, T. Koslowski, and F. Mercati, “The solution to the problem of time in shape dynamics,” Classical and Quantum Gravity, vol. 31, p. 155001, July 2014.
  * [54] J. Barbour, T. Koslowski, and F. Mercati, “A gravitational origin of the arrows of time,” 2013.


# Stopping to Reflect: Asymptotic Static Moving Mirrors as Quantum Analogs of
Classical Radiation

Michael R.R. Good1,2,3<EMAIL_ADDRESS>Eric V. Linder3,4
<EMAIL_ADDRESS>1Physics Department, Nazarbayev University, Astana,
Kazakhstan
2 Leung Center for Cosmology and Particle Astrophysics, National Taiwan
University, Taipei, Taiwan
3Energetic Cosmos Lab, Nazarbayev University, Astana, Kazakhstan
4Berkeley Center for Cosmological Physics & Berkeley Lab, University of
California, Berkeley, CA, USA

###### Abstract

Radiation from an accelerating charge is a basic process that can serve as an
intersection between classical and quantum physics. We present two exactly
soluble electron trajectories that permit analysis of the radiation emitted,
exploring its time evolution and spectrum by analogy with the moving mirror
model of the dynamic Casimir effect. These classical solutions are finite
energy, rectilinear (nonperiodic), asymptotically zero velocity worldlines
with corresponding quantum analog beta Bogolyubov coefficients. One of them
has an interesting connection to uniform acceleration and Leonardo da Vinci’s
water pitcher experiment.

moving mirrors, black hole evaporation, acceleration radiation, Larmor power,
point charge

###### pacs:

41.60.-m (Radiation by moving charges), 04.70.Dy (Quantum aspects of black
holes)

##  I Introduction

The mechanism of particle creation proposed by Hawking Hawking (1975), whereby
the gravitational field of a collapsing star in curved spacetime amplifies
vacuum fluctuations into particle emission, bears striking resemblance to the
radiation of particles from a perfect mirror in flat spacetime accelerated
through the vacuum DeWitt (1975); Fulling and Davies (1976); Davies and
Fulling (1977). Particles of a massless quantum scalar field in $1+1$
dimensions Birrell and Davies (1984); Fabbri and Navarro-Salas (2005) are
created due to the acceleration of the mirror, which is an ideal point and
boundary condition on the field Walker (1985); Ford and Vilenkin (1982);
Carlitz and Willey (1987); Chen and Mourou (2017); Chen  _et al._ (2022); Good
_et al._ (2021); Chen and Mourou (2020), essentially a dynamical Casimir
effect Moore (1970). In this study, we demonstrate a functional duality and
analog to an accelerated point charge in ordinary 3+1 space-time and its non-
thermal radiation spectrum, revealing the particle creation correspondence.

Accelerating point charge radiation has been a subject of interest in physics
for over a century Larmor (1897), and it is of particular interest as a simple
example of nonthermal radiation. Nonthermal radiation is ubiquitous in
astrophysical phenomena, for example, and the particle number and angular
spectral distribution may not be apparent. Furthermore, even evaporating black
holes might emit non-thermal radiation, e.g. the recent Svidzinsky (2023).
Therefore a concrete relation between accelerated particle nonthermal
radiation and the moving mirror “slicing” of the vacuum Lin and Chen (2021);
Reyes (2021); Lin and Chen (2022); Akal  _et al._ (2021), especially in light
of the well-established correspondence between moving mirrors and black hole
horizons, is of interest.

The discovery of a clear association (generalized to non-thermal emissions)
between the radiation from an electron and from a moving mirror became
apparent via radiation reaction derived by Ford and Vilenkin in 1982 Ford and
Vilenkin (1982). In 1995, Nikishov and Ritus Nikishov and Ritus (1995)
established a formal link through particle count, which further strengthened
this connection. Ritus Ritus (1998, 2002, 2003, 2022) later provided
additional development on the Bogolyubov-current association. The relationship
was next confirmed via Larmor power in Zhakenuly et al Zhakenuly  _et al._
(2021). One of the present authors has exploited the electron-mirror
connection using explicit solutions; for instance, the connection between
radiation power loss and kinetic power loss for an electron approaching the
speed of light was demonstrated in Good  _et al._ (2022), and in Good and Ong
(2023) an electron was treated as a mirror for a trajectory that
asymptotically approaches a constant velocity. This article focuses on the
interesting results for the electron-mirror relation for trajectories that
come to a complete stop, giving finite energy, finite particle creation, and
unitary evolution.

In Sec. II, we review some elements of acceleration radiation for relativistic
moving point charges, including Larmor power, Feynman power, and their
connection to total energy emitted. We present the spectra for two different
motions of point charges and the quantum analogs that have desirable
properties and analytic Bogolyubov coefficients in Sec. III and Sec. IV. In
Sec. V we show the general correspondence between the classical bremsstrahlung
and dynamical Casimir effect in energy, particle count, and spectral
distribution. We summarize and discuss further areas for study in Sec. VI.

##  II Acceleration Radiation Elements

In this section, we set up the various elements needed to compute the radiated
power, energy, and spectral distribution of both an accelerating charge and
from a moving mirror dynamical Casimir effect. Throughout we use natural
units, $\hbar=c=\mu_{0}=\epsilon_{0}=1$, thus $e^{2}=4\pi\alpha_{\rm fs}$
where $\alpha_{\rm fs}$ is the fine structure constant. However, for
simplicity, when exclusively in the context of classical electrodynamics we
switch units and employ unit charge $e=1$ ($\hbar=1/4\pi\alpha_{\rm fs}$).

###  II.1 Power and Force

In classical electrodynamics Jackson (1999), the power radiated and the
radiation reaction force,

$P=\frac{\alpha^{2}}{6\pi}\ ,\qquad F=\frac{\alpha^{\prime}(\tau)}{6\pi}\ ,$
(1)

are given by the relativistically covariant Larmor formula and the (magnitude
of the) Lorentz-Abraham-Dirac (LAD) force. Here $\alpha$ is the proper
acceleration, and the prime is a derivative with respect to the argument, in
this case proper time $\tau$.

###  II.2 Energy Integrals

When the charged particle accelerates, energy is radiated, with the total
energy found by integrating over coordinate time. That is, for particle
velocity $v(t)$ the integrals

$E=\int_{-\infty}^{\infty}P\mathop{}\\!\mathrm{d}{t}=-\int_{-\infty}^{\infty}F\cdot
v\mathop{}\\!\mathrm{d}{t},$ (2)

demonstrate that the Larmor power, $P=\alpha^{2}/6\pi$, and what we call the
‘Feynman power’ Feynman (1996), $F\cdot v$, associated with the self-force
(radiation reaction force), directly tell an observer the total energy emitted
by a point charge along its time-like worldline. The total energy is finite as
long as the proper acceleration is asymptotically zero; that is, the worldline
must possess asymptotic inertia. We restrict ourselves to this case.

The negative sign demonstrates that the total work against the LAD force
represents the total energy loss. That is, the total energy loss from
radiation resistance due to Feynman power must equal the total energy radiated
by Larmor power. We will demonstrate that the Larmor and Feynman powers
themselves – the integrands – are not the same. Separately, it is a subtle
matter that these powers are not applicable for asymptotically non-inertial
rectilinear trajectories (which we do not consider here); see e.g. Good  _et
al._ (2022); Singal (2020).

A third expression for the total energy can be employed to establish a link to
quantum physics and verify consistency. This spectral consistency integrates
over spectral modes,

$E=\int_{0}^{\infty}\int_{0}^{\infty}p\,|\beta_{pq}|^{2}\mathop{}\\!\mathrm{d}{p}\mathop{}\\!\mathrm{d}{q}\
,$ (3)

using the quantum analog moving mirror model, generalized to 3+1 dimensions
using both sides of the 1+1 dimensional moving mirror, see e.g. Zhakenuly  _et
al._ (2021); Nikishov and Ritus (1995).

The quantity $\beta_{pq}$ is the beta Bogolyubov coefficient related to the
creation/annihilation operators and $p$ and $q$ are the out-going and in-going
frequencies, respectively, that describe the modes used to expand the field
subject to the accelerating boundary.

###  II.3 Spectral Distribution

The spectral distribution Schwinger (1949) of the total radiation energy $E$
with respect to frequency $\omega$ and solid angle $\Omega$ is

$\frac{\mathop{}\\!\mathrm{d}{I(\omega)}}{\mathop{}\\!\mathrm{d}{\Omega}}\coloneqq\frac{\mathop{}\\!\mathrm{d}^{2}{E}}{\mathop{}\\!\mathrm{d}{\omega}\mathop{}\\!\mathrm{d}{\Omega}}\
,$ (4)

see also Jackson (1999). For the radiation of a moving point charge (in
natural units with unit charge – see e.g. Eq. 23.89 on page 911 of Zangwill
Zangwill (2013) in SI units or Eq. 14.67 on page 701 of Jackson Jackson (1999)
in Gaussian units) this is given by the motion as

$\frac{\mathop{}\\!\mathrm{d}I(\omega)}{\mathop{}\\!\mathrm{d}\Omega}=\frac{\omega^{2}}{16\pi^{3}}\,\left|\
\bm{\hat{n}}\times\int\displaylimits_{-\infty}^{\infty}dt\,\bm{\dot{r}}(t)e^{i\phi}\
\right|^{2}.$ (5)

Here $\omega$ is the frequency, $\bm{k}=\omega\bm{\hat{n}}$ the wave vector,
$\operatorname{d}\\!{\Omega}$ the solid angle, $\bm{r}$ the charge trajectory
with velocity vector $\bm{\dot{r}}$, and $\phi=\omega t-\bm{k}\cdot\bm{r}(t)$.
Defining $\bm{\hat{n}}\cdot\bm{\hat{r}}=\cos\theta$ and assuming straight line
motion, we have

$\frac{\mathop{}\\!\mathrm{d}I(\omega)}{\mathop{}\\!\mathrm{d}\Omega}=\frac{\omega^{2}}{16\pi^{3}}\,\sin^{2}\theta\,\left|\
\int\displaylimits_{-\infty}^{\infty}dt\,\dot{r}(t)e^{i\phi}\ \right|^{2}\ .$
(6)

Integrating this over solid angle $d\Omega=\sin\theta d\theta d\varphi$ and
frequency $\omega$ will yield the total energy emitted.

We can also interpret the trajectory as not that of a point charge but an
accelerating mirror (boundary) and compare the horizon radiation from this
dynamical Casimir effect. Thus we can test that the classical energy emitted
agrees with the quantum result from the Bogolyubov creation/annihilation
coefficients, and also, contrast the Larmor and Feynman powers. This further
provides a way to derive the spectrum angular distribution for particle
production from a moving mirror trajectory.

###  II.4 Asymptotic Rest

To pursue an understanding of the spectrum angular dependence for the quantum
analog, we consider moving mirror trajectories that deliver finite total
energy and particle count (ensuring all integrals are convergent).
Asymptotically inertial mirrors have finite total energy, while mirrors that
also are asymptotically static (eventually coming to rest with zero velocity)
have finite particle count, entropy, and have unitary evolution (seen
geometrically since all light rays reflect off the mirror and none are lost).
Therefore we consider only cases with asymptotic rest.

The following list summarizes the only known trajectories possessing
asymptotic rest with solved Bogolyubov coefficients.

  * •

Walker-Davies Walker and Davies (1982): but noninvertible $t(x)$.

  * •

Arctx Good  _et al._ (2013): but nonfunctional particle count.

  * •

Self-Dual Good and Linder (2017): time symmetric.

  * •

betaK Good and Linder (2019): time antisymmetric.

  * •

Schwarzschild-Planck Good  _et al._ (2020); Good and Linder (2021) (also see
Moreno-Ruiz and Bermudez (2022)): fully evaporating black hole with unitarity.

None of these have previously had published solutions for the beta Bogolyubov
coefficients using both mirror sides to obtain the 3+1 D analog (and hence
classical particle motion). In the next two sections we present solutions for
the two boldface trajectories – in particular as examples of time-symmetric vs
antisymmetric motion, and the associated spectral distributions.

##  III Self-Dual Trajectory

The self-dual mirror trajectory Good and Linder (2017)

$x(t)=\frac{-v}{\kappa}\,\ln(\kappa^{2}t^{2}+1)\ ,$ (7)

is even in time, and the self-dual nature means that the particle emission
spectrum is equal on both sides of the mirror. The quantity $v$ is the maximum
speed of the mirror, occurring at $\kappa t=1$. The quantity $\kappa$ sets the
scale of the acceleration (and the surface gravity of the black hole analog in
the accelerating boundary correspondence).

The analog Larmor power radiated is

$P_{L}=\frac{2\kappa^{2}v^{2}\left(\kappa^{4}t^{4}-1\right)^{2}}{3\pi\left[\left(\kappa^{2}t^{2}+1\right)^{2}-4\kappa^{2}t^{2}v^{2}\right]^{3}}\
.$ (8)

As expected, no power is radiated by a stationary particle, $v=0$, and none at
the moment of maximum velocity when the acceleration is zero (i.e. when
$\kappa t=1$, as well as at asymptotically early and late times).

The Feynman force can be similarly calculated analytically but the expression
is long. Figure 1 plots the Larmor and Feynman powers vs time. The Larmor
power is of course always positive, while the Feynman power from the radiation
reaction force can be both positive and negative. The Feynman power crosses
zero at maxima of the Larmor power. Both types of power asymptotically vanish
rapidly.

Figure 1: The Larmor and Feynman powers for the self-dual trajectory are
plotted vs time, with $v=0.9$. A higher maximum velocity squeezes and
heightens the peaks for both powers. The Feynman power plotted is
$P_{F}=-F\cdot v$ so that the total area under the curve is positive, $E=\int
P_{F}\mathop{}\\!\mathrm{d}{t}$, see Eq. (2). Note the integrals under the
curves are equal, giving the total energy radiated, Eq. (9).

Integrating over all time, Eq. (2), the total energy emitted is

$E=\frac{\kappa}{24}\gamma v^{2}\left(\gamma^{2}+3\right)\ ,$ (9)

where $\gamma=(1-v^{2})^{-1/2}$ is the Lorentz factor. Figure 2 plots the
total energy as a function of the maximum velocity. As the velocity approaches
the speed of light, the Lorentz factor greatly increases the energy emitted.

Figure 2: The total energy as a function of maximum velocity parameter is
plotted for the self-dual trajectory (Eq. 9) and the betaK trajectory (Eq.
19).

For the Bogolyubov spectrum as found from the double-sided moving mirror, the
result (see e.g. Good  _et al._ (2013) for the details of the steps) is

$|\beta_{pq}|^{2}=\frac{16vpq}{\pi^{2}\kappa^{2}\sigma\omega}\,\sinh\left(\frac{\pi
v\sigma}{\kappa}\right)\,\left|K_{iv\frac{\sigma}{\kappa}+\frac{1}{2}}\left(\frac{\omega}{\kappa}\right)\right|^{2}\
,$ (10)

where $\sigma=p-q$ and $\omega=p+q$. The particle spectrum $N_{p}=\int
dq\,|\beta_{pq}|^{2}$ is non-thermal, and has finite particle production, as
seen in Figure 3.

Figure 3: A plot of particle spectrum $N(p)$ from the mirrors. This is the
particle count as a function of the outgoing mirror mode frequency, $p$. Here
the maximum velocity of each mirror is $v=v_{0}=0.9$.

For the spectral (angular) distribution, we use the self dual trajectory in
Eq. (6), giving

$\frac{\mathop{}\\!\mathrm{d}{I}}{\mathop{}\\!\mathrm{d}{\Omega}}=\frac{v\omega^{2}}{\kappa^{2}\pi^{3}}\frac{1-T^{2}}{2T}\sinh\left(\frac{\pi
vT\omega}{\kappa}\right)\left|K_{\frac{1}{2}+\frac{ivT\omega}{\kappa}}\left(\frac{\omega}{\kappa}\right)\right|^{2}\
,$ (11)

where $T\equiv\cos\theta$. Some details of the derivation are given in
Appendix A. Note the similarity to the form of the beta Bogolyubov
coefficients, but with added angular dependence (see the next subsection for
further discussion).

Figures 4 and 5 plot the spectral distribution in a 3D view. Notice there is
no radiation in the forward or backward $T\to\pm 1$ ($\theta\to[0,\pi]$)
directions. This is expected of straight-line bremsstrahlung Griffiths (2013).
The spectral distribution in the $T\to 0$ ($\theta\to\pi/2$) limit is:

$\lim_{T\to
0}\,\frac{\mathop{}\\!\mathrm{d}{I}}{\mathop{}\\!\mathrm{d}{\Omega}}=\frac{v^{2}\omega^{2}}{4\pi\kappa^{2}}e^{-2\omega/\kappa}\,,$
(12)

which demonstrates a radiation allotment in directions perpendicular to the
motion that is exponentially suppressed at high frequencies. The spectrum,
$I(\omega)$, can be numerically found by integrating the spectral
distribution, Eq. (11), over solid angle. See Figure 6 for an illustration.

Figure 4: 3D view of the radiated spectrum angular distribution
$\mathop{}\\!\mathrm{d}I/\mathop{}\\!\mathrm{d}\Omega$ from motion
corresponding to the self dual trajectory. Here we use unit charge, natural
units, and $\omega=\kappa=1$. The maximum speed of the charge is $v=0.95$.
Note the expected property of zero radiation directly in the forward
direction.  Figure 5: As Figure 4 but for $\omega=4$, $\kappa=1$, showing the
high-frequency exponential suppression.  Figure 6: A plot of energy spectrum
$I(\omega)$, which numerically integrates the spectral distributions for the
self-dual, Eq. (11), and betaK, Eq. (21), cases over solid angle $\Omega$. The
vertical axis has been multiplied by $10^{3}$ for readability. Here the
maximum velocity of each case is $v=v_{0}=0.9$.

The spectral distribution can be directly integrated over solid angle and
frequency to obtain the total energy

$\displaystyle E$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}\mathop{}\\!\mathrm{d}{\omega}\int_{-1}^{1}\mathop{}\\!\mathrm{d}{T}\int_{0}^{2\pi}\mathop{}\\!\mathrm{d}{\varphi}\;\frac{\mathop{}\\!\mathrm{d}{I}}{\mathop{}\\!\mathrm{d}{\Omega}}$
(13) $\displaystyle=$ $\displaystyle\frac{\kappa}{24}\,\gamma
v^{2}\left(\gamma^{2}+3\right)\ .$ (14)

This indeed agrees with Eq. (9).

##  IV betaK Trajectory

The betaK trajectory Good and Linder (2019)

$x(t)=\frac{-v_{0}}{\kappa}\,\sinh^{-1}\kappa t\ ,$ (15)

by contrast is odd in time, and gives more tractable solutions than the
Walker-Davies or Arctx models. Furthermore it has an interesting relation to
uniform acceleration in 3+1 D (though not in the 1+1 D mirror case)111We thank
Ahmad Shariati for pointing this out.. Its name arises because this trajectory
has exactly solvable beta Bogolyubov coefficients involving a modified Bessel
function $K$ in the moving mirror model, giving finite energy and finite
particle production.

This trajectory equation arises as well for a particle shot horizontally from
the origin with an initial velocity $v_{0}$ (which is also the maximum
velocity) encountering a constant vertical acceleration. Indeed, this is
similar to the recently rediscovered “Leonardo da Vinci’s water pitcher” that
moves horizontally at constant speed $v$ spilling water in a uniform
gravitational field Gharib  _et al._ (2022) – but here we consider
relativistic speeds. The derivation appears in Appendix B.

Note that in the relativistic case, despite no horizontal force the particles
(water drops) do not have constant horizontal velocity: due to the coupling of
horizontal and vertical motions through the Lorentz factor a horizontal
acceleration is induced as made clear in Appendix B.

The Larmor power radiated by a charge with the betaK trajectory is

$P_{L}=\frac{\alpha^{2}}{6\pi}=\frac{\kappa^{2}}{6\pi}\gamma^{6}\left(v_{0}^{2}-V^{2}\right)\frac{V^{4}}{v_{0}^{4}}\
,$ (16)

where the velocity is

$V(t)\equiv\dot{x}(t)=\frac{-v_{0}}{\sqrt{\kappa^{2}t^{2}+1}}\ .$ (17)

The speed $|V|\leq|v_{0}|$ so the power always remains nonnegative. For this
time antisymmetric trajectory, the power has only one maximum on each side of
$t=0$ and no zeros for finite $t\neq 0$. The Feynman power is

$P_{F}=\frac{\alpha^{2}}{6\pi}\,\left[2-\frac{V^{2}(1-v_{0}^{2})}{v_{0}^{2}-V^{2}}\right]\
.$ (18)

The total energy, using Eq. (2), is

$E=\frac{\kappa}{48}\gamma_{0}^{3}v_{0}^{2}\ .$ (19)

See Figure 2 for the energy and Figure 7 for the Larmor and Feynman powers.

Figure 7: The Larmor and Feynman powers for the betaK trajectory are plotted
vs time, with $v_{0}=0.9$. Like the self-dual trajectory, a higher $v_{0}$
narrows and heightens the peaks for both powers. For illustration, the Feynman
power plotted is $P_{F}=-F\cdot v$ so that the total area under the curve is
positive. The areas under the curves are equal, giving the total energy
radiated, Eq. (19).

The Bogolyubov spectrum as found from the double-sided moving mirror is

$|\beta_{pq}|^{2}=\frac{8v_{0}^{2}pq}{\pi^{2}\kappa^{2}\omega^{2}}\cosh\left(\pi
v_{0}\frac{\sigma}{\kappa}\right)\left|K_{iv_{0}\frac{\sigma}{\kappa}}\left(\frac{\omega}{\kappa}\right)\right|^{2}\
,$ (20)

where $\sigma=p-q$ and $\omega=p+q$. This spectrum is not thermal. Note the
similarities, but also subtle differences with the self-dual case, Eq. (10).
The energy is confirmed by associating a quantum $\hbar p$ (where $p$ is the
outgoing frequency mode) and integrating using Eq. (3), which yields Eq. (19).
The particle spectrum $N_{p}=\int dq\,|\beta_{pq}|^{2}$ is shown in Figure 3.

Using the betaK trajectory within classical electrodynamics Jackson (1999), we
find the spectral distribution,

$\frac{\mathop{}\\!\mathrm{d}I}{\mathop{}\\!\mathrm{d}\Omega}=\frac{v_{0}^{2}\omega^{2}}{4\kappa^{2}\pi^{3}}(1-T^{2})\cosh\left(\pi
v_{0}T\frac{\omega}{\kappa}\right)\left|K_{iv_{0}T\frac{\omega}{\kappa}}\left(\frac{\omega}{\kappa}\right)\right|^{2}.$
(21)

where $T=\cos\theta$. Again a relation between the classical spectral
distribution and quantum beta Bogolyubov coefficient is apparent; we address
this in Section V.

The energy spectrum $I(\omega)$ is shown in Figure 6. Integration of Eq. (21)
over $\mathop{}\\!\mathrm{d}{\omega}\mathop{}\\!\mathrm{d}{\Omega}$ agrees
with the total energy of Eq. (19). Like the self-dual case, there is no
radiation in the forward or backward $T\to\pm 1$ ($\theta\to[0,\pi]$)
directions, as expected. See Figure 8 for a 3D view of the spectral
distribution. The spectral distribution in the $T\to 0$ ($\theta\to\pi/2$)
limit is:

$\displaystyle\lim_{T\to
0}\,\frac{\mathop{}\\!\mathrm{d}{I}}{\mathop{}\\!\mathrm{d}{\Omega}}$
$\displaystyle=$
$\displaystyle\frac{v_{0}^{2}\omega^{2}}{4\pi^{3}\kappa^{2}}\,\left[K_{0}\left(\frac{\omega}{\kappa}\right)\right]^{2}$
(22) $\displaystyle\approx$
$\displaystyle\frac{v_{0}^{2}\omega}{8\pi^{2}\kappa}\,e^{-\frac{2\omega}{\kappa}}\
,$ (23)

again showing the high-frequency exponential suppression, wherein the second
line we have expanded around large $\omega/\kappa$.

The betaK trajectory is well-motivated, physically intuitive, and potentially
realizable in the laboratory as it is straightforwardly the horizontal
component of an electron’s motion subject to an initial horizontal velocity
and constant vertical force. In the following section, we use betaK’s analytic
tractability to help confirm the duality between the classical point charge
and the quantum moving mirror.

Figure 8: 3D view of the radiated spectrum angular distribution
$\mathop{}\\!\mathrm{d}I/\mathop{}\\!\mathrm{d}\Omega$ from motion
corresponding to the betaK trajectory. Here we use unit charge, natural units,
and $\omega=\kappa=1$. The maximum speed of the charge is $v_{0}=0.95$.

##  V Classical-Quantum Correspondence

We have seen that at the level of total energy there is agreement between the
charge radiation approach and the moving mirror Bogolyubov coefficient
approach,

$E=\int_{0}^{\infty}\mathop{}\\!\mathrm{d}{\omega}\int_{-1}^{1}\mathop{}\\!\mathrm{d}{T}\int_{0}^{2\pi}\mathop{}\\!\mathrm{d}{\varphi}\;\frac{\mathop{}\\!\mathrm{d}{I}}{\mathop{}\\!\mathrm{d}{\Omega}}\Leftrightarrow\int_{0}^{\infty}\int_{0}^{\infty}p\,|\beta_{pq}|^{2}\mathop{}\\!\mathrm{d}{p}\mathop{}\\!\mathrm{d}{q}\
.$ (24)

We can further see that the agreement extends to the particle count,

$N=\int\frac{1}{\omega}\frac{\mathop{}\\!\mathrm{d}{I}}{\mathop{}\\!\mathrm{d}{\Omega}}\mathop{}\\!\mathrm{d}{\Omega}\mathop{}\\!\mathrm{d}{\omega}\Leftrightarrow\frac{1}{2}\int\int|\beta_{pq}|^{2}\mathop{}\\!\mathrm{d}{p}\mathop{}\\!\mathrm{d}{q}\
.$ (25)

The factor $1/\omega$ converts particle energy to particle number, and the
factor 1/2 arises because while both sides of the mirror are employed in the
correspondence, an observer could only see one side See Figure 9 for an
illustration of particle count.

Figure 9: A plot of total finite particle count of the radiation particles
created by the two mirrors, using Eq. (25), for maximum velocity ranging from
0.05 to 0.99.

As mentioned in Section III and Section IV the connection persists at the
level directly between the spectral distribution and the beta Bogolyubov
coefficient, i.e. the integrands. The steps to obtain the exact relation are
as follows. First, the Jacobian going from $\\{p,q\\}$ coordinates to
$\\{\omega,T\\}$ coordinates is $\omega/2$. Recall that
$\mathop{}\\!\mathrm{d}{\Omega}=\sin\theta\mathop{}\\!\mathrm{d}{\theta}\mathop{}\\!\mathrm{d}{\varphi}$
and that
$\mathop{}\\!\mathrm{d}{T}\equiv\mathop{}\\!\mathrm{d}{(\cos\theta)}=\sin\theta\mathop{}\\!\mathrm{d}{\theta}$
and the $\mathop{}\\!\mathrm{d}{\varphi}$ integral simply contributes $2\pi$.
Finally, the parity is reversed on opposite sides of the mirror so that one
side is related to the other by $T\leftrightarrow-T$, so we write

$\int_{-1}^{+1}dT\,\frac{dI}{d\Omega}=\frac{1}{2}\left[\int_{-1}^{+1}dT\,\frac{dI(T)}{d\Omega}+\int_{-1}^{+1}dT\,\frac{dI(-T)}{d\Omega}\right]\
.$ (26)

Putting all the elements together delivers the correspondence

$|\beta_{pq}|^{2}\ \leftarrow\
\frac{4\pi}{\omega^{2}}\left[\frac{\mathop{}\\!\mathrm{d}{I}}{\mathop{}\\!\mathrm{d}{\Omega}}(\omega,\cos\theta)+\frac{\mathop{}\\!\mathrm{d}{I}}{\mathop{}\\!\mathrm{d}{\Omega}}(\omega,-\cos\theta)\right]\
.$ (27)

This can be verified directly for the solutions given for the two
trajectories. Note that the correspondence formally goes in only one
direction, from charge radiation to moving mirror, as the beta Bogolyubov
coefficient has no angular information on the in-going and out-going modes.
Only once we introduce an angle $\theta$ such that $p=\omega(1+\cos\theta)/2$
and $q=\omega(1-\cos\theta)/2$, hence $p+q=\omega$ and $\sigma\equiv
p-q=\omega\cos\theta$, can we go the other way.

Such a classical-quantum correspondence is very useful, but we emphasize that
it does not capture all quantum effects. While the particle production can be
computed classically, this neglects quantum effects when the radiation
(photon) energy becomes comparable to the particle (electron) energy, e.g. the
radiation wavelength is smaller than the charge de Broglie wavelength.

##  VI Conclusions

We have solved for the accelerating point charge radiation – its energy,
particle count, and spectral angular distribution – of two trajectories that
asymptotically come to complete stop, compatible with finite total particle
emission. As Feynman Feynman (1996) has emphasized,

Larmor’s power is only valid for cyclic motions, or at least motions which do
not grow forever in time.

The betaK and Self-Dual trajectories fulfill that condition, and these two
solutions inspired by the accelerating boundary (moving mirror) analog are the
only known rectilinear solutions with exactly soluble spectra, finite energy,
and finite particle count. This allows comparison of classical and quantum
systems directly.

The main results presented include:

  * •

We have found the time dependence of radiative solutions. One utility of an
exact solution for moving point charge radiation is that in QED, time-
dependent computations are notoriously difficult. Here the dynamics are
explicit in the applicable Larmor and Feynman powers.

  * •

We have demonstrated consistency between the total energy derived in terms of
the Larmor power, the Feynman power, and the quantum Bogolyubov coefficients.

  * •

We have derived the spectral distributions of these two accelerating, but
asymptotically static, motions analytically, and further shown consistency
with the total energy emission and total particle count. In addition to 3D
plots of the radiation angular distribution we discussed the angular limits
(e.g. the forward and transverse emission) and high frequency limits.

  * •

We have laid out explicitly a quantum-classical correspondence to the moving
mirror model, mapping between the classical spectral distribution and the
quantum Bogolyubov coefficients.

The demonstrated consistency and explicit correspondence enhances the utility
of the moving mirror model by showing its role as a point charge analog. Thus
the accelerated boundary correspondence of the moving mirror to black hole
radiation may potentially point to a connection to accelerating charge
radiation via a Hawking-Feynman-Larmor correspondence.

This is an exciting prospect for future directions. It may be tractable to
link directly these electron trajectories to curved spacetime counterparts,
revealing spacetime metrics that radiate with similar nonthermal spectra (or
reveal charge motions that could show a period of thermal emission). Further,
given that a connection for beta decay to a moving mirror analog has been made
Good and Davies (2023); Lynch and Good (2023); Good (2023), other well-known
QED scattering processes might correspond at lowest order to one of the
solutions given. Asymptotic rest, with its finite particles and unitarity,
could be a powerful tool, and it would be interesting to develop further
solutions, such as the Schwarzschild-Planck radiation Good  _et al._ (2020);
Good and Linder (2021); Moreno-Ruiz and Bermudez (2022) to compare
accelerating electron and black hole radiation in the thermal limit.

###### Acknowledgements.

Funding comes in part from the FY2021-SGP-1-STMM Faculty Development
Competitive Research Grant No. 021220FD3951 at Nazarbayev University. This
work is supported in part by the Energetic Cosmos Laboratory, and in part by
the U.S. Department of Energy, Office of Science, Office of High Energy
Physics, under contract no. DE-AC02-05CH11231.

##  Appendix A Spectral Distribution Calculation

To show how one can go from the formula for the spectral distribution, Eq.
(6), to the modified Bessel function result we illustrate the steps for the
self dual case. The integral has the form

$A\equiv\int_{-\infty}^{+\infty}dt\,\dot{x}\,e^{i\omega(t-x\cos\theta)}\ .$
(28)

Substituting in the self dual expressions for $x(t)$ from Eq. (7), and
$\dot{x}$, and writing $T\equiv\cos\theta$ we have

$\displaystyle A$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{+\infty}dt\,\frac{-2v\kappa
t}{\kappa^{2}t^{2}+1}\,e^{i\omega[t+(vT/\kappa)\ln(\kappa^{2}t^{2}+1)]}$ (29)
$\displaystyle=$
$\displaystyle\frac{-2v}{\kappa}\int_{-\infty}^{+\infty}ds\,s(s^{2}+1)^{-1+i\omega
vT/\kappa}\,e^{i\omega t}$ (30) $\displaystyle=$
$\displaystyle\frac{-4iv}{\kappa}\int_{0}^{\infty}ds\,s(s^{2}+1)^{-1+i\omega
vT/\kappa}\,\sin\frac{\omega s}{\kappa}\ .$ (31)

In the second line we have taken the exponential of the log term, and defined
$s=\kappa t$, while in the third line we have used that we must take the odd
part of the remaining exponential to give an even integrand over the symmetric
range of integration.

This integral can be evaluated through Gradshteyn & Ryzhik 3.771.5 Gradshteyn
and Ryzhik (1994), resulting in

$\displaystyle A$ $\displaystyle=$
$\displaystyle\frac{4v}{\kappa\sqrt{\pi}}\,\left(\frac{\omega}{2\kappa}\right)^{1/2-i\omega
vT/\kappa}\,\sinh(\pi\omega vT/\kappa)\,\Gamma\left(\frac{i\omega
vT}{\kappa}\right)$ (32) $\displaystyle\times K_{1/2+i\omega
vT/\kappa}\left(\frac{\omega}{\kappa}\right)\ .$

The modulus squared, using that $|\Gamma(ix)|^{2}=\pi/(x\sinh x)$, is

$|A|^{2}=\frac{8v}{\kappa^{2}T}\,\sinh(\pi\omega
vT/\kappa)\,\left|K_{1/2+i\omega
vT/\kappa}\left(\frac{\omega}{\kappa}\right)\right|^{2}\ .$ (33)

For the betaK case we proceed similarly, noting that since $\dot{x}$ is even
in time in that case we must take the even part of the exponential (i.e.
cosine).

##  Appendix B Leonardo’s Pitcher: From Electron to betaK

The motion of a relativistic particle with unit mass subject to an external
force comes from the action222This is a first prototypical system of a
relativistic Lagrangian (see e.g. page 323 of Goldstein (1980)).

$S=-\int dt\;\left(\sqrt{1-v^{2}}+Fx\right)\ .$ (34)

For a force dependent only on position the equations of motion are simply

$\displaystyle\alpha$ $\displaystyle=$
$\displaystyle\frac{d}{dt}\frac{v}{\sqrt{1-v^{2}}}\equiv\frac{d(\gamma
v)}{dt}$ (35) $\displaystyle=$ $\displaystyle(0,\alpha_{y},0)\ ,$ (36)

where the last line holds for purely vertical force, and we will take
$\alpha_{y}=\,$const (e.g. gravity in Leonardo’s water pitcher experiment).
Finally, we take the initial velocity to be purely horizontal,
$v=(v_{0},0,0)$.

The results are simple – nonuniform motion in the horizontal direction due to
the relativistic boost factor $\gamma$, and hyperbolic motion under constant
acceleration in the vertical direction – but worth quickly going through to
reveal the form of nonuniformity.

The $z$ direction is trivial: as there is no initial velocity, nor subsequent
acceleration, in this direction then Eq. (35) guarantees that $z(t)=z(0)$ and
we can ignore this dimension. In the $x$ (horizontal) direction, Eq. (35)
gives

$\gamma(t)v_{x}(t)=\gamma_{0}v_{0}\ ,$ (37)

and the key point is that while nonrelativistically one would simply have
$v_{x}(t)=v_{0}$, i.e. uniform motion, the Lorentz factor $\gamma$ couples in
the $y$ motion (recall $\gamma=1/\sqrt{1-v_{x}^{2}-v_{y}^{2}}$), which is
accelerated. This results in nonuniform motion horizontally.

We can relate $v_{x}$ and $v_{y}$, and solve for both motions by squaring Eq.
(37) to get

$v_{x}^{2}=(1-v_{y}^{2})v_{0}^{2}\ .$ (38)

This immediately tells us that $v_{x}$ has its maximum value at the initial
time, so $v_{y}(t)<v_{0}=v_{y}(0)$. That is, the vertical acceleration
effectively causes a horizontal deceleration!

In the $y$ (vertical) direction, the equation of motion gives $\gamma
v_{y}=\alpha_{y}t$ so

$v_{y}=\frac{\kappa t}{\sqrt{1+(\kappa t)^{2}}}\ .$ (39)

At late times this approaches the speed of light. To presage the betaK mirror
analogy we have written $\kappa\equiv\alpha_{y}/\gamma_{0}$. Finally, with Eq.
(38) we obtain the horizontal velocity

$v_{x}=\frac{v_{0}}{\sqrt{1+(\kappa t)^{2}}}\ ,$ (40)

which indeed decelerates from its initial value to zero. Again presaging the
mirror analog, we will end up with an asymptotically static mirror defined by
the 1D horizontal motion.

Integrating the velocities gives the trajectories, with

$y(t)=\kappa^{-1}\sqrt{1+\kappa^{2}t^{2}}-\kappa^{-1}\ ,$ (41)

revealing hyperbolic motion in the vertical direction. In the horizontal
direction,

$x(t)=\frac{v_{0}}{\kappa}\,\sinh^{-1}\kappa t\ ,$ (42)

exactly (after a trivial sign flip on initial velocity) the betaK trajectory,
Eq. (15).

## References

  * Hawking (1975) S. Hawking, Commun. Math. Phys. 43, 199 (1975).
  * DeWitt (1975) B. S. DeWitt, Phys. Rept. 19, 295 (1975).
  * Fulling and Davies (1976) S. A. Fulling and P. C. W. Davies, Proc. R. Soc. Lond. A 348, 393 (1976).
  * Davies and Fulling (1977) P. Davies and S. Fulling, Proc. R. Soc. Lond. A A356, 237 (1977).
  * Birrell and Davies (1984) N. Birrell and P. Davies,  _Quantum Fields in Curved Space_, Cambridge Monographs on Mathematical Physics (Cambridge Univ. Press, Cambridge, UK, 1984).
  * Fabbri and Navarro-Salas (2005) A. Fabbri and J. Navarro-Salas,  _Modeling Black Hole Evaporation_ (Imperial College Press, 2005).
  * Walker (1985) W. R. Walker, Phys. Rev. D 31, 767 (1985).
  * Ford and Vilenkin (1982) L. Ford and A. Vilenkin, Phys. Rev. D 25, 2569 (1982).
  * Carlitz and Willey (1987) R. D. Carlitz and R. S. Willey, Phys. Rev. D 36, 2327 (1987).
  * Chen and Mourou (2017) P. Chen and G. Mourou, Phys. Rev. Lett. 118, 045001 (2017), arXiv:1512.04064 [gr-qc] .
  * Chen  _et al._ (2022) P. Chen _et al._ (AnaBHEL), (2022), arXiv:2205.12195 [gr-qc] .
  * Good  _et al._ (2021) M. R. R. Good, A. Lapponi, O. Luongo, and S. Mancini, Phys. Rev. D 104, 105020 (2021), arXiv:2103.07374 [gr-qc] .
  * Chen and Mourou (2020) P. Chen and G. Mourou, Phys. Plasmas 27, 123106 (2020), arXiv:2004.10615 [physics.plasm-ph] .
  * Moore (1970) G. T. Moore, J. of Math. Phys. 11, 2679 (1970).
  * Larmor (1897) J. Larmor, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 44, 503 (1897).
  * Svidzinsky (2023) A. A. Svidzinsky, Phys. Rev. D 107, 024038 (2023), arXiv:2301.13131 [gr-qc] .
  * Lin and Chen (2021) K.-N. Lin and P. Chen, (2021), arXiv:2107.09033 [gr-qc] .
  * Reyes (2021) I. A. Reyes, Phys. Rev. Lett. 127, 051602 (2021), arXiv:2103.01230 [hep-th] .
  * Lin and Chen (2022) K.-N. Lin and P. Chen, (2022), arXiv:2212.00550 [gr-qc] .
  * Akal  _et al._ (2021) I. Akal, Y. Kusuki, N. Shiba, T. Takayanagi, and Z. Wei, Phys. Rev. Lett. 126, 061604 (2021), arXiv:2011.12005 [hep-th] .
  * Nikishov and Ritus (1995) A. Nikishov and V. Ritus, J. Exp. Theor. Phys. 81, 615 (1995).
  * Ritus (1998) V. Ritus, J. Exp. Theor. Phys. 87, 25 (1998), arXiv:hep-th/9903083 .
  * Ritus (2002) V. Ritus, Int. J. Mod. Phys. A 17, 1033 (2002).
  * Ritus (2003) V. Ritus, J. Exp. Theor. Phys. 97, 10 (2003), arXiv:hep-th/0309181 .
  * Ritus (2022) V. I. Ritus, Usp. Fiz. Nauk 192, 507 (2022).
  * Zhakenuly  _et al._ (2021) A. Zhakenuly, M. Temirkhan, M. R. R. Good, and P. Chen, Symmetry 13, 653 (2021), arXiv:2101.02511 [gr-qc] .
  * Good  _et al._ (2022) M. R. R. Good, C. Singha, and V. Zarikas, Entropy 24, 1570 (2022), arXiv:2211.00264 [gr-qc] .
  * Good and Ong (2023) M. R. R. Good and Y. C. Ong, Physics 5, 131 (2023), arXiv:2302.00266 [gr-qc] .
  * Jackson (1999) J. D. Jackson,  _Classical electrodynamics; 3rd ed._ (Wiley, New York, NY, 1999).
  * Feynman (1996) R. P. Feynman,  _Feynman lectures on gravitation_ , edited by F. B. Morinigo, W. G. Wagner, and B. Hatfield (1996).
  * Singal (2020) A. K. Singal, Symmetry 12, 1833 (2020).
  * Schwinger (1949) J. S. Schwinger, Phys. Rev. 75, 1912 (1949).
  * Zangwill (2013) A. Zangwill,  _Modern electrodynamics_ (Cambridge Univ. Press, Cambridge, 2013).
  * Walker and Davies (1982) W. R. Walker and P. C. W. Davies, Journal of Physics A: Mathematical and General 15, L477 (1982).
  * Good  _et al._ (2013) M. R. R. Good, P. R. Anderson, and C. R. Evans, Phys. Rev. D 88, 025023 (2013), arXiv:1303.6756 [gr-qc] .
  * Good and Linder (2017) M. R. R. Good and E. V. Linder, Phys. Rev. D 96, 125010 (2017), arXiv:1707.03670 [gr-qc] .
  * Good and Linder (2019) M. R. Good and E. V. Linder, Phys. Rev. D 99, 025009 (2019), arXiv:1807.08632 [gr-qc] .
  * Good  _et al._ (2020) M. R. Good, E. V. Linder, and F. Wilczek, Phys. Rev. D 101, 025012 (2020), arXiv:1909.01129 [gr-qc] .
  * Good and Linder (2021) M. R. R. Good and E. V. Linder, New J. Phys. 23, 043007 (2021), arXiv:2003.01333 [gr-qc] .
  * Moreno-Ruiz and Bermudez (2022) A. Moreno-Ruiz and D. Bermudez, Class. Quant. Grav. 39, 145001 (2022), arXiv:2112.00194 [gr-qc] .
  * Griffiths (2013) D. J. Griffiths,  _Introduction to electrodynamics; 4th ed._ (Pearson, Boston, MA, 2013) re-published by Cambridge University Press in 2017\.
  * Gharib  _et al._ (2022) M. Gharib, C. Roh, and F. Noca, Leonardo , 21 (2022).
  * Good and Davies (2023) M. R. R. Good and P. C. W. Davies, (2023), arXiv:2206.07291 [gr-qc] .
  * Lynch and Good (2023) M. H. Lynch and M. R. R. Good, (2023), arXiv:2211.14774 [nucl-ex] .
  * Good (2023) M. R. R. Good, (2023), arXiv:2211.00946 [gr-qc] .
  * Gradshteyn and Ryzhik (1994) I. Gradshteyn and I. Ryzhik,  _Table of Integrals, Series, and Products_ , 7th ed. (Academic Press, San Diego, CA, 1994).
  * Goldstein (1980) H. Goldstein,  _Classical Mechanics_ (Addison-Wesley, 1980).