# Extreme Electron Acceleration with Fixed Radiation Energy

Michael R. R. Good1,2<EMAIL_ADDRESS>Chiranjeeb Singha3
<EMAIL_ADDRESS>Vasilios Zarikas4<EMAIL_ADDRESS>1 Energetic
Cosmos Laboratory & Physics Department, Nazarbayev University, Astana 010000,
Kazakhstan
2 Leung Center for Cosmology and Particle Astrophysics, National Taiwan
University, Taipei 10617, Taiwan
3 Theory Division, Saha Institute of Nuclear Physics, Kolkata 700064, India
4 Department of Mathematics, University of Thessaly, Lamia 35100, Greece

###### Abstract

We examine the extreme situation of radiation from an electron that is
asymptotically accelerated to the speed of light, resulting in finite emission
energy. The analytic solution explicitly demonstrates the difference between
radiation power loss and kinetic power loss (null).

moving mirrors, black hole evaporation, acceleration radiation

###### pacs:

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

Introduction. \- Sending an electron to light speed usually results in a blast
of infinite energy. Can this be avoided? Here, we demonstrate how to push an
electron to its speed limit without a divergence in the emission energy. We
compute the Larmor power, ‘Feynman power’, angular distribution, and by
analogy with the moving mirror model, the Bogoliubov spectrum. The total
energy radiated is computed and shown to be analytic and finite. The
disconnect between kinetic energy and radiation energy is sharp. There are
three motivations for this work:

  * •

Black hole evaporation Hawking (1975) leading to explosions Hawking (1974)
have been unobserved thus far Carr  _et al._ (2010); insight into the process
may be gained by studying asymptotic light speed acceleration radiation from
non-exploding electrons.

  * •

While moving point charge radiation is a staple of electrodynamics Griffiths
(2013); Jackson (1999); Zangwill (2013), global, continuous, and dynamically
accelerated rectilinear solutions are virtually absent in the literature.

  * •

Although asymptotic inertia is a key requirement for unitarity, in the
presence of a horizon it fails to conserve information Good and Abdikamalov
(2020); studying an asymptotic inertial and horizon-free trajectory could
provide insight into quantum purity.

Furthermore, the lack of horizon formation coupled to asymptotic light speed
is an interesting direction; see the early past of Carlitz and Willey Carlitz
and Willey (1987a)). A tractable global trajectory without a horizon and
asymptotic light speed is rare. Only two known solutions exist: (1) the
‘proex’ trajectory Good  _et al._ (2013), which is the limiting light speed
case of the experimentally observed sublight speed drift of an electron
emitting inner bremsstrahlung from beta decay Good and Davies (2022); and (2)
the Light and Airy trajectory Good and Linder (2021), which is the only known
analytic solution for time evolution of particle creation from the quantum
vacuum. The first solution, ‘proex’, suffers from infinite emission energy if
used to describe an electron in 3 + 1 dimensions Good (2017). For this reason,
we focus on the second solution. We are able to extend the 1 + 1 dimensional
mirror model to the more realistic situation of an electron accelerating in 3
+ 1 dimensions.

For clarity, we consider uniform acceleration; while very mathematically
simple, it is not physically realistic as a global trajectory. For instance, a
single electron emits an infinite amount of radiation energy when travelling
eternally at uniform acceleration. Of course, while this makes sense because
of the infinite energy required to accelerate it forever, it is easy to see
this visually for the hyperbola of uniform acceleration because there are
horizons, i.e., in null coordinates $u=t-x$ and $v=t~{}+~{}x$, a horizon is
when $u(v)$ goes to infinity at some finite value of $v$ or $v(u)$ goes to
infinity at some finite value of $u$.

This long-persistent global definition problem associated with horizons has
given rise to theoretical misunderstandings, and will continue to lead
physical interpretation astray if regularization of the emitted radiation is
not taken into account when global context matters. Infinite radiation energy
plagues worldline examination of physical connections between kinetic power
loss and radiation power loss.

Collective attention should be paid to directly addressing radiation energy
independent of kinetic energy considerations. As is already well-understood,
non-uniform accelerated trajectories that drop to zero are perfectly capable
of regularizing the global total emitted radiation energy. The trade-off is
the lack of simplicity, and consequently, tractability (for instance, when
determining the global radiation spectrum).

We present a result in this article that provides a well-defined, analytic,
simple, consistent, and complete solution for finite radiation energy and its
corresponding spectrum. Uncommon solutions of this type can be immediately
used to investigate questions of interest wherever globally continuous
equations of motion are needed. For instance, the solution is well-suited for
applications such as study via an Unruh–DeWitt detector trajectory Cong  _et
al._ (2020) or the non-uniform Davies–Fulling–Unruh effect Doukas  _et al._
(2013).

Additionally, this work provides a simple conceptual and quantitative analog
approach to understanding the radiation emitted by an electron that
asymptotically advances to light speed, in a similar way as a uniformly
accelerated electron asymptotically approaches light speed; however, the
differences are crucial to analytic tractability: there is no horizon, the
acceleration drops to zero, and the radiation energy is finite (though it
nonetheless travels to light speed!). We emphasize that the analog approach is
general enough to be applied to any rectilinear trajectory that emits finite
radiation energy; however, it cannot be applied to uniform acceleration
because of above aforementioned problems associated with the horizons.

We analytically compute the relevant quantities for the specific extreme
solution and demonstrate consistency between the various ways of computing the
radiation energy. The analog approach treats the electron as a tiny double-
sided moving mirror; interestingly, this reveals previously unknown electron
acceleration radiation spectra as well as a general connection between
classical acceleration radiation and quantum acceleration radiation.

The outline of the paper is as follows. First we provide context and consider
the trajectory. Second, we compute the power and the self-force, examining the
time-evolution. Third, we compute the ‘local acceleration’, determine the
conditions required for avoiding a divergence in the emission energy, and find
the angular distribution of power. Fourth, we propose a candidate Bogoliubov
spectrum, and lastly, we perform consistency checks and compute the total
energy emitted.

##  I Infinite Emission Energy in the Causal Limit

Imagine one has an arbitrarily large (infinite) amount of energy to accelerate
an electron, ignoring the mechanism that is doing the acceleration while
assuming nonetheless that the electron accelerates along a specified sub-light
speed trajectory in a time-like fashion to asymptotically approach the speed
of light. This would be the case, for instance, when attempting to push an
electron along an eternal uniformly accelerated trajectory. Such a motion
would properly result in an infinite emission of radiation energy.

This is the case when considering the scaling of radiated energy from
trajectories that do not even have horizons. For instance, let us examine the
case of emission energy associated with lowest-order inner bremsstrahlung
Jackson (1999) and the scaling with respect to the final electron speed $s$
(see Good and Davies (2022)):

$E\sim\frac{1}{s}\ln\left(\frac{1~{}+~{}s}{1-s}\right)-2.$ (1)

When the charge approaches the speed of light, i.e., when $s\to 1$, then
$E\to\infty$. A mathematical divergence in the emission energy occurs despite
the lack of horizon. Of course, we must make it absolutely clear that an
electron cannot be accelerated to the speed of light in the real world, due to
limitations on the finite amount of available energy to give to the electron.
However, without even considering the nonphysical nature of the infinite
amount of energy required to push a massive charged particle, it is physically
clear from Equation (1) that a radiating electron always travels at sub-light
speeds (even asymptotically); otherwise, an infinite amount of radiation
energy occurs in the expression for the emitted energy.

Thus, equipped with large amounts of available energy, could it be possible to
push an electron to the speed of light without a divergence appearing in the
total emission energy?

##  II Trajectory

Allow us to consider an electron following the light-cone advanced time
$v=t~{}+~{}z$, trajectory Good and Linder (2021)

$v(u)=u~{}+~{}\frac{\kappa^{2}}{3}u^{3},$ (2)

with a single parameter $\kappa$ which sets the acceleration scale of the
system. Here, $u=t-z$ is retarded time. The electron moves in a straight-line
path along one spatial direction (rectilinear), $z$, where

$z=\frac{v-u}{2}=\frac{\kappa^{3}u^{3}}{6\kappa},\quad\textrm{and,}\quad
u=t-z=\frac{1}{\kappa}(6\kappa z)^{1/3}.$ (3)

The rapidity $\eta(u)$, using $2\eta(u)\equiv\ln v^{\prime}(u)$, where the
prime is a derivative with respect to the argument, is

$\eta(u)=\frac{1}{2}\ln\left(\kappa^{2}u^{2}~{}+~{}1\right).$ (4)

From the rapidity, the velocity along the direction of motion is easily found
to be $\beta\equiv\tanh\eta$. Plugging in Equation (4) provides

$\beta(u)=\frac{\kappa^{2}u^{2}}{\kappa^{2}u^{2}~{}+~{}2}\,,$ (5)

and the proper acceleration Lorentz scalar, which follows from
$\alpha(u)\equiv e^{-\eta(u)}\eta^{\prime}(u)$, is

$\alpha(u)=\frac{\kappa^{2}u}{\left(\kappa^{2}u^{2}~{}+~{}1\right)^{3/2}}.$
(6)

The asymptotic approach to the speed of light is explicit: $\beta(u)\to 1$, as
$u\to\pm\infty$. Interestingly, the brakes are being applied, as $\alpha(u)\to
0$ as well. Notice that $\alpha=\beta=0$ at $u=0$. The position $z(t)$,
velocity $\beta(t)$, and the proper acceleration $\alpha(t)$ are plotted in
Figure 1 to illustrate the dynamical properties of the motion. For further
visual clarity, a parity-reversed plot of the trajectory from Equation (2) is
provided in a spacetime plot, and a (1 + 1) dimensional Penrose diagram is
provided via Figures 1 and 2 of Good and Linder (2021).

Figure 1: The position $z(t)$, velocity $\beta(t)$, and proper acceleration
$\alpha(t)$ plotted to demonstrate the position (red) approaching a near 45
degree angle (diagonal null-ray is gray) as the velocity (blue) approaches the
speed of light (black), while the acceleration (green) drops to zero. Here,
$\kappa=1$. The key takeaway is that an this motion has an asymptotic end
state with drifting or coasting velocity at the speed of light, and is coupled
to asymptotic zero acceleration.

##  III Larmor power

By working with unit charge and written as $\bar{P}=\frac{dE}{du},$ such that
$\bar{P}=P\frac{dt}{du}=P/(1-\beta)$, where $P=\alpha^{2}/6\pi$, we find the
Larmor power $\bar{P}(u)$, formulated in terms of retarded time $u$, providing
us with

$\bar{P}=\frac{\kappa^{4}u^{2}\left(\kappa^{2}u^{2}~{}+~{}2\right)}{12\pi\left(\kappa^{2}u^{2}~{}+~{}1\right)^{3}}.$
(7)

Key takeaways from the Larmor power include the asymptotic end to the
radiation at early and late times, symmetry in time, and zero emission at
$u=0$. Notice that the Larmor power emission is always positive. This is in
stark contrast to the negative energy flux commonly associated with asymptotic
inertial mirrors in 1 + 1 dimensions from the quantum stress tensor energy
flux, e.g., Walker (1985a). A plot of $\bar{P}(u)$ illustrates the power in
Figure 2. We make special note that the Larmor formula describes radiation in
the ‘far zone’, and does not tell the story closer to the radiation charge.
For example, if a (classical) electric dipole starts to collapse at a certain
time, the Larmor formula describes the radiation in the far zone; however,
close to the dipole the energy flows from that initially stored in the
electromagnetic field onto the collapsing dipole; see, e.g., Schantz (1995).

Figure 2: A plot of the Larmor power, $\bar{P}(u)$, from Equation (7). From
Equation (22), the integral under the curve is the energy,. Note that the
Larmor power is always positive, asymptotically zero, and time-symmetric about
$u=0$, where $\bar{P}=0$. Here, $\kappa=1$.

##  IV Feynman power

The Larmor power, $P=\alpha^{2}/6\pi$, or $\bar{P}$, Equation (7), is not
sufficient to understand the full story of the radiation. For instance, it is
often supposed that the Larmor power tells us ‘when’ the energy is radiated;
however, this is not true for all motions. Larmor’s formula can only be valid
for certain types of motions, such as horizonless trajectories with finite
global energy emission. In fact, this insufficiency has led to considerable
confusion when dealing with the equivalence principle and the question of
whether an electron radiates while sitting in a gravitational field Feynman
(1996).

To better understand the radiated energy, it is beneficial to determine the
force of radiation resistance and the work against this force, which is a
measure of the energy loss due to radiation emission.

Therefore, we now turn to the self-force, $F=\alpha^{\prime}(\tau)/6\pi$
Myrzakul  _et al._ (2021); Ford and Vilenkin (1982); we find it convenient to
dub this the ‘Feynman power’ Feynman (1996). This is an analogous ‘self-force
measure’ $\bar{F}(u)=F\frac{dr}{du}=F\beta/(1-\beta)$ as a function of
retarded time $u$:

$\bar{F}=\frac{\kappa^{4}u^{2}(1-2\kappa^{2}u^{2})}{12\pi\left(\kappa^{2}u^{2}~{}+~{}1\right)^{3}}.$
(8)

Unlike the Larmor power, the Feynman power can be negative. Indeed, it must be
negative for asymptotic inertial motions, as non-trivial changes in
acceleration that asymptotically approach zero are characterized by sign flips
(‘what goes up must come down’). Similar to the Larmor power, the self-force
and Feynman power are asymptotically zero at early and late times: $\bar{F}=0$
at $u=0$. A plot of the Feynman power is shown in Figure 3. Notice the stark
contrast between the Larmor power in Figure 2 and the sometimes negative
Feynman power in Figure 3, which obviously tell very different, yet globally
consistent, stories about the power of the emission.

Figure 3: The above graph is of the Feynman power, or ‘self-force measure’,
from Equation (8). Here, $\kappa=1$. The integral under the curve is the sign-
flipped radiated energy, or the work done by the self-force, from Equation
(22). Note that the Feynman power possesses negative dips, which contrasts
with the always-positive Larmor power shown in Figure 2.

##  V Local acceleration

Consider the so-called ‘local acceleration’ Carlitz and Willey (1987b),
$\bar{\kappa}(u)=\partial_{u}\ln v^{\prime}(u)=v^{\prime\prime}/v^{\prime}$,
or in spacetime coordinates,

$\kappa(t)=\frac{2\ddot{z}}{(1-\dot{z})^{2}(1~{}+~{}\dot{z})}.$ (9)

The local acceleration is a useful measure of the dynamical trajectory; in our
case its value, as a function of retarded time $u$, is,

$\bar{\kappa}(u)=\frac{2\kappa^{2}u}{\kappa^{2}u^{2}~{}+~{}1}.$ (10)

The local acceleration is a particularly good quantity for gauging the
thermality of the system, i.e., when it asymptotically approaches a constant
$\bar{\kappa}\to\kappa$, the radiation is Planck-distributed with
$T=\kappa/2\pi$.

However, even though in our case $\bar{\kappa}(u)=\kappa$ when $u=1/\kappa$,
our trajectory is non-thermal (non-Planckian), and thus the local acceleration
can help us. This provides a tool to understand how to avoid a divergence in
the radiation energy. Consider the analog between mirrors and electrons
(namely, powers Good and Linder (2022a); Zhakenuly  _et al._ (2021) and self-
forces Myrzakul  _et al._ (2021); Ford and Vilenkin (1982)) outlined in the
last four equations of Section VII. A connection between moving mirrors and
moving charges has been established by Ritus Ritus (2003, 2002, 1998);
Nikishov and Ritus (1995), who found symmetries linking the 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. Walker Walker (1985b) has pointed out how the local acceleration
$v^{\prime\prime}/v^{\prime}$ must vanish, as it is the boundary term for an
integration by parts when computing the total radiated energy. As can be seen,
in Equation (10) our $\bar{\kappa}(u)$ vanishes as $u\to\infty$. Thus, despite
the asymptotic approach to light-speed $\beta\to 1$ (in the forward direction
toward the far away observer), the proper acceleration $\alpha\to 0$ dies off
more quickly., that is, there is tension between the speed and the
acceleration:

$\bar{\kappa}(u)\sim
e^{\eta}\alpha=\sqrt{\frac{1~{}+~{}\beta}{1-\beta}}\alpha.$ (11)

The asymptotic inertia characterized by the acceleration must be strong enough
to overcome the infinite rapidity. Figuratively, it is as if the observer
finishes detection of the radiated energy before the electron smashes into the
equipment. In order to avoid a divergence in the radiation energy, it is not
enough to have asymptotic inertia when approaching to the speed of light; one
must have extreme braking acceleration as well, and for the overall local
acceleration one must have $v^{\prime\prime}/v^{\prime}\to 0$. A plot of the
local acceleration is shown in Figure 4.

Figure 4: Plot of the ‘local’ or ‘radiation’ acceleration $\bar{\kappa}(u)$
from Equation (10), which is a measure of the asymptotic inertia. The
asymptotic approach to zero insures no divergence in the radiated energy. At
$u=1/\kappa$, $\bar{\kappa}(u)=\kappa$. Here, $\kappa=1$. Note the time
asymmetry of the local acceleration, in contrast to Figures 2 and 3.

##  VI Distribution

To highlight the 3 + 1 dimensional character of the solution, we compute the
power distribution for the charge in relativistic motion. The power
distribution has a well-known form (e.g., Jackson Jackson (1999) or Griffiths
Griffiths (2013)) for rectilinear motion. We can find the distribution with
straightforward vector algebra (see, e.g., the procedure in Good  _et al._
(2019) and the notation details in Appendix A.1), consistent with rectilinear
motions,

$\frac{d\bar{P}}{d\Omega}=\frac{\kappa^{2}\beta(1-\beta)^{4}\sin^{2}\theta}{8\pi^{2}(1-\beta\cos\theta)^{5}},$
(12)

where $\beta=\beta(u)$ is the time-dependent velocity from Equation (5). The
$\hat{z}$ axis points along $\vec{\beta}$. A spherical plot is shown in Figure
5. Integration over solid angle $d\Omega=\sin\theta d\theta d\phi$ provides

$\bar{P}=\frac{\kappa^{2}\beta(1-\beta)^{4}}{3\pi\left(1-\beta^{2}\right)^{3}},$
(13)

which is identical to Equation (7). Integration of the distribution from
Equation (12) over time provides the time-independent distribution or angular
differential distribution of the radiated energy:

$\frac{dE}{d\Omega}=\frac{5\kappa}{512\sqrt{2}\pi}\frac{\sin^{2}\theta}{(1-\cos\theta)^{3/2}}.$
(14)

This can be integrated over the solid angle to find the total energy,
resulting in Equation (24).

Figure 5: The power distribution in Equation (12) illustrated in a spherical
plot. The shape, as expected for rectilinear braking radiation, is about the
forward direction, not in the forward direction. Here, $\kappa=1$ and
$\beta=0.90$.

##  VII Bogoliubov spectrum

To determine particle count and particle spectrum in a quantum context, it is
worthwhile to compute the Bogoliubov coefficients in the style that Hawking
Hawking (1975) did for black hole particle creation. In the same way that
particle creation by black holes can be found with these coefficients,
particle creation by a moving mirror can be found with the beta Bogoliubov
coefficients Birrell and Davies (1984).

Here, the beta Bogoliubov coefficients $\beta_{\omega\omega^{\prime}}$, in the
spirit of the analogy that a moving charge has with a moving mirror, are
tractable for the spectrum dependent on the acceleration parameter $\kappa$.
Here, $\omega^{\prime}$ is the incoming frequency mode and $\omega$ is the
outgoing frequency mode. We use the same procedure and result of Good and
Linder (2021), combining the results for each side of the mirror in Good and
Linder (2021), both left and right, by adding the squares of the beta
Bogoliubov coefficients. The beta coefficient can be found by

$\beta_{\omega\omega^{\prime}}=\frac{-1}{2\pi}\sqrt{\frac{\omega}{\omega^{\prime}}}\int_{-\infty}^{~{}+~{}\infty}du\,e^{-i\omega
u-i\omega^{\prime}v(u)}\ .$ (15)

The overall spectrum is

$|\beta_{\omega\omega^{\prime}}|^{2}=\frac{\omega^{\prime}\text{Ai}\left(\frac{\omega~{}+~{}\omega^{\prime}}{\kappa^{2/3}\sqrt[3]{\omega}}\right)^{2}}{\kappa^{4/3}\omega^{5/3}}~{}+~{}\frac{\omega\text{Ai}\left(\frac{\omega~{}+~{}\omega^{\prime}}{\kappa^{2/3}\sqrt[3]{\omega^{\prime}}}\right)^{2}}{\kappa^{4/3}\omega^{\prime
5/3}}.$ (16)

Considering the close association between accelerating mirrors and charges, we
postulate that the electron spectrum has the same form as Equation (16). See
Figure 6 for an illustration of the symmetry between the modes, and see Figure
7 for a plot of the spectrum after integration over $\omega^{\prime}$,

$N_{\omega}=\int_{0}^{\infty}|\beta_{\omega\omega^{\prime}}|^{2}d\omega^{\prime}.$
(17)

Figure 6: The $|\beta_{\omega\omega^{\prime}}|^{2}$ spectrum of Equation (16)
in a contour plot. The symmetry demonstrates that the summing of either
$\omega$ quanta (Equation (23)) or $\omega^{\prime}$ quanta (Equation (24))
results in the same energy being radiated,. Figure 7: The spectrum
$N_{\omega}$ (Equation (17)), which is number of particles at any given
frequency $\omega$, demonstrating the usual infrared divergence that signals a
divergent total particle count $N$. Such a divergence is absent with
asymptotic resting motions.

The reason for the agreement between emitted energy is related to the Lorentz
symmetry group of Maxwell’s equations in classical electrodynamics and the
Möbius symmetry group PSL(2,R) of the quantum moving mirror model. The
quotient PSL(2,R) group is the restricted Lorentz group of 3 + 1-dimensional
Minkowski space. The moving mirror model can be seen to obey Möbius symmetry,
most explicitly by examining the Schwarzian derivative for the energy flux
emission Birrell and Davies (1984), although the particle count and
entanglement entropy are invariant under Möbius transforms of the trajectory
as well Good and Linder (2022b).

From a scaling perspective, the classical electron Larmor power has exactly
the same scaling on proper acceleration as the quantum mirror power. The same
is true for the self-force of both the electron and mirror.

The analog method works in part because of Lorentz invariance being the main
underlying postulate of symmetry. The invariance manifests itself in analogies
relating to mirrors, electrons and black holes with respect to both quantum
and classical phenomena Zhakenuly  _et al._ (2021). For a mirror moving in 3 +
1D, the radiation obeys the quantum power formula Good and Linder (2022a)

$P_{\textrm{mirror}}=\frac{\hbar\alpha^{2}}{6\pi
c^{2}}\to\frac{\alpha^{2}}{6\pi}\,,$ (18)

where $\alpha$ is the frame-invariant proper acceleration. Independent of
dimension, i.e., the speed of light, this measure of power is Lorentz
invariant. This can be compared to the classical counterpart, as we have seen,
with the same relativistic covariant scaling that is well-studied in graduate
classical electrodynamics Jackson (1999) of the Larmor formula,

$P_{\textrm{electron}}=\frac{2}{3}\frac{q^{2}\alpha^{2}}{4\pi\epsilon_{0}c^{3}}\to\frac{q^{2}\alpha^{2}}{6\pi},$
(19)

where the left side is in SI units, “$\to$” implies conversion to natural
units, where $\epsilon_{0}=c=\hbar=1$, and $\alpha$ is the magnitude of the
proper acceleration of the moving point charge.

The Lorentz invariance cross-over is present in the quantum
Lorentz–Abraham–Dirac (LAD) force discovered by Ford and Vilenkin Ford and
Vilenkin (1982) (for a derivation see Myrzakul  _et al._ (2021)), the
magnitude of which is a Lorentz scalar invariant jerk

$F_{\textrm{mirror}}=\frac{\hbar\alpha^{\prime}}{6\pi
c^{2}}\to\frac{\alpha^{\prime}}{6\pi},$ (20)

where the prime denotes a derivative with respect to proper time. For
convenience, this is compared to the classical LAD self-force with scalar
magnitude, as we have seen:

$F_{\textrm{electron}}=\frac{q^{2}\alpha^{\prime}}{6\pi\epsilon_{0}c^{3}}\to\frac{q^{2}\alpha^{\prime}}{6\pi}.$
(21)

These are quantum and classical forces (and power), tying together mirror and
electrons in close analogy. The moving mirror system, as an analog, thus
serves to provide a new way of looking at the electron acceleration radiation
problem, permitting novel application of the ideas in the quantum system to
the corresponding classical system.

##  VIII Consistency checks

Using the retarded time clock of the observer, the following integration
holds:

$E=\int_{-\infty}^{\infty}\bar{P}(u)du=-\int_{-\infty}^{\infty}\bar{F}(u)du,$
(22)

demonstrating that both the Larmor and Feynman powers from Equations (7) and
(8) are consistent with the conservation of globally radiated energy. The
total radiated energy from the Larmor power equals the total energy loss from
the radiation reaction, as provided by the total energy associated with the
Feynman power. Despite $\bar{F}\neq\bar{P}$, there is no discrepancy between
the total mechanical energy lost and the total radiated energy lost, in full
agreement with the conclusions of Singal Singal (2020, 2018, 2016). The
Bogoliubov spectrum from Equation (16) is consistent with the above results as
well, because

$E=\int_{0}^{\infty}\int_{0}^{\infty}\omega|\beta_{\omega\omega^{\prime}}|^{2}d\omega
d\omega^{\prime}.$ (23)

We may also sum $\omega^{\prime}$ quanta by replacing the above integrand with
$\omega^{\prime}|\beta_{\omega\omega^{\prime}}|^{2}$ and, by mode symmetry,
obtaining the total radiated energy $E$. The quanta summing in Equation (23)
demonstrates that the radiated particles carry the energy. Thus, the total
emitted energy is analytic:

$E=\frac{5\kappa}{96}.$ (24)

This establishes that the total emitted energy of the radiation is finite,
positive, and dependent on detector scale sensitivity range set by the
acceleration scale $\kappa$. Despite the asymptotic approach to the speed of
light and the ever-increasing infinite energy used to accelerate the massive
point charge, communication is severed between the acceleration energy and the
radiation energy. The electron does not explode.

##  IX Discussion

Kinetic processes responsible for converting the outflow energy of blazars,
pulsar wind nebulae, and gamma ray bursters into electron acceleration
radiation allow electrons to be accelerated to very high Lorentz factors
before radiation damping is appreciable; see, e.g., Liang and Noguchi (2009).
The electric fields lines experience a tight stretching effect due to their
non-uniform acceleration relative to uniform acceleration (see Figures 2 and 3
of Ruhlandt  _et al._ (2020)). An illustration of the electric field lines is
shown in Figure 8.

Figure 8: Electric field (density plot) and electric field lines (red) for the
accelerated point charge of Equation (2) with $\kappa=1$. Here, $p(u)=0$
before $u<0$ in order to reproduce the field lines of a static charge at large
distances. The lines of the non-uniformly moving charge at small distances
from the particle are tightly stretched ($u=3$). The acceleration-related
transition zone in between is outpaced by the charge itself. The numerical
calculations for this figure were performed with the Ruhlandt–Mühle–Enderlein
code Ruhlandt  _et al._ (2020).

The Light and Airy trajectory from Equation (2) is the extreme case of ‘high
Lorentz factor’, where the kinetic energy completely outpaces the radiation
energy. The acceleration radiation experiences an asymptotic stop to further
outflow. Using the scale set by IB beta decay in Good and Davies (2022), where
$\kappa=48\alpha c\omega_{\gamma}$,
$\omega_{\gamma}=\omega_{\textrm{max}}-\omega_{\textrm{min}}$ and
$E_{\gamma}=\hbar\omega_{\gamma}$, we can convert Equation (24) to SI units,
resulting in $E=5\alpha E_{\gamma}/2$, where $\alpha$ is the fine structure
constant. Thus, in this scale the energy emitted is dependent on the
sensitivity range of the detector, suggesting independence from the
responsible kinetic processes. This makes sense, as conservation of energy is
not in line with the total divergent kinetic energy, at least in the sense
that the frequencies for appreciable radiation should be limited to the total
energy $E_{T}$ provided to the electron, $\omega_{\textrm{max}}<E_{T}/\hbar$.

The Light and Airy trajectory works for finite radiation energy because of
strong asymptotic inertia coupled to motion near the speed of light;
interestingly, the reality of the beta Bogoliubov coefficient Good and Linder
(2021) leads to fixed radiation that is analytic and simple.

The simplest examples of non-trivial beta Bogoliubov transformations are those
in which the Bogoliubov coefficients are real-valued. The simplicity of a real
non-zero beta Bogoliubov coefficient for particle creation is likely connected
to the simplicity of the associated trajectory and finite radiation energy.

To guarantee a real-valued beta Bogoliubov coefficient, the trajectory $v(u)$
must be an odd function to ensure that the exponential over the symmetric
interval for the beta coefficient turns into a cosine of the argument, i.e., a
real valued function. The simplest odd function that accelerates with
asymptotic inertia and accelerates to the speed of light is $v(u)\sim
u~{}+~{}u^{3}$ (Equation (2)).

##  X Conclusion

In this note, we have avoided the radiation catastrophe that accompanies
asymptotic light speed. With strong asymptotic inertia, our proposed solution
demonstrates that infinite energy radiance need not occur in conjunction with
asymptotic light speed approach. Despite the nonphysical requirement of
infinite kinetic energy necessary to push an electron along the proposed
motion, the total radiated energy is fixed and physically consistent with the
radiation reaction, power emission, and acceleration.

The trajectory demonstrates the physical separation and interplay between
kinetic energy loss and radiative energy loss. Moreover, the motion
demonstrates how to avoid the formation of a horizon despite the advance
toward the speed of light. Mapping to curved spacetime and finding the analog
geometry would be an interesting topic for future study.

## Acknowledgments

Funding comes in part from the FY2021-SGP-1-STMM Faculty Development
Competitive Research Grant No. 021220FD3951 at Nazarbayev University.

## References

  * Hawking (1975) S. Hawking, Commun. Math. Phys. 43, 199 (1975).
  * Hawking (1974) S. Hawking, Nature 248, 30 (1974).
  * Carr  _et al._ (2010) B. J. Carr, K. Kohri, Y. Sendouda, and J. Yokoyama, Phys. Rev. D 81, 104019 (2010), arXiv:0912.5297 [astro-ph.CO] .
  * Griffiths (2013) D. J. Griffiths,  _Introduction to electrodynamics; 4th ed._ (Pearson, Boston, MA, 2013) re-published by Cambridge University Press in 2017\.
  * Jackson (1999) J. D. Jackson,  _Classical electrodynamics; 3rd ed._ (Wiley, New York, NY, 1999).
  * Zangwill (2013) A. Zangwill,  _Modern electrodynamics_ (Cambridge Univ. Press, Cambridge, 2013).
  * Good and Abdikamalov (2020) M. R. R. Good and E. Abdikamalov, Universe 6, 131 (2020), arXiv:2008.08776 [gr-qc] .
  * Carlitz and Willey (1987a) R. D. Carlitz and R. S. Willey, Phys. Rev. D 36, 2327 (1987a).
  * 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 Davies (2022) M. R. R. Good and P. C. W. Davies, (2022), arXiv:2206.07291 [gr-qc] .
  * Good and Linder (2021) M. R. R. Good and E. V. Linder, Universe 7, 60 (2021), arXiv:2101.10576 [gr-qc] .
  * Good (2017) M. R. R. Good,  _Reflecting at the Speed of Light_ (World Scientific, Singapore, 2017).
  * Cong  _et al._ (2020) W. Cong, C. Qian, M. R. R. Good, and R. B. Mann, JHEP 10, 067 (2020), arXiv:2006.01720 [gr-qc] .
  * Doukas  _et al._ (2013) J. Doukas, S.-Y. Lin, B. L. Hu, and R. B. Mann, JHEP 11, 119 (2013), arXiv:1307.4360 [gr-qc] .
  * Walker (1985a) W. Walker, Class. Quant. Grav. 2, L37 (1985a).
  * Schantz (1995) H. G. Schantz, American Journal of Physics 63, 513 (1995), https://doi.org/10.1119/1.17862 .
  * Feynman (1996) R. P. Feynman,  _Feynman lectures on gravitation_ , edited by F. B. Morinigo, W. G. Wagner, and B. Hatfield (CRC Press, Pasadena, Caltech, CA, 1996).
  * Myrzakul  _et al._ (2021) A. Myrzakul, C. Xiong, and M. R. R. Good, Entropy 23, 1664 (2021), arXiv:2101.08139 [gr-qc] .
  * Ford and Vilenkin (1982) L. Ford and A. Vilenkin, Phys. Rev. D 25, 2569 (1982).
  * Carlitz and Willey (1987b) R. D. Carlitz and R. S. Willey, Phys. Rev. D 36, 2336 (1987b).
  * Good and Linder (2022a) M. R. R. Good and E. V. Linder, Eur. Phys. J. C 82, 204 (2022a), arXiv:2111.15148 [gr-qc] .
  * Zhakenuly  _et al._ (2021) A. Zhakenuly, M. Temirkhan, M. R. R. Good, and P. Chen, Symmetry 13, 653 (2021), arXiv:2101.02511 [gr-qc] .
  * Ritus (2003) V. Ritus, J. Exp. Theor. Phys. 97, 10 (2003), arXiv:hep-th/0309181 .
  * Ritus (2002) V. Ritus, Int. J. Mod. Phys. A 17, 1033 (2002).
  * Ritus (1998) V. Ritus, J. Exp. Theor. Phys. 87, 25 (1998), arXiv:hep-th/9903083 .
  * Nikishov and Ritus (1995) A. Nikishov and V. Ritus, J. Exp. Theor. Phys. 81, 615 (1995).
  * Walker (1985b) W. R. Walker, Phys. Rev. D 31, 767 (1985b).
  * Good  _et al._ (2019) M. R. Good, M. Temirkhan, and T. Oikonomou, Int. J. Theor. Phys. 58, 2942 (2019), arXiv:1907.01751 [gr-qc] .
  * 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).
  * Good and Linder (2022b) M. R. R. Good and E. V. Linder, Class. Quant. Grav. 39, 105003 (2022b), arXiv:2108.07451 [gr-qc] .
  * Singal (2020) A. K. Singal, Symmetry 12, 1833 (2020).
  * Singal (2018) A. K. Singal, J. Phys. Comm. 2, 031002 (2018).
  * Singal (2016) A. K. Singal, Found. Phys. 46, 554 (2016), arXiv:1411.4456 [physics.class-ph] .
  * Liang and Noguchi (2009) E. Liang and K. Noguchi, Astrophys. J. 705, 1473 (2009), arXiv:0905.0160 [astro-ph.HE] .
  * Ruhlandt  _et al._ (2020) D. Ruhlandt, S. Mühle, and J. Enderlein, Am. J. Phys. 88, 5 (2020), arXiv:1809.05868 [physics.class-ph] .