# Relativistic quantum information as radiation reaction: entanglement entropy
and self-force of a moving mirror analog to the CGHS black hole

Aizhan Myrzakul1<EMAIL_ADDRESS>Chi Xiong2,3<EMAIL_ADDRESS>Michael R.R. Good1<EMAIL_ADDRESS>1Physics Department & Energetic
Cosmos Laboratory, Nazarbayev University,
53 Kabanbay Batyr, Nur-Sultan, 010000, Kazakhstan
2School of Mathematical and Physical Sciences, Nanyang Technological
University,
50 Nanyang Avenue, 639798, Singapore
3Department of Mathematics and Applied Mathematics,
Xiamen University Malaysia,
Selangor, 43900, Malaysia

###### Abstract

The CGHS black hole has a spectrum and temperature that corresponds to an
accelerated reflecting boundary condition in flat spacetime. The beta
coefficients are identical to a moving mirror model where the acceleration is
exponential in laboratory time. The center and the event horizon of the black
hole are at the same location modeled by the perfectly reflecting regularity
condition that red-shifts the field modes. In addition to computing the energy
flux, we find the corresponding parameter associated with the black hole mass
and the cosmological constant in the gravitational analog system. Generalized
to any mirror trajectory we derive the self-force (Lorentz-Abraham-Dirac) and
express it and the power (Larmor) in connection with entanglement entropy,
inviting an interpretation of acceleration radiation in terms of information
flow. The mirror self-force and radiative power are applied to the particular
CGHS black hole analog moving mirror which reveals the physics of information
at the horizon during asymptotic approach to thermal equilibrium.

##  I Introduction

Three decades ago, several (1+1)-dimensional black hole models were introduced
to gain insight into the quantum nature of black hole radiation, with one of
the most prominent and physically interesting models being the Callan-
Giddings-Harvey-Strominger (CGHS) system Callan  _et al._ (1992). Simplified
CGHS models, albeit with certain limitations, are exactly soluble and lead to
many associated discoveries. New surprises related to complexity,
temperatures, and entropy are still being found Strominger (1994); Giddings
(1994); Thorlacius (1995); Fabbri and Navarro-Salas (2005); Ashtekar  _et al._
(2011).

Moving mirrors are accelerated boundaries that create energy, particles, and
entropy. They are simplified (1+1)-dimensional versions of the dynamical
Casimir effect Moore (1970); DeWitt (1975). Interesting in their own right,
they also act as toy models for black hole evaporation Hawking (1975); Unruh
(1976); Fulling and Davies (1976); Davies and Fulling (1977); Juárez-Aubry and
Louko (2018); Cong  _et al._ (2020); Wilson  _et al._ (2020). The general and
physically relevant connections of moving mirrors to black hole physics can be
found in canonical textbooks Fabbri and Navarro-Salas (2005); Birrell and
Davies (1984) and also in recent works, e.g. Good  _et al._ (2017a); Myrzakul
and Good (2018); Good  _et al._ (2019); Lin  _et al._ (2020).

There have been a number of studies that relate different specific black hole
models (e.g. the Schwarzschild Good  _et al._ (2016) case) and their analog
moving mirrors, including the extremal Reissner-Nordström Liberati  _et al._
(2000); Good (2020a), extremal Kerr Rothman (2000), Reissner-Nordström Good
and Ong (2020), Taub-NUT Foo  _et al._ (2020) and Kerr Good  _et al._ (2020a)
black holes. In addition, de Sitter and anti-de Sitter cosmologies Good  _et
al._ (2020b) are also modeled by moving mirror trajectories. For appropriately
chosen trajectories Good and Linder (2019), close comparisons can be made with
the radiation emitted from dynamic spacetimes Wilczek (1993); Good and Linder
(2017). Such an equivalence between a mirror and a curved spacetime is called
an accelerated boundary correspondence (ABC).

Our motivation in this paper is to synthesize and strongly link the well-known
and important CGHS black hole model with its analog moving mirror counterpart.
In the process we want to derive the spectrum of particle production exactly
and analytically, drawing close parallels between the two systems via the
temperature, horizons and parameter analogs associated with CGHS black hole
mass and cosmological constant. Furthermore, we aim to initiate an
investigation into the entanglement entropy of a generalized mirror system and
its relationship to the self-force on the mirror and power of the emitted
vacuum radiation. As we shall see, this link of inquiry reveals a close
connection between the seemingly distinct concepts of self-force and
information. Application of the results for the CGHS mirror reveals the
divergent self-force is directly a consequence of information loss.

Additionally, we push the correspondence further, by considering the close
connection to classical electrodynamic analogies. The moving mirror is found
to behave almost like a neutral particle coupled to the massless scalar field
(similar to a charged particle coupled to the electromagnetic field). Hence
some of the familiar radiation results in classical electrodynamics has direct
correspondence in the mirror case.

The paper is organized in the following: in Sec. II we briefly review the CGHS
action, the corresponding field equations of motion, and the formation of the
CGHS black hole. In Sec. III, the details of the CGHS metric and the
transformation of this coordinate system to an accelerated mirror trajectory
are investigated. In Sec. IV, we derive the particle flux radiated from the
exponentially accelerated mirror in laboratory time and demonstrate its
thermal characters for late-times. In Sec. V, the particle flux radiated from
the CGHS black hole is reviewed. Sec. VI and VII are dedicated to an overview
of the quantum stress tensor and mirror entanglement entropy, respectively. In
Sec. VIII, the moving mirror Larmor formula and Lorentz-Abraham-Dirac (LAD)
force analogs are derived with an emphasis on entanglement entropy. In Sec.
IX, the formulas derived in the previous section are used to find the
radiative power and radiation reaction force for our particular moving mirror
(CGHS). In the last section, Sec. X, the properties of the correspondence
between the CGHS black hole and the exponentially accelerated mirror as well
as the relation between the moving mirror and electrodynamics are summarized.
In addition, some insight into future directions are provided. We use
$\hbar=c=1$ except in the results of Eq. (111) and Eq. (112) of Sec. IX.

##  II Action & Field Equation

In this section we will briefly summarize the action and field equations of
the Callan–Giddings–Harvey–Strominger model in which a linear dilaton vacuum
evolves into a black hole from matter injection. The CGHS action reads as,

$S=\frac{1}{2}\int
d^{2}x\sqrt{-g}\left[e^{-2\phi}\left(R+4(\nabla\phi)^{2}+4\Lambda^{2}\right)-|\nabla\chi|^{2}\right],$
(1)

where $g$ is the metric tensor, $\phi$ is the dilaton field, $\Lambda$ is the
cosmological constant, and $\chi$ are the matter fields. To obtain the
equations of motion one may vary the action, Eq. (1), with respect to the
metric $g^{ab}$ and the dilaton field $\phi$, respectively,

$\displaystyle
2e^{-2\phi}\left[\nabla_{a}\nabla_{b}\phi+g_{ab}((\nabla\phi)^{2}-\nabla^{2}\phi-\Lambda^{2})\right]$
$\displaystyle=$ $\displaystyle T_{ab},$ (2) $\displaystyle
e^{-2\phi}\left[-R+4(\nabla\phi)^{2}-4\nabla^{2}\phi-4\Lambda^{2})\right]$
$\displaystyle=$ $\displaystyle 0.$ (3)

where
$T_{ab}\equiv\nabla_{a}\chi\nabla_{b}\chi-\frac{1}{2}g_{ab}(\nabla\chi)^{2}$.
Following Fabbri and Navarro-Salas (2005), one can readily solve Eq. (2) in
the conformal gauge, $ds^{2}=-e^{2\rho}dx^{+}dx^{-}$. The solution is
$\rho=\phi$ (gauge fixing) and

$e^{-2\rho}=\frac{M(x^{+})}{\Lambda}-\Lambda^{2}x^{+}\left(x^{-}+\frac{\mathcal{C}(x^{+})}{\Lambda^{2}}\right),$
(4)

where the functions $M(x^{+})$ and $\mathcal{C}(x^{+})$ are integrals
depending on the stress-energy tensor of the matter field, connected to the
mass of CGHS black hole and the event horizon, respectively. Assume that we
start with a linear dilaton vacuum, then turn on the matter flux injected to
the system at some time $x^{+}_{i}$ and turn it off after the time
$x^{+}_{f}$. Then when $x^{+}>x^{+}_{f}$ the geometry of the system will
approach and finally settle down to the static CGHS black hole background. The
value $M(x^{+}_{f})$ becomes the mass of the black hole and
$\mathcal{C}(x^{+}_{f})$ gives the curve of the event horizon. Therefore one
can observe how the linear dilaton vacuum (before the time $x^{+}_{i}$)
evolves into a CGHS black hole (after the time $x^{+}_{f}$) due to the matter
injection.

##  III CGHS black hole and matching condition

In this section we concentrate on the CGHS black hole solution and some of the
significant observable quantities within this model. The relevant metric for
the CGHS black hole system can be cast in the form Giddings (2016) (c.f.
Schwarzschild gauge Fabbri and Navarro-Salas (2005)),

$ds^{2}=-f(r)dt^{2}+f(r)^{-1}dr^{2},$ (5)

where

$f(r)=1-\frac{M}{\Lambda}e^{-2\Lambda r},$ (6)

with $\Lambda>0$ as the cosmological constant parameterization scale of the
spacetime and $M>0$ as the mass of the CGHS black hole. The curve of the event
horizon function $\mathcal{C}(x^{+})$ is set to zero in Eq. (4) in order to
obtain the metric for stationary CGHS black hole, Eq. (5). Notice that when
$M=\Lambda$, Eq. (5) possesses a singularity at the radial coordinate $r=0$
which also reduces to the event horizon, in contrast to the coordinate
singularity of the Schwarzschild black hole event horizon (see e.g. the
Schwarzschild mirror Good  _et al._ (2016); Good (2017); Anderson  _et al._
(2017); Good  _et al._ (2017b)). For general $M$ and $\Lambda$, the horizon is
at $r_{H}=\frac{1}{2\Lambda}\ln\frac{M}{\Lambda}$. The surface gravity of the
CGHS black hole can be obtained as Derek and Edwin (2009),

$\displaystyle\kappa$
$\displaystyle=\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}r}f(r)\Big{|}_{r=r_{H}}=\Lambda,$
(7)

where $r_{H}=0$, so that consistency with the laws of black hole
thermodynamics dictates the temperature is $T=\Lambda/(2\pi)$. For a double
null coordinate system ($u,v$) with $u=t-r^{\star}$ and $v=t+r^{\star}$, the
associated tortoise coordinate $r^{\star}$ can be obtained in the usual way
Fabbri and Navarro-Salas (2005), via

$\displaystyle r^{\star}$ $\displaystyle=\int\frac{\mathrm{d}r}{f(r)},$ (8)

which yields

$r^{*}=\frac{1}{2\Lambda}\ln\left|\frac{M}{\Lambda}-e^{2\Lambda r}\right|.$
(9)

The absolute brackets are critical for real coordinate values.

Following Wilczek Wilczek (1993), let us coincide the inner and outer regions
of a collapsing null shell to form a black hole, where the exterior background
is given by the CGHS metric,

$ds^{2}=\begin{cases}-dt_{in}^{2}+dr^{2},&\text{for $t_{in}+r\leq v_{0}$},\\\
-fdt_{out}^{2}+f^{-1}dr^{2},&\text{for $t_{out}+r\geq v_{0}$}.\end{cases}$
(10)

and $v_{0}$ is a light-like shell. In null coordinates, the system Eq. (10)
can be rewritten as,

$ds^{2}=\begin{cases}-dUdV,&\text{where $U=t_{in}-r$, $V=t_{in}+r$},\\\
-fdudv,&\text{where $u=t_{out}-r_{*}$, $v=t_{out}+r_{*}$}.\end{cases}$ (11)

So, the metric for the geometry describing the outside region of a collapsing
shell takes the simplified form, $\mathrm{d}s^{2}=-fdudv$.

The matching condition (see Fabbri and Navarro-Salas (2005); Wilczek (1993);
Good  _et al._ (2020a)) with the flat interior geometry, described by the
interior coordinates $(U,V)$ is the trajectory corresponding to $r=0$,
expressed in terms of the exterior function $u(U)$. We can obtain this
matching via the association $r=r^{\star}$, and taking
$r^{\star}(r=(v_{0}-U)/2)=(v_{0}-u)/2$ along the light ray, $v_{0}$. We set
$v_{0}=0$ for simplicity without loss of generality. This matching condition,

$u(U)=-\frac{1}{\Lambda}\ln\left|\frac{M}{\Lambda}-e^{-\Lambda U}\right|,$
(12)

is the outside $u$ trajectory of the origin as a function of the inside
coordinate $U$. We can write this as,

$u(U)=U-\frac{1}{\Lambda}\ln\left(1-\frac{M}{\Lambda}e^{\Lambda U}\right),$
(13)

where $U<0$ and $\Lambda>0$. The regularity condition of the modes requires
that they vanish at $r=0$, which acts as a reflecting boundary in the black
hole system. In the accelerated boundary correspondence (ABC) of the mirror
system, the origin of the black hole functions as the mirror trajectory in
flat spacetime. The position of the origin is a dynamic function $u$ with
independent variable $U$. Since the field vanishes (does not exist for $r<0$),
the form of the field modes can be determined, allowing for the identification
$U\Leftrightarrow v$ (where $v$ is the flat spacetime advanced time in the
moving mirror model) for the Doppler-shifted field modes. In the next section
we will define the analog mirror trajectory for the CGHS spacetime by making
the identification $u(U)\Leftrightarrow f(v)$, which is a known function of
the advanced time $v$.

##  IV Exponentially Accelerated Mirror

In this section we focus on the trajectory and particle flux radiation of the
exponentially accelerated mirror in coordinate time to demonstrate their
equivalence with the corresponding quantities in the CGHS black hole model.

In line with previous accelerated boundary correspondences, consider the
exponentially accelerated mirror trajectory with proper acceleration Good and
Linder (2018):

$\alpha(t)=-\frac{\kappa}{2}e^{\kappa(t-v_{H})},$ (14)

where $\kappa>0$ is a parameter of the acceleration and $v_{H}$ is the horizon
in advanced time, $v=t+x$. This $x$ and $t$ are the usual lab coordinates of
flat (1+1)-dimensional Minkowski spacetime. The trajectory in light cone
coordinates as a function of advanced time is

$f(v)=v-\frac{1}{\kappa}\ln\left(1-e^{\kappa(v-v_{H})}\right),$ (15)

where, identifying Eq. (15) with Eq. (13) as usual (see prior ABCs), the
associated parameters in the CGHS system define the moving mirror’s null-ray
horizon,

$v_{H}=\frac{1}{\Lambda}\ln\left(\frac{\Lambda}{M}\right),$ (16)

which is the location that the last incoming left-moving ray reflects off the
mirror. Past this position, there is no more reflection and left-mover modes
never make it to an observer at right null-infinity, $\mathscr{I}^{+}_{R}$.
The mirror horizon couples the parameters $\Lambda$ and $M$, which are the
cosmological constant and mass of the black hole, respectively, in the CGHS
system. The fact that $v_{H}$, which is the finite $v$ for the mirror horizon,
is also closely related to the CGHS black hole horizon, through
$2r_{H}=-v_{H}$, further corroborates a correspondence between the CGHS black
hole and the exponentially accelerated mirror. A spacetime plot of this
asymptotic light-like moving mirror is given in Fig. 1. A Penrose conformal
diagram is given in Fig. 2. Notice when $v_{H}=0$, Eq. (15) is Eq. (13), i.e.
$u(U)\Leftrightarrow f(v)$, when $\Lambda=M$.

It is interesting to note that in Schwarzschild black hole case the
singularity is located at the center, that corresponds to the mirror, and the
event horizon is located at $r=2M$. Unlike the Schwarzschild black hole, for
the CGHS black hole the singularity happens at $r=0$ which is the location of
event horizon as well when $\Lambda=M$. So, the mirror mimics both the event
horizon and the center of the black hole, simplifying physical interpretation
and giving a straightforward answer to the origin of particle creation in the
CGHS system.

Now we will derive the thermal Planck distribution of the exponentially
accelerated moving mirror particles by use of the beta Bogolubov coefficient.
The beta coefficient can be found via an integration Good  _et al._ (2017a) by
parts where we ignore non-contributing surface terms,

$\beta_{\omega\omega^{\prime}}=\frac{1}{2\pi}\sqrt{\frac{\omega^{\prime}}{\omega}}\int_{-\infty}^{v_{H}=0}dv\;e^{-i\omega^{\prime}v}e^{-i\omega
f(v)},$ (17)

to get

$\beta_{\omega\omega^{\prime}}=\frac{1}{2\pi\kappa}\sqrt{\frac{\omega^{\prime}}{\omega}}B\left[-\frac{i\omega_{+}}{\kappa},1+\frac{i\omega}{\kappa}\right],$
(18)

where we utilize the Euler integral of the first kind as a Beta function,
$B(a,b)=\frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}$, and
$\omega_{+}=\omega+\omega^{\prime}$. Multiplying by its complex conjugate
gives the particle count per $\omega^{\prime}$ mode, per $\omega$ mode:

$|\beta_{\omega\omega^{\prime}}|^{2}=\frac{1}{4\pi^{2}\kappa^{2}}\frac{\omega^{\prime}}{\omega}\left|B\left[\frac{i\omega_{+}}{\kappa},1-\frac{i\omega}{\kappa}\right]\right|^{2},$
(19)

or, equivalently expressed,

$|\beta_{\omega\omega^{\prime}}|^{2}=\frac{e^{\frac{2\pi\omega}{\kappa}}\left(e^{\frac{2\pi\omega^{\prime}}{\kappa}}-1\right)}{2\pi\kappa\omega_{+}\left(e^{\frac{2\pi\omega}{\kappa}}-1\right)\left(e^{\frac{2\pi\omega_{+}}{\kappa}}-1\right)}.$
(20)

Thermal character results in the high frequency limit
$\omega^{\prime}\gg\omega$ approximation (a good explanation for how this
corresponds to late times is given by Hawking Hawking (1975)). We can see by
inspection that,

$|\beta_{\omega\omega^{\prime}}|^{2}\approx\frac{1}{2\pi\kappa\omega^{\prime}}\frac{1}{e^{2\pi\omega/\kappa}-1}\quad\text{for}\;\omega^{\prime}\gg\omega$
(21)

so that $T=\kappa/(2\pi)$ at late times.

Figure 1: Spacetime diagram with $M=\Lambda=2$, of Eq. (15), $f(v)$. The
mirror starts asymptotically timelike and finishes asymptotically lightlike
with infinite acceleration along the horizon $v_{H}=0$.  Figure 2: Penrose
diagram with Blue, Green, Red, Black, parametrized respectively with
$\kappa=1/2,1,2,4$ of Eq. (15), $f(v)$. As in Figure 1 (the Red line is the
same trajectory) the mirror starts asymptotically timelike and finishes
asymptotically lightlike with infinite acceleration along the horizon
$v_{H}=0$.

##  V CGHS particle radiation

In this section we calculate the particle flux radiated by the CGHS black hole
and the corresponding beta Bogolubov coefficients, following the standard
procedure (see e.g. Fabbri and Navarro-Salas (2005)). Remarkably, the beta
Bogolubov coefficients match those corresponding ones obtained from the
particle radiation of the exponentially accelerating mirror (Section IV).

The standard Bogolubov procedure for calculating the Hawking radiation
considers two relevant regions as mentioned in Sec. II: the linear dilaton
vacuum and the CGHS black hole, described by the corresponding “in” and “out”
coordinates which can be connected via the Kruskal coordinates (see Fabbri and
Navarro-Salas (2005) for details and elaboration on the standard notation).
One then identifies plane wave modes for ingoing and outgoing sectors using
null Minkowski coordinates $(\sigma_{in}^{\pm},\sigma_{out}^{\pm})$. They are
related as,

$\sigma_{in}^{+}=\sigma_{out}^{+}=\sigma^{+},$ (22)
$\sigma_{in}^{-}=-\frac{1}{\Lambda}\ln\left(e^{-\Lambda\sigma_{out}^{-}}+\frac{\mathcal{C}(x_{f}^{+})}{\Lambda}\right).$
(23)

So, using $\sigma^{-}$ sector the plane wave modes have the following forms,

$g_{\omega^{\prime}}^{in}=\frac{1}{\sqrt{4\pi\omega^{\prime}}}e^{-i\omega^{\prime}\sigma_{in}^{-}},$
(24)
$g_{\omega}^{out}=\frac{1}{\sqrt{4\pi\omega}}e^{-i\omega\sigma_{out}^{-}},$
(25)

designated by $g$ (sometimes $u$ is used but here $u$ is already a retarded
time null coordinate). The next step is to evaluate the beta Bogolubov
coefficients by calculating scalar product between the plane wave modes. For
our particular case the corresponding integral is,

$\beta_{\omega\omega^{\prime}}=(g_{\omega}^{out},g_{\omega^{\prime}}^{in*})=2i\int_{-\infty}^{+\infty}d\sigma_{in}^{-}g_{\omega}^{out}\frac{\partial
g_{\omega^{\prime}}^{in}}{\partial\sigma_{in}^{-}}.$ (26)

Substituting Eq. (24) and Eq. (25) into Eq. (26), and also using Eq. (23), the
above integral becomes,

$\beta_{\omega\omega^{\prime}}=\frac{1}{2\pi}\sqrt{\frac{\omega^{\prime}}{\omega}}\int_{-\infty}^{\sigma_{in,H}^{-}}d\sigma_{in}^{-}\frac{e^{\frac{i\omega}{\Lambda}\ln\left(e^{-\Lambda\sigma_{in}^{-}-\frac{\mathcal{C}(x_{f}^{+})}{\Lambda}}\right)}}{e^{i\omega^{\prime}\sigma_{in}^{-}}},$
(27)

where

$\sigma_{in,H}^{-}=\frac{1}{\Lambda}\ln\frac{\Lambda}{\mathcal{C}(x_{f}^{+})}$
(28)

is the black hole event horizon location that is formed when injecting matter
into linear dilaton vacuum. Note that Eq. (28) is similar to Eq. (16), i.e.
$v_{H}\Leftrightarrow\sigma_{in,H}^{-}$. Calculation of Eq. (27) gives,

$\beta_{\omega\omega^{\prime}}=\frac{1}{2\pi\Lambda}\sqrt{\frac{\omega^{\prime}}{\omega}}\left(\frac{\mathcal{C}(x_{f}^{+})}{\Lambda}\right)^{\frac{i\omega_{+}}{\Lambda}}B\left[-\frac{i\omega_{+}}{\Lambda},1+\frac{i\omega}{\Lambda}\right].$
(29)

The complex conjugate squared of Eq. (29) yields,

$|\beta_{\omega\omega^{\prime}}|^{2}=\frac{1}{4\pi^{2}\Lambda^{2}}\frac{\omega^{\prime}}{\omega}\left|B\left[\frac{i\omega_{+}}{\Lambda},1-\frac{i\omega}{\Lambda}\right]\right|^{2},$
(30)

which is exactly Eq. (19) given that $|\Lambda|=|\kappa|$. Interestingly, the
mass of the CGHS black hole, $M(x^{+}_{f})$, does not appear in
$\beta_{\omega\omega^{\prime}}$, and $\mathcal{C}(x^{+}_{f})$ disappears when
calculating $|\beta_{\omega\omega^{\prime}}|^{2}$. This is because the
spectrum of the CGHS black hole does not explicitly depend on its mass. The
correspondence between the black hole mass and the curve of the event horizon,
$\mathcal{C}(x_{f}^{+})$, can be seen from the comparison of Eq. (16) and Eq.
(28). The curve of the event horizon is defined by the so-called “apparent
horizon”, $x^{-}=-\mathcal{C}(x^{+})/\Lambda^{2}$, which is spacelike or null
and coincides with the event horizon after the matter injection has finished
at time $x^{+}_{f}$, i.e. after time $x^{+}_{f}$ when the geometry of the CGHS
black hole is settled and the apparent horizon becomes the event horizon.

Hereinafter, due to the identical particle production Eq. (19) between the
exponentially accelerated mirror in coordinate laboratory time and the CGHS
system Eq. (30), for short, we refer to this specific perfectly reflecting
boundary trajectory, Eq. (15), as the CGHS mirror.

##  VI Quantum stress-tensor

In this section we will briefly review the quantum stress tensor of the moving
mirror following Davies and Fulling Fulling and Davies (1976); Davies and
Fulling (1977). We will need this to specialize to the CGHS mirror and compute
its energy flux which we do in Section IX. In (1+1)-dimensional flat spacetime
the energy-momentum tensor is determined by the following $2\times 2$ matrix,

$T_{\mu\nu}=\frac{1}{2}\begin{bmatrix}\left(\frac{\partial\phi}{\partial
t}\right)^{2}+\left(\frac{\partial\phi}{\partial
x}\right)^{2}&\frac{\partial\phi}{\partial x}\frac{\partial\phi}{\partial
t}+\frac{\partial\phi}{\partial t}\frac{\partial\phi}{\partial x}\\\
\frac{\partial\phi}{\partial t}\frac{\partial\phi}{\partial
x}+\frac{\partial\phi}{\partial x}\frac{\partial\phi}{\partial
t}&\left(\frac{\partial\phi}{\partial
t}\right)^{2}+\left(\frac{\partial\phi}{\partial x}\right)^{2}\end{bmatrix}.$
(31)

Here $\phi(t,x)$ is technically a free field that obeys the massless scalar
wave equation,

$\frac{\partial^{2}\phi}{\partial t^{2}}-\frac{\partial^{2}\phi}{\partial
x^{2}}=0,$ (32)

but there are effective interactions with the boundary condition,

$\phi(t,x)|_{z}=0,$ (33)

which is imposed on the field equation of motion, where $x=z(t)$ is the
trajectory of the moving mirror. In this two dimensional case, the mirror is a
perfectly reflecting point moving along a timelike worldline $z(t)$.

In the usual quantum field theory, $\phi(t,x)$ is an operator defined by field
modes as,

$\phi(t,x)=\int_{0}^{\infty}\left[\hat{a}_{\omega}^{in}\phi_{\omega}+\hat{a}_{\omega}^{in\dagger}\phi_{\omega}^{*}\right]d\omega,$
(34)

where $\hat{a}_{\omega}$ and $\hat{a}_{\omega}^{\dagger}$ are ladder
operators, and $\phi_{\omega}^{*}$ is a complex conjugate of $\phi$. After
inserting Eq. (34) into the matrix Eq. (31), the stress-tensor can be written
as,

$T_{\mu\nu}=:T_{\mu\nu}:+\braket{T_{\mu\nu}},$ (35)

where the first term has normal ordering of the ladder operators, i.e.

$:\hat{a}_{\omega}^{in}\hat{a}_{\omega}^{in\dagger}:=\hat{a}_{\omega}^{in\dagger}\hat{a}_{\omega}^{in}.$
(36)

The second term in Eq. (35) is the expectation value of the operator in
vacuum, which is defined as,

$\braket{T_{\mu\nu}}=\int_{0}^{\infty}T_{\mu\nu}(\phi_{\omega},\phi_{\omega}^{*})d\omega.$
(37)

The stress tensor here, as it stands, is of significant interest, but
unfortunately the above integral is divergent. In order to make the integral
in Eq. (37) finite and extract useful information about the emitted radiation,
point-splitting regularization is utilized. The key idea is to evaluate the
field modes at different times: $\phi$ at $(t,x)$ and $\phi^{*}$ at
$(t+\epsilon,x)$, respectively, where $\epsilon$ is an infinitesimally small
quantity. So, the field modes at corresponding points are:

$\begin{cases}\begin{matrix}\frac{\partial\phi_{\omega}}{\partial t}\\\
\frac{\partial\phi_{\omega}}{\partial
x}\end{matrix}\end{cases}=\sqrt{\frac{\omega}{4\pi}}\left[e^{-i\omega v}\mp
p^{\prime}(u)e^{-i\omega p(u)}\right],$ (38)
$\begin{cases}\begin{matrix}\frac{\partial\phi_{\omega}^{*}}{\partial t}\\\
\frac{\partial\phi_{\omega}^{*}}{\partial
x}\end{matrix}\end{cases}=\sqrt{\frac{\omega}{4\pi}}\left[e^{i\omega(v+\epsilon)}\mp
p^{\prime}(u+\epsilon)e^{i\omega p(u+\epsilon)}\right],$ (39)

where $p(u)\equiv 2t(u)-u$ and $u\equiv t-z(t)$. Inserting Eqs. (38) and (39)
into Eq. (37) leads to,

$\begin{cases}\begin{matrix}\braket{T_{00}}=\braket{T_{11}}\\\
\braket{T_{10}}=\braket{T_{01}}\end{matrix}=\frac{1}{4\pi}\int_{0}^{\infty}\omega\left[e^{i\omega\epsilon}\pm\frac{p^{\prime}(u)p^{\prime}(u+\epsilon)}{e^{-i\omega(p(u+\epsilon)-p(u))}}\right]d\omega.\end{cases}$
(40)

Calculation of the above integrals results in,

$\begin{cases}\begin{matrix}\braket{T_{00}}\\\
\braket{T_{01}}\end{matrix}=-\frac{1}{4\pi\epsilon^{2}}\mp\frac{1}{4\pi}\frac{p^{\prime}(u)p^{\prime}(u+\epsilon)}{[p(u)-p(u+\epsilon)]^{2}}.\end{cases}$
(41)

Consequently,

$\braket{T_{00}}=-\frac{1}{2\pi\epsilon^{2}}-\braket{T_{01}}.$ (42)

The result for $\braket{T_{01}}$ in Eq. (41) is Taylor expanded in $\epsilon$
to give,

$\braket{T_{01}}=\frac{1}{24\pi}\left[\frac{p^{\prime\prime\prime}}{p^{\prime}}-\frac{3}{2}\left(\frac{p^{\prime\prime}}{p^{\prime}}\right)^{2}\right]+O(\epsilon).$
(43)

In the limit $\epsilon\rightarrow 0$, the first term in Eq. (42) becomes
divergent. Since the second term is independent of $\epsilon$, it does not
vanish like the higher order terms. This second term is of particular physical
interest, corresponding to the energy flux radiated by the mirror,

$\braket{T_{00}}=\mathcal{F}(u)=-\frac{1}{24\pi}\left[\frac{p^{\prime\prime\prime}}{p^{\prime}}-\frac{3}{2}\left(\frac{p^{\prime\prime}}{p^{\prime}}\right)^{2}\right].$
(44)

The first component of the renormalized stress tensor expectation value gives
energy flux radiated by the mirror into the vacuum, characterizing the
amplified quantum fluctuations due to the presence of the accelerating
boundary.

Using the relations between spacetime $(t,x)$ and null $(u,v)$ coordinates as,

$u\equiv t-z(t),~{}~{}~{}\quad v\equiv t+z(t),$ (45)

and

$p(u)=2t(u)-u,~{}~{}~{}\quad f(v)=2t(v)-v,$ (46)

the energy flux can also be expressed with care, using straightforward
differential algebra,

$\mathcal{F}(t)=\dfrac{\dddot{z}(\dot{z}^{2}-1)-3\dot{z}\ddot{z}^{2}}{12\pi(\dot{z}+1)^{2}(\dot{z}-1)^{4}},$
(47) $\mathcal{F}(x)=\dfrac{t^{\prime\prime\prime}(t^{\prime
2}-1)-3t^{\prime}t^{\prime\prime
2}}{12\pi(t^{\prime}+1)^{2}(t^{\prime}-1)^{4}},$ (48)
$\mathcal{F}(v)=\dfrac{1}{24\pi}\bigg{[}\dfrac{f^{\prime\prime\prime}}{f^{\prime}}-\dfrac{3}{2}\bigg{(}\dfrac{f^{\prime\prime}}{f^{\prime}}\bigg{)}^{2}\bigg{]}\dfrac{1}{f^{\prime
2}},$ (49)

where dots and primes denote derivatives with respect to arguments $t$, $x$
and $v$, respectively. Eq. (48) will be used to find the flux of the CGHS
moving mirror in Sec. IX.

##  VII Mirror Entanglement Entropy

In this section we will review the derivation of (1+1)-dimensional
entanglement (geometric) entropy in conformal field theory (CFT) and its
connection to the rapidity of the moving mirror (see other derivations e.g.
Good (2020b); Fitkevich  _et al._ (2020)).

Consider the entropy of a system in (1+1)-D CFT Holzhey  _et al._ (1994),

$S=\frac{1}{6}\ln\frac{L}{\epsilon},$ (50)

where $L$ is the size of the system in general (and in our case it is the
mirror trajectory which measures the size of the system by the spacetime
traversed accessible to the quantum field), and $\epsilon$ is a UV cut-off.

For a general arbitrary moving mirror,

$L\equiv p(u)-p(u_{0}),$ (51)

where $u$ and $u_{0}$ are null coordinates that form the region in the system
which we are considering, and $\epsilon$ is asymmetrically smeared, i.e.
$\epsilon^{2}\equiv\epsilon_{p}\epsilon_{p_{0}}$. Here $p(u)$ is the
trajectory of the mirror in null coordinates (it is a function of retarded
time $u$). The smearing and dynamics of the mirror are related as,

$\epsilon_{p}=p^{\prime}(u)\epsilon_{u},\qquad\epsilon_{p_{0}}=p^{\prime}(u_{0})\epsilon_{u_{0}}.$
(52)

Substituting Eqs. (52) into Eq. (50) yields the bare entropy of the system,

$S_{bare}=\frac{1}{12}\ln\frac{\left[p(u)-p(u_{0})\right]^{2}}{p^{\prime}(u)p^{\prime}(u_{0})\epsilon_{u}\epsilon_{u_{0}}}.$
(53)

The vacuum entropy of the system can be found by considering a static mirror
where $L=u-u_{0}$ and $\epsilon^{2}=\epsilon_{u}\epsilon_{u_{0}}$. Thus,

$S_{vac}=\frac{1}{12}\ln\frac{(u-u_{0})^{2}}{\epsilon_{u}\epsilon_{u_{0}}}.$
(54)

Even though the entropies above are defined in terms of smearing, this
dependence can be removed by an intuitive renormalization via,

$S_{ren}=S_{bare}-S_{vac}=\frac{1}{12}\ln\frac{[p(u)-p(u_{0})]^{2}}{p^{\prime}(u)p^{\prime}(u_{0})(u-u_{0})^{2}}.$
(55)

Further simplification proceeds by a Taylor expansion of our arbitrary
function $p(u)$ around $u=u_{0}$ up to first order, that is,

$p(u)=p(u_{0})+p^{\prime}(u_{0})(u-u_{0})+\mathcal{O}(u-u_{0})^{2}.$ (56)

Substituting Eq. (56) into Eq. (55) brings us to,

$S_{ren}=\frac{1}{12}\ln\frac{p^{\prime}(u_{0})}{p^{\prime}(u)}.$ (57)

Moreover, for a static mirror $p(u)=u$ and $p(u_{0})=u_{0}$, therefore
$p^{\prime}(u_{0})=1$. As a result, Eq. (57) reduces to $S_{ren}\rightarrow
S(u)$ where,

$S(u)=-\frac{1}{12}\ln p^{\prime}(u).$ (58)

Eq. (58) is valid for any moving mirror that starts asymptotically static
(zero velocity). Notable exceptions are the eternally thermal Carlitz-Willey
mirror Carlitz and Willey (1987) and the eternally uniformly accelerated
mirror Birrell and Davies (1984); however, most of the solved mirrors in the
literature, by construction, do start static as they are often used to model
gravitational collapse. The CGHS mirror is no exception.

Eq. (58) is more intuitively written in spacetime coordinates using the
relation between null and spacetime trajectories of the mirror as,

$p^{\prime}(u)=\frac{1+\dot{z}(t)}{1-\dot{z}(t)}.$ (59)

Applying this relation into the Eq. (58) yields,

$S(t)=-\frac{1}{6}\tanh^{-1}[\dot{z}(t)]=-\frac{1}{6}\eta(t),$ (60)

where $\eta(t)\equiv\tanh^{-1}[\dot{z}(t)]$ is the time-dependent rapidity. It
is simple to see that the magnitude of the entropy increases as the mirror
moves faster. Unitarity in this context Bianchi and Smerlak (2014a) strictly
requires that the entropy must achieve a constant value in the far past and
far future.

This von Neumann entropy measure of the degree of quantum entanglement between
the two subsystems (past & future) constitutes a two-part composite quantum
system. It explicitly reveals the connection of information of entanglement to
the dynamics (rapidity) of the moving mirror system. Allow us to speculate a
thermodynamic treatment of the system, and the corresponding macroscopic state
of the entanglement is characterized by a distribution of its microstates,
then it may be appropriate to coincide the Boltzmann entropy with the von
Neumann entropy. In this conjectural case, a discrete speed based on the basic
smallest unit of an operable binary digit of information results.111This
discreteness necessarily leads to a smallest non-zero speed for the moving
mirror. In SI units, this is $17.2\;\textrm{fm/s}$, from $v=c\tanh(6k_{B}\ln
2)$ which is about the diameter of a gold nucleus in one second, or about 5
centimeters in 100,000 years.

In the next section we will ultimately apply the above entanglement-rapidity
relationship, Eq. (60), which can be expressed independently of coordinates or
its argument, $-6S=\eta$, to gain insight into the self-force and Larmor power
by reformulating them in terms of entropy.

##  VIII Mirror Larmor formula and LAD force

###  VIII.1 Quantum relativistic Larmor formula

In this subsection we derive the quantum relativistic power radiated by the
moving mirror and find it has the same form as the classical relativistic
Larmor formula of electrodynamics. We account for the power radiated to both
sides of the moving mirror, utilizing the quantum stress tensor derived in
Section VI.

Let us consider the total energy radiated as derived from the Davies-Fulling
quantum stress tensor, to the right side of the mirror, expressed as Equation
(2.34) of Good  _et al._ (2013),

$E^{R}=\frac{1}{12\pi}\int_{-\infty}^{\infty}\alpha^{2}(1+\dot{z})\,\operatorname{d}\\!{t}.$
(61)

An observer at $\mathscr{I}^{+}_{R}$ measures $E^{R}$ energy emitted, but the
mirror also radiates energy to $\mathscr{I}^{+}_{L}$ for an observer on the
left. The energy radiated to the left, $E^{L}$, is found by the same
expression but with a parity flip, $\dot{z}\rightarrow-\dot{z}$, so that the
total radiated energy is,

$E=E^{R}+E^{L}=\frac{1}{6\pi}\int_{-\infty}^{\infty}\alpha^{2}\,\operatorname{d}\\!{t}.$
(62)

We define the quantity, $E$, without an average giving us a measure of the
total radiation, independent of observer. This allows us to identify and
define a quantum relativistic Larmor power analog for the moving mirror,
$P=\operatorname{d}\\!{E}/\operatorname{d}\\!{t}$,

$E=\int_{-\infty}^{\infty}\frac{dE}{dt}\operatorname{d}\\!{t}\equiv\int_{-\infty}^{\infty}P\operatorname{d}\\!{t},$
(63)

which gives the familiar relativistic Larmor scaling for proper acceleration:

$P=\frac{\alpha^{2}}{6\pi}.$ (64)

The quantum power radiated by the mirror takes the same form as that of a
classical point charge in electrodynamics (see Zhakenuly  _et al._ (2020) for
the derivation and distribution in (3+1) dimensions). Recall here that
$\alpha$ is the scalar invariant which is defined as the proper time
derivative of rapidity, $\alpha=\eta^{\prime}(\tau)$, even though the integral
in which we define the power with respect to uses ordinary coordinate time.
Eq. (64) is in harmony with Ford-Vilenkin Ford and Vilenkin (1982) who found
that the self-force of the moving mirror also has the same form as radiation-
reaction of a point charge in classical electrodynamics.

###  VIII.2 Relativistic entanglement-power

In this subsection we apply the entanglement-rapidity relationship, Eq. (60),
to the power derived in the previous subsection, Eq. (64). Motivated to
understand the radiated emission in terms of information flow, we find the
quantum power is expressed as the square of the first derivative of the
entanglement.

The covariant Larmor power $P$ is a Lorentz scalar invariant since it is
proportional to the square of the proper acceleration,

$P=\frac{\alpha^{2}}{6\pi}=\frac{\eta^{\prime}(\tau)^{2}}{6\pi}.$ (65)

The rapidity-entanglement relationship, given by $\eta=-6S$ (see also Section
VII and further references Good  _et al._ (2017a); Bianchi and Smerlak
(2014b); Good (2020a)), can be expressed with independent variable as proper
time, $\eta^{\prime}(\tau)=-6S^{\prime}(\tau)$, and thus the power is
formulated as,

$P=\frac{6}{\pi}S^{\prime}(\tau)^{2}.$ (66)

This entanglement-power relationship characterizes one-dimensional
transmission of entropy or information for non-thermal radiation. At thermal
equilibrium, this can be compared to Pendry’s maximum entropy rate for power,
$\dot{S}=\sqrt{\pi P/3}$, also called the noiseless quantum channel capacity,
investigated by Bekenstein-Mayo in the context of black holes information flow
as (1+1)-dimensional Bekenstein and Mayo (2001). The factor of 2 is accounted
for by a uni-directional flow to a single observer, conventionally taken to be
situated at $\mathscr{I}^{+}_{R}$ future null right infinity. The result Eq.
(66) compliments the celebrated Bianchi-Smerlak formula Bianchi and Smerlak
(2014b) revealing the energy-entanglement connection,

$2\pi\mathcal{F}(u)=6S^{\prime}(u)^{2}+S^{\prime\prime}(u),$ (67)

which implies that the outgoing flux is completely determined by the structure
of entanglement at future null infinity, and vice versa, the entanglement
entropy is completely determined by the flux through the second order
differential equation from the values of $S(u)$ and $S^{\prime}(u)$ at a
single point at $\mathscr{I}^{+}_{R}$. Eq. (67) involves derivatives with
respect to retarded time $u$ rather than proper time $\tau$ as is the case in
Eq. (66).

###  VIII.3 Averaging Radiation Reaction

Turning our attention to the self-force in this subsection, we derive the
formula for radiation reaction from our previously derived mirror Larmor
power, Eq. (64), using an average over proper time. Our results in this
subsection confirm those of Ford-Vilenkin Ford and Vilenkin (1982), building
confidence in the overall theme by connecting the self-force to the result for
Larmor power.

Energy lost by radiation for an accelerating point charge tends to slow it
down. This is because there is a backreaction of the radiation on the particle
itself. In the case of a moving mirror this is in practise not so, because the
trajectory is usually just assumed apriori. Allow us to try to find this
radiation reaction on the mirror without assuming locally any trajectory and
by use of an averaging over proper time. Starting with the Larmor mirror
power, Eq. (64), expressed in rapidity Eq. (65),

$P=\frac{\eta^{\prime}(\tau)^{2}}{6\pi},$ (68)

the reaction must be, averaged over proper time, the work done on the mirror
equal to the negative of the energy lost to the vacuum radiation:

$\overline{F\eta}=-\frac{\overline{\eta^{\prime}(\tau)^{2}}}{6\pi}.$ (69)

We are still in natural units, $c=1$, and so $\overline{F\eta}$ is an average
radiation reaction power linearly and proportionally dependent on the
rapidity, $\eta$, of the mirror. Writing the proper acceleration as,

$\eta^{\prime}(\tau)^{2}=\frac{d}{d\tau}(\eta\eta^{\prime})-\eta\eta^{\prime\prime},$
(70)

where the total derivative with respect to proper time vanishes due to our
averaging procedure (or in the case of assuming global asymptotic inertial
trajectory222This is also equivalently accomplished by assuming periodicity.),
we then have

$\overline{F\eta}=+\frac{\overline{\eta\eta^{\prime\prime}(\tau)}}{6\pi}.$
(71)

This allows us to identify the radiation reaction force as,

$F=\frac{\eta^{\prime\prime}(\tau)}{6\pi}.$ (72)

We will show that this result is in agreement with the magnitude, Eq. (88), of
the covariant LAD 4-force Eq. (78), derived using the Davies-Fulling stress
tensor, in the next subsection.

###  VIII.4 Confirmation of LAD magnitude

A rigorous non-averaging derivation of Eq. (72) is accomplished by
relativistic covariance. In this subsection we derive the mirror LAD force
using electromagnetic 4-vector formulation and quantities known in special
relativity, e.g. proper acceleration, celerity and rapidity.

Before turning to the derivation of the force, let us briefly review some
known formulations. The point charge in SI units to moving mirror natural
units ($\mu_{0}=\epsilon_{0}=1$) has a coupling which can be expressed by the
substitution:

$\frac{2}{3}\left(\frac{q^{2}}{4\pi\epsilon_{0}c^{3}}\right)=\frac{q^{2}}{6\pi\epsilon_{0}c^{3}}=\frac{\mu_{0}q^{2}}{6\pi
c}\quad\Rightarrow\quad\frac{1}{6\pi}.$ (73)

Notice Gaussian units are $4\pi\epsilon_{0}=1$ and $\mu_{0}=4\pi$. We will
need the proper acceleration, $\alpha$, that is a Lorentz invariant, defined
by:

$\alpha^{2}\equiv-\frac{d^{2}x^{\mu}}{d\tau^{2}}\frac{d^{2}x_{\mu}}{d\tau^{2}}.$
(74)

It will also be helpful to have the velocity $v$, the Lorentz factor $\gamma$,
and the celerity $w$, which are defined through the rapidity $\eta$,

$v=\tanh\eta,\quad\gamma=\cosh\eta,\quad w=\sinh\eta,$ (75)

or similarly,

$v=\frac{dx}{dt},\qquad\gamma=\frac{dt}{d\tau},\qquad w=\frac{dx}{d\tau}.$
(76)

The self-force scalar invariant, $F$, can be written via 4-vectors as,

$F^{2}\equiv-F^{\mu}F_{\mu},$ (77)

which will be the final object that we obtain in this subsection.

Let us now confirm the mirror self-force by substituting Eqs. (73)-(77) into
the radiative force introduced by Ford-Vilenkin Ford and Vilenkin (1982),

$6\pi
F^{\mu}=\frac{d^{3}x^{\mu}}{d\tau^{3}}-\alpha^{2}\frac{dx^{\mu}}{d\tau}.$ (78)

Here the 4-vector has time $F^{0}=F^{t}$ and space $F^{1}=F^{x}$ components
respectively for $6\pi F^{\mu}$:

$\gamma^{\prime\prime}(\tau)-\alpha^{2}\gamma=\alpha^{\prime}(\tau)w,$ (79)
$w^{\prime\prime}(\tau)-\alpha^{2}w=\alpha^{\prime}(\tau)\gamma,$ (80)

where $\alpha=\eta^{\prime}(\tau)$. So, the time and space components are
expressed in rapidity as:

$6\pi F^{t}=\eta^{\prime\prime}(\tau)\sinh\eta,$ (81) $6\pi
F^{x}=\eta^{\prime\prime}(\tau)\cosh\eta.$ (82)

With signature $(+,-,-,-)$ or just $(+,-)$ for our (1+1)-dimensional context,
the magnitude can be found by,

$\displaystyle F$ $\displaystyle=$ $\displaystyle\sqrt{-F^{\mu}F_{\mu}}$ (83)
$\displaystyle=$ $\displaystyle\sqrt{-(|F^{0}|^{2}-|F^{1}|^{2})}$ (84)
$\displaystyle=$ $\displaystyle\sqrt{|F^{x}|^{2}-|F^{t}|^{2}}$ (85)
$\displaystyle=$
$\displaystyle\frac{\eta^{\prime\prime}}{6\pi}\sqrt{\cosh^{2}\eta-\sinh^{2}\eta}$
(86) $\displaystyle=$ $\displaystyle\frac{1}{6\pi}\eta^{\prime\prime}(\tau),$
(87)

giving us the simple relationship for jerk,

$\alpha^{\prime}(\tau)=6\pi F,$ (88)

which coincides with the Eq. (72) taking into account the change of rapidity
with respect to proper time, $\alpha=\eta^{\prime}(\tau)$.

###  VIII.5 Derivation of LAD formula

Let us now move to the derivation of the self-force from the moving mirror
point of view explicitly. Consider the total energy-momentum emitted to the
right of the mirror,

$E^{R}=\int_{-\infty}^{\infty}\mathcal{F}^{R}\operatorname{d}\\!{u},$ (89)

where the Schwarzian derivative defines the quantum stress tensor Fulling and
Davies (1976),

$\mathcal{F}^{R}(u)=-\frac{1}{24\pi}\\{p(u),u\\},$ (90)

which we convert to proper time Good and Linder (2018),

$\mathcal{F}^{R}(\tau)=-\frac{1}{12\pi}\eta^{\prime\prime}(\tau)e^{+2\eta(\tau)},$
(91)

so that with Jacobian $du=e^{-\eta}d\tau$,

$\displaystyle E^{R}$ $\displaystyle=$
$\displaystyle-\frac{1}{12\pi}\int_{-\infty}^{\infty}\eta^{\prime\prime}e^{+2\eta}(e^{-\eta}\operatorname{d}\\!{\tau}),$
(92) $\displaystyle=$
$\displaystyle-\frac{1}{12\pi}\int_{-\infty}^{\infty}\eta^{\prime\prime}e^{+\eta}\operatorname{d}\\!{\tau}.$
(93)

Similarly, with a parity flip, the energy-momentum emitted to left of the
mirror is,

$E^{L}=+\frac{1}{12\pi}\int_{-\infty}^{\infty}\eta^{\prime\prime}e^{-\eta}\operatorname{d}\\!{\tau}.$
(94)

We are looking for the difference in energy-momentum between the left and
right sides of the mirror to construct the 4-vector radiation reaction self-
force. The time component, $F^{t}=Fw$, is constructed from the energy,

$\Delta U=\int F\operatorname{d}\\!{x}=\int Fv\operatorname{d}\\!{t}=\int
Fv\gamma\operatorname{d}\\!{\tau}=\int Fw\operatorname{d}\\!{\tau}.$ (95)

On the other hand, the energy is defined as,

$\Delta
U=U^{L}-U^{R}=\int_{-\infty}^{\infty}\frac{dU}{d\tau}\operatorname{d}\\!{\tau}=\int_{-\infty}^{\infty}F^{t}\operatorname{d}\\!{\tau},$
(96)

where

$F^{t}=\frac{dU}{d\tau}=\frac{dU}{dx}\frac{dx}{d\tau}=F\sinh\eta=Fw.$ (97)

The space component of the force, $F^{x}=F\gamma$, is constructed from the
momentum,

$\Delta\mathcal{P}=\int F\operatorname{d}\\!{t}=\int
F\gamma\operatorname{d}\\!{\tau}.$ (98)

The difference in momentum radiated between the two sides is expressed as,

$\Delta\mathcal{P}=\mathcal{P}^{L}-\mathcal{P}^{R}=\int_{-\infty}^{\infty}\frac{d\mathcal{P}}{d\tau}\operatorname{d}\\!{\tau}=\int_{-\infty}^{\infty}F^{x}\operatorname{d}\\!{\tau},$
(99)

which defines the space component piece of the radiation reaction on the
mirror. This is explicitly,

$F^{x}=\frac{\eta^{\prime\prime}}{12\pi}[e^{-\eta}-(-e^{+\eta})]=\frac{\eta^{\prime\prime}}{6\pi}\cosh\eta=F\gamma,$
(100)

where one can already see that $F=\eta^{\prime\prime}/6\pi$. The radiation
reaction force,

$F^{\mu}=(F^{t},F^{x})=\gamma(Fv,F)=(wF,\gamma F),$ (101)

or, equivalently written in covariant notation as in Eq. (78), has a Lorentz
scalar invariant jerk,

$F=\frac{\eta^{\prime\prime}(\tau)}{6\pi}.$ (102)

While our derivation in this subsection for the radiation-reaction force
ostensibly relies on the expression for finite conserved energy, the
integration is not explicitly taken. For finite energy one requires the
acceleration to vanish asymptotically. It is safe to assume the self-force
does not actually require this constraint. This is congruent with the usual
LAD expression in electrodynamics which holds even in the mathematical case
where a charged point is accelerated in both asymptotic limits.

We have derived the moving mirror LAD formula for the radiation reaction using
conservation of energy (the difference in energy-momentum between the right
and left sides of the mirror), but made no effort to identify the mechanism
responsible for the force. In the case of a point charge, one imagines the
force as the recoil effect of the particle’s own field acting back on the
charge, but in the case of a mirror we see it is not the source of a field
whatsoever.

In the electromagnetic case, one has the problem of the field blowing up right
at the point charge. But in the mirror model we know the field is identically
zero at the mirror. So what then, is the mechanism? The accepted answer in the
case of charge is that an extended charge distribution divided into
infinitesimal pieces gives rise to a net force of the charge on itself - the
self-force - as a consequence of the breakdown of Newton’s third law within
the structure of the particle. Perhaps a distributed boundary condition
calculation could also give rise to a self-force on the mirror. In other
words, the net force exerted by the scalar field generated by different pieces
of the distributed boundary condition acts on each other to produce a mirror
self-force. Such a calculation is beyond the scope of this paper.

###  VIII.6 Entanglement & Radiative Force

In the relativistic entanglement-power Section VIII.2, we applied the
entanglement-rapidity relationship to the power. This result gave us insight
into the information flow distributed by the radiation. In this subsection we
apply the entanglement-rapidity relationship to the radiation reaction itself.
We find the self-force is proportional to the second derivative of the von
Neumann entanglement entropy in a simple entanglement-force relationship.

Using the Davies-Fulling exact relativistic quantum stress tensor, expressed
in proper time $\tau$ as $T_{00}=\mathcal{F}(\tau)$ Good and Linder (2018)
where,

$12\pi\mathcal{F}(\tau)=-\eta^{\prime\prime}(\tau)e^{2\eta(\tau)},$ (103)

as well as the Lorentz-Abraham-Dirac 4-force in one dimension, we have
demonstrated that $F^{t}=Fw$, where the celerity is $w=dx/d\tau$, and
$F^{x}=F\gamma$, where the Lorentz factor is $\gamma=dt/d\tau$, (see the
closely related results of Higuchi-Martin Higuchi and Martin (2005)). These
results ultimately led to,

$6\pi F(\tau)=\alpha^{\prime}(\tau).$ (104)

The derivation reveals Eq. (104) as the most simple interpretation of the LAD
force, with magnitude $F$ of Eq. (78) as the jerk of the mirror, i.e. the
proper time derivative of the proper acceleration. The source of this force is
the reaction of the scalar field to the presence of the accelerating mirror in
vacuum. The proper time derivative of the proper acceleration determines the
force and can be nonzero even when the acceleration itself of the mirror is
instantaneously zero, and the mirror is not radiating particles. The
disturbing implications of the Lorentz-Abraham-Dirac formula which are still
not entirely understood in classical electrodynamics (see e.g. Rohrlich
(1997); Yaghjian (2006)), carry over in analog, to the quantum scalar field of
the moving mirror model.

Using the rapidity-entanglement relationship, $\eta=-6S$ in Eq. (102), it is
easy to find the von Nuemann entanglement entropy in terms of the radiative
reaction force,

$S^{\prime\prime}(\tau)=-\pi F(\tau).$ (105)

This relationship connects information flow in the system to the self-force on
the mirror. In a similar vein to the interesting features like negativity and
thermodynamic interpretations of entropic forces Visser (2011), this
entanglement self-force also assumes negative values and demonstrates an
information interpretation of the radiative reaction force: it is the second
proper time derivative of the von Neumann entanglement entropy.333The sign in
$S^{\prime\prime}=-\pi F$ tells us that when the force on the mirror is to the
left, away from the observer at $\mathscr{I}^{+}_{R}$, then $S^{\prime\prime}$
is positive. The sign is by convention because the observer is chosen to be
located on the right.

It would be interesting to know whether Eq. (105) holds outside the moving
mirror model considering the closely related accelerated boundary
correspondences (ABCs) with cosmologies and black holes. Regardless, advances
in general relativity, like the maximum force conjecture, may play a role in
better understanding the moving mirror model and associated entanglement
entropy. There have been a number of works in the last decades indicating that
$F_{\max}=c^{4}/(4G)$ is the limiting force (see Barrow and Gibbons (2015);
Good and Ong (2015) for more references) in general relativity. Taking into
account entropy-force relation of Eq. (105), this implies a constraint on the
rate of change of the entanglement entropy, i.e. if the force, or jerk
$\alpha^{\prime}(\tau)$, has this maximum, then the second derivative of
entropy has a minimum possible value,
$S^{\prime\prime}_{\textrm{min}}=-\pi/4$.

##  IX CGHS Larmor power and self-force

Having derived the power and self-force for any moving mirror in general, we
now specialize to the exponentially accelerated mirror that has particle
production which corresponds to the CGHS system. We apply the Larmor power and
LAD force derived in previous sections to our particular CGHS mirror and find
that a simple entanglement-over-distance relationship is revealed connecting
the entanglement-rapidity relationship to the space traversed. In addition,
the loss of unitarity is explicitly manifest in the divergence of the power
and self-force at the time the horizon forms in the proper frame.

Let us start from the trajectory of the CGHS mirror in spacetime coordinates
Good and Linder (2018),

$z(t)=-\frac{1}{\kappa}\sinh^{-1}\left(\frac{e^{\kappa(t-v_{H})}}{2}\right).$
(106)

The Larmor power and self-force for the CGHS mirror are found using Eq. (64)
and Eq. (88), where $\alpha(\tau)$ is the acceleration in proper time. The
procedure of defining and deriving $\alpha(\tau)$ is given in Good and Linder
(2018). Let us start from the connection between proper and coordinate times.
For CGHS mirror it is obtained to be,

$\tau(t)=\int\frac{dt}{\gamma(t)}=\frac{1}{2\kappa}\ln\left|\frac{\sqrt{4+e^{2\kappa(t-v_{H})}}-2}{\sqrt{4+e^{2\kappa(t-v_{H})}}+2}\right|.$
(107)

The inverse of Eq. (107) yields,

$t(\tau)=v_{H}+\frac{1}{2\kappa}\ln\left|\frac{16e^{2\kappa\tau}}{(1-e^{2\kappa\tau})^{2}}\right|.$
(108)

Applying it into Eq. (106) leads to the trajectory in proper time,

$z(\tau)=\frac{1}{\kappa}\sinh^{-1}\left(\csch(\kappa\tau)\right).$ (109)

The next step to obtain $\alpha(\tau)$ is to find celerity and then rapidity.
Using the rapidity, the proper acceleration is found to be,

$\alpha(\tau)=\frac{d\eta(\tau)}{d\tau}=\kappa\csch(\kappa\tau).$ (110)

This result, Eq. (110), is found using a different method in Juárez-Aubry
Juárez-Aubry (2017) and is in agreement. Substituting Eq. (110) into Eq. (64)
and Eq. (88), we obtain corresponding Larmor power and radiation reaction
force for the CGHS mirror as,

$P=\frac{\alpha^{2}}{6\pi}=\frac{\hbar}{c^{2}}\frac{\kappa^{2}\csch^{2}(\frac{\kappa\tau}{c})}{6\pi},$
(111)

and

$F=\frac{\alpha^{\prime}}{6\pi}=-\frac{\hbar}{c^{3}}\frac{\kappa^{2}}{6\pi}\coth(\frac{\kappa\tau}{c})\csch(\frac{\kappa\tau}{c}).$
(112)

The terms on the right of Eq. (111) and Eq. (112) have reinstated $\hbar$ and
$c$, noting that $\kappa$ has units of an acceleration, in order to emphasize
they are a quantum Larmor power and quantum self-force, respectively. The
dependences of the CGHS mirror Larmor power and self-force on proper time and
$\kappa$, Eq. (111) and Eq. (112), are demonstrated graphically in Figs. 3 and
4.

Figure 3: Larmor power for CGHS mirror, Eq. (111), where plots from left to
the right correspond to $\kappa=1,2,3,4,5$ cases, respectively, $\hbar=c=1$,
and power is normalised by $10$. The power increases asymptotically as time
approaches $\tau=0$. The key takeaway is this divergence at a finite proper
time when the horizon forms. Figure 4: Radiative reaction or the self-force
for CGHS mirror, Eq. (112), where blue, orange, green, red and purple lines
correspond to $\kappa=1,2,3,4,5$ cases, respectively, $\hbar=c=1$, and force
is normalised by $10^{2}$. The self-force has opposite direction with respect
to the Larmor power and demonstrates left-hand side trend as the mirror is
moving to the left.

Fig. 4 has lines corresponding to different values of $\kappa$ which
intersect. This is explained by the fact that the dependence of the CGHS self-
force, Eq. (112), on the single parameter of the system $\kappa$ is non-
trivially different from the dependence of the power, Eq. (111).

Let us now consider the timespace trajectory of the mirror in natural units,

$t(x)=v_{H}+\frac{1}{\kappa}\ln[-2\sinh(\kappa x)].$ (113)

Using this form of the trajectory we find rapidity in terms of space
coordinate $x$,

$\eta(x)=\kappa x.$ (114)

So, the rapidity, or the information defining dynamical quantity, in terms of
$x$ has surprisingly simple form: it linearly depends on the space coordinate.
Eq. (114) is the simplest way to express the trajectory of the CGHS mirror.

The last interesting quantity we compute is the energy flux in terms of $x$.
Using Eq. (48), the CGHS mirror flux is found to be,

$\mathcal{F}(x)=\frac{\kappa^{2}}{48\pi}(1-e^{4\kappa x}).$ (115)

This form immediately clarifies that at late times (far-left positions,
$x\to-\infty$), the energy flux is a constant associated with thermal emission
that is in agreement with the thermal behaviour of the CGHS black hole
radiation, $\mathcal{F}=\kappa^{2}/(48\pi)$. The graphical illustration of
this flux is shown in Fig. 5.

Figure 5: This graph should be read from right to left mapping the initial
motion of the mirror in the past to the future. This is the energy flux for
the CGHS mirror, Eq. (115), with $\kappa=\sqrt{48\pi}$ for illustration so
that at thermality the flux is equal to one. The mirror moves to the left,
while the flux ascends from $x=0$, then monotonically approaches constant
thermal emission at late positions, $x=-\infty$.

##  X Conclusions & Future Work

Overall, the equivalence between the CGHS black hole and the exponentially
accelerating moving mirror in lab time can be seen from several explicit
matching quantities: the matching condition for the CGHS black hole and the
trajectory of the mirror in null coordinates, the spectra, and consequently,
the temperatures. A one-to-one correspondence is ensured as long as
$|\Lambda|=|\kappa|$ requirement is met. The correspondence is summarized in
Table 1.

An important physical result of this correspondence is the demonstration that
the moving mirror horizon corresponds in the black hole system to a quantity
determined by the black hole parameters $\Lambda$ and $M$. This is understood
by seeing that the horizon location of the mirror is associated with the CGHS
black hole spacetime geometry as determined by the global non-zero curvature.
This geometry, in turn, is determined by the cosmological constant and
curvature caused by the black hole mass. Interestingly, it has been found that
unlike Schwarzschild black hole case, where the singularity and the event
horizon are located at different positions, the CGHS mirror mimics both the
event horizon and the center of the CGHS black hole as they happen at the same
location, $r=0$. This overlap is almost assuredly responsible for the
particular simplicity of the mathematics in the mirror case and the ease and
utility in describing this exact spectrum via a simple accelerating trajectory
in a flat-spacetime background. It also highlights no conflict between the
origin singularity and event horizon as the location of particle production,
since they are both one and the same.

More general considerations have given us the Larmor power radiated by an
arbitrary moving mirror and the LAD formula for the radiation reaction. The
derivations utilize general dynamics of the mirror expressed in terms of
proper acceleration and rapidity, and lead naturally to an information
interpretation by expressing the rapidity in terms of entanglement entropy.
The power and force are found to have the same dynamic form as that in
classical electrodynamics for a moving point charge. In terms of information,
the entanglement power and entanglement self-force are interpreted in terms of
first and second derivatives of the von-Neuman entanglement entropy,
respectively.

Specializing to our particular CGHS moving mirror, the Larmor power is found
to diverge as $\tau\to 0^{-}$. As proper time ticks to $\tau=0$, the mirror is
infinitely accelerating, reaching the speed of light. Consistently, the
direction of the self-force is opposite the direction of the radiated Larmor
power. It is worth emphasizing that as a guide, in SI units both the Larmor
power and LAD force for the CGHS mirror are proportional to $\hbar$,
underscoring the fact that the power and self-force are quantum (not
classical) measures.

Lastly, the CGHS mirror has two simplifying results when expressed in terms of
space rather than time or light-cone coordinates: the trajectory rapidity is
simply proportional to the distance travelled, $\eta=\kappa x$ and the
radiative flux emitted by the CGHS mirror is seen by eye as thermal (constant
emission) at far-left positions (late times). In summary list-form, the
salient features of this work are:

  * •

CGHS mirror $\Leftrightarrow$ EH & BH center; IV

  * •

Mirror Larmor power; $P\sim\alpha^{2}(\tau)$; VIII.1

  * •

Entanglement-power; $P\sim S^{\prime}(\tau)^{2}$; VIII.2

  * •

Larmor to LAD Averaging; $P\rightarrow F$; VIII.3

  * •

Mirror LAD self-force; $F\sim\alpha^{\prime}(\tau)$; VIII.4 & VIII.5

  * •

Entanglement-force; $F\sim S^{\prime\prime}(\tau)$; VIII.6

  * •

CGHS self-force and CGHS power; IX

Future extensions of this work are foreseen. Hawking Hawking (1975) pointed
out that at very early times of gravitational collapse, a star cannot be
described by the no-hair theorem. So in this context, a variety of different
collapse situations corresponds to different mirror trajectories. It is likely
to be fruitful to consider modifications to the mirror trajectory (one of
which has already been done to CGHS e.g. Good  _et al._ (2013)) made to
provide different early time approaches to a thermal distribution,
particularly those modifications that can afford unitarity and finite
evaporation energy, modeling more realistic situations congruent with finite
mass black holes and quantum purity.

The modifications that can take into account energy conservation like those of
the dilaton gravity models and moving mirror models have had significant
success as a laboratory for studying black hole evaporation. The physical
problem in (1+1) dilaton gravity of the evaporating black hole and its
modified emission extends to complete evaporation for the Russo, Susskind, and
Thorlacius (RST) model Russo  _et al._ (1992) and to partial evaporation
leaving a remnant for the Bose, Parker, and Peleg (BPP) model Bose  _et al._
(1995). The two-dimensional RST model for evaporating black holes is locally
equivalent - at the full quantum level - to Jackiw-Teitelboim (JT) gravity
that was recently shown to be unitary Fitkevich  _et al._ (2020).

The similarity of drifting moving mirrors (see e.g. Good  _et al._ (2017a)) to
the BPP model is striking in several qualitative aspects: NEF emission as a
thunderpop (NEF emission from evaporating black holes, at least in the case of
a (1+1)-dimensional dilaton gravity, has already been known in the literature
for over 20 years Bose  _et al._ (1996)), a left over remnant, and finite
total energy emission. It is also interesting that the mass of the remnant in
the BPP model is independent of the mass $M$ of the infalling matter, since
with respect to the issue of energy conservation, there is no known physical
analog for $M=1/(4\kappa)$, the initial mass of the shockwave, in the mirror
model.

We hope this work offers insight for future direction, using the formulas for
mirror radiative power and radiation reaction force. Investigations of
behavior of the Larmor power and LAD force of other existing moving mirror
models will be used to compare results to better understand the specific
physics of moving mirrors and the general physics of acceleration radiation
from the quantum vacuum.

Table 1: Matching quantities for CGHS black hole and exponentially accelerating mirror. Quantity | black hole | moving mirror
---|---|---
Trajectory | $u(U)=U-\frac{1}{\Lambda}\ln\left(1-e^{\Lambda(U-v_{0})}\right)$ | $f(v)=v-\frac{1}{\kappa}\ln\left(1-e^{\kappa(v-v_{H})}\right)$
Spectrum | $|\beta_{\omega\omega^{\prime}}|^{2}=\frac{1}{4\pi^{2}\Lambda^{2}}\frac{\omega^{\prime}}{\omega}\left|B\left[\frac{i\omega_{+}}{\Lambda},1-\frac{i\omega}{\Lambda}\right]\right|^{2}$ | $|\beta_{\omega\omega^{\prime}}|^{2}=\frac{1}{4\pi^{2}\kappa^{2}}\frac{\omega^{\prime}}{\omega}\left|B\left[\frac{i\omega_{+}}{\kappa},1-\frac{i\omega}{\kappa}\right]\right|^{2}$
Temperature | $T=\frac{\Lambda}{2\pi}$ | $T=\frac{\kappa}{2\pi}$

###### Acknowledgements.

Funding from state-targeted program “Center of Excellence for Fundamental and
Applied Physics” (BR05236454) by the Ministry of Education and Science of the
Republic of Kazakhstan is acknowledged. MG is also funded by the
FY2018-SGP-1-STMM Faculty Development Competitive Research Grant No.
090118FD5350 at Nazarbaev University.

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