# Vacuum energy around particles

Danilo T. Alves<EMAIL_ADDRESS>Faculdade de Física, Universidade Federal do
Pará, 66075-110, Belém, Brazil

(August 28, 2024)

###### Abstract

The creation of particles by the excitation of the quantum vacuum in a cavity
with a moving mirror was predicted in 1969. Here, we investigate that, in
addition to real particles, the excitation of the quantum vacuum in a
dynamical cavity can also result in the creation of a certain amount of
positive vacuum energy around these particles. We show that while in the case
of a single mirror moving in a free space, with the field in the vacuum state,
all the energy transferred from the mirror to the field is converted into real
particles, on the other hand, when considering a second and static mirror
forming a cavity with the first, the same movement of the first mirror can
lead to less energy being converted into real particles, with the difference
being converted into positive vacuum energy around these particles.

##  I Introduction

Considering a real massless scalar field in a two-dimensional spacetime, Moore
[1] predicted in 1969 the creation of particles by the excitation of the
quantum vacuum in a cavity with a moving mirror. This effect, usually known as
the dynamical Casimir effect (DCE), was investigated in other pioneering
articles (for example, in Refs. [2, 3, 4, 5, 6, 7]) and, since then, hundreds
of articles have been written on this issue (see Refs. [8, 9] and those
therein), with some experiments already done [10, 11, 12, 13]. Beyond the DCE,
there are other effects of particle creation from the quantum vacuum in non-
stationary systems as, for example, in models of expanding universe and
rotating black holes [14].

In the present paper, we investigate that, in addition to real particles, the
excitation of the quantum vacuum in a dynamical cavity can also produce an
amount (a “cloud”) of positive vacuum energy around these particles.
Considering the same model of a real massless scalar field in a two-
dimensional spacetime used by Moore [1], we show that when one has only one
mirror moving in a free space, with the field in the vacuum state, all the
energy dissipated from the mirror is converted into real particles [Fig.
1(a)], but, when considering a second and static mirror, the same movement of
the first mirror, with the same dissipated energy, can lead to less energy
being converted into real particles, with the difference being converted into
positive vacuum energy around the particles [Fig. 1(b)]. The basis of our
investigation is an expanding one-dimensional cavity, in which one of the
mirrors, initially at rest, starts moving smoothly, acquires constant velocity
for a certain time interval, after which it smoothly stops, returning to rest.
We focus on this configuration because one can define in- and out-states,
separate the vacuum regions from those where the particles can be present [2,
3, 15], and the field energy density is well defined for all spacetime points.

##  II Basic ideas

We start by reviewing some fundamental formulas and techniques of calculation
found in Refs. [1, 3, 16, 17, 18, 19]. We also present original discussions,
useful for the purposes of the present paper. We consider a real massless
scalar field in a cavity, satisfying the wave equation
$\left(\partial_{t}^{2}-\partial_{x}^{2}\right)\phi\left(t,x\right)=0$ (we
assume throughout this paper $\hbar=c=1$), and obeying the Dirichlet boundary
condition imposed at a static mirror located at $x=0$, and also at the moving
mirror’s position at $x=L(t)$ (i.e. $\phi\left(t,0\right)=0$ and
$\phi[t,L(t)]=0$), where $L(t)>0$ is an arbitrary prescribed law for the
moving mirror (hereafter named right mirror), with $L(t<0)=L_{0}$, where
$L_{0}$ is the length of the cavity in the initial ($t<0$) static situation
(in-state). The exact formula for the expected value ${\cal T}(t,x)$ of the
energy density operator $\hat{T}_{00}(t,x)$, i.e. ${\cal
T}(t,x)=\langle\hat{T}_{00}(t,x)\rangle_{\text{in}}$ (the averages
$\langle...\rangle_{\text{in}}$ are taken over a vacuum in-state in the
cavity, which is our interest in this paper), can be written recursively by
the equations (see Refs. [16, 17, 18, 19] and Appendix A):

$\displaystyle{\cal
T}(t,x)={\cal{T}}_{{\cal\tilde{A}}}\left(t,x\right)+{\cal{T}}_{{\cal\tilde{B}}}\left(t,x\right),$
(1)
$\displaystyle{\cal{T}}_{{\cal\tilde{A}}}\left(t,x\right)=-{\pi}/({48L_{0}^{2}})[{\cal\tilde{A}}(u)+{\cal\tilde{A}}(v)],$
(2)
$\displaystyle{\cal{T}}_{{\cal\tilde{B}}}\left(t,x\right)=-{\cal\tilde{B}}(u)-{\cal\tilde{B}}(v),$
(3) $\displaystyle{\cal\tilde{A}}(z)=\prod_{i=1}^{n(z)}{\cal A}[t_{i}(z)],$
(4) $\displaystyle{\cal\tilde{B}}(z)=\sum_{j=1}^{n(z)}{\cal
B}[t_{j}(z)]\prod_{i=1}^{j-1}{\cal A}[t_{i}(z)],$ (5)
$\displaystyle{\cal{A}}(t)=\left[\frac{1-\dot{L}(t)}{1+\dot{L}(t)}\right]^{2},$
(6) $\displaystyle{\cal
B}(t)=\frac{-\frac{\dddot{L}\left(t\right)}{12\pi}-\frac{1}{4\pi}\frac{\ddot{L}^{2}\left(t\right)\dot{L}\left(t\right)}{\left[1-\dot{L}\left(t\right)^{2}\right]}}{\left[1+\dot{L}\left(t\right)\right]^{3}\left[1-\dot{L}\left(t\right)\right]},$
(7)

where $u=t-x$, $v=t+x$, $n(z)$ is the number of reflections off the moving
mirror, necessary to connect a null line $v=z$ (or $u=z$) to a null line in
the static zone [ $v<L_{0}$, where $f(v)={\pi}/({{48}L_{0}^{2}})$], and
$t_{1},...,t_{n}$ are such that the points $[t_{i},L(t_{i})]$ are those where
the null lines $v=t_{i}+L(t_{i})$ intersect the right mirror, and are
calculated via $z=t_{1}+L(t_{1})$, and
$t_{i+1}+L(t_{i+1})=t_{i}-L(t_{i}),\;i=1,2,3,...$ (for more details, see Refs.
[16, 17, 18, 19]).

When the right mirror stops its motion at an instant $t=\beta$, with the
cavity reaching a final length $L_{\text{fin}}$ (out-state), we have the total
energy $E$ given by $E=E_{\cal\tilde{A}}+E_{\cal\tilde{B}}$, with
$E_{\cal\tilde{A}}$ $=$
$\int_{0}^{L_{\text{fin}}}{\cal{T}}_{{\cal\tilde{A}}}\left(t,x\right)dx$, and
$E_{\cal\tilde{B}}=\int_{0}^{L_{\text{fin}}}{\cal{T}}_{{\cal\tilde{B}}}\left(t,x\right)dx$.
On the other hand, $E$ can also be written as (see Ref. [18] and Appendix A)
$E=E^{\text{(cas)}}+E^{\text{(par)}}$, where $E^{\text{(cas)}}$ is the Casimir
energy in the out-state (out-Casimir energy), given by
$E^{\text{(cas)}}=-{\pi}/({24L_{\text{fin}}})$ [hereafter, the superscripts
“cas” and “cas-in” mean Casimir energy in out- and in-states, respectively],
and the total energy of real particles $E^{\text{(par)}}$ is given by
$E^{\text{(par)}}=\sum_{n}\omega_{n}{N}_{n}$, with
$\omega_{n}=n\pi/L_{\text{fin}}$, and
${N}_{n}=\left\langle\hat{a}_{n}^{{\dagger}}\hat{a}_{n}\right\rangle_{\text{in}}$
is the mean number of particles related to the $n$th field mode in a cavity
with length $L_{\text{fin}}$, $\hat{a}_{n}^{{\dagger}}$ and $\hat{a}_{n}$ are,
respectively, the creation and annihilation operators related to the $n$th
field mode. We highlight that, independently of the field state, the total
vacuum energy in the static cavity is the Casimir one. Using both formulas for
$E$, we have

$\displaystyle
E_{\cal\tilde{A}}+E_{\cal\tilde{B}}=E^{\text{(cas)}}+E^{\text{(par)}}.$ (8)

This formula is fundamental to the discussion presented here.

(a) (b)

Figure 1:  In (a), we illustrate the case of a free space, where all the
energy dissipated from the moving mirror is converted into real particles
(small circles). In (b), we illustrate the case of a cavity with the same
movement of the moving mirror and the same amount of dissipated energy as in
(a), but with only a part of this energy converted into real particles
(smaller number of small circles), with the difference being converted into
positive vacuum energy (big circle) propagating together with these particles.

##  III Missing Casimir energy and positive vacuum energy around particles

Fulling and Davies [3] investigated the model of a real massless scalar field
in 1+1 dimensions in a cavity in which one of the mirrors remains at rest at
$x=0$, and the other starts moving abruptly at $t=0$, acquiring a constant
velocity $\alpha$ $(0<\alpha<1)$, and stopping abruptly at $t=\beta$. The
vacuum was considered the initial field state $(t<0)$. The worldlines of the
mirrors are shown, for $\beta<2L_{0}/\left(1-\alpha\right)$, in Fig. 2(a)
(which was adapted from Fig. 4(b) found in Ref. [3]). The dashed lines, which
have slopes $1$ or $-1$, are the worldlines (null lines) of the massless
particles emitted from the abrupt accelerations of the right mirror. These
idealized accelerations, proportional to Dirac $\delta$ functions, generate an
undefined value of the field energy density in these lines, and it also can be
shown that this motion produces a divergent expected value of the total number
of created particles [15]. On the other hand, regions $A$, $B$, $C$ and $D$,
separated by the dashed lines [Fig. 2(a)], are vacuum regions (since there is
no particle production during the uniform motion of the mirror [2, 3, 15])
with well defined field energy densities given by [3]: ${\cal T}_{A}={\cal
T}_{B}=-\left[\pi/(24L_{0}^{2})\right]\left[\left(1+\alpha^{2}\right)/\left(1+\alpha\right)^{2}\right]$,
${\cal T}_{C}$$={\cal T}^{\text{(cas-in)}}=-\pi/(24L_{0}^{2})$, ${\cal
T}_{D}=$$-\left[\pi/(24L_{0}^{2})\right]\left[\left(1-\alpha\right)/\left(1+\alpha\right)\right]^{2}$,
where ${\cal T}^{\text{(cas-in)}}$ is the Casimir field energy density in the
in-state. These equations can be recovered using Eqs. (1)-(7) (see Appendix
B).

In order to work with a well defined ${{\cal T}}\left(t,x\right)$
$\forall(t,x)$, we will discuss models of expanding cavities which do not
generate the mentioned divergences occurring at $t=0$ and $t=\beta$. Thus, we
propose laws of motion that smooth the jumps in the derivatives of $L(t)$
occurring at these instants, so that $\dot{L}\left(t\right)$,
$\ddot{L}\left(t\right)$, and $\dddot{L}\left(t\right)$ are well defined
functions $\forall\;t$. In this way, the energy density given in Eq. (1) is
well defined in all spacetime points, so that singularities are no more
present in the regions where the created particles can be. Trajectories
$x=L(t)$ that do not lead to singularities are, for example, given by (see
Appendix C)

$L\left(t\right)=\begin{cases}L_{0}&\left(t<0\right),\\\
L_{1}\left(t\right)&\left(0\leq t\leq\tau\right),\\\ L_{0}+\alpha
t&\left(\tau<t<\beta-\tau\right),\\\ L_{2}\left(t\right)&\left(\beta-\tau\leq
t\leq\beta\right),\\\ L_{0}+\alpha\beta&\left(\beta<t\right),\end{cases}$ (9)

where
$L_{1}\left(t\right)=\;e^{t}-1+L_{0}-t-\frac{1}{2}t^{2}-\frac{1}{6}t^{3}+a_{1}t^{4}+b_{1}t^{5}+c_{1}t^{6}+d_{1}t^{7}$,
and
$L_{2}\left(t\right)=e^{\left(t-\beta\right)}-1+L_{0}+\alpha\beta-\left(t-\beta\right)-\frac{1}{2}\left(t-\beta\right)^{2}-\frac{1}{6}\left(t-\beta\right)^{3}+a_{2}\left(t-\beta\right)^{4}+b_{2}\left(t-\beta\right)^{5}+c_{2}\left(t-\beta\right)^{6}+d_{2}\left(t-\beta\right)^{7}$.
Note that, for $\tau=0$, we recover the mentioned law of motion investigated
by Fulling and Davies [3].

Hereafter, our discussions assume (but are not restricted to) functions $L(t)$
according to the law of motion in Eq. (9), which is illustrated in Fig. 2(b).
We assume that $\beta<2L_{0}/\left(1-\alpha\right)$, which means that
particles can be created [3, 15] (see also Appendix B), and that a field mode,
after being affected by the moving mirror, never meets this mirror in motion
again, which allows us to exclude the effect of reinteraction with the moving
mirror on the particle creation process [20].

For any instant $t>\beta$, the cavity is static again, with length
$L_{\text{fin}}=L_{0}+\alpha\beta$, and the field is in an out-state different
from vacuum. One can see that $C$ regions [Figs. 2(a) and 2(b)] are vacuum,
but with an energy density smaller than the out-Casimir one: ${\cal
T}_{C}<{\cal T}^{\text{(cas)}}$, where ${\cal
T}^{\text{(cas)}}=-\pi/(24L_{\text{fin}}^{2})$. On the other hand, ${\cal
T}_{D}>{\cal T}^{\text{(cas)}}$, so that $D$ regions are vacuum, but with an
energy density greater than the out-Casimir one. Regions $A$ and $B$ are also
vacuum, where ${\cal T}_{A}$ and ${\cal T}_{B}$ can be smaller, equal or
greater than ${\cal T}^{\text{(cas)}}$, depending on the parameters $\alpha$
and $\beta$. It is also interesting to highlight that in the out-state the $C$
regions can be viewed as a “fossil”, or an explicit signature, of the in-state
of the field in the cavity.

Before we go on, let us define some conventions to be used hereafter. The
superscript “vac” indicates that the energy (or density) comes from vacuum (in
out-state). This superscript, combined with the subscripts “vac” and “par”,
indicates that this vacuum energy (or density) is located in vacuum or
particle regions (those where real particles can be found), respectively. The
superscript “par” indicates that the energy (or density) comes from the real
particles (out-state), which is just different from zero in the particle
regions. $\overline{{\cal T}}$ means the average (in space) of a given energy
density, in a given region of the cavity.

(a) (b)

Figure 2:  In (a) and (b): the solid lines are worldlines of two mirrors, with
one of them remaining static at $x=0$, and the other following the law of
motion given in Eq. (9), with $\tau=0$ (a) and $\tau\neq 0$ (b) (figure (a)
was adapted from that found in Ref. [3]); the lighter regions $A$, $B$, $C$
and $D$ are vacuum regions. In (a), the dashed lines (particle regions) are
the worldlines (null lines) of the massless particles emitted from the abrupt
accelerations of the right mirror. In (b), the darker regions (particle
regions) are regions where the created particles can be found.

Now, let us define the difference between the vacuum energy in the vacuum
regions, and the corresponding out-Casimir energy in these regions (both in
the out-state): $\Delta
E_{\text{(vac)}}^{\text{(vac)}}(t,\tau)=E_{\text{(vac)}}^{\text{(vac)}}(t,\tau)-E_{\text{(vac)}}^{\text{(cas)}}(t,\tau)$.
For $\tau=0$ [Fig. 2(a)], we have $\Delta
E_{\text{(vac)}}^{\text{(vac)}}(t,0)=-{\pi}/{(24L_{0}^{2})}{[\alpha^{2}\beta\left(\alpha\beta+2{L_{0}}-\beta\right)]}/[\left(1+\alpha\right)\left(\alpha\beta+{L_{0}}\right)]$,
with $-\pi/(24L_{0})<$ $\Delta E_{\text{(vac)}}^{\text{(vac)}}(t,0)$ $<0$ (see
the dotted line in Fig. 3). Since $E^{\text{(cas)}}_{\text{(par)}}(t,0)=0$
and, as already discussed, the total vacuum energy is the out-Casimir one,
this implies that an amount of positive vacuum energy $-\Delta
E_{\text{(vac)}}^{\text{(vac)}}(t,0)$ must be in the particle regions [the
null lines in Fig. 2(a)], so that
$E_{\text{(par)}}^{\text{(vac)}}(t,0)=-\Delta
E_{\text{(vac)}}^{\text{(vac)}}(t,0)$. The amount $\Delta
E_{\text{(vac)}}^{\text{(vac)}}(t,0)$, we call “missing Casimir energy” in the
vacuum regions. Defining $x_{a}(t)$ and $x_{b}(t)$ the $x$ coordinates of the
particle regions, we can write $\overline{{\cal
T}}_{\text{(par)}}^{\text{(vac)}}\left(t,x\right)=-\frac{1}{2}\Delta
E_{\text{(vac)}}^{\text{(vac)}}\left\\{\delta\left[x-x_{a}\left(t\right)\right]+\delta\left[x-x_{b}\left(t\right)\right]\right\\}$.

Now, we investigate the missing Casimir energy $\Delta
E_{\text{(vac)}}^{\text{(vac)}}(t,\tau)$ for $\tau\neq 0$. In these cases (and
for any $\tau>0$), the length of the vacuum regions can vary in time, and,
consequently, the vacuum energies $E_{\text{(vac)}}^{\text{(vac)}}(t,\tau)$
can also vary in these regions. This can be calculated directly using the
formulas for ${\cal T}_{A}$,…,${\cal T}_{D}$, and the equations for the null
lines in Fig. 2(b) (see Appendix D). Let us consider $L_{0}=1$, $\alpha=0.55$,
and $\beta=2.56$. In Fig. 3, the dash-dotted and dashed lines correspond to
the cases $\tau=10^{-1}$ and $\tau=3\times 10^{-1}$, respectively. One can see
that $\Delta E_{\text{(vac)}}^{\text{(vac)}}(t,\tau)<0$, so that an amount of
positive vacuum energy $-\Delta E_{\text{(vac)}}^{\text{(vac)}}(t,\tau)$ needs
to be found in the particle regions [darker regions in Fig. 4], added to
$E^{\text{(cas)}}_{\text{(par)}}(t,\tau)$. This, in turn, implies a mean
energy density $\overline{{\cal
T}}_{\text{(par)}}^{\text{(vac)}}\left(t,x\right)>{\cal T}^{\text{(cas)}}$.

In Fig. 4, we show the behavior of ${\cal
T}_{\text{(vac)}}^{\text{(vac)}}\left(t,x\right)$, $\overline{{\cal
T}}^{\text{(vac)}}_{\text{(par)}}\left(t,x\right)$, and ${\cal
T}^{\text{(cas)}}$, for $t=\beta=2.56$. In Figs. 4(a) and 4(b), we have
$\tau=3\times 10^{-1}$ and $\tau=10^{-1}$, respectively. Thus, in the particle
regions, there is the presence of a vacuum energy greater than the
corresponding out-Casimir ones. However, note that $\overline{{\cal
T}}_{\text{(par)}}^{\text{(vac)}}\left(\beta,x\right)$ remains negative in
Fig. 4(a), whereas it becomes positive in Fig. 4(b). In the limit $\tau\to 0$,
$\overline{{\cal T}}_{\text{(par)}}^{\text{(vac)}}\to\infty$, so that
$\overline{{\cal T}}_{\text{(par)}}^{\text{(vac)}}$ can be written in terms of
Dirac delta functions, as already discussed.

Figure 3:  Missing Casimir energy $\Delta
E_{\text{(vac)}}^{\text{(vac)}}(t,\tau)$ (vertical axes) versus $t$
(horizontal axes), for $L_{0}=1$, $\alpha=0.55$, and $\beta=2.56$. The dotted,
dash-dotted, and dashed lines show the cases $\tau=0$, $\tau=10^{-1}$, and
$\tau=3\times 10^{-1}$, respectively.

(a) (b)

Figure 4:  Energy densities (vertical axes) at $t=\beta$, versus $x$
(horizontal axes), for $0\leq x\leq L_{0}+\alpha\beta$, considering $L_{0}=1$,
$\alpha=0.55$, and $\beta=2.56$. In both figures lighter regions are vacuum
regions, whereas darker regions are particle regions. The solid red lines in
lighter regions show ${\cal
T}_{\text{(vac)}}^{\text{(vac)}}\left(\beta,x\right)$, the solid green lines
in darker regions show the mean values $\overline{{\cal
T}}^{\text{(vac)}}_{\text{(par)}}\left(\beta,x\right)$, and the dashed lines
show ${\cal T}^{\text{(cas)}}\approx-0.023$. In (a) and (b), we have
$\tau=3\times 10^{-1}$ and $\tau=10^{-1}$, respectively. In (a),
$\overline{{\cal
T}}_{\text{(par)}}^{\text{(vac)}}\left(\beta,x\right)\approx-0.016$, whereas
in (b) $\overline{{\cal
T}}_{\text{(par)}}^{\text{(vac)}}\left(\beta,x\right)\approx 0.033$.

##  IV What is behind the positive vacuum energy

Let us start splitting $E_{\cal\tilde{A}}$ and $E_{\cal\tilde{B}}$ as follows:
$E_{\cal\tilde{A}}=E_{\cal\tilde{A}\text{(vac)}}^{\text{(vac)}}+E_{\cal\tilde{A}\text{(par)}}^{\text{(vac)}}+E_{\cal\tilde{A}}^{\text{(par)}}$;
$E_{\cal\tilde{B}}=E_{\cal\tilde{B}\text{(vac)}}^{\text{(vac)}}+E_{\cal\tilde{B}\text{(par)}}^{\text{(vac)}}+E_{\cal\tilde{B}}^{\text{(par)}}$.
The energy $E_{{\cal\tilde{A}}}$ is negative [from Eqs. (2), (4), and (6)],
and it involves only the functions ${\cal A}$ [Eq. (6)], which, by means of a
Doppler-shifted reflection of the incoming vacuum in-fields, only contributes
to the rearrangement of the vacuum energy, giving no contribution to the
particle production ($E_{\cal\tilde{A}}^{\text{(par)}}=0$). The energy
$E_{\cal\tilde{B}}$ involves the functions ${\cal B}$ [Eq. (7)], which are
zero in vacuum regions [$\ddot{L}(t_{i})=0$ and $\dddot{L}(t_{i})=0$ in this
case], so that $E_{\cal\tilde{B}\text{(vac)}}^{\text{(vac)}}=0$. Then, we have
(see Sec. II)
$E_{\cal\tilde{A}\text{(vac)}}^{\text{(vac)}}+E_{\cal\tilde{A}\text{(par)}}^{\text{(vac)}}+E_{\cal\tilde{B}\text{(par)}}^{\text{(vac)}}=E^{\text{(cas)}}$,
and $E_{\cal\tilde{B}}^{\text{(par)}}=E^{\text{(par)}}$.

From Eqs. (4) and (6), one can see that
$E_{\cal\tilde{A}\text{(par)}}^{\text{(vac)}}<0$, and, taking $L_{0}\to\infty$
(free space), we have: $E_{\cal\tilde{A}\text{(vac)}}^{\text{(vac)}}\to 0$,
$E_{\cal\tilde{A}\text{(par)}}^{\text{(vac)}}\to 0$, and $E^{\text{(cas)}}\to
0$ (the vacuum energy in free space is renormalized to zero); we also have
$E_{\cal\tilde{B}\text{(par)}}^{\text{(vac)}}\to 0$. From Eqs. (5) and (7),
one can see that $E_{\cal\tilde{B}}$ does not depend on $L_{0}$, so that it
does not matter whether we consider a cavity or a free space, it has the same
value. Thus, for $L_{0}\to\infty$, we have
$E_{\cal\tilde{B}}=E_{\cal\tilde{B}}^{\text{(par)}}|_{L_{0}\to\infty}=\int_{0}^{\infty}d\omega\omega{\cal
N}\left(\omega\right),$ where ${\cal N}\left(\omega\right)$ is the mean
density of number of particles in the mode $\omega$.

In the particle regions in Fig. 4(b), we have a positive vacuum energy, this
means that
$E_{\cal\tilde{A}\text{(par)}}^{\text{(vac)}}+E_{\cal\tilde{B}\text{(par)}}^{\text{(vac)}}>0$.
Since $E_{\cal\tilde{A}\text{(par)}}^{\text{(vac)}}<0$, the only way to
achieve this is with $E_{\cal\tilde{B}\text{(par)}}^{\text{(vac)}}>0$. In
other words, it is the presence of
$E_{\cal\tilde{B}\text{(par)}}^{\text{(vac)}}>0$ that is behind the positive
vacuum energy in the particle regions shown in Fig. 4(b).

Putting the ideas of the last two paragraphs together, we have that when there
is only a single mirror moving in a free space, all the energy
$E_{\cal\tilde{B}}$ transferred (in this case, dissipated) from the mirror to
the field is converted into real particles [left hand side of Eq. (10)]. On
the other hand, when considering a second and static mirror forming a cavity,
and the same movement of the first mirror dissipating the same amount of
energy $E_{\cal\tilde{B}}$, we have that only a part of this energy is
converted into real particles [right hand side of Eq. (10)], with the
difference [$E_{\cal\tilde{B}\text{(par)}}^{\text{(vac)}}$] being converted
into positive vacuum energy propagating together with these particles [right
hand side of Eq. (10)]. Thus, we have

$\int_{0}^{\infty}d\omega\omega{\cal
N}\left(\omega\right)=\sum_{n}\omega_{n}{N}_{n}+E_{\cal\tilde{B}\text{(par)}}^{\text{(vac)}},$
(10)

with the left and right hand sides of this equation being illustrated in Figs.
1(a) and 1(b), respectively.

##  V Application to a concrete law of motion

Although we have worked with exact values for the vacuum energy density in
vacuum regions (lighter regions in Fig. 4), we only worked with mean values of
the vacuum energy density in the particle regions (darker regions in Fig. 4),
and nothing has been said about the particle energy itself. In this section,
we discuss the exact behavior of ${\cal{T}}\left(t,x\right)$, examining its
parts ${\cal{T}}_{{\cal\tilde{A}}}\left(t,x\right)$ and
${\cal{T}}_{{\cal\tilde{B}}}\left(t,x\right)$, in whole space, including in
the particle regions. Although ${\cal{T}}\left(t,x\right)$ is given exactly by
Eq. (1), this formula can require a numerical treatment. In this section, our
results are found taking as basis the computer routines described in Ref.
[19].

Let us focus on the case corresponding to Fig. 4(b), where $L_{0}=1$,
$\alpha=0.55$, $\beta=2.56$, and $\tau=10^{-1}$. Our calculations show that,
for these values, the functions $L_{1}(t)$ and $L_{2}(t)$ have the following
parameters (see Appendix C): $a_{1}\approx 10999.96$; $b_{1}\approx-2.48\times
10^{5}$; $c_{1}\approx 1.98\times 10^{6}$; $d_{1}\approx-5.50\times 10^{6}$;
$a_{2}\approx-5.50\times 10^{6}$; $b_{2}\approx-1.98\times 10^{6}$;
$c_{2}\approx-2.48\times 10^{5}$; $d_{2}\approx-11000.04$. We show the exact
behavior of ${\cal{T}}_{{\cal\tilde{A}}}\left(\beta,x\right)$ in Fig. 5(a),
and ${\cal{T}}_{{\cal\tilde{B}}}\left(\beta,x\right)$ in Fig. 5(b).

After integrating the energy densities in Fig. 5, in each of their respective
regions, and using previous formulas, we obtain that (see Appendix E):
$E_{\cal\tilde{B}}\left(\beta\right)\approx 2.699$,
$E_{\cal\tilde{B}\text{(par)}}^{\text{(vac)}}\left(\beta\right)\approx 0.025$,
and $E_{\cal\tilde{B}}^{\text{(part)}}\left(\beta\right)\approx 2.674$. Thus,
as discussed in Sec. IV, here we find that not all of the energy
$E_{\cal\tilde{B}}$ is converted into the energy
$E_{\cal\tilde{B}}^{\text{(part)}}$ of real particles, with the difference
being converted into the positive vacuum energy
$E_{\cal\tilde{B}\text{(par)}}^{\text{(vac)}}$, which is located in the same
regions where the particles can be. From our calculations we also recover (see
Appendix E) $\overline{{\cal
T}}_{\text{(par)}}^{\text{(vac)}}\left(t,x\right)\approx 0.033$, as indicated
in Fig. 4(b). The ratio
$E_{\cal\tilde{B}\text{(par)}}^{\text{(vac)}}(\beta)/E_{\cal\tilde{B}}\approx
0.9\%$, so $0.9\%$ of the energy $E_{\cal\tilde{B}}$ dissipated from the
mirror is converted into vacuum positive energy. The ratio
$[E_{\cal\tilde{A}\text{(par)}}^{\text{(vac)}}+E_{\cal\tilde{B}\text{(par)}}^{\text{(vac)}}]/[E_{\cal\tilde{A}\text{(par)}}^{\text{(vac)}}+E_{\cal\tilde{B}}]\approx
0.4\%$, which means that an average value of $0.4\%$ of the total energy in
the particle regions is positive vacuum energy.

(a) (b)

Figure 5:  Energy densities (vertical axes) ${\cal T}_{{\cal\tilde{A}}}$
[shown in (a)] and ${\cal T}_{{\cal\tilde{B}}}$ [in (b)], at $t=\beta$, versus
$x$ (horizontal axes), for $0\leq x\leq L_{0}+\alpha\beta$, considering
$L_{0}=1$, $\alpha=0.55$, $\beta=2.56$, and $\tau=10^{-1}$. In both figures,
lighter regions are vacuum regions, whereas darker regions are particle
regions.

##  VI Final remarks

We showed that, in addition to real particles, the excitation of the quantum
vacuum in a dynamical cavity can also produce a cloud of positive vacuum
energy around these particles. In the same way that the creation of particles
by the excitation of the quantum vacuum, as predicted by Moore [1], has been
also predicted in the context of other fields, boundary conditions, and
spacetime dimensions [9], we expect that the possibility of creation of
positive vacuum energy around particles, predicted here, can also be
generalized to any confined field under dynamic boundary conditions. Moreover,
since, in addition to moving mirrors, models with non-static spacetime
(expanding universe) are another way to disturb the quantum vacuum and create
particles [14], it is also interesting to investigate if such a vacuum
positive energy can be present around massive particles created in these
models, and if the equivalent in mass of this vacuum energy could be of any
relevance to the understanding of some aspect of the dark matter around the
visible matter. Finally, considering that some experiments have already
observed photons created via excitation of the quantum vacuum [10, 11, 12,
13], an observation of a signature of the vacuum positive energy predicted
here, namely the difference between the energy of the spectrum of created
particles in a free space and in a cavity [Eq. (10)], seems not unfeasible.

###### Acknowledgements.

The author thanks: A. L. C. Rego, A. N. Braga, C. Farina, E. R. Granhen, J. D.
L. Silva, L. Oliveira, L. Queiroz, O. J. N. Neto, P. A. M. Neto, S. S. Coelho,
T. R. Alves, T. R. Alves, V. S. Alves, and V. V. Dodonov for valuable
discussions; A. L. C. Rego, E. R. Granhen, J. D. L. Silva, L. Queiroz, and V.
S. Alves for a careful reading of this manuscript; L. Queiroz and O. J. N.
Neto for the support in improving some figures presented here. V. V. Dodonov
and A. V. Dodonov for the invitation which enabled the presentation of a
preliminary version of this work in the Second International Workshop on
Nonstationary Systems - Brasília (2023), with support of the FAPDF - Brazil;
E. C. Marino for the invitation which enabled the presentation of another
version of this work in the workshop Quantum Field Theory Approach to
Condensed Matter Physics - Rio de Janeiro (2024). This work was partially
supported by CNPq - Brazil, Processo 408735/2023-6 CNPq/MCTI.

##  Appendix A Dynamical and static cavities: basic formulas

The real scalar field field operator, solution of the wave equation in 1+1
dimensions, is given by (see, for instance, Ref. [1, 3, 18])

$\hat{\phi}(t,x)=\sum^{\infty}_{n=1}\left[\hat{a}^{\text{(in)}}_{n}\phi_{n}\left(t,x\right)+H.c.\right],$
(11)

where $\hat{a}_{n}^{{\text{(in)}}{\dagger}}$ and $\hat{a}^{\text{(in)}}_{n}$
are, respectively, the creation and annihilation operators related to the
$n$th field mode in the cavity with length $L_{0}$, and the field modes
$\phi_{n}(t,x)$ are given by

$\displaystyle\phi_{n}(t,x)=\frac{i}{2\sqrt{n\pi}}\left[\varphi_{n}(v)-\varphi_{n}(u)\right],$
(12)

with $\varphi_{n}(z)=e^{-in\pi R(z)}$, $u=t-x$, $v=t+x$, and $R$ satisfying
Moore’s functional equation

$R[t+L(t)]-R[t-L(t)]=2.$ (13)

The exact formulas for ${\cal T}(t,x)$, the expected value of the energy
density operator $\hat{T}_{00}(t,x)$, so that ${\cal
T}(t,x)=\langle\hat{T}_{00}(t,x)\rangle_{\text{in}}$, can be split as (see,
for instance, Refs. [18, 19])

${\cal T}(t,x)={\cal T}_{\text{from-vac}}(t,x)+{\cal T}_{\text{from-
par}}(t,x),$ (14)

where

$\displaystyle{\cal T}_{\text{from-
vac}}(t,x)=\frac{\pi}{4}\sum_{n=1}^{\infty}n\left[R^{\prime
2}\left(v\right)+R^{\prime 2}\left(u\right)\right],$ (15)

and

${\cal T}_{\text{ from-par}}(t,x)={\cal
T}_{\left\langle\hat{a}^{{\text{(in)}}{\dagger}}\hat{a}^{\text{(in)}}\right\rangle_{\text{in}}}(t,x)+{\cal
T}_{\left\langle\hat{a}^{\text{(in)}}\hat{a}^{\text{(in)}}\right\rangle_{\text{in}}}(t,x),$
(16)

with

$\displaystyle{\cal
T}_{\left\langle\hat{a}^{{\text{(in)}}{\dagger}}\hat{a}^{\text{(in)}}\right\rangle_{\text{in}}}(t,x)=g_{1}(v)+g_{1}(u),$
(17) $\displaystyle{\cal
T}_{\left\langle\hat{a}^{\text{(in)}}\hat{a}^{\text{(in)}}\right\rangle_{\text{in}}}(t,x)=g_{2}(v)+g_{2}(u),$
(18) $\displaystyle
g_{1}(z)=\frac{\pi}{2}\sum_{n,n^{\prime}=1}^{\infty}\sqrt{nn^{\prime}}$
$\displaystyle\times{\mbox{R}e}\left\\{e^{i\left(n-n^{\prime}\right)\pi
R\left(z\right)}\left[R^{\prime}\left(z\right)\right]^{2}\left\langle\hat{a}^{\text{(in)}{\dagger}}_{n}\hat{a}^{\text{(in)}}_{n^{\prime}}\right\rangle_{\text{in}}\right\\},$
(19) $\displaystyle
g_{2}(z)=\frac{\pi}{2}\sum_{n,n^{\prime}=1}^{\infty}\sqrt{nn^{\prime}}$
$\displaystyle\times{\mbox{R}e}\left\\{e^{-i\left(n+n^{\prime}\right)\pi
R\left(z\right)}\left[R^{\prime}\left(z\right)\right]^{2}\left\langle\hat{a}^{\text{(in)}}_{n}\hat{a}^{\text{(in)}}_{n^{\prime}}\right\rangle_{\text{in}}\right\\}.$
(20)

The averages $\langle...\rangle_{\text{in}}$, are here taken over any initial
field state (further on, as mentioned in Sec. II, we apply our calculation
considering the vacuum as the initial field state, the focus of the present
paper). The term ${\cal T}_{\text{from-vac}}$ has its origin in the vacuum
fluctuations, whereas ${\cal T}_{\text{ from-par}}$ in the presence of real
particles in the initial field state [see Eqs. (15)-(20)].

The term ${\cal T}_{\text{ from-vac}}$ is divergent, but adopting the point-
splitting regularization method [3] one has ${\cal T}_{\text{ from-vac}}$, now
redefined as the renormalized local energy density, given by [3]

${\cal T}_{\text{ from-vac}}(t,x)=-f(v)-f(u),$ (21)

where

$f=\frac{1}{24\pi}\left[\frac{R^{\prime\prime\prime}}{R^{\prime}}-\frac{3}{2}\left(\frac{R^{\prime\prime}}{R^{\prime}}\right)^{2}+\frac{\pi^{2}}{2}{R^{\prime}}^{2}\right],$
(22)

with the derivatives taken with respect to the argument of the $R$ function.
The value of the energy density at each point of the spacetime [Eq. (1)], can
be calculated recursively, as discussed in Ref. [18] and shown in Eqs.
(1)-(7).

For a static “in” situation with an arbitrary initial field state, we can
write the total energy density ${\cal T}(t,x)$, now relabeled as ${\cal
T}^{\text{(in)}}(t,x)$, as given by (see Ref. [18])

${\cal T}^{\text{(in)}}(t,x)={\cal T}^{\text{(cas-in)}}+{\cal
T}^{\text{(in)}}_{\left\langle\hat{a}^{{\text{(in)}}{\dagger}}\hat{a}^{\text{(in)}}\right\rangle_{\text{in}}}(t,x)+{\cal
T}^{\text{(in)}}_{\left\langle\hat{a}^{\text{(in)}}\hat{a}^{\text{(in)}}\right\rangle_{\text{in}}}(t,x),$
(23)

where ${\cal T}_{\text{cas}}^{\text{(in)}}=-{\pi}/({24L_{0}^{2}})$ is the
Casimir energy density, and the functions ${\cal
T}^{\text{(in)}}_{\left\langle\hat{a}^{{\text{(in)}}{\dagger}}\hat{a}^{\text{(in)}}\right\rangle_{\text{in}}}(t,x)$
and ${\cal
T}^{\text{(in)}}_{\left\langle\hat{a}^{\text{(in)}}\hat{a}^{\text{(in)}}\right\rangle_{\text{in}}}(t,x)$
are obtained by using $R(z)={z}/{L_{0}}$ in Eqs. (17) and (18), respectively.
For an arbitrary field state, we have [18]:

$\displaystyle\int_{0}^{L_{0}}{\cal T}^{\text{(cas-in)}}\;dx=E^{\text{(cas-
in)}},\;\;\int_{0}^{L_{0}}{\cal
T}_{\left\langle\hat{a}^{\text{(in)}}\hat{a}^{\text{(in)}}\right\rangle_{\text{in}}}(t,x)\;dx=0,$
(24) $\displaystyle\int_{0}^{L_{0}}{\cal
T}_{\left\langle\hat{a}^{{\text{(in)}}{\dagger}}\hat{a}^{\text{(in)}}\right\rangle_{\text{in}}}(t,x)\;dx=\sum_{n=1}^{\infty}\omega_{n}^{\text{(in)}}{N}^{\text{(in)}}_{n},$
(25)

where the Casimir energy is $E^{\text{(cas-in)}}=-{\pi}/({24L_{0}})$,
$\omega_{n}^{\text{(in)}}=n\pi/L_{0}$, and
${N}_{n}^{\text{(in)}}=\left\langle\hat{a}_{n}^{\text{(in)}{\dagger}}\hat{a}^{\text{(in)}}_{n}\right\rangle_{\text{in}}$
is the mean number of particles in the $n$th mode. We highlight that the
function ${\cal
T}_{\left\langle\hat{a}^{\text{(in)}}\hat{a}^{\text{(in)}}\right\rangle_{\text{in}}}(t,x)$
can give contribution for the local behavior of the energy density ${\cal
T}(t,x)$, but not for the total energy stored in the cavity. From Eqs.
(23)-(25), the total energy
$E^{\text{(in)}}=\int_{0}^{L_{0}}{\cal{T}}(t,x)\;dx$ can be written as [18]

$E^{{\text{(in)}}}=E^{\text{(cas-in)}}+E^{\text{(par-in)}},$ (26)

where $E^{\text{(par-
in)}}=\sum_{n}\omega_{n}^{{\text{(in)}}}{N}_{n}^{\text{(in)}}$ is the total
energy corresponding to the real particles. We highlight that, independently
of the field state, the total vacuum energy in the static cavity is the
Casimir one. Also note that, despite the fact that ${\cal
T}_{\left\langle\hat{a}^{{\text{(in)}}{\dagger}}\hat{a}^{\text{(in)}}\right\rangle_{\text{in}}}$
and ${\cal
T}_{\left\langle\hat{a}^{\text{(in)}}\hat{a}^{\text{(in)}}\right\rangle_{\text{in}}}$
can depend on time, from the principle of energy conservation, the total
energy is a constant in time for the static situation.

When the mirror stops its motion at a instant $\beta$, with the cavity
reaching a final length $L_{\text{fin}}$, we have an “out” state, in general
different from vacuum (even when considering the vacuum as the initial field
state). In the “out” state, the field operator $\hat{\phi}(t,x)$ can be
written as in Eq. (11) or as (see, for instance, Refs. [21, 22])

$\hat{\phi}(t,x)=\sum^{\infty}_{n=1}\left[\hat{a}_{n}\phi^{(0)}_{n}\left(t,x\right)+H.c.\right],$
(27)

where $\hat{a}_{n}^{{\dagger}}$ and $\hat{a}_{n}$ are, respectively, the
creation and annihilation operators related to the $n$th field mode in the
cavity with length $L_{\text{fin}}$, and the field modes
$\phi^{(0)}_{n}\left(t,x\right)$ are obtained by replacing $R(z)\to
z/L_{\text{fin}}$ in Eq. (12). The energy density ${\cal T}(t,x)$, now
relabeled as ${\cal T}^{\text{(out)}}(t,x)$, is given by (see Ref. [18])

${\cal T}^{\text{(out)}}(t,x)={\cal T}^{\text{(cas)}}+{\cal
T}^{\text{(out)}}_{\left\langle\hat{a}^{{\dagger}}\hat{a}\right\rangle_{\text{in}}}(t,x)+{\cal
T}^{\text{(out)}}_{\left\langle\hat{a}\hat{a}\right\rangle_{\text{in}}}(t,x),$
(28)

where ${\cal T}^{\text{(cas)}}=-\pi/(24L_{\text{fin}}^{2})$ is the Casimir
energy density, and the functions ${\cal
T}^{\text{(out)}}_{\left\langle\hat{a}^{{\dagger}}\hat{a}\right\rangle_{\text{in}}}(t,x)$
and ${\cal
T}^{\text{(out)}}_{\left\langle\hat{a}\hat{a}\right\rangle_{\text{in}}}(t,x)$
are obtained from Eqs. (17) and (18), respectively, by making in Eqs. (19) and
(20): $R(z)={z}/{L_{\text{fin}}}$,
$\hat{a}_{n}^{{\text{(in)}}{\dagger}}\to\hat{a}_{n}^{{\dagger}}$, and
$\hat{a}_{n}^{\text{(in)}}\to\hat{a}_{n}$. After performing integrations
similar to those in Eqs. (24) and (25), we find that the total energy in the
cavity in “out” situation is given as written in Sec. II.

##  Appendix B Recovering Fulling-Davies formulas

The particular case of a dynamical cavity subjected to the law of motion given
in Eq. (9) with $\tau=0$, with $\beta=2L_{0}/\left(1-\alpha\right)$, was
considered by Moore [1]. In this case, the null line corresponding to energy
density produced by the discontinuity in the mirror velocity at $t=0$, after
reflection from the static mirror at $x=0$, intersects the moving mirror
exactly when it instantaneously decelerates, which implies in no particle
creation in this process [1, 3, 15]. Here, our focus is on situations where
$\beta<2L_{0}/\left(1-\alpha\right)$, which means that particles can be
created [3, 15].

Due to the presence, in the law of motion given by Eq. (9) with $\tau=0$, of
sudden jumps in the derivatives of $L(t)$ at $t=0$ and $t=\beta$, from Eq.
(1), one can see that the energy density is not defined on the null lines
[dashed lines in Fig.2(a)]. By means of the same equations, we recover the
vacuum energy densities in regions $A$, $B$, $C$ and $D$, as found in Ref.
[3]. At any point $(t,x)$ in the region $A$, the function
$f(u)=\pi/(48L_{0}^{2})$, and
$f\left(v\right)=\left[\pi/(48L_{0}^{2})\right]\left[\left(1-\alpha\right)/\left(1+\alpha\right)\right]^{2}$.
Note that this expression for $f\left(v\right)$ can be viewed as the Doppler-
shifted reflection of the incoming constant field
$f(u)=f^{(s)}=\pi/(48L_{0}^{2})$, which is modified by the factor
$\left[\left(1-\alpha\right)/\left(1+\alpha\right)\right]^{2}$ [17]. Also note
that ${\cal B}(t)=0$, since there is no acceleration in the mirror. Thus,
using these expressions for $f(u)$ and $f(v)$ in Eq. (21), we recover the
vacuum energy density ${\cal T}_{A}$ given in Sec. III. In region $B$, the
function $f(v)=\pi/(48L_{0}^{2})$, and
$f\left(u\right)=\left[\pi/(48L_{0}^{2})\right]\left[\left(1-\alpha\right)/\left(1+\alpha\right)\right]^{2}$.
Thus, we recover ${\cal T}_{B}={\cal T}_{A}$. In region $C$, we have
$f(u)=f(v)=\pi/(48L_{0}^{2})$, so that ${\cal T}_{C}={\cal
T}^{\text{(cas,in)}}=-\pi/(24L_{0})$, as shown in Sec. III. In region $D$, we
have
$f(u)=f(v)=\left[\pi/(48L_{0}^{2})\right]\left[\left(1-\alpha\right)/\left(1+\alpha\right)\right]^{2}$,
and thus, using Eq. (21), we recover ${\cal T}_{D}$ given in Sec. III.

##  Appendix C Expanding cavities with continuous derivatives of ${L}(t)$

One way to avoid the divergences in the energy density occurred at $t=0$ and
$t=\beta$ when considering the law of motion as shown in Fig. 2(a), is looking
for a law of motion such that:

$L\left(t\right)=\begin{cases}L_{0}&\left(t<0\right),\\\
L_{1}\left(t\right)&\left(0\leq t\leq\tau_{1}\right),\\\
L_{1}\left(\tau_{1}\right)+\alpha\left(t-\tau_{1}\right)&\left(\tau_{1}<t<\beta-\tau_{2}\right),\\\
L_{2}\left(t\right)&\left(\beta-\tau_{2}\leq t\leq\beta\right),\\\
L_{\text{fin}}&\left(\beta<t\right),\end{cases}$ (29)

where $\dot{L}_{1}\left(t\right)$, $\ddot{L}_{1}\left(t\right)$,
$\dddot{L}_{1}\left(t\right)$, $\dot{L}_{2}\left(t\right)$,
$\ddot{L}_{2}\left(t\right)$, and $\dddot{L}_{2}\left(t\right)$ are well
defined functions, and:

$\displaystyle L_{1}\left(0\right)=L_{0};\
\dot{L}_{1}\left(0\right)=\ddot{L}_{1}\left(0\right)=\dddot{L}_{1}\left(0\right)=0;$
(30)
$\displaystyle\dot{L}_{1}\left(\tau_{1}\right)=\alpha;\;\ddot{L}_{1}\left(\tau_{1}\right)=\dddot{L}_{1}\left(\tau_{1}\right)=0;$
(31) $\displaystyle
L_{2}\left(\beta-\tau_{2}\right)=L_{1}\left(\tau_{1}\right)+\alpha\left(\beta-\tau_{2}-\tau_{1}\right);\
$ (32)
$\displaystyle\dot{L}_{2}\left(\beta-\tau_{2}\right)=\alpha;\ddot{L}_{2}\left(\beta-\tau_{2}\right)=\dddot{L}_{2}\left(\beta-\tau_{2}\right)=0;\;$
(33) $\displaystyle L_{2}\left(\beta\right)=L_{\text{fin}};\
\dot{L}_{2}\left(\beta\right)=\ddot{L}_{2}\left(\beta\right)=\dddot{L}_{2}\left(\beta\right)=0.$
(34)

These conditions guarantee that the energy density given in Eq. (1) is well
defined in all spacetime points.

Now, for simplicity, we choose

$\displaystyle\tau_{1}=\tau_{2}=\tau<\beta/2,$ (35) $\displaystyle
L_{1}(\tau)=L_{0}+\alpha\tau,$ (36) $\displaystyle
L_{\text{fin}}=L_{0}+\alpha\beta.$ (37)

Thus, we propose $L(t)$ as given in Eq. (9). The function $L_{1}$ naturally
satisfies Eq. (30), and $L_{2}$ satisfies Eqs. (34) and (37). The constants
$a_{1}$, $b_{1}$, $c_{1}$ and $d_{1}$ are found by solving Eqs. (31) and (36),
whereas $a_{2}$, $b_{2}$, $c_{2}$ and $d_{2}$, by solving Eqs. (32) and (33).
The solution found is valid if

$|\dot{L}\left(t\right)|<1.$ (38)

We show in Sec. V that valid solutions for $L(t)$ can be found for appropriate
choices of $L_{0}$, $\alpha$, $\beta$ and $\tau$.

##  Appendix D Vacuum and Casimir energy in vacuum regions

In this section, we focus our discussion on situations like the one
illustrated in Fig. 2(b). Let us define the following instants:
$t_{a}^{(1)}=L_{0}$, $t_{a}^{\prime(1)}=L_{0}+\tau(1+\alpha)$,
$t_{\text{enc}}=L_{0}+(1/2)(\beta-\tau)(1+\alpha)$,
$t_{N}=L_{0}+(1/2)\beta(1+\alpha)$,
$t_{\text{des}}=L_{0}+(1/2)(\beta-\tau)(1+\alpha)+\tau(1+\alpha)$,
$t_{a}^{(2)}=2L_{0}+\alpha\beta$,
$t_{a}^{\prime(2)}=2L_{0}+\tau(1+\alpha)+\alpha\beta$,
$t_{b}^{\prime(1)}=L_{0}+(\beta-\tau)(1+\alpha)$,
$t_{b}^{(1)}=L_{0}+\beta(1+\alpha)$. We consider $t_{b}^{\prime(1)}\geq
t_{a}^{\prime(2)}$, which implies that $\beta\geq
L_{0}+2\tau\left(\alpha+1\right)$ (this also implies that $\beta\geq
t_{a}^{\prime(1)}$). In addition, we also consider a value of $\beta$ chosen
in such a way that in the time interval $[0,\beta]$ there is no superposition
between the pulses of the excited field, which implies that
$\beta\leq{[2L_{0}-\tau(\alpha+1)]}/{(1-\alpha)}$ [for $\tau\neq 0$, this
means that $\beta<{2L_{0}}/{(1-\alpha)}$, in agreement with our previous
statement]. Both conditions lead to
$L_{0}+2\tau(1+\alpha)\leq\beta\leq[2L_{0}-\tau(\alpha+1)]/(\alpha-1)$. The
formula for $E_{\text{(vac)}}^{(\text{vac})}\left(t\right)$, for $\beta\leq
t<t_{b}^{(1)}$, is given by

$E^{\text{(vac)}}_{(\text{vac})}\left(t\right)=\begin{cases}E^{\text{(vac)}}_{(\text{DAC})}&\left(\beta\leq
t<t_{\text{enc}}\right),\\\
E^{\text{(vac)}}_{(\text{DC}1)}(t)&\left(t_{\text{enc}}\leq t<t_{N}\right),\\\
E^{\text{(vac)}}_{(\text{DC}2)}(t)&\left(t_{N}^{(1)}\leq
t<t_{\text{des}}\right),\\\
E^{\text{(vac)}}_{(\text{DBC})}&\left(t_{\text{des}}\leq
t<t_{a}^{(2)}\right),\\\
E^{\text{(vac)}}_{(\text{DB}1)}(t)&\left(t_{a}^{(2)}\leq
t<t_{N}^{(2)}\right),\\\
E^{\text{(vac)}}_{(\text{DB}2)}(t)&\left(t_{N}^{(2)}\leq
t<t_{a}^{\prime(2)}\right),\\\
E^{\text{(vac)}}_{(\text{DBD})}&\left(t_{a}^{\prime(2)}\leq
t<t_{b}^{\prime(1)}\right),\\\
E^{\text{(vac)}}_{(\text{BD1})}(t)&\left(t_{b}^{\prime(1)}\leq
t<t_{N}^{(2)}\right),\\\
E^{\text{(vac)}}_{(\text{BD2})}(t)&\left(t_{N}^{(2)}\leq
t<t_{b}^{(1)}\right),\end{cases}$ (39)

where:

$\displaystyle E^{\text{(vac)}}_{(\text{DAC})}=$
$\displaystyle\frac{\pi}{24L_{0}^{2}\left(1+\alpha\right)}\left[\alpha^{2}\left(\beta-2\tau\right)\right.$
$\displaystyle\left.+\alpha\left(-\beta+2\tau+L_{0}\right)-2\tau+L_{0}\right],$
(40) $\displaystyle E^{\text{(vac)}}_{(\text{DC}1)}(t)=$
$\displaystyle-\frac{\pi}{24L_{0}^{2}}\left[\frac{\left(1-\alpha\right)^{2}\left(t-L_{0}-\tau\left(1+\alpha\right)\right)}{\left(1+\alpha\right)^{2}}\right]$
$\displaystyle-\frac{\pi}{24L_{0}^{2}}\left(\beta+t\right),$ (41)
$\displaystyle E^{\text{(vac)}}_{(\text{DC}2)}(t)=$
$\displaystyle\frac{\pi}{24L_{0}^{2}}\frac{\left(1-\alpha\right)^{2}\left[-t+L_{0}+\left(\beta-\tau\right)\left(1+\alpha\right)\right]}{\left(1+\alpha\right)^{2}}$
$\displaystyle-\frac{\pi}{24L_{0}^{2}}\left(\alpha\beta+2{\it
L_{0}}-t\right),$ (42) $\displaystyle
E^{\text{(vac)}}_{(\text{DBC})}=E^{\text{(vac)}}_{(\text{DAC})},$ (43)
$\displaystyle E^{\text{(vac)}}_{(\text{DB1})}(t)=$
$\displaystyle\frac{\pi}{24L_{0}^{2}\left(1+\alpha\right)}\left[-t\left(\alpha+1\right)+2\alpha^{2}\tau+L_{0}\alpha\right.$
$\displaystyle\left.+2\alpha\beta-2\tau\alpha+L_{0}+2\tau\right],$ (44)
$\displaystyle E^{\text{(vac)}}_{(\text{DB2})}(t)=$
$\displaystyle-\frac{\pi\left(1-\alpha\right)^{2}}{24L_{0}^{2}\left(1+\alpha\right)^{2}}\left[-t+L_{0}+\left(\beta-\tau\right)\left(1+\alpha\right)\right]$
$\displaystyle-\frac{\pi\left(\alpha^{2}+1\right)}{24L_{0}^{2}\left(1+\alpha\right)^{2}}\left[2L_{0}+\beta\left(-1+\alpha\right)\right],$
(45) $\displaystyle E^{\text{(vac)}}_{(\text{DBD})}=$
$\displaystyle-\frac{\pi}{24L_{0}^{2}\left(1+\alpha\right)}\left[\left(\alpha-1\right)\alpha\beta-2\alpha\tau+L_{0}\alpha\right.$
$\displaystyle\left.+4\tau\alpha+L_{0}-2\tau\right],$ (46) $\displaystyle
E^{\text{(vac)}}_{(\text{BD1})}(t)=$
$\displaystyle-\frac{\pi}{24L_{0}^{2}\left(1+\alpha\right)^{2}}\left[\left(\alpha^{2}-1\right)\beta-\alpha^{3}\tau+\alpha^{2}t\right.$
$\displaystyle\left.+\alpha^{2}\tau+4L_{0}\alpha-2\alpha
t+\tau\alpha+t-\tau\right],$ (47) $\displaystyle
E^{\text{(vac)}}_{(\text{BD2})}(t)=$
$\displaystyle-\frac{\pi}{24L_{0}^{2}\left(1+\alpha\right)}\left[\left(2\alpha^{2}+\alpha+1\right)\beta-2\alpha^{2}\tau\right.$
$\displaystyle\left.+2L_{0}\alpha-\alpha t+2\tau\alpha+2L_{0}-t-2\tau\right].$
(48)

The formula for $E^{\text{(cas)}}_{(\text{vac})}\left(t\right)$, for
$\beta\leq t<t_{b}^{(1)}$, is given by

$E^{\text{(cas)}}_{(\text{vac})}\left(t\right)=\begin{cases}E^{\text{(cas)}}_{\text{(DAC)}}&\left(\beta\leq
t<t_{\text{enc}}\right),\\\
E^{\text{(cas)}}_{(\text{DC}1)}(t)&\left(t_{\text{enc}}\leq t<t_{N}\right),\\\
E^{\text{(cas)}}_{(\text{DC}2)}(t)&\left(t_{N}^{(1)}\leq
t<t_{\text{des}}\right),\\\
E^{\text{(cas)}}_{(\text{DBC})}&\left(t_{\text{des}}\leq
t<t_{a}^{(2)}\right),\\\
E^{\text{(cas)}}_{(\text{DB}1)}(t)&\left(t_{a}^{(2)}\leq
t<t_{N}^{(2)}\right),\\\
E^{\text{(cas)}}_{(\text{DB}2)}(t)&\left(t_{N}^{(2)}\leq
t<t_{a}^{\prime(2)}\right),\\\
E^{\text{(cas)}}_{(\text{DBD})}&\left(t_{a}^{\prime(2)}\leq
t<t_{b}^{\prime(1)}\right),\\\
E^{\text{(cas)}}_{(\text{BD1})}(t)&\left(t_{b}^{\prime(1)}\leq
t<t_{N}^{(2)}\right),\\\
E^{\text{(cas)}}_{(\text{BD2})}(t)&\left(t_{N}^{(2)}\leq
t<t_{b}^{(1)}\right),\end{cases}$ (49)

where:

$\displaystyle
E^{\text{(cas)}}_{(\text{DAC})}=-\frac{\pi}{24}\frac{(\alpha\beta+L_{0}-2\tau-2\tau\alpha)}{(\alpha\beta+L_{0})^{2}},$
(50) $\displaystyle
E^{\text{(cas)}}_{(\text{DC1})}(t)=\frac{\pi}{24}\frac{(\alpha\tau+L_{0}+\beta-2t+\tau)}{(\alpha\beta+L_{0})^{2}},$
(51) $\displaystyle
E^{\text{(cas)}}_{(\text{DC2})}(t)=-\frac{\pi}{24}{\frac{\left(2\alpha\beta-\alpha\tau+3{L_{0}}+\beta-2t-\tau\right)}{\left(\alpha\beta+{L_{0}}\right)^{2}}},$
(52) $\displaystyle
E^{\text{(cas)}}_{(\text{DBC})}=E^{\text{(cas)}}_{(\text{DAC})},$ (53)
$\displaystyle
E^{\text{(cas)}}_{(\text{DB1})}(t)=\frac{\pi}{24}{\frac{\left(2\alpha\tau+{L_{0}}-t+2\tau\right)}{\left(\alpha\beta+{L_{0}}\right)^{2}}},$
(54) $\displaystyle
E^{\text{(cas)}}_{(\text{DB2})}(t)=-\frac{\pi}{24}{\frac{\left(2\alpha\beta-\alpha\tau+3{L_{0}}-t-\tau\right)}{\left(\alpha\beta+{L_{0}}\right)^{2}}},$
(55) $\displaystyle
E^{\text{(cas)}}_{(\text{DBD})}=-\frac{\pi}{24}{\frac{\left(\alpha\beta-2\alpha\tau+{L_{0}}-2\tau\right)}{\left(\alpha\beta+{L_{0}}\right)^{2}}},$
(56) $\displaystyle
E^{\text{(cas)}}_{(\text{BD1})}(t)=-\frac{\pi}{24}{\frac{\left(\alpha\tau+\beta-t+\tau\right)}{\left(\alpha\beta+{L_{0}}\right)^{2}}},$
(57) $\displaystyle
E^{\text{(cas)}}_{(\text{BD2})}(t)=-\frac{\pi}{24}{\frac{\left(2\alpha\beta-2\alpha\tau+2{L_{0}}+\beta-t-2\tau\right)}{\left(\alpha\beta+{L_{0}}\right)^{2}}}.$
(58)

##  Appendix E Some details of the application to a concrete law of motion

Let us divide the regions in each Fig. 5(a) and 5(b) into: vacuum regions I
(vac-I) and II (vac-II), corresponding, respectively, to the left and right
lighter regions in each one of these figures; in a similar way, particle
regions I (par-I) and II (par-II), corresponding, respectively, to the left
and right darker regions. After integrating the energy densities in Fig. 5, in
each of their respective regions, we have:
$E_{\cal\tilde{A}\text{(vac-I)}}^{\text{(vac)}}\left(\beta\right)\approx-0.016$;
$E_{\cal\tilde{A}\text{(vac-
II)}}^{\text{(vac)}}\left(\beta\right)\approx-0.049$;
$E_{\cal\tilde{A}\text{(vac)}}^{\text{(vac)}}\left(\beta\right)=E_{\cal\tilde{A}\text{(vac-I)}}^{\text{(vac)}}\left(\beta\right)+E_{\cal\tilde{A}\text{(vac-
II)}}^{\text{(vac)}}\left(\beta\right)\approx-0.065$;
$E_{\cal\tilde{A}\text{(par-I)}}^{\text{(vac)}}\left(\beta\right)\approx-0.002$;
$E_{\cal\tilde{A}\text{(par-
II)}}^{\text{(vac)}}\left(\beta\right)\approx-0.012$;
$E_{\cal\tilde{A}\text{(par)}}^{\text{(vac)}}\left(\beta\right)=E_{\cal\tilde{A}\text{(par-I)}}^{\text{(vac)}}\left(\beta\right)+E_{\cal\tilde{A}\text{(par-
II)}}^{\text{(vac)}}\left(\beta\right)\approx-0.014$;
$E_{\cal\tilde{B}\text{(par-I)}}\left(\beta\right)\approx 1.349$;
$E_{\cal\tilde{B}\text{(par-II)}}\left(\beta\right)\approx 1.349$;
$E_{\cal\tilde{B}}\left(\beta\right)=E_{\cal\tilde{B}\text{(par-I)}}\left(\beta\right)+E_{\cal\tilde{B}\text{(par-
II)}}\left(\beta\right)\approx 2.699$;
$E^{\text{(cas)}}\left(\beta\right)\approx-0.054$. Considering the length $l$
of each particle region $l=0.155$, we have $\overline{{\cal
T}}_{\text{(par)}}^{\text{(vac)}}\left(t,x\right)=(E_{\cal\tilde{A}\text{(par)}}^{\text{(vac)}}\left(\beta\right)+E_{\cal\tilde{B}\text{(par)}}^{\text{(vac)}})/(2l)\approx
0.033$.

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