# Joint evolution of a
Lorentz-covariant massless scalar field
and its point-charge source
in one space dimension

Lawrence Frolov∗, Samuel Leigh†, and Shadi Tahvildar-Zadeh∗
∗ Department of Mathematics, Rutgers University (New Brunswick)
† Department of Physics, University of Washington

(May 2024)

###### Abstract

In this paper we prove that the static solution of the Cauchy problem for a
massless real scalar field that is sourced by a point charge in $1+1$
dimensions is asymptotically stable under perturbation by compactly-supported
radiation. This behavior is due to the process of back-reaction. Taking the
approach of Kiessling, we rigorously derive the expression for the force on
the particle from the principle of total energy-momentum conservation. We
provide a simple, closed form for the particle’s self-action, and show that it
is restorative in this model , i.e. proportional to negative velocity, and
causes the charge to return to rest after the radiation passes through. We
establish these results by studying the joint evolution problem for the
particle-scalar field system, and proving its global well-posedness and the
claimed asymptotic behavior.

##  1 Introduction and Statement of Main Results

###  1.1 Background

Consider the dynamics of a vibrating point charge and the electromagnetic
field it is sourcing. As the particle oscillates it radiates electromagnetic
waves which propagate away from the charge at the speed of light. These waves
carry both energy and momentum, so to conserve the total energy-momentum of
the system, the particle must undergo some dampening through an interaction
with its own field. This dampening self-interaction is one example of back-
reaction, the process in which charge distributions source fields which then
“re-act” on the charges.

As it is well-known, attempting to study the process of back-reaction in the
framework of Maxwell-Lorentz electrodynamics leads to a fundamental
inconsistency. The Lorentz force needed to calculate the path of the particle
requires us to evaluate the electromagnetic field along the particle’s path,
and yet the field sourced by the particle is undefined precisely on this path!
Resolving this inconsistency has been an open problem for more than a century,
and has been worked on by many notable figures such as Abraham [15], Poincaré
[17] and Dirac [6]. A gripping account of this endeavor can be found in [19],
which also includes an excellent review of the main approach mathematicians
have taken to successfully resolve the inconsistency, namely smearing the
point charge into a smooth charge distribution. In this approach however, it
is not possible to take the smearing away, once it is introduced. The
introduction of smearing also complicates the task of keeping the system of
equations fully Lorentz-covariant, leading to many of the results obtained
thus far being restricted to non-relativistic motions of the particle. (See
also [12], Section 2.3, for more recent results in this direction.)

Previous techniques of studying these field-particle systems directly, either
without smearing or with smearing that is put in and then taken away, has left
something to be desired. Outlined in a recent review [18], they include an
infinite bare-mass renormalization that is mathematically ill-defined, or an
ad-hoc averaging of the fields in a neighborhood of the charge. A breakthrough
occurred in 2019 when, following up on the work of Poincaré [17], Kiessling
[9, 10] showed that postulating energy-momentum conservation of the field-
particle system yields a unique and admissible force law, provided that the
field’s momentum density is locally integrable around the particle. Although
this integrability assumption rules out the classical vacuum law of Maxwell,
given by $E=D$, $B=H$, it admits others, such as the Bopp-Landé-Thomas-
Podolsky (BLTP) vacuum law [2, 3, 13, 14, 16].

In three dimensions, the Maxwell electromagnetic field sourced by a point
charge is not only too singular to evaluate along the charge’s path, but also
its energy and momentum densities are not locally integrable around the
source. By contrast BLTP, which is a higher order modification of Maxwell,
does have the regularity necessary to derive a unique force law from the
assumption of energy-momentum conservation. In the context of BLTP, Kiessling
and Tahvildar-Zadeh [11] have successfully applied this force law to prove
local well-posedness of the joint field-particle dynamics, and Hoang et al.
[8] proved global existence for the scattering problem of a single charge
interacting with a smooth potential. However, the complex form of the BLTP
energy-momentum conserving force law has so far resisted a clear analysis of
the asymptotic behavior of the particle.

This paper studies the dynamics of a relativistic point charge coupled to a
massless scalar field on flat $1+1$ dimensional space-time. Working in one
space dimension provides us with the regularity needed to derive the energy-
momentum conserving force law without making any higher order modifications to
the theory, thus allowing a much simpler analysis to be performed. We choose
to study scalar charges because electromagnetism is not viable in one space
dimension. Our model closely resembles the one studied in [1], with the
exception that our dynamics are fully relativistic. To isolate the effects of
the interaction between the scalar charge and its own field, we will be
focusing on the case of a single particle perturbed by scalar radiation.

###  1.2 Main Results

Taking Kiessling’s approach, we show that the 2-force which acts on a single
charged particle at $z=(z^{0},z^{1})\in\mathbb{R}^{1,1}$ is given by

$F^{\mu}(z^{0},z^{1})=-\bigg{[}n_{\nu}T^{\mu\nu}_{S}(z^{0},x^{1})\bigg{]}_{x^{1}=z^{1}},$
(1.1)

where $n_{\mu}$ is the unit covector that is annihilated by the particle’s
two-velocity, $T_{S}^{\mu\nu}$ is the energy-momentum tensor of the field, and
$[\cdot]_{x^{1}=z^{1}}$ denotes the jump in space at $z^{1}$. The force law
given by equation (1.1) is derived from the principle of energy-momentum
conservation

$\partial_{\nu}T_{p}^{\mu\nu}+\partial_{\nu}T_{S}^{\mu\nu}=0,$ (1.2)

where $T_{p}^{\mu\nu}$ is the energy-momentum tensor of the particle, which is
concentrated on the particle’s world-line, and the derivatives are taken
weakly, to account for singularities in the two energy tensors.

Guided by Weyl’s “agens theory” of matter [20], according to which the world-
lines of matter particles are simply the locus of singularities of the
underlying spacetime and/or the fields defined on that spacetime, we are led
to the study of a joint evolution problem in which the path of the particle
$z(\tau)$ appears as a jump discontinuity in the derivatives of the scalar
field $U$, with the jumps showing up inside the force term in the equation for
$\ddot{z}$. The equation of motion for a scalar field with a point charge
source is

$\eta^{\mu\nu}\partial_{\mu}\partial_{\nu}U(x)=a\int\delta^{(2)}(x-z(\tau))d\tau$
(1.3)

where $\tau$ and $a$ are the particle’s proper time and scalar charge
respectively. We consider a stationary field-particle system which is
perturbed by some incoming scalar “radiation.” The joint evolution problem
corresponding to this is given by the initial value problem for the unknown
field $U=U(x)$ and unknown trajectory $z=(z^{\mu}(\tau))$ satisfying

$\left\\{\begin{array}[]{rcl}\eta^{\mu\nu}\partial_{\mu}\partial_{\nu}U(x)&=&a\int\delta^{(2)}(x-z(\tau))d\tau\\\
U(0,x^{1})&=&-\frac{a}{2}|x^{1}|+V_{0}(x^{1})\\\
\partial_{0}U(0,x^{1})&=&V_{1}(x^{1}),\end{array}\right.$ (1.4)
$\begin{cases}\frac{dz^{\mu}}{d\tau}(\tau)=:u^{\mu}(\tau)\\\
\frac{dp^{\mu}}{d\tau}(\tau)=F^{\mu}(z(\tau)),\qquad
p^{\mu}(\tau):=(\tilde{m}-aU(z(\tau)))u^{\mu}(\tau).\end{cases}$ (1.5)

where $\tilde{m}$ is the bare mass of the particle and $F$ is as in (1.1). We
work in the fixed Lorentz frame where the particle remained stationary at the
origin for all time $x^{0}<0$. Thus the data for (1.5) is $z(\tau)=(0,0)$ and
$p(\tau)=(\tilde{m},0)$ for all $\tau\leq 0$. The motion of the charge is
perturbed by incoming radiation, which is represented by $V_{0}$, $V_{1}$ in
the initial data for $U$. The following is an informal statement of the first
main result for this paper (for the precise statement, see Theorem 3.1):

######  Theorem.

For any set of particle parameters with positive bare mass and non-zero real
scalar charge, and for any set of small, smooth, compactly supported functions
$V_{0}(x^{1})$, $V_{1}(x^{1})$, the joint initial value problem given by (1.4)
and (1.5) admits a unique, global-in-time solution.

To prove this, we explicitly compute the forces in equation (1.1), and from
that we extract a closed form for the force that scalar point charges exert on
themselves. We show that the scalar self-force resulting from this back-
reaction is proportional to the negative of the particle’s velocity, see
equation (3.6).

######  Remark 1.1.

We were able to determine this self-force because scalar fields sourced by
point charges are regular enough in one space dimension for radiation to be
emitted at a finite rate. However, the interaction between scalar fields and
their charges generate mass (see (2.6)), and we show that there is
cancellation between said self-force and the particle’s loss in mass (see
(4.1)). Hence, there is a sense in which the radiation-reaction in this model
does not contribute to the proper acceleration of the point charge, only the
back-reaction on the particle of the static field in its own past does.

The following is an informal statement of the second main result for this
paper (for the precise statement, see Theorem 4.1):

######  Theorem.

Let $z$, $U$ satisfy the joint IVP given by (1.4–1.5) for a single scalar
particle with positive bare mass and non-zero real scalar charge, and $V_{0}$,
$V_{1}$ small, smooth, and compactly supported. Then
$\lim_{x^{0}\to\infty}u^{1}(x^{0})=0$.

Accordingly (by Lorentz covariance), a charged particle undergoing uniform
motion in some frame, when perturbed by some compactly supported radiation,
asymptotically returns to that uniform motion. This stability is not a result
of radiation-reaction, for reasons explained in Remark 1.1. It is instead a
consequence of the charge’s interaction with its own freely evolving field
coming from the $-\frac{a}{2}|x|$ term in the initial data of (1.4).

######  Remark 1.2.

In forthcoming work [7], we show that such an asymptotic stability result does
not hold for charges coupled to massive scalar fields in one space dimension.
This is because the stationary particle’s massive field has finite energy, as
a result of which the charge’s interaction with the field decays sufficiently
rapidly in time, so that the motion may asymptote to a different steady state,
which is a form of orbital stability. Asymptotic stability may thus be
specific to charges coupled to massless scalar fields in one dimension.
Indeed, if the dynamics of smeared-out charges [19] can be a guide, it is
likely that this will not hold in general for electromagnetic point charges,
or in higher dimensions.

In section 2 we derive the equations of motion for a scalar field from the
principle of stationary action, and derive the force law from the assumption
of energy conservation. In section 3 we present a proof of the global well-
posedness result for the joint evolution of a single scalar particle and its
field. In section 4 we study the asymptotics of our joint evolution, and
provide a proof of the asymptotic stability of the stationary solution.

##  2 Derivation of Equations of Motion

###  2.1 Field Equations From Principle of Stationary Action

We derive the equations of motion for our scalar field using the principle of
stationary action. The action for a point charge coupled to a massless scalar
field $U$ defined on $1+1$ dimensional flat space-time111We work with
signature $\eta_{\mu\nu}=\text{diag}(1,-1)$. is given by

$S[U,z]=\int\mathcal{L}\sqrt{-\eta}d^{2}x,$ (2.1)

where the Lagrangian density $\mathcal{L}$ is defined via

$\begin{split}\mathcal{L}(x):=&\frac{1}{\sqrt{-\eta}}\int-(\tilde{m}-aU(z))\sqrt{\eta_{\mu\nu}\dot{z}^{\mu}\dot{z}^{\nu}}\delta^{(2)}(x-z)d\theta\\\
&+\frac{1}{2}\eta^{\mu\nu}\partial_{\mu}U\partial_{\nu}U(x).\end{split}$ (2.2)

Here $z^{\mu}$, $\tilde{m}$, and $a$ represent the particle’s space-time
position, bare mass, and scalar charge respectively, while $\theta$ is an
arbitrary parameterization of the particle’s worldline. Extremizing the action
by taking variations with respect to $U$ returns

$\eta^{\mu\nu}\partial_{\mu}\partial_{\nu}U=a\int\sqrt{\eta_{\mu\nu}\dot{z}^{\mu}\dot{z}^{\nu}}\delta^{(2)}(x-z)d\theta=a\int\delta^{(2)}(x-z(\tau))d\tau,$
(2.3)

where $\tau$ is the particle’s proper time defined by
$d\tau=\sqrt{\eta_{\mu\nu}\dot{z}^{\mu}\dot{z}^{\nu}}d\theta$. Equation (2.3)
shows that the point charge is acting as a singularity in the derivatives of
the scalar field $U$. Given a world-line for our particle along with some
specified initial data for $U$, we could solve for the evolution of $U$ by
solving the associated initial value problem. However, we are interested in
studying the joint evolution problem of our field-particle system. So the
motion of the charge must be in accordance with all the scalar forces that act
on it. We may naively derive the forces acting on our charges by taking
variations of the action with respect to $z^{\mu}$. This leads to a familiar
law of motion that is inconsistent with the field equations, namely:

$\frac{dp^{\mu}}{d\tau}=-a\partial^{\mu}U(z),$ (2.4)

where $p$ is the dynamical momentum defined by

$p^{\mu}:=(\tilde{m}-aU(z))u^{\mu},\quad u^{\mu}:=\frac{dz^{\mu}}{d\tau}.$
(2.5)

The inconsistency arises since the force law (2.4) derived from the principle
of stationary action requires one to evaluate the first derivatives of the
scalar field along the particle’s world-line. However, the field equations
derived from the same principle imply that the first derivatives of the field
are undefined precisely along this world-line. So there can be no joint
evolution of the field-particle system which extremizes the action.

In non-rigorous settings, one is typically taught to ignore the ill-defined
self-interactions terms. But the dynamics given by ignoring singular self-
forces is off-shell. In particular, it does not conserve the total energy-
momentum of the system, as we will show later. The problems we face here are
not unique to scalar point particles, and we take inspiration from the work of
Kiessling on force laws for electromagnetic particles in three space-
dimensions [9], to derive a rigorous force law for our scalar point charges.

We conclude this section by defining the dynamical mass $m$ which depends on
the field $U$ evaluated along the particle’s world-line

$m(x^{0}):=\frac{p^{\mu}}{u^{\mu}}=\tilde{m}-aU(x^{0},z^{1}(x^{0})),$ (2.6)

where

$z^{1}(x^{0}):=z^{1}(\tau^{-1}(x^{0})),$ (2.7)

and $\tau^{-1}(x^{0})$ is the unique value for which
$z^{0}(\tau^{-1}(x^{0}))=x^{0}$.

######  Remark 2.1.

The dynamical mass and momentum will frequently appear throughout our
calculations, taking the place of mass and momentum wherever one would expect
the latter two to appear.

###  2.2 Force Law From Conservation of Energy-Momentum

In this section we will derive our force law from the assumption of the local
conservation of energy-momentum. From the action (2.1) we derive the following
energy density-momentum density-stress tensor (energy tensor, for short) for
our system:

$\begin{split}T^{\mu\nu}(x)=T_{p}^{\mu\nu}(x)+T_{S}^{\mu\nu}(x),\end{split}$
(2.8)

where

$T_{p}^{\mu\nu}:=m(x^{0})\int
u^{\mu}u^{\nu}\delta^{(2)}(x-z(\tau))d\tau,\hskip
14.22636ptT_{S}^{\mu\nu}:=(\partial^{\mu}U)(\partial^{\nu}U)-\frac{1}{2}\eta^{\mu\nu}(\partial_{\alpha}U)(\partial^{\alpha}U).$
(2.9)

The total energy tensor is singular along the particle’s world-line, so we
will take conservation of energy-momentum in the weak sense of distributions
to derive our force law.

######  Theorem 2.1.

Suppose $U$ is such that

$\forall x^{0}\geq 0,\ \lim_{\epsilon\rightarrow
0}\int_{z^{1}(x^{0})-\epsilon}^{z^{1}(x^{0})+\epsilon}T^{0\nu}_{S}(x^{0},x^{1})dx^{1}=0.$
(2.10)

(This requires $T_{S}^{0\nu}$ be locally integrable222While this condition
does not hold for fields sourced by scalar point charges in three space
dimensions, it holds in one space dimension. around $z$.) Then, assuming
energy-momentum conservation in the weak sense of

$\int_{\Omega}\partial_{\mu}T^{\mu 0}dV=0=\int_{\Omega}\partial_{\mu}T^{\mu
1}dV,$ (2.11)

for all tubular regions $\Omega$ around the particle’s world-line, yields the
unique force law

$\frac{dp^{\nu}}{d\tau}=-\bigg{[}n_{\mu}T^{\mu\nu}_{S}(z^{0},x^{1})\bigg{]}_{x^{1}=z^{1}},$
(2.12)

where $n_{\mu}=(-u^{1},u^{0})$ is the space-like unit covector annihilated by
$u^{\mu}$.

###### Proof.

Fix $\nu$. By the definition of weak derivatives,

$\int_{\Omega}\partial_{\mu}T^{\mu\nu}dV=\int_{\partial\Omega}T^{\alpha\nu}N_{\alpha}dS,$
(2.13)

where $N$ are the unit normal vectors to our boundary $\partial\Omega$ and
$dS$ is the surface element induced by $\eta$. The boundary $\partial\Omega$
consists of four parts (see Figure 1): Two space-like curves given by

$\begin{split}&\bar{T}_{1}=\\{x\in\mathbb{R}^{1,1}|z^{1}(T_{1})-\epsilon<x^{1}<z^{1}(T_{1})+\epsilon,x^{0}=T_{1}\\}\\\
&\bar{T}_{2}=\\{x\in\mathbb{R}^{1,1}|z^{1}(T_{2})-\epsilon<x^{1}<z^{1}(T_{2})+\epsilon,x^{0}=T_{2}\\}\end{split}$
(2.14)

Figure 1:  Region of integration and its normals

and two time-like curves given by

$\begin{split}&\bar{C}_{1}=\\{x\in\mathbb{R}^{1,1}|T_{1}<x^{0}<T_{2},x^{1}=z^{1}(x^{0})-\epsilon\\}\\\
&\bar{C}_{2}=\\{x\in\mathbb{R}^{1,1}|T_{1}<x^{0}<T_{2},x^{1}=z^{1}(x^{0})+\epsilon\\}.\end{split}$
(2.15)

The unit normal covectors for these boundaries are given by

$N^{T_{1}}_{\mu}=-\begin{pmatrix}1\\\ 0\end{pmatrix},\quad
N^{T_{2}}_{\mu}=\begin{pmatrix}1\\\ 0\end{pmatrix},\quad
N^{C_{1}}_{\mu}=-\begin{pmatrix}-\frac{\partial z^{1}}{\partial\tau}\\\
\frac{\partial z^{0}}{\partial\tau}\end{pmatrix},\quad
N^{C_{2}}_{\mu}=\begin{pmatrix}-\frac{\partial z^{1}}{\partial\tau}\\\
\frac{\partial z^{0}}{\partial\tau}\end{pmatrix}.$ (2.16)

Notice that $N_{\mu}^{C_{2}}=-N_{\mu}^{C_{1}}=n_{\mu}$ is the unit covector
annihilated by $u^{\mu}$, up to a sign. Since $T_{p}^{\mu\nu}$ has no support
on the boundaries $\bar{C}_{1}$ and $\bar{C}_{2}$ their contribution to the
surface integral vanishes, and we are left with

$\int_{\partial\Omega}T^{\alpha\nu}_{p}N_{\alpha}dS=\int_{z^{1}(T_{2})-\epsilon}^{z^{1}(T_{2})+\epsilon}T^{0\nu}_{p}dx^{1}-\int_{z^{1}(T_{1})-\epsilon}^{z^{1}(T_{1})+\epsilon}T^{0\nu}_{p}dx^{1}.$
(2.17)

Recall that

$T^{0\nu}_{p}=m(x^{0})\int
u^{0}u^{\nu}\delta^{(2)}(x-z(\tau))d\tau=p^{\nu}\delta(x^{1}-z^{1}).$ (2.18)

Plugging this into (2.17), we arrive at

$\int_{\partial\Omega}T^{\alpha\nu}_{p}N_{\alpha}dS=p^{\nu}(T_{2})-p^{\nu}(T_{1})=\int_{\tau_{1}}^{\tau_{2}}\frac{dp^{\nu}}{d\tau}d\tau.$
(2.19)

where $z^{0}(\tau_{i})=T_{i}$. So our assumption of weak conservation of
energy-momentum grants us the following equation

$\int_{\tau_{1}}^{\tau_{2}}\frac{dp^{\nu}}{d\tau}d\tau=-\int_{\partial\Omega}T_{S}^{\alpha\nu}N_{\alpha}dS,$
(2.20)

which states that the total change in the particle’s momentum is equal to the
total energy flux of the field through a tube around the world-line of the
particle. The L.H.S of this equation has no dependence on the width of our
tubular region $\epsilon$, so it follows that the R.H.S should also have no
dependence on $\epsilon$. Taking the limit as $\epsilon$ goes to zero returns

$\int_{\tau_{1}}^{\tau_{2}}\frac{dp^{\nu}}{d\tau}d\tau=-\lim_{\epsilon\to
0}\int_{\partial\Omega}T_{S}^{\alpha\nu}N_{\alpha}dS.$ (2.21)

By our assumptions two of the boundary contributions vanish:

$\lim_{\epsilon\rightarrow
0}\int_{\bar{T}_{2}}T^{\alpha\nu}_{S}N_{\alpha}dS=\lim_{\epsilon\rightarrow
0}\int_{z^{1}(T_{2})-\epsilon}^{z^{1}(T_{2})+\epsilon}T^{0\nu}_{S}dx^{1}=0,$
(2.22)

and

$\lim_{\epsilon\rightarrow
0}\int_{\bar{T}_{1}}T^{\alpha\nu}_{S}N_{\alpha}dS=\lim_{\epsilon\rightarrow
0}\int_{z^{1}(T_{1})-\epsilon}^{z^{1}(T_{1})+\epsilon}-T^{0\nu}_{S}dx^{1}=0.$
(2.23)

When we parameterize the remaining space-like boundaries by $\tau$, we obtain

$\int_{\bar{C}_{2}}T^{\alpha\nu}_{S}N^{\bar{C}_{2}}_{\alpha}dS=\int_{\tau_{1}}^{\tau_{2}}N_{\alpha}^{C_{2}}T^{\alpha\nu}_{S}(z^{0}(\tau),z^{1}(\tau)+\epsilon)d\tau,$
(2.24)

and

$\int_{\bar{C}_{1}}T^{\alpha\nu}_{S}N^{\bar{C}_{1}}_{\alpha}dS=-\int_{\tau_{1}}^{\tau_{2}}N_{\alpha}^{C_{2}}T^{\alpha\nu}_{S}(z^{0}(\tau),z^{1}(\tau)-\epsilon)d\tau_{.}$
(2.25)

Plugging these back into equation (2.21) returns

$\int_{\tau_{1}}^{\tau_{2}}\frac{dp^{\nu}}{d\tau}d\tau=-\int_{\tau_{1}}^{\tau_{2}}\bigg{[}n_{\mu}T^{\mu\nu}_{S}(z^{0},x^{1})\bigg{]}_{x^{1}=z^{1}}d\tau.$
(2.26)

Since this holds for all $\tau_{1},\tau_{2}$, Equation (2.12) immediately
follows. ∎

##  3 Well-Posedness of Joint Evolution

In this section we present the global well-posedness result for a scalar
particle perturbed by some smooth radiation. We work in the fixed Lorentz
frame where the scalar charge was stationary and remained at the origin for
all time $x^{0}<0$. The radiation will appear as smooth initial data in the
IVP for $U$.

###  3.1 Joint IVP Set Up

We begin by setting up the joint initial value problem for our field-particle
system. The Cauchy problem for the scalar field $U$ is given by

$\left\\{\begin{array}[]{rcl}\eta^{\mu\nu}\partial_{\mu}\partial_{\nu}U(x)&=&a\int\delta^{(2)}(x-z(\tau))d\tau\\\
U(0,x^{1})&=&-\frac{a}{2}|x^{1}|+V_{0}(x^{1})\\\
\partial_{0}U(0,x^{1})&=&V_{1}(x^{1}),\end{array}\right.$ (3.1)

where $V_{0}$ and $V_{1}$ are smooth functions which represent the external
radiation. The term $-\frac{a}{2}|x^{1}|$ is included in the initial data to
represent the field that was sourced by the stationary charge. It is
compatible with the particle being at the origin for all $x^{0}<0$. Since our
evolution equation is linear, it is natural to split the Cauchy problem into
three parts by setting

$U=V+U_{\text{stat}}+U_{\text{source}}$ (3.2)

where

$\left\\{\begin{array}[]{rcl}\eta^{\mu\nu}\partial_{\mu}\partial_{\nu}V&=&0\\\
V(0,x^{1})&=&V_{0}(x^{1})\\\
\partial_{0}V(0,x^{1})&=&V_{1}(x^{1})\end{array}\right.\quad\left\\{\begin{array}[]{rcl}\eta^{\mu\nu}\partial_{\mu}\partial_{\nu}U_{\text{stat}}&=&0\\\
U_{\text{stat}}(0,x^{1})&=&-\frac{a}{2}|x^{1}|\\\
\partial_{0}U_{\text{stat}}(0,x^{1})&=&0\end{array}\right.$ (3.3)
$\left\\{\begin{array}[]{rcl}\eta^{\mu\nu}\partial_{\mu}\partial_{\nu}U_{\text{source}}(x)&=&\int
a\delta^{(2)}(x-z(\tau))d\tau\\\ U_{\text{source}}(0,x^{1})&=&0\\\
\partial_{0}U_{\text{source}}(0,x^{1})&=&0\end{array}\right.$ (3.4)

The solutions to (3.3) are given by d’Alembert’s formula while the solution to
(3.4) is given by Duhamel’s principle. $U_{\text{source}}$ can be written as
an integral equation which depends on the trajectory of the charge. From these
solutions it is easy to calculate the R.H.S of our force law:

$\frac{dp^{1}}{d\tau}=-\bigg{[}n_{\mu}T_{S}^{\mu
1}(x^{0},x^{1})\bigg{]}_{x^{1}=z^{1}}=-\frac{a^{2}}{2}u^{1}+a\partial_{1}V(x^{0},z^{1}).$
(3.5)

(See [1] for a similar calculation.)

######  Remark 3.1.

The force law derived from the conservation of energy is similar to the one
derived from the principle of stationary action. However, the singular “self-
force” term has been determined rather than ignored, and the expression for it
guarantees the conservation of the system’s total energy-momentum.

Written in terms of the dynamical mass and momentum, the initial value problem
for the charge’s trajectory is

$\displaystyle\frac{dz^{1}}{dx^{0}}$ $\displaystyle=$
$\displaystyle\frac{p^{1}}{\sqrt{m^{2}+(p^{1})^{2}}},\quad$ (3.6)
$\displaystyle\frac{dp^{1}}{dx^{0}}$ $\displaystyle=$
$\displaystyle-\frac{a^{2}}{2}\frac{p^{1}}{\sqrt{m^{2}+(p^{1})^{2}}}+a\frac{m}{\sqrt{m^{2}+(p^{1})^{2}}}\partial_{1}V(x^{0},z^{1}),$
(3.7)

with

$z^{1}(0)=0,\qquad p^{1}(0)=0.$ (3.8)

We now state and prove the first main result of this paper.

######  Theorem 3.1.

For any set of particle parameters
$\\{\tilde{m}>0,a\in\mathbb{R}\setminus\\{0\\}\\}$, and for any set of
sufficiently small, smooth, compactly supported functions $V_{0}(x^{1})$,
$V_{1}(x^{1})$, the joint initial value problem given by (3.6), (3.7), and
(3.1) admits a unique global-in-time solution with $U$ belonging to the space
of Lipschitz continuous scalar fields on $\mathbb{R}_{\geq 0}\times\mathbb{R}$
and $z$ a Lipschitz continuous world-line.

######  Remark 3.2.

The smallness condition taken on the initial data is
$||V_{0}||_{L^{\infty}}+\frac{1}{2}||V_{1}||_{L^{1}}\leq\frac{\tilde{m}}{|a|}$,
and is placed to ensure that the mass $m(x^{0})$ is bounded from below.

######  Remark 3.3.

In fact, the regularity of the world-line $z$ can be improved to $C^{\infty}$
by applying a bootstrap argument to the system of differential equations
(3.6), (3.7). However, the singularity appearing in (3.1) clearly marks that
we cannot expect anything better than Lipschitz regularity for the scalar
field $U$.

###  3.2 Proof of Well-Posedness

Strategy for the proof: Imagine that instead of solving for the joint
evolution of the particle and the field, we solve for the dynamics of one when
the other is given. For a given charge trajectory one can solve for $U$ by
plugging the trajectory into equation (3.1). Conversely, given the dynamics of
the field one can solve for a test charge’s trajectory via (3.5). Consider
what happens if one were to recursively define a sequence of trajectories and
field solutions by using the $i$th trajectory to solve for the $(i+1)$th field
solution via (3.1), and then using the $(i+1)$th field to solve for the
$(i+1)$th trajectory via (3.5), ad infinitum. If this sequence converges, it
would converge to a trajectory which sources the same field solution that
guides it. This is the key idea behind our proof, and we will see that this
process does converge to a unique joint evolution.

In this section we provide a proof of the well-posedness of the joint IVP. We
will do this by transforming our joint IVP into a set of integral equations.
The solution to (3.4) can be written in the form of an integral equation using
Duhamel’s principle

$\begin{split}U_{\text{source}}(x^{0},x^{1})&=\frac{1}{2}\int_{0}^{x^{0}}\int_{x^{1}-(x^{0}-t)}^{x^{1}+(x^{0}-t)}a\int\delta^{(2)}(x-z(\tau))d\tau
dsdt\\\
&=\frac{1}{2}\int_{0}^{x^{0}}\frac{a}{u^{0}(t)}\chi_{[x^{1}-(x^{0}-t),x^{1}+(x^{0}-t)]}(z^{1}(t))dt.\end{split}$
(3.9)

Since our system’s dynamics depend only on the field evaluated along the
world-line of the particle, it suffices to consider

$\begin{split}U_{\text{source}}(x^{0},z^{1}(x^{0})):=\frac{a}{2}\int_{0}^{x^{0}}\frac{1}{u^{0}(t)}dt=\frac{a}{2}\int_{0}^{x^{0}}\frac{m(t)}{\sqrt{m^{2}(t)+p^{2}(t)}}dt.\end{split}$
(3.10)

######  Theorem 3.2.

Let $U_{\text{stat}}$ and $V$ be the solutions to (3.3). Given the conditions
of (3.1), there exists a unique, global in time solution to the following set
of integral equations

$\displaystyle Q(x^{0})$ $\displaystyle=$
$\displaystyle\int_{0}^{x^{0}}\frac{p}{\sqrt{m^{2}+p^{2}}}dt$ (3.11)
$\displaystyle p(x^{0})$ $\displaystyle=$
$\displaystyle\int_{0}^{x^{0}}-\frac{a^{2}}{2}\frac{p}{\sqrt{m^{2}+p^{2}}}+a\frac{m}{\sqrt{m^{2}+p^{2}}}\partial_{1}V(t,Q(t))\
dt$ (3.12) $\displaystyle W(x^{0})$ $\displaystyle=$
$\displaystyle\int_{0}^{x^{0}}\frac{a}{2}\frac{m}{\sqrt{m^{2}+p^{2}}}dt,$
(3.13)

where $m(t,Q(t),W(t)):=\tilde{m}-aU_{\text{stat}}(t,Q(t))-aV(t,Q(t))-aW(t)$.

###### Proof.

We treat this a fixed-point problem. Introduce
$q(\cdot)=\begin{pmatrix}Q(\cdot)\\\ p(\cdot)\\\ W(\cdot)\end{pmatrix}$ so
that we may write (3.11), (3.12), and (3.13) as

$q(x^{0})=\int_{0}^{x^{0}}f(q(t),t)dt:=F_{x^{0}}(q(\cdot)).$ (3.14)

We seek to prove the existence and uniqueness of a fixed point for the
function $F_{*}(q(\cdot))$, which maps a given curve in $\mathbb{R}^{3}$ to
another curve in $\mathbb{R}^{3}$. Notice that any such fixed point will have
a bounded first derivative in all three components. Define
$L_{k,\gamma}(\mathbb{R}_{\geq 0})$ to be the normed set of Lipschitz
continuous functions $l(\cdot):\mathbb{R}_{\geq 0}\rightarrow\mathbb{R}$ with
Lipschitz constant $k$, and which satisfy $l(0)=0$. We equip this space with
the metric induced by the weighted $L^{\infty}$ norm

$||l(\cdot)||_{\gamma}=\sup_{t\geq 0}e^{-\gamma t}|l(t)|<\infty.$ (3.15)

We extend this definition for maps from $\mathbb{R}_{\geq
0}\rightarrow\mathbb{R}^{n}$ by defining

$L_{\vec{k},\gamma}(\mathbb{R}_{\geq
0})=\prod_{i=1}^{n}L_{k_{i},\gamma}(\mathbb{R}_{\geq 0})$ (3.16)

(Cartesian product), where $\vec{k}=(k_{1},k_{2},...k_{n})$.

######  Remark 3.4.

For each fixed $\vec{k}$, $L_{\vec{k},\gamma}$ is a one-parameter family of
metric spaces that share the same elements but differ in their equipped
metric.

######  Lemma 3.1.

Fixing both $\vec{k}$ and $\gamma>0$, we have that $L_{\vec{k},\gamma}$
equipped with the metric induced by $||\cdot||_{\gamma}$ is a complete metric
space.

###### Proof.

Corollary (A.1) in the appendix. ∎

Letting $K$ be an upper bound for
$\frac{a^{2}}{2}+||a\partial_{1}V_{0}||_{L^{\infty}}+||aV_{1}||_{L^{\infty}}$,
and $\vec{k}=(1,K,\frac{|a|}{2})$, it becomes clear that our desired fixed
point should, if it exists, reside in $L_{\vec{k},\gamma}$.

######  Proposition 3.1.

$F_{*}(q(\cdot)):L_{\vec{k},\gamma}\rightarrow L_{\vec{k},\gamma}$ is a well
defined mapping.

###### Proof.

Let $q=\begin{pmatrix}Q(\cdot)\\\ p(\cdot)\\\ W(\cdot)\end{pmatrix}\in
L_{\vec{k},\gamma}$, and write $F_{*}(x(\cdot))=\begin{pmatrix}P(*)\\\
\rho(*)\\\ S(*)\end{pmatrix}$. Notice

$\begin{split}\forall\theta,\theta^{\prime}\geq
0,\quad|\rho(\theta)-\rho(\theta^{\prime})|&=\left|\int_{\theta}^{\theta^{\prime}}-\frac{a^{2}}{2}\frac{p}{\sqrt{m^{2}+p^{2}}}+a\frac{m}{\sqrt{m^{2}+p^{2}}}\partial_{1}V(t,Q(t))\
dt\right|\\\ &\leq K|\theta-\theta^{\prime}|.\end{split}$ (3.17)

So $\rho(*)\in L_{K,\gamma}(\mathbb{R}_{\geq 0})$. The proofs for $P(*)$ and
$S(*)$ are analogous. ∎

In order to prove the existence of a unique fixed point of $F_{*}(q(\cdot))$,
we need to show it is a contraction mapping on $L_{\vec{k},\gamma}$. Towards
this, we prove two lemmas.

######  Lemma 3.2.

For $|Q(t)|\leq t$ and $|W(t)|\leq\frac{|a|}{2}t$, the quantity
$m_{V}:=\tilde{m}-|a|(||V_{0}||_{L^{\infty}}+\frac{1}{2}||V_{1}||_{L^{1}})$ is
a lower bound for $m(Q(t),W(t))$, and is positive by our smallness assumption
on $V_{0}$, $V_{1}$.

###### Proof.

(3.2). Since $|Q(t)|\leq t$, (3.3) lets us compute that

$U_{\text{stat}}(t,Q(t))=-\frac{a}{2}t.$ (3.18)

Thus,

$\begin{split}m(W(t),Q(t))&=\tilde{m}-aV(Q(t))+\frac{a^{2}}{2}t-aW(t)\\\
&\geq\tilde{m}-aV(Q(t))\geq\tilde{m}-|a|||V||_{L^{\infty}}\geq
m_{V}.\end{split}$ (3.19)

∎

######  Lemma 3.3.

Let $Y_{\gamma}$ be a complete metric space equipped with the metric induced
by $||\cdot||_{\gamma}$, such that $F_{*}(q(\cdot)):Y_{\gamma}\rightarrow
Y_{\gamma}$ is a well defined mapping. Suppose there exists an $L<\gamma$ such
that for all $q_{1}(\cdot),q_{2}(\cdot)\in Y_{\gamma}$

$||f(q_{2}(\tau),\tau)-f(q_{1}(\tau),\tau)||_{X}\leq
L||q_{2}(\tau)-q_{1}(\tau)||_{X}.$ (3.20)

Then $F_{*}(q(\cdot)):Y_{\gamma}\rightarrow Y_{\gamma}$ is a contraction
mapping.

###### Proof.

Theorem (A.1) in the appendix. ∎

######  Proposition 3.2.

There exists an $L<\infty$ independent of $\gamma$ such that for all
$q_{1}(\cdot),q_{2}(\cdot)\in L_{\vec{k},\gamma}$,

$||f(q_{2}(t),t)-f(q_{1}(t),t)||\leq L||q_{2}(t)-q_{1}(t)||.$ (3.21)

###### Proof.

Fix $t\geq 0$. We omit arguments of $t$ for ease of notation, and write
$q_{1}=\begin{pmatrix}Q\\\ p\\\ W\end{pmatrix},q_{2}=\begin{pmatrix}P\\\
\rho\\\ S\end{pmatrix}$. By the triangle inequality we have

$\begin{split}&||f(q_{2})-f(q_{1})||\leq\\\
&\leq\sum_{i=1}^{3}|f_{i}\begin{pmatrix}Q\\\ p\\\
W\end{pmatrix}-f_{i}\begin{pmatrix}P\\\ p\\\
W\end{pmatrix}|+|f_{i}\begin{pmatrix}P\\\ p\\\
W\end{pmatrix}-f_{i}\begin{pmatrix}P\\\ \rho\\\
W\end{pmatrix}|+|f_{i}\begin{pmatrix}P\\\ \rho\\\
W\end{pmatrix}-f_{i}\begin{pmatrix}P\\\ \rho\\\ S\end{pmatrix}|,\end{split}$
(3.22)

where $f_{i}$ denotes the components of $f$. There are nine terms that we wish
to bound. For the sake of brevity, we will restrict ourselves to the least
trivial bound. By the mean value theorem there exists an $O$ between $Q$ and
$P$ such that

$|f_{2}\begin{pmatrix}Q\\\ p\\\ W\end{pmatrix}-f_{2}\begin{pmatrix}P\\\ p\\\
W\end{pmatrix}|=|Q-P||D_{1}f_{2}\begin{pmatrix}O\\\ p\\\ W\end{pmatrix}|.$
(3.23)

We wish to bound $D_{1}f_{2}$. Recall

$f_{2}\begin{pmatrix}O\\\ p\\\
W\end{pmatrix}=-\frac{a^{2}}{2}\frac{p}{\sqrt{m^{2}+p^{2}}}+a\frac{m}{\sqrt{m^{2}+p^{2}}}\partial_{1}V(O).$
(3.24)

Since $O$ is between $Q$ and $P$, $m(O,W)$ is bounded from below by $m_{V}$.
Also, $|\frac{dm(Q,W)}{dQ}(O,W)|=|a\partial_{1}V(O)|$ which is bounded from
above by $K$. Let $K^{\prime}$ be the Lipschitz constant of $\partial_{1}V$.
Tedious calculations then yield the following bound

$|D_{1}f_{2}\begin{pmatrix}O\\\ p\\\
W\end{pmatrix}|\leq\frac{a^{2}}{2}\frac{K}{m_{V}}+\frac{K^{2}}{m_{V}}+|a|K^{\prime}=:M_{1,2}.$
(3.25)

Performing a similar analysis on the other 8 terms yields a bound of the form

$\begin{split}||f(q_{2})-f(q_{1})||&\leq|Q-P|\sum_{j=1}^{3}M_{1,j}+|p-\rho|\sum_{j=1}^{3}M_{2,j}+|W-S|\sum_{j=1}^{3}M_{3,j}\\\
&\leq||q_{2}-q_{1}||\sqrt{(\sum_{j=1}^{3}M_{1,j})^{2}+(\sum_{j=1}^{3}M_{2,j})^{2}+(\sum_{j=1}^{3}M_{3,j})^{2}}.\end{split}$
(3.26)

∎

######  Corollary 3.1.

Applying Lemma (3.3) with $Y_{\gamma}=L_{\vec{k},\gamma}$ and $\gamma>L$, we
find that there exists a unique $q\in L_{\vec{k},\gamma}(\mathbb{R}_{\geq 0})$
satisfying

$q(x^{0})=F_{x^{0}}(q(\cdot))=\int_{0}^{x^{0}}f(q(t),t)dt$ (3.27)

for all $x^{0}\geq 0$. This concludes our proof of Theorem (3.2).

∎

######  Porism 3.1.

Letting $q(x^{0})=\begin{pmatrix}Q(x^{0})\\\ p(x^{0})\\\
W(x^{0})\end{pmatrix}$, and defining the function
$W_{\text{source}}(x^{0},x^{1})$ via

$W_{\text{source}}(x^{0},x^{1})=\int_{0}^{x^{0}}\frac{a}{2}\sqrt{1-(\frac{dQ}{dx^{0}})^{2}}\chi_{[x^{1}-(x^{0}-t),x^{1}+(x^{0}-t)]}(Q(t))dt,$
(3.28)

we find that $Q(x^{0})$, $p(x^{0})$, and
$U(x^{0},x^{1}):=V(x^{0},x^{1})+U_{\text{stat}}(x^{0},x^{1})+W_{\text{source}}(x^{0},x^{1})$
are unique global solutions to the joint IVP given by equations (3.6), (3.7),
and (3.1). This concludes our proof of Theorem (3.1).

##  4 Asymptotic Stability of Scalar Particle

The force law that describes the charge’s dynamics given by (3.5) is very
similar to the one given by the principle of stationary action (2.4), with the
exception of a well-defined self-force. We derived this from the assumption
that the total energy-momentum of our system is conserved, so it is not
unexpected that the self-force is restorative, in magnitude proportional to
the particle’s velocity but with the opposite sign. This animation333Link to
animation:
https://sites.math.rutgers.edu/l̃af230/Motion_of_Perturbed_Scalar_Particle_V2.gif
of a scalar particle’s sourced field showcases how as the particle’s motion is
disturbed so too is the field that it is sourcing. These disturbances in the
sourced field carry energy, so to preserve the total energy of the system, the
field disturbance’s energy must have been sourced by the particle’s kinetic
energy. This is interpreted as an effective “self-force” which was conjectured
to sap away the particle’s kinetic energy until all motion ceases.

We will show that, as expected, in the case of a scalar particle perturbed by
some incoming radiation, the self-force causes the charge to asymptotically
return towards rest.

###  4.1 Asymptotic behavior of self-force

Take $V_{0}$, $V_{1}$ sufficiently small, smooth, and compactly supported.
Although the form of (3.5) was beneficial for proving well-posedness, we will
find it best to convert the equation to one for $\frac{du^{1}}{dx^{0}}$ to
better study the charge’s motion. Recalling that the dynamical mass of the
charge is given by equation (2.6), we obtain

$m\frac{du^{1}}{d\tau}=au^{1}\frac{dU(z(\tau))}{d\tau}-\frac{a^{2}}{2}u^{1}+\partial_{1}V.$
(4.1)

It is easy to verify from (3.9) that

$\frac{dU_{\text{source}}(z(\tau))}{d\tau}=\frac{a}{2},$

which implies via (3.2) a distinct cancellation in (4.1) between the
radiation-reaction self-force and the mass loss generated by
$U_{\text{source}}$.

######  Remark 4.1.

We note that a similar cancellation holds for point charges coupled to massive
scalar fields in one space dimension [7].

Despite this cancellation, our final equation of motion still contains a
restoring term from the mass contribution of the freely evolving stationary
state solution $U_{\text{stat}}$. We are left with

$m\frac{du^{1}}{dx^{0}}=-\frac{a^{2}}{2}u^{1}+au^{1}\frac{dV(x^{0},z^{1}(x^{0}))}{dx^{0}}+\frac{a}{u^{0}}\partial_{1}V(x^{0},z^{1}).$
(4.2)

The terms in (4.2) can be split up into two categories, those which depend on
the external radiation and those which do not. We write

$m\frac{du^{1}}{dx^{0}}=F_{\text{self}}+F_{\text{ext}},$ (4.3)

where

$F_{\text{self}}=-\frac{a^{2}}{2}u^{1},$ (4.4)

and

$F_{\text{ext}}=au^{1}\frac{dV(x^{0},z^{1}(x^{0}))}{dx^{0}}+\frac{a}{u^{0}}\partial_{1}V.$
(4.5)

Since the radiation was compactly supported and is propagating at the speed of
light, we expect it to perturb the motion of the particle and then propagate
away. Eventually we expect only the self-forces to remain, and it is clear
that the particle will asymptotically tend towards rest as long as the
dynamical mass does not grow too quickly.

We begin by proving that the external radiation does eventually propagate away
from the particle.

######  Lemma 4.1.

Let $z^{\mu}$, $p^{\mu}$, and $U$ satisfy the joint IVP given by (3.6), (3.7),
and (3.1) with the same conditions as (3.1). Let $v=\frac{dz^{1}}{dx^{0}}$.
Then

$\int_{0}^{\infty}1-|v(t)|\ dt=\infty.$ (4.6)

###### Proof.

Suppose not. Then

$\int_{0}^{\infty}1-|v(t)|\ dt=D<\infty.$ (4.7)

Recall that

$v=\frac{p^{1}}{\sqrt{m^{2}+(p^{1})^{2}}}.$ (4.8)

We proceed by proving estimates about $p^{1}$ and $m$ which will contradict
the rate of growth of $|v|$. The first will be the rough linear estimate for
$p^{1}$. From (3.5) we have

$\frac{dp^{1}}{dx^{0}}=-\frac{a^{2}}{2}v+a\sqrt{1-v^{2}}\partial_{1}V.$ (4.9)

So $|p^{1}(x^{0})|\leq Rx^{0}$ where
$R\geq\frac{a^{2}}{2}+||a\partial_{1}V_{0}||_{L^{\infty}}+||aV_{1}||_{L^{\infty}}$.
To estimate $m(t)$, recall

$m(x^{0})=\tilde{m}-aV(x^{0},z^{1}(x^{0}))+\frac{a^{2}}{2}x^{0}-aU_{\text{source}}(x^{0},z^{1}(x^{0})).$
(4.10)

We bounded $m$ from below in the previous section by showing that
$|aU_{\text{source}}(x^{0},z^{1}(x^{0}))|\leq\frac{a^{2}}{2}x^{0}$. Using our
contradiction hypothesis, we will now prove a stronger bound. For $x^{0}\geq
0$ we have

$\begin{split}|aU_{\text{source}}(x^{0},z^{1}(x^{0}))|&=\frac{a^{2}}{2}\int_{0}^{x^{0}}\sqrt{1-v(t)^{2}}dt\\\
&\leq\frac{a^{2}}{2}\bigg{(}\sqrt{\int_{0}^{x^{0}}1-|v(t)|dt}\bigg{)}\cdot\bigg{(}\sqrt{\int_{0}^{x^{0}}1+|v(t)|dt}\bigg{)}\\\
&\leq\frac{a^{2}}{2}\sqrt{D}\sqrt{2x^{0}}.\end{split}$ (4.11)

By the conditions on the radiation we see that the linear term will dominate
$m(x^{0})$. So, there exists a time $T>0$, and a slope $M>0$ such that for all
$x^{0}>T$ we have that $m(x^{0})\geq Mx^{0}$. Using our estimates for $p^{1}$
and $m$, we can conclude with a final estimate for $v$. For all $t>T$

$|v(t)|=\frac{|p^{1}|}{\sqrt{m^{2}+(p^{1})^{2}}}\leq\frac{Rt}{\sqrt{(Mt)^{2}+(Rt)^{2}}}=\frac{R}{\sqrt{M^{2}+R^{2}}}<1.$
(4.12)

But this implies that $1-|v(t)|$ is bounded from below, so

$\int_{0}^{\infty}1-|v(t)|\ dt=\infty.$ (4.13)

as desired. ∎

######  Corollary 4.1.

Since $V_{0}$ and $V_{1}$ are compactly supported, it follows that all
external force terms in $F_{\text{ext}}$ will vanish in finite time.

###### Proof.

It suffices to show that $\partial_{1}V$, $\partial_{0}V$ evaluated along the
particle’s world-line vanish after finite time. By d’Alambert’s formula we
have

$\begin{split}\partial_{1}V(x^{0},z^{1}(x^{0}))=\frac{1}{2}[V_{0}(z^{1}(x^{0})+x^{0})+V_{0}(z^{1}(x^{0})-x^{0})+V_{1}(z^{1}(x^{0})+x^{0})-V_{1}(z^{1}(x^{0})-x^{0})].\\\
\partial_{0}V(x^{0},z^{1}(x^{0}))=\frac{1}{2}[V_{0}(z^{1}(x^{0})+x^{0})-V_{0}(z^{1}(x^{0})-x^{0})+V_{1}(z^{1}(x^{0})+x^{0})+V_{1}(z^{1}(x^{0})-x^{0})].\end{split}$
(4.14)

Lemma (4.1) implies that the quantities $z^{1}(x^{0})\pm x^{0}$ grow
arbitrarily large, so each term in (4.14) will vanish once $z^{1}(x^{0})\pm
x^{0}$ leaves the support of $V_{0},V_{1}$. ∎

We conclude this section by stating and proving the second main result of this
paper: that the perturbed charge asymptotically returns towards rest.

######  Theorem 4.1.

Let $z$, $U$ satisfy the joint IVP given by (3.6) (3.7), and (3.1) for a
single scalar particle with positive bare mass and non-zero real scalar
charge, where $V_{0}$ and $V_{1}$ satisfy the same conditions as in Theorem
(3.1). Then for every $\epsilon>0$, there exists a $T_{\epsilon}>0$ such that
$|u^{1}(x^{0})|<\epsilon$ for all $x^{0}>T_{\epsilon}$.

###### Proof.

By Corollary 4.1, there exists a time $T$ such that for all $x^{0}>T$,
$u^{1}(x^{0})$ satisfies

$\frac{du^{1}}{dx^{0}}=-\frac{a^{2}u^{1}}{2m}.$ (4.15)

If at some time $T^{\prime}>T$ we have that $u^{1}=0$, then it will remain 0
for all time afterwards and the theorem holds. Suppose $|u^{1}|>0$ for all
time after $T$. Dividing both sides of (4.15) by $u^{1}$ and integrating
returns

$\ln(|u^{1}(x^{0})|)=\ln(|u^{1}(T)|)-\frac{a^{2}}{2}\int_{T}^{x^{0}}\frac{1}{m}dt.$
(4.16)

Recall that $m$ is strictly positive and increases at most linearly, thus
there exists an $A$ such that for all $x^{0}>T>0$, $m<Ax^{0}$. It follows that

$\ln(|\frac{u^{1}(x^{0})}{u^{1}(T)}|)<-\frac{a^{2}}{2A}(\ln(\frac{x^{0}}{T})).$
(4.17)

So

$|u^{1}(x^{0})|<|u^{1}(T)|\big{(}\frac{x^{0}}{T}\big{)}^{-\frac{a^{2}}{2A}}.$
(4.18)

Since $\frac{a^{2}}{2A}>0$, we see that $u^{1}$ decays to zero asymptotically.
∎

##  5 Summary and Outlook

In this paper we studied the joint evolution of a point particle coupled to a
scalar field in $1+1$ dimensions. A rigorous derivation for the force on a
charged particle was given from the assumption of conservation of energy-
momentum, and from this we calculated the self-force which result from back-
reaction. The scalar self-force that charged particles experience is
restorative, proportional to negative velocity, and causes the charge to
asymptotically return towards rest after being perturbed by radiation. Written
explicitly:

$m\frac{du^{1}}{dx^{0}}=F_{\text{self}}+F_{\text{ext}},$ (5.1)

where

$F_{\text{self}}=-\frac{a^{2}}{2}u^{1},$ (5.2)

and $a$ denotes the scalar charge. We also proved the well-posedness for the
joint evolution problem.

Although our study was meant to act as a toy model, it has produced some
fruitful results. They indicate that for universes in which fields coupled to
point particles evolve with a certain level of regularity, the joint evolution
problem of fields and particles is well-posed under the condition that their
dynamics preserve the total energy-momentum. The Maxwell-Lorentz theory of
point charges in $3+1$ dimensions does not give rise to the level of
regularity sufficient for such joint evolutions to exist. BLTP, a higher-order
modification to Maxwell-Lorentz theory has found success in $3+1$ dimensions
because of the regularity of its field’s evolution. However, the complexity of
the BLTP electromagnetic self-force in $3+1$ dimensions does not admit an easy
way to investigate whether or not the process of back-reaction stabilizes a
perturbed particle in that case (see [4] for an interesting preliminary result
in this direction.) Working in $1+1$ dimensions has provided us with a much
simpler set of dynamics, and we were able to prove asymptotic stability in the
case of a scalar particle. The mathematical simplicity afforded to us in this
lower-dimensional setting makes our results approachable and easily
understood, while we hope many of the physically intuitive arguments still
carry over to their higher-dimensional counterparts.

There are multiple avenues that we will take to further investigate the
process of back-reaction, still within this simplified framework: In ongoing
work [7] we generalize our results to the case of massive scalar fields, which
we believe will provide researchers with a clearer picture of back-reaction in
higher-order modifications of $3+1$ dimensional field theories. We would like
to study a scattering problem for this model, one analogous to [8], in which
the initial data is prescribed at the infinite past. Another possibility is to
study the general initial value problem with arbitrary data for the field that
is compatible with a point charge (although in that case one might expect
additional singularities to form in the field on the light cone of the initial
data position, as was shown in [5] for Maxwell-Lorentz electrodynamics.) We
also plan to pursue a project investigating gravitational back-reaction, to
see whether a gravitational self-force arises when deriving the force law from
the principle of energy-momentum conservation.

##  6 Acknowledgment

We are grateful the anonymous referees for their sharp eyes and insightful
remarks. We thank Aditya Agashe and Ethan Lee for their assistance in drawing
Figure 1.

##  Appendix A Well-Posedness of Integral Equations

In this section we will provide brief proofs of well-known results regarding
the well-posedness of integral equations. We will do this with the help of
Banach fixed point theorem.

We first recall a standard result:

######  Lemma A.1.

Let $X$ be a metric space, and let $C_{\gamma}^{0}(\mathbb{R}_{\geq 0},X)$ be
the normed set of continuous maps from $\mathbb{R}_{\geq 0}\rightarrow X$ such
that for all $x(\cdot)\in C_{\gamma}^{0}(\mathbb{R}_{\geq 0},X)$ we have

$||x(\cdot)||_{\gamma}=\sup_{t\geq 0}e^{-\gamma t}||x(t)||_{X}<\infty.$ (A.1)

Then $C_{\gamma}^{0}(\mathbb{R}_{\geq 0},X)$ is a Banach space.

Let $f:X\times\mathbb{R}_{\geq 0}\rightarrow X$. Define

$F_{t}(q(\cdot)|x_{0}):=x_{0}+\int_{0}^{t}f(q(\tau),\tau)d\tau.$ (A.2)

######  Theorem A.1.

Let $Y_{\gamma}$ be a closed subset of $C_{\gamma}^{0}(\mathbb{R}_{\geq 0},X)$
such that $F_{*}(x(\cdot)|x_{0}):Y_{\gamma}\rightarrow Y_{\gamma}$ is a well
defined mapping. Suppose there exists an $L<\gamma$ such that for all
$q_{1}(\cdot),q_{2}(\cdot)\in Y_{\gamma}$

$||f(q_{2}(\tau),\tau)-f(q_{1}(\tau),\tau)||_{X}\leq
L||q_{2}(\tau)-q_{1}(\tau)||_{X}.$ (A.3)

Then $F_{*}(x(\cdot)|x_{0})$ restricted to $Y_{\gamma}$ is a contraction
mapping on a complete metric space. Thus, $Y_{\gamma}$ contains a unique fixed
point $q(\cdot)$ which satisfies

$q(t)=F_{t}(q(\cdot)|x_{0})=x_{0}+\int_{0}^{t}f(q(\tau),\tau)d\tau.$ (A.4)

for all $t\geq 0$.

###### Proof.

We wish to show that for all $x_{1}(\cdot),x_{2}(\cdot)\in Y_{\gamma}$,

$||F_{*}(x_{2}(\cdot)|x_{0})-F_{*}(x_{1}(\cdot)|x_{0})||_{\gamma}\leq\frac{L}{\gamma}||x_{2}(\cdot)-x_{1}(\cdot)||_{\gamma}.$
(A.5)

By definition

$\begin{split}||F_{*}(x_{2}(\cdot)|x_{0})-F_{*}(x_{1}(\cdot)|x_{0})||_{\gamma}&=\sup_{t\geq
0}e^{-\gamma
t}||\int_{0}^{t}f(x_{2}(\tau),\tau)-f(x_{1}(\tau),\tau)d\tau||_{X}\\\
&\leq\sup_{t\geq 0}e^{-\gamma
t}\int_{0}^{t}||f(x_{2}(\tau),\tau)-f(x_{1}(\tau),\tau)||_{X}d\tau\\\
&\leq\sup_{t\geq 0}e^{-\gamma
t}\int_{0}^{t}L||x_{2}(\tau)-x_{1}(\tau)||_{X}d\tau\\\ &=\sup_{t\geq
0}\int_{0}^{t}e^{-\gamma(t-\tau)}e^{-\gamma\tau}L||x_{2}(\tau)-x_{1}(\tau)||_{X}d\tau\\\
&\leq\sup_{t\geq
0}\int_{0}^{t}e^{-\gamma(t-\tau)}L||x_{2}(\cdot)-x_{1}(\cdot)||_{\gamma}d\tau\\\
&\leq\frac{L}{\gamma}||x_{2}(\cdot)-x_{1}(\cdot)||_{\gamma}.\end{split}$ (A.6)

∎

In cases where $f:\mathbb{R}\times\mathbb{R}_{\geq 0}\rightarrow\mathbb{R}$ is
bounded it is useful to consider $L_{k,\gamma}(\mathbb{R}_{\geq 0})\subset
C_{\gamma}^{0}(\mathbb{R}_{\geq 0},\mathbb{R})$, the normed set of all
Lipschitz continuous functions $l(\cdot)$ with Lipschitz constant $k$ and
$l(0)=0$.

######  Theorem A.2.

For all $\gamma>0$, $L_{k,\gamma}(\mathbb{R}_{\geq 0})$ is a closed subset of
$C_{\gamma}^{0}(\mathbb{R}_{\geq 0},\mathbb{R})$, and therefore a complete
metric space.

###### Proof.

Let $\\{l_{n}(\cdot)\\}$ be a Cauchy sequence in
$L_{k,\gamma}(\mathbb{R}_{\geq 0})$. This has a limit $l(\cdot)$ in
$C_{\gamma}^{0}(\mathbb{R}_{\geq 0},\mathbb{R})$. Also, we have that for every
$t,t^{\prime}\geq 0$

$k|t-t^{\prime}|-|l_{n}(t)-l_{n}(t^{\prime})|\geq 0.$ (A.7)

The convergence is pointwise, so taking the limit of both sides as $n$ goes to
infinity yields

$k|t-t^{\prime}|-|l(t)-l(t^{\prime})|\geq 0.$ (A.8)

Taking the infimum over $t,t^{\prime}$ yields that $l(\cdot)\in
L_{k,\gamma}(\mathbb{R}_{\geq 0})$ as desired. ∎

######  Corollary A.1.

For all $\gamma>0$, $\vec{k}\in\mathbb{R}^{n}$, it follows that
$L_{\vec{k},\gamma}(\mathbb{R}_{\geq
0}):=\prod_{i=1}^{n}L_{k_{i},\gamma}(\mathbb{R}_{\geq 0})$ is a closed subset
of $C_{\gamma}^{0}(\mathbb{R}_{\geq 0},\mathbb{R}^{n})$, and therefore a
complete metric space.

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