Radiation reaction as a singular model reduction

Work in one named laboratory chart with spatial coordinate space ${\mathbb{R}}^3$ and time coordinate $t\in \mathbb{R}$. Let $q\in \mathbb{R}\setminus \{0\}$ be the framework charge parameter, let $m>0$ be an externally supplied mass parameter, and take the SI constants $c>0$ and ${\epsilon }_0>0$ of the electromagnetic framework. Define the positive time constant

$\tau :=\frac{q^2}{6\pi {\epsilon }_{0}m{c}^3}>0.$

The units are $[q]=\mathrm{C}$, $[m]=\mathrm{k}\mathrm{g}$, $[c]=\mathrm{m}{\mathrm{s}}^{-1}$, and $[{\epsilon }_0]=\mathrm{F}{\mathrm{m}}^{-1}$, so $[\tau ]=\mathrm{s}$. A prescribed external force is a continuous map $f:\mathbb{R}\to {\mathbb{R}}^3$ with units $\mathrm{N}$. A particle trajectory in this model is a map $x:\mathbb{R}\to {\mathbb{R}}^3$ with three continuous time derivatives; its velocity and acceleration are $v(t):=\dot{x}(t)$ and $a(t):=\ddot{x}(t)$, with units $\mathrm{m}{\mathrm{s}}^{-1}$ and $\mathrm{m}{\mathrm{s}}^{-2}$. The term point particle means that the model assigns one position $x(t)$ and no spatial extent.

Adopt, as an additional effective model postulate, the equation

$m(a(t)-\tau \dot{a}(t))=f(t)(t\in \mathbb{R})\mathrm{.}$

Both $ma$ and $m\tau \dot{a}$ have units of force. The dot on $a$ means differentiation with respect to the laboratory time. This equation is not obtained by evaluating the particle's singular Maxwell self-field in the adopted Lorentz force law. The previously developed prescribed-source radiation model does not define that self-interaction; this postulate introduces a separate effective equation.

For physical use on a compact interval $I\subset \mathbb{R}$, assume $f$ is continuously differentiable there, choose dimensionless tolerances $\eta ,\delta \in (0,1)$ and a positive force scale $F_{\ast }$ with units $\mathrm{N}$, and state the chosen values. In this item, low-speed at tolerance $\eta$ means ${\sup }_{t\in I}\mathrm{\mid }v(t)\mathrm{\mid }\le \eta c$, and slowly varying relative to $\tau$ at tolerance $\delta$ means $\tau {\sup }_{t\in I}\mathrm{\mid }\dot{f}(t)\mathrm{\mid }\le \delta F_{\ast }$. These are explicit regime conventions for an application, not universal empirical thresholds. The ODE is an exact adopted equation on its mathematical domain; its physical interpretation is restricted to applications that state their regime and tolerances.

Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields

Framework-local SI units, the vacuum constants and the exact/measured status of their values

Charge parameter, frame charge and current densities, volume charge and flux, and the four-current

The Lorentz coupling: external force on a test charge and the continuum force and power densities

Self-force, runaways and finite-size regularizations are not supplied by this page

Use the equation and constants of the preceding postulate, with $m,\tau >0$. Let $f:\mathbb{R}\to {\mathbb{R}}^3$ be bounded and continuous, and let a solution mean a continuously differentiable map $a:\mathbb{R}\to {\mathbb{R}}^3$ satisfying $m(a-\tau \dot{a})=f$ at every real time. Bounded for a vector map means each of its three real coordinates is bounded; in ${\mathbb{R}}^3$ this is equivalent to bounded Euclidean norm. The vector improper Riemann integral below is taken componentwise, with ${\int }_t^{\mathrm{\infty }}h(s)ds:={\lim }_{R\to \mathrm{\infty }}{\int }_t^{R}h(s)ds$ when that limit exists. Then the integral converges absolutely, and

$a_{\ast }(t):=\frac{1}{m\tau }{\int }_t^{\mathrm{\infty }}\exp \text{\hspace{0.17em}⁣}(-\frac{s-t}{\tau })f(s)ds$

is a bounded solution. For any fixed $t_0\in \mathbb{R}$, every solution has the form

$a(t)=a_{\ast }(t)+C\exp \text{\hspace{0.17em}⁣}\left(\frac{t-t_0}{\tau }\right),C\in {\mathbb{R}}^3$

Here $C$ is a constant acceleration vector with units $\mathrm{m}{\mathrm{s}}^{-2}$.

The only solution bounded on a future half-line $[T,\mathrm{\infty })$, for any finite $T$, is $a_{\ast }$. A nonzero $C$ gives an exponentially growing acceleration branch, called a runaway branch in this item. Moreover, there is a bounded smooth force that vanishes for $t\le 0$, is nonzero for some $t\in (0,1)$, and for which $a_{\ast }(t)\mathrm{\ne }0$ for every $t<0$. Preacceleration means this nonzero acceleration before the force begins; it is a property of the selected ODE branch, not an observation about a physical particle.

The nonrelativistic Abraham–Lorentz equation as a restricted effective model

The real exponential function and the number $e$ by a power series

The exponential function is smooth and $(\exp {)}^{\mathrm{\prime }}=\exp$

The exponential is positive and satisfies $\exp (-x)=1\mathrm{/}\exp (x)$

The mean value theorem, as the case $g(x)=x$ of Cauchy's: for $f$ continuous on $[a,b]$ with $a<b$ and differentiable on $(a,b)$ there is $c\in (a,b)$ with $f(b)-f(a)=f^{\mathrm{\prime }}(c)(b-a)$

The first fundamental theorem: if $f$ is integrable on $[a,b]$ and continuous at $c$, then $F^{\mathrm{\prime }}(c)=f(c)$; in particular a continuous $f$ has $F$ as a primitive

A smooth bump between concentric Euclidean balls

Assume the equation and bounded-force hypotheses of the preceding theorem: $m,\tau >0$, $f:\mathbb{R}\to {\mathbb{R}}^3$ is bounded and twice continuously differentiable, and the unique bounded-future solution $a_{\ast }$ is used. Suppose

$M_2:={\sup }_{t\in \mathbb{R}}\mathrm{\mid }{f}^{\mathrm{\prime }\mathrm{\prime }}(t)\mathrm{\mid }<\mathrm{\infty }\mathrm{.}$

Since $f$ has units $\mathrm{N}$ and $t$ has units $\mathrm{s}$, $M_2$ has units $\mathrm{N}{\mathrm{s}}^{-2}$.

Define the first-order reduction-of-order approximation by

$a_{\mathrm{R}\mathrm{O}}(t):=\frac{f(t)+\tau f^{\mathrm{\prime }}(t)}{m}\mathrm{.}$

Its residual when substituted into the adopted equation is exactly

$m(a_{\mathrm{R}\mathrm{O}}(t)-\tau a_{\mathrm{R}\mathrm{O}}^{\mathrm{\prime }}(t))-f(t)=-{\tau }^2{f}^{\mathrm{\prime }\mathrm{\prime }}(t)\mathrm{.}$

On the bounded-future branch the acceleration error obeys

$\mathrm{\mid }{a}_{\ast }(t)-a_{\mathrm{R}\mathrm{O}}(t)\mathrm{\mid }\le \frac{{\tau }^2{M}_2}{m}(t\in \mathbb{R})\mathrm{.}$

The residual and bound are exact conditional statements about the prescribed-force ODE; they do not assert that a nonlinear self-consistent particle trajectory is approximated by $a_{\mathrm{R}\mathrm{O}}$.

Runaway and bounded-future branches of the Abraham–Lorentz equation

Sums, scalar multiples, products and quotients: $(f+g{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)+g^{\mathrm{\prime }}(c)$, $(\alpha f{)}^{\mathrm{\prime }}(c)=\alpha f^{\mathrm{\prime }}(c)$, $(fg{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)g(c)+f(c)g^{\mathrm{\prime }}(c)$, and $(f\mathrm{/}g{)}^{\mathrm{\prime }}(c)=(f^{\mathrm{\prime }}(c)g(c)-f(c)g^{\mathrm{\prime }}(c))\mathrm{/}g(c{)}^2$ when $g(c)\mathrm{\ne }0$

The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g(c)$, then $f\circ g$ is differentiable at $c$ with $(f\circ g{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(g(c))g^{\mathrm{\prime }}(c)$

The first fundamental theorem: if $f$ is integrable on $[a,b]$ and continuous at $c$, then $F^{\mathrm{\prime }}(c)=f(c)$; in particular a continuous $f$ has $F$ as a primitive

Cauchy-Schwarz $\mid ⟨x,y⟩\mid \le \parallel x{\parallel }_2\parallel y{\parallel }_2$ with its equality case, the triangle inequality for $\parallel \cdot {\parallel }_2$, the parallelogram law and polarisation