Microscopic Foundations of Kinetic Plasma Theory: The Relativistic Vlasov–Maxwell Equations and Their Radiation-Reaction-Corrected Generalization

Abstract

It is argued that the relativistic Vlasov–Maxwell equations of the kinetic theory of plasma approximately describe a relativistic system of N charged point particles interacting with the electromagnetic Maxwell fields in a Bopp–Landé–Thomas–Podolsky (BLTP) vacuum, provided the microscopic dynamics lasts long enough. The purpose of this work is not to supply an entirely rigorous vindication, but to lay down a conceptual road map for the microscopic foundations of the kinetic theory of special-relativistic plasma, and to emphasize that a rigorous derivation seems feasible. Rather than working with a BBGKY-type hierarchy of n-point marginal probability measures, the approach proposed in this paper works with the distributional PDE of the actual empirical 1-point measure, which involves the actual empirical 2-point measure in a convolution term. The approximation of the empirical 1-point measure by a continuum density, and of the empirical 2-point measure by a (tensor) product of this continuum density with itself, yields a finite-N Vlasov-like set of kinetic equations which includes radiation-reaction and nontrivial finite-N corrections to the Vlasov–Maxwell–BLTP model. The finite-N corrections formally vanish in a mathematical scaling limit $N\rightarrow \infty$ in which charges $\propto 1/\surd {N}$. The radiation-reaction term vanishes in this limit, too. The subsequent formal limit sending Bopp’s parameter $\varkappa \rightarrow \infty$ yields the Vlasov–Maxwell model.

Appendix

1.1 A The Radiation-Reaction Force Term

In this appendix we collect the pertinent formulas needed to compute the radiation-reaction force terms.

We write $\varvec{\Pi }^{\mathrm{field}}_{\alpha ,n}(t,{\varvec{{ s}} })$ as a sum of three terms, sorted by their singularities,

$$\begin{aligned} \varvec{\Pi }^{\mathrm{field}}_{\alpha ,n}(t,{\varvec{{ s}} }) = \frac{e_\alpha ^2}{16\pi ^2 c} \;{\sum \limits _{k=0}^2}\; \frac{\varvec{\pi }^{[k]}_{\varvec{\xi }_n^\alpha }(t,{\varvec{{ s}} })}{\big |{{\varvec{{ s}} }-{\varvec{{ q}} }_n\big (t^{\mathrm {ret}}_{\varvec{\xi }_n^\alpha } (t,{\varvec{{ s}} })\big )}\big |^k}, \end{aligned}$$

(A.1)

where the suffix $\varvec{\xi }_n^\alpha$ indicates the vector function $t\mapsto \varvec{\xi }_n^\alpha (t) \equiv ({\varvec{{ q}} }_n^\alpha ,{\varvec{{ v}} }_n^\alpha ,\varvec{{ a}} _n^\alpha )(t)$. Here, and now dropping ${}_n$ and ${}^\alpha$ indices,

$$\begin{aligned} \varvec{\pi }_{\varvec{\xi }}^{[0]}(t,{\varvec{{ s}} })&= - \varkappa ^4 \frac{1}{4}\left[ {\textstyle { \frac{ \left( {{\varvec{{ n}} }({\varvec{{ q}} },{\varvec{{ s}} })}_{}-\frac{1}{c}{{\varvec{{ v}} }}\right) {\varvec{\times }}\left( { \frac{1}{c}{{\varvec{{ v}} }}{\varvec{\times }}{{\varvec{{ n}} }({\varvec{{ q}} },{\varvec{{ s}} })}_{} }\right) }{ \bigl ({1-\frac{1}{c} {{\varvec{{ v}} }}\cdot {\varvec{{ n}} }({\varvec{{ q}} },{\varvec{{ s}} })}\bigr )^{\!2} } }}\right] _{\mathrm {ret}}\nonumber \\&\quad + \varkappa ^4\frac{1}{2}\left[ {\textstyle { \frac{ {{\varvec{{ n}} }({\varvec{{ q}} },{\varvec{{ s}} })}_{}-\frac{1}{c}{{\varvec{{ v}} }}}{ {1-\frac{1}{c} {{\varvec{{ v}} }}\cdot {\varvec{{ n}} }({\varvec{{ q}} },{\varvec{{ s}} })} } }}\right] _{\mathrm {ret}} {\varvec{\times }}\! \int _{-\infty }^{t^\mathrm {ret}_{\varvec{\xi }}(t,{\varvec{{ s}} })}\!\!\!\! {{\varvec{{ v}} }(t')}{\varvec{\times }}\mathbf {K}_{\varvec{\xi }}(t',t,{\varvec{{ s}} })\mathrm {d}{t'}\nonumber \\&\quad - \varkappa ^4\frac{1}{2}\left[ {\textstyle {\frac{ \frac{1}{c}{{\varvec{{ v}} }}{\varvec{\times }}{\varvec{{ n}} }({\varvec{{ q}} },{\varvec{{ s}} })}{ 1-\frac{1}{c} {{\varvec{{ v}} }}\cdot {\varvec{{ n}} }({\varvec{{ q}} },{\varvec{{ s}} })}}}\right] _{\mathrm {ret}} {\varvec{\times }}\int _{-\infty }^{t^\mathrm {ret}_{\varvec{\xi }}(t,{\varvec{{ s}} })}\!\!\!\! c\mathbf {K}_{\varvec{\xi }}(t',t,{\varvec{{ s}} })\mathrm {d}{t'}\nonumber \\&\quad - \varkappa ^4 \int _{-\infty }^{t^\mathrm {ret}_{\varvec{\xi }}(t,{\varvec{{ s}} })} \!\!\!\! c\mathbf {K}_{\varvec{\xi }}(t',t,{\varvec{{ s}} })\mathrm {d}{t'} {\varvec{\times }}\int _{-\infty }^{t^\mathrm {ret}_{\varvec{\xi }}(t,{\varvec{{ s}} })} \!\!\!\! {{\varvec{{ v}} }(t')}{\varvec{\times }}\mathbf {K}_{\varvec{\xi }}(t',t,{\varvec{{ s}} })\mathrm {d}{t'}\nonumber \\&\quad - \varkappa ^4 c\int _{-\infty }^{t^\mathrm {ret}_{\varvec{\xi }}(t,{\varvec{{ s}} })} \!\!\!\! \mathrm {K}_{\varvec{\xi }}(t',t,{\varvec{{ s}} })\mathrm {d}{t'} \int _{-\infty }^{t^\mathrm {ret}_{\varvec{\xi }}(t,{\varvec{{ s}} })} \!\!\!\!\mathrm {K}_{\varvec{\xi }}(t',t,{\varvec{{ s}} }) {{\varvec{{ v}} }}(t')\mathrm {d}{t'} \end{aligned}$$

(A.2)

$$\begin{aligned} \varvec{\pi }_{\varvec{\xi }}^{[1]}(t,{\varvec{{ s}} })&= - \varkappa ^2 \left[ {\textstyle { { {{\varvec{{ n}} }({\varvec{{ q}} },{\varvec{{ s}} })}\frac{\left( {{\varvec{{ n}} }({\varvec{{ q}} },{\varvec{{ s}} })}{\varvec{\times }}[{ \left( {{\varvec{{ n}} }({\varvec{{ q}} },{\varvec{{ s}} })}-\frac{1}{c}{{\varvec{{ v}} }}\right) {\varvec{\times }}\varvec{{ a}} }]\right) \cdot \frac{1}{c}{\varvec{{ v}} }}{ c^2 \bigl ({1-\frac{1}{c} {{\varvec{{ v}} }}\cdot {\varvec{{ n}} }({\varvec{{ q}} },{\varvec{{ s}} })}\bigr )^{\!4} } } + {{\varvec{{ n}} }({\varvec{{ q}} },{\varvec{{ s}} })}_{}{\varvec{\times }}\frac{ \left( {{\varvec{{ n}} }({\varvec{{ q}} },{\varvec{{ s}} })}_{}-\frac{1}{c}{{\varvec{{ v}} }}\right) {\varvec{\times }}\varvec{{ a}} }{ 2 c^2 \bigl ({1-\frac{1}{c} {{\varvec{{ v}} }}\cdot {\varvec{{ n}} }({\varvec{{ q}} },{\varvec{{ s}} })}\bigr )^{\!3} } }}\right] _{\mathrm {ret}}\nonumber \\&\quad - \varkappa ^2\left[ {\textstyle { {{\varvec{{ n}} }({\varvec{{ q}} },{\varvec{{ s}} })}_{}{\varvec{\times }}\frac{ \left( {{\varvec{{ n}} }({\varvec{{ q}} },{\varvec{{ s}} })}_{}-\frac{1}{c}{{\varvec{{ v}} }}\right) {\varvec{\times }}\varvec{{ a}} }{ c^2 \bigl ({1-\frac{1}{c} {{\varvec{{ v}} }}\cdot {\varvec{{ n}} }({\varvec{{ q}} },{\varvec{{ s}} })}\bigr )^{\!3} } }}\right] _{\mathrm {ret}} \!\! {\varvec{\times }}\! \int _{-\infty }^{t^\mathrm {ret}_{\varvec{\xi }}(t,{\varvec{{ s}} })}\!\!\!\! {{\varvec{{ v}} }(t')}{\varvec{\times }}\mathbf {K}_{\varvec{\xi }}(t',t,{\varvec{{ s}} })\mathrm {d}{t'}\nonumber \\&\quad + \varkappa ^2\left[ {\varvec{{ n}} }({\varvec{{ q}} },{\varvec{{ s}} }){\varvec{\times }}\biggl [{\textstyle {{\varvec{{ n}} }({\varvec{{ q}} },{\varvec{{ s}} }){\varvec{\times }}\frac{\left( {{\varvec{{ n}} }({\varvec{{ q}} },{\varvec{{ s}} })}_{}-\frac{1}{c}{{\varvec{{ v}} }}\right) {\varvec{\times }}\varvec{{ a}} }{ c^2\bigl ({1-\frac{1}{c} {{\varvec{{ v}} }}\cdot {\varvec{{ n}} }({\varvec{{ q}} },{\varvec{{ s}} })}\bigr )^{\!3} } }}\biggr ]\right] _{\mathrm {ret}} \!\!\! {\varvec{\times }}\! \int _{-\infty }^{t^\mathrm {ret}_{\varvec{\xi }}(t,{\varvec{{ s}} })} \!\!\!\! c\mathbf {K}_{\varvec{\xi }}(t',t,{\varvec{{ s}} })\mathrm {d}{t'}\nonumber \\&\quad + \varkappa ^3 \left[ \textstyle \frac{1}{{1-\frac{1}{c} {{\varvec{{ v}} }}\cdot {\varvec{{ n}} }({\varvec{{ q}} },{\varvec{{ s}} })} }\right] _{\mathrm {ret}} \int _{-\infty }^{t^\mathrm {ret}_{\varvec{\xi }}(t,{\varvec{{ s}} })}\!\!\!\! \mathrm {K}_{\varvec{\xi }}(t',t,{\varvec{{ s}} })\left[ {{\varvec{{ v}} }} ({t^\mathrm {ret}_{\varvec{\xi }}(t,{\varvec{{ s}} })}))+{{\varvec{{ v}} }}(t')\right] \mathrm {d}{t'} \end{aligned}$$

(A.3)

$$\begin{aligned} \varvec{\pi }_{\varvec{\xi }}^{[2]}(t,s)&= - \varkappa ^2 \left[ \textstyle \frac{1}{\bigl ({1-\frac{1}{c} {{\varvec{{ v}} }}\cdot {\varvec{{ n}} }({\varvec{{ q}} },{\varvec{{ s}} })} \bigr )^{\!2} }\frac{1}{c}{{\varvec{{ v}} }}{- \Big [\!{1-\tfrac{1}{c^2}\big |{\varvec{{ v}} }\big |^2}\!\Big ] \frac{ \left( {{\varvec{{ n}} }({\varvec{{ q}} },{\varvec{{ s}} })}_{}-\frac{1}{c}{{\varvec{{ v}} }}\right) {\varvec{\times }}\left( \frac{1}{c}{\varvec{{ v}} }{\varvec{\times }}{\varvec{{ n}} }({\varvec{{ q}} },{\varvec{{ s}} })\right) }{ \bigl ({1-\frac{1}{c} {{\varvec{{ v}} }}\cdot {\varvec{{ n}} }({\varvec{{ q}} },{\varvec{{ s}} })}\bigr )^4 }} \right] _{\mathrm {ret}} \nonumber \\&\quad +\varkappa ^2 \left[ \Big [\!{1-\tfrac{1}{c^2}\big |{\varvec{{ v}} }\big |^2}\!\Big ]{\varvec{{ n}} }({\varvec{{ q}} },{\varvec{{ s}} }){\varvec{\times }}{\textstyle { \frac{ {{\varvec{{ n}} }({\varvec{{ q}} },{\varvec{{ s}} })}_{}-\frac{1}{c}{{\varvec{{ v}} }} }{ \bigl ({1-\frac{1}{c} {{\varvec{{ v}} }}\cdot {\varvec{{ n}} }({\varvec{{ q}} },{\varvec{{ s}} })}\bigr )^{\!3} } }}\right] _{\mathrm {ret}} {\varvec{\times }}\int _{-\infty }^{t^\mathrm {ret}_{\varvec{\xi }}(t,{\varvec{{ s}} })} \!\!\!\!c\mathbf {K}_{\varvec{\xi }}(t',t,{\varvec{{ s}} })\mathrm {d}{t'} \nonumber \\&\quad - \varkappa ^2\left[ \Big [\!{1-\tfrac{1}{c^2}\big |{\varvec{{ v}} }\big |^2}\!\Big ] {\textstyle { \frac{ {{\varvec{{ n}} }({\varvec{{ q}} },{\varvec{{ s}} })}_{}-\frac{1}{c}{{\varvec{{ v}} }} }{ \bigl ({1-\frac{1}{c} {{\varvec{{ v}} }}\cdot {\varvec{{ n}} }({\varvec{{ q}} },{\varvec{{ s}} })}\bigr )^{\!3} } }}\right] _{\mathrm {ret}} {\varvec{\times }}\! \int _{-\infty }^{t^\mathrm {ret}_{\varvec{\xi }}(t,{\varvec{{ s}} })} \!\!\!\! {{\varvec{{ v}} }(t')}{\varvec{\times }}\mathbf {K}_{\varvec{\xi }}(t',t,{\varvec{{ s}} })\mathrm {d}{t'} , \end{aligned}$$

(A.4)

with the abbreviations $\mathrm {K}_{\varvec{\xi }}, \mathbf {K}_{\varvec{\xi }}$ defined as in (2.19) and (2.24), and where $\big |_\mathrm {ret}$ means that ${\varvec{{ q}} }(\tilde{t})$, ${\varvec{{ v}} }(\tilde{t})$, and $\varvec{{ a}} (\tilde{t})$ are evaluated with $\tilde{t} = {t^\mathrm {ret}_{\varvec{\xi }}}(t,{\varvec{{ s}} })$.

With $\overline{\varvec{\xi }}\equiv ({\overline{\varvec{{ q}} }},{\varvec{{ v}} }_{\!0},\varvec{{ 0}} )$, we thus have

$$\begin{aligned}&{\displaystyle \int _{B_{ct}({\varvec{{ q}} }_n(0))}\!\! \left( \varvec{\Pi }^{\mathrm{field}}_{\alpha ,n}(t,{\varvec{{ s}} }) -{\varvec{\Pi }}^{\mathrm{field}}_{\alpha ,n}(0,{\varvec{{ s}} }-{\overline{\varvec{{ q}} }}_n(t)) \right) \!\mathrm {d}^3{s}}\nonumber \\&\quad = \frac{e_\alpha ^2}{16\pi ^2 c} \;{\sum \limits _{k=0}^2}\; {\displaystyle \int _{B_{ct}({\varvec{{ q}} }_0)}\!\!\left( \tfrac{\textstyle \varvec{\pi }_{\varvec{\xi }_n^\alpha }^{[k]} (t,{\varvec{{ s}} })}{\left| {\varvec{{ s}} }-{\varvec{{ q}} }_n\big (t^{\mathrm {ret}}_{\varvec{\xi }_n}(t,{\varvec{{ s}} })\big ) \right| ^k}- \tfrac{\textstyle \varvec{\pi }_{\overline{\varvec{\xi }}_n^\alpha }^{[k]} (t,{\varvec{{ s}} })}{\left| {\varvec{{ s}} }-{\overline{\varvec{{ q}} }}_n\big (t^{\mathrm {ret}}_{{\overline{\varvec{{ q}} }}_n}(t,{\varvec{{ s}} })\big ) \right| ^k} \right) \mathrm {d}^3{s}}. \end{aligned}$$

(A.5)

Note that the acceleration $t\mapsto \varvec{{ a}} _n(t)$ only features linearly, in the $k=1$ term.

To carry out the integrations at r.h.s. (A.5) we in [54] switch to “retarded spherical co-ordinates” with the colatitude $\vartheta$ defined with respect to ${\varvec{{ s}} }- {\varvec{{ q}} }(t^{\mathrm {r}})$. Let $\mathrm {d}\omega _r({\varvec{{ s}} })$ denote the uniform measure on $\partial B_r\left( {\varvec{{ q}} }(t^{\mathrm {r}})\right)$. For any ${\mathfrak {C}}^{1,1}$ map $t^{\mathrm {r}}\mapsto {\varvec{{ q}} }(t^{\mathrm {r}})$ with ${{\dot{{\varvec{{ q}} }}}}(t^{\mathrm {r}})=:{\varvec{{ v}} }(t^{\mathrm {r}})$ we then have

$$\begin{aligned} \mathrm {d}^3{s} = \frac{\mathrm {d}{\omega _r({\varvec{{ s}} })}}{\big \vert \partial _{{\varvec{{ s}} }}{\left| {\varvec{{ s}} }-{\varvec{{ q}} }(t^{\mathrm {r}}) \right| } \big \vert } \biggl .\biggr |_{\partial B_r\left( {\varvec{{ q}} }(t^{\mathrm {r}})\right) }^{}\mathrm {d}{r} = \left( 1-\tfrac{1}{c}\big \vert {\varvec{{ v}} }(t-\tfrac{1}{c} r) \big \vert \cos \vartheta \right) r^2\sin \vartheta \mathrm {d}{\vartheta }\mathrm {d}{\varphi }\mathrm {d}{r}. \end{aligned}$$

(A.6)

This gives us

$$\begin{aligned}&\int _{B_{ct}({\varvec{{ q}} }_0)} \biggl [ \tfrac{\textstyle \varvec{\pi }_{\varvec{\xi }_n}^{[k]} (t,{\varvec{{ s}} })}{\left| {\varvec{{ s}} }-{\varvec{{ q}} }_n\big (t^{\mathrm {ret}}_{\varvec{\xi }_n}(t,{\varvec{{ s}} })\big ) \right| ^k}- \tfrac{\textstyle \varvec{\pi }_{\varvec{\xi }_n^\circ }^{[k]} (t,{\varvec{{ s}} })}{\left| {\varvec{{ s}} }-{\overline{\varvec{{ q}} }}_n\big (t^{\mathrm {ret}}_{{\overline{\varvec{{ q}} }}_n}(t,{\varvec{{ s}} })\big ) \right| ^k} \biggr ]\mathrm {d}^3{s} \nonumber \\&\quad = \int _0^{ct}\! \int _0^{2\pi }\!\! \int _0^{\pi }\! \Bigl [\left( 1-\tfrac{1}{c}\big \vert {\varvec{{ v}} }_n^{}(t-\tfrac{1}{c} r) \big \vert \cos \vartheta \right) \varvec{\pi }_{\varvec{\xi }_n}^{[k]}\big (t,{\varvec{{ s}} }_{\varvec{\xi }_n} (r,\vartheta ,\varphi )\big )\biggr .\nonumber \\&\qquad -\biggl . \left( 1-\tfrac{1}{c}\big \vert {\varvec{{ v}} }_{\!n}(0) \big \vert \cos \vartheta \right) \varvec{\pi }_{\varvec{\xi }_n^\circ }^{[k]}\big (t,{\varvec{{ s}} }_{{\overline{\varvec{{ q}} }}_n} (r,\vartheta ,\varphi )\big )\Bigr ] \sin \vartheta \mathrm {d}{\vartheta }\mathrm {d}{\varphi }\, r^{2-k} \mathrm {d}{r} \,. \end{aligned}$$

(A.7)

Next we register the following important facts about the angular integrations over the spheres $\partial B_r\left( {\varvec{{ q}} }(t^{\mathrm {r}})\right)$ (where ${\varvec{{ q}} }$ stands for either ${\varvec{{ q}} }_n$ or ${\overline{\varvec{{ q}} }}_n$; ${\varvec{{ v}} }$ and $\varvec{{ a}}$ similarly):

• on $\partial B_{r}({\varvec{{ q}} }(t^{\mathrm{r}}))$ the retarded time $t^{\mathrm{r}}_{{\varvec{{ q}} }}(t,{\varvec{{ s}} }) = t-\frac{1}{c} r = t^{\mathrm{r}}$ is constant;

• the vectors ${\varvec{{ q}} }(t^{\mathrm{r}})$, ${\varvec{{ v}} }(t^{\mathrm{r}})$, and $\varvec{{ a}} (t^{\mathrm{r}})$ are constant during the integration over $\partial B_{r}({\varvec{{ q}} }(t^{\mathrm{r}}))$;

• for ${\varvec{{ s}} }\in \partial B_{r}({\varvec{{ q}} }(t^{\mathrm{r}}))$, we have ${\varvec{{ s}} }= r\, {\varvec{{ n}} }({\varvec{{ q}} }(t^{\mathrm{r}}),{\varvec{{ s}} }) + {\varvec{{ q}} }\big (t^{\mathrm{r}}\big )$;

• for ${\varvec{{ s}} }\in \partial B_{r}({\varvec{{ q}} }(t^{\mathrm{r}}))$ the unit vector ${\varvec{{ n}} }({\varvec{{ q}} }(t^{\mathrm{r}}),{\varvec{{ s}} })=\big (\sin \vartheta \cos \varphi , \sin \vartheta \sin \varphi , \cos \vartheta \big )$.

It follows that on $\partial B_{r}({\varvec{{ q}} }(t^{\mathrm{r}}))$ the vectors $\varvec{\pi }_{\varvec{\xi }}^{[k]}\big (t,{\varvec{{ s}} }_{{\varvec{{ q}} }}(r,\vartheta ,\varphi )\big )$ are bounded continuous functions of $\vartheta$ and $\varphi$. Thus the angular integrations at r.h.s. (A.7) can be carried out to yield

$$\begin{aligned} \displaystyle \int _{B_{ct}({\varvec{{ q}} }_0)} \left[ \tfrac{\textstyle \varvec{\pi }_{\varvec{\xi }_n}^{[k]}(t,{\varvec{{ s}} })}{\left| {\varvec{{ s}} }-{\varvec{{ q}} }_n\big (t^{\mathrm {ret}}_{\varvec{\xi }_n}(t,{\varvec{{ s}} })\big ) \right| ^k}- \tfrac{\textstyle \varvec{\pi }_{\varvec{\xi }_n^\circ }^{[k]} (t,{\varvec{{ s}} })}{\left| {\varvec{{ s}} }-{\overline{\varvec{{ q}} }}_n\big (t^{\mathrm {ret}}_{{\overline{\varvec{{ q}} }}_n}(t,{\varvec{{ s}} })\big ) \right| ^k} \right] \mathrm {d}^3{s} = \!\! \displaystyle \int _0^{ct}\! \Bigl [ \widetilde{\mathbf {Z}}_{\varvec{\xi }_n}^{[k]}\big (t,r\big ) -\! \widetilde{\mathbf {Z}}_{\varvec{\xi }_n^\circ }^{[k]}\big (t,r\big )\Bigr ] r^{2-k} \mathrm {d}{r} . \end{aligned}$$

(A.8)

The vector functions $\widetilde{\mathbf {Z}}_{\varvec{\xi }}^{[k]}(t,r)$ depend on the maps $t'\mapsto {\varvec{{ q}} }(t')$ and $t'\mapsto {\varvec{{ v}} }(t')$ between $t'=0$ and $t'=t^{\mathrm{ret}}_{{\varvec{{ q}} }}(t,{\varvec{{ s}} })=t^{\mathrm{r}}$, through the $\mathrm {d}{t'}$ integrals over the Bessel function kernels, through which they also depend on the initial data ${\varvec{{ q}} }(0)$ and ${\varvec{{ v}} }(0)$ (recall that the integrations over $t'<0$ involve an unaccelerated auxiliary motion determined by the initial data ${\varvec{{ q}} }(0)$ and ${\varvec{{ v}} }(0)$). They also depend on t and r, explicitly through the Bessel function kernels (2.24) and (2.19) (recall also the third bullet point above), and implicitly through the dependence on $t^{\mathrm{r}} = t-r/c$ in: (a) the vectors ${\varvec{{ q}} }(t^{\mathrm{r}})$, ${\varvec{{ v}} }(t^{\mathrm{r}})$, and (when $k=1$) $\varvec{{ a}} (t^{\mathrm{r}})$, and (b) the upper limit of integration of the integrals over the Bessel function kernels.

Next, by a change of integration variable from r to $t^{\mathrm{r}}= t-r/c$ we eliminate the implicit t dependence from the integrands at r.h.s.(A.8). Writing $\widetilde{\mathbf {Z}}_{\varvec{\xi }}^{[k]}\big (t,r\big ) = {\mathbf {Z}}_{\varvec{\xi }}^{[k]}\big (t,t^{\mathrm{r}}\big )$, this yields (2.25).

We thus arrive at the “self”-field force on the n-th point charge source of the MBLTP field equations, given for $t>0$ by the negative t derivative of (2.25), viz.

$$\begin{aligned}&-\tfrac{16\pi ^2}{e_\alpha ^2}{\textstyle \frac{\mathrm {d}}{\mathrm {d}t}}{\displaystyle \int _{B_{ct}({\varvec{{ q}} }_n(0))}\!\! \left( \varvec{\Pi }^{\mathrm{field}}_{\alpha ,n}(t,{\varvec{{ s}} }) -{\varvec{\Pi }}^{\mathrm{field}}_{\alpha ,n} (0,{\varvec{{ s}} }-{\overline{\varvec{{ q}} }}^\alpha _n(t))\right) \!\mathrm {d}^3{s}} \;\nonumber \\&\quad = {\mathbf {Z}}_{\varvec{\xi }_n}^{[2]}(t,t) + {\mathbf {Z}}_{\varvec{\xi }_n^\circ }^{[2]}(t,t) \,\nonumber \\&\qquad - \;{\textstyle \sum \limits _{0\le k\le 1}}\; c^{2-k}(2-k) \displaystyle \int _0^{t}\! \Bigl [{\mathbf {Z}}_{\varvec{\xi }_n}^{[k]}\big (t,t^{\mathrm{r}}\big ) -\!{\mathbf {Z}}_{\varvec{\xi }_n^\circ }^{[k]}\big (t, t^{\mathrm{r}}\big )\Bigr ] (t- t^{\mathrm{r}})^{1-k} \mathrm {d}{t^{}r} \nonumber \\&\qquad - \;{\textstyle \sum \limits _{0\le k\le 2}}\; c^{2-k} \displaystyle \int _0^{t}\! \Bigl [{\textstyle {\frac{\partial }{\partial t}}}{\mathbf {Z}}_{\varvec{\xi }_n}^{[k]} \big (t,t^{\mathrm{r}}\big ) -\!{\textstyle {\frac{\partial }{\partial t}}}{\mathbf {Z}}_{\varvec{\xi }_n^\circ }^{[k]} \big (t, t^{\mathrm{r}}\big )\Bigr ] (t- t^{\mathrm{r}})^{2-k} \mathrm {d}{t^{}\mathrm{r}} . \end{aligned}$$

(A.9)

Remark A.1

Thus the integrals in the second and third line at r.h.s.(A.9) are well defined Riemann integrals.

Remark A.2

Since $t^{\mathrm {r}}$ is a mute integration variable at r.h.s.(A.9), the $\frac{\partial }{\partial t}$ derivative in the third line at r.h.s(A.9) does not act on it. In particular, it does not act on ${\varvec{{ q}} }_n(t^\mathrm {r})$, ${\varvec{{ v}} }_n(t^\mathrm {r})$, or $\varvec{{ a}} _n(t^\mathrm {r})$ in any of the ${\mathbf {Z}}_{\varvec{\xi }_n}^{[k]}\big (t,t^{\mathrm {r}}\big )$. This leads to the important observation that the “self” force does not involve higher-than-second-order time derivates of $t\mapsto {\varvec{{ q}} }(t)$.

Remark A.3

The derivative $\frac{\partial }{\partial t}$ also does not act on ${\varvec{{ q}} }_n(t^\prime )$ or ${\varvec{{ v}} }_n(t^\prime )$ in the kernels $\mathbf {K}_{\varvec{\xi }}(t',t,{\varvec{{ s}} }_{\varvec{{ q}} })$ and $\mathrm {K}_{\varvec{\xi }}(t',t,{\varvec{{ s}} }_{\varvec{{ q}} })$. Thus the acceleration enters the “self” force only linearly, as $\varvec{{ a}} _n(t^{\mathrm {r}})$, through ${\mathbf {Z}}_{\varvec{\xi }_n}^{[1]}\big (t,t^{\mathrm {r}}\big )$ in the second, and $\frac{\partial }{\partial t}{\mathbf {Z}}_{\varvec{\xi }_n}^{[1]}\big (t,t^{\mathrm {r}}\big )$ in the third line at r.h.s.(A.9). This gives rise to the Volterra integral equation for the accelerations, mentioned in 2.3.

There is no purely light-like contribution to the third line at r.h.s.(A.9). The $\frac{\partial }{\partial t}$ in the third line at r.h.s.(A.9) acts only on the explicit t-dependence in $\mathbf {K}_{\varvec{\xi }}$ and in $\mathrm {K}_{\varvec{\xi }}$ in the mixed light- & time-like and purely time-like contributions to ${\mathbf {Z}}_{\varvec{\xi }_n}^{[k]}\big (t,t^{\mathrm {r}}\big )$. The identity

$$\begin{aligned} \frac{1}{x} \frac{\mathrm {d}}{\mathrm {d}x}\frac{J_\nu (x)}{x^\nu } = - \frac{J_{\nu +1}(x)}{x^{\nu +1}} \end{aligned}$$

(A.10)

(see §10.6(ii) in [17]) yields these derivatives easily as

$$\begin{aligned} {\frac{\partial }{\partial t}}\mathrm {K}_{\varvec{\xi }}(t',t,{\varvec{{ s}} }_{\varvec{{ q}} })&= -\varkappa c^2 \tfrac{J_2\!\bigl (\varkappa \sqrt{c^2(t-t')^2-|{\varvec{{ s}} }_{\varvec{{ q}} }-{\varvec{{ q}} }(t')|^2 }\bigr )}{{c^2(t-t')^2-|{\varvec{{ s}} }_{\varvec{{ q}} }-{\varvec{{ q}} }(t')|^2}^{}} (t-t'), \end{aligned}$$

(A.11)

$$\begin{aligned} {\textstyle \frac{\partial }{\partial t}}\mathbf {K}_{\varvec{\xi }}(t',t,{\varvec{{ s}} }_{\varvec{{ q}} })&=-\varkappa c^2\tfrac{J_3\!\bigl (\varkappa \sqrt{c^2(t-t')^2 -|{\varvec{{ s}} }_{\varvec{{ q}} }-{\varvec{{ q}} }(t')|^2 }\bigr )}{(c^2(t-t')^2-|{\varvec{{ s}} }_{\varvec{{ q}} }-{\varvec{{ q}} }(t')|^2)^{3/2} } (t-t') \left( {\varvec{{ s}} }_{\varvec{{ q}} }-{\varvec{{ q}} }(t')- {\varvec{{ v}} }(t')(t-t')\right) \nonumber \\&\quad - {\varvec{{ v}} }(t') \tfrac{J_2\!\bigl (\varkappa \sqrt{c^2(t-t')^2-|{\varvec{{ s}} }_{\varvec{{ q}} }-{\varvec{{ q}} }(t')|^2 }\bigr )}{{c^2(t-t')^2-|{\varvec{{ s}} }_{\varvec{{ q}} }-{\varvec{{ q}} }(t')|^2}^{} }, \end{aligned}$$

(A.12)

where ${\varvec{{ s}} }_{\varvec{{ q}} }= r{\varvec{{ n}} }+{\varvec{{ q}} }(t^\mathrm {r})$.