Radiation-Reaction on the Straight-Line Motion of a Point Charge accelerated by a constant applied Electric Field in an Electromagnetic Bopp-Landé-Thomas-Podolsky vacuum

Abstract

The radiation-reaction problem of standard Lorentz electrodynamics with point charges is pathological, standing in contrast to Bopp--Landé--Thomas--Podolsky (BLTP) electrodynamics where it is in fact well-defined and calculable, as reported in a previous publication. To demonstrate the viability of BLTP electrodynamics, we consider the BLTP analogue of the radiation reaction of a classical point charge accelerated from rest by a static homogeneous capacitor plate field, and calculate it up to $O(\varkappa^4)$ in a formal expansion about $\varkappa=0$ in powers of $\varkappa$, Bopp's reciprocal length, a new electrodynamics parameter introduced by BLTP theory. In a paper by Carley and Kiessling (arXiv:2303.01720 [physics.class-ph]) the radiation-reaction corrections to test-particle motion were explicitly computed to $O(\varkappa^3)$, the first non-vanishing order. In this article a crucial question regarding this ``small-$\varkappa$'' expansion, raised by Carley and Kiessling, is answered as follows: The motions computed with terms $O(\varkappa^3)$ included are mathematically accurate approximations to {physically reasonable} solutions of the actual BLTP initial value problem for short times $t$, viz. when $\varkappa c t \ll 1$, where $c$ is the speed of light in vacuo, but their unphysical behavior over {much} longer times does not accurately approximate the actual BLTP solutions even when the dimensionless parameter $\varkappa e^2 / |m_b| c^2 \ll 1$, where $e$ is the elementary charge and $m_b$ the bare rest mass of the electron. This has the important implication that BLTP electrodynamics remains a viable contender for an accurate classical electrodynamics with point charges that does not suffer from the infinite self-interaction problems of textbook Lorentz electrodynamics with point charges.

Figures (4)

• Figure 1: The velocity of a point charge, starting from rest in a constant applied electrostatic field $\pmb{{\cal E}}^{\hbox{\tiny{hom}}}=10e\varkappa^2$, vs. time, as per test particle theory (dashed curve), resp. BLTP electrodynamics with radiation-reaction included to $O(\varkappa^3)$ (dotted curve), resp. to $O(\varkappa^4)$ (continuous curve), when $\varkappa e^2/m_{\text{b}} c^2 =0.01$. The period of the velocity of the $O(\varkappa^3)$ BLTP motion is $\varkappa c T = 160$. The test particle's velocity and the BLTP particle's $O(\varkappa^4)$ velocity asymptote to $c$.
• Figure 2: Same as Fig. 1, but now for parameter values $\pmb{{\cal E}}^{\hbox{\tiny{hom}}}=10e\varkappa^2$ and $\varkappa e^2/m_{\text{b}} c^2 =0.001$.
• Figure 3: Same as Fig. 1, but now for parameter values $\pmb{{\cal E}}^{\hbox{\tiny{hom}}}=100e\varkappa^2$ and $\varkappa e^2/m_{\text{b}} c^2 =0.001$.
• Figure 4: The velocity of a point charge with negative bare mass $m_{\text{b}}<0$, starting from rest in a constant applied electrostatic field $\pmb{{\cal E}}^{\hbox{\tiny{hom}}}=0.1e\varkappa^2$, vs. time, as per test particle theory (dashed curve), resp. BLTP electrodynamics with radiation-reaction included to $O(\varkappa^3)$ (dotted curve), resp. to $O(\varkappa^4)$ (continuous curve), when $\varkappa e^2/m_{\text{b}} c^2 = -2$. Also shown is the radiation-free motion of a "charge soliton," identical to charged test particle motion with effective mass $m_{\text{e}} = m_{\text{b}} + E^{\hbox{\tiny{field}}}/c^2 >0$, though with $m_{\text{e}}$ about 1000 times larger than it should, for optical purposes.