Controlled prescribed-field trajectory limit

Fix a reference speed $c_{\ast }>0$, mass $m>0$, charge $q\in \mathbb{R}$, cutoff $0<\delta <1$, finite time $T>0$, an open time interval $I\supseteq [0,T]$, and an open spatial region $\mathrm{\Omega }\subseteq {\mathbb{R}}^3$. In a fixed inertial chart with $x^0=ct$, let $\mathbf{E},\mathbf{B}:I\times \mathrm{\Omega }\to {\mathbb{R}}^3$ be prescribed jointly $C^1$ fields in SI units, with $\mathbf{E}$ in volts per metre and $\mathbf{B}$ in teslas. The same fields, mass, and charge are held fixed in the following formal family of model equations indexed by $c>0$; no Maxwell equations or empirical premise are added.

Choose reference scales $L_{\ast }>0$ metres and $P_{\ast }>0$ kilogram metres per second and define the dimensionless state norm

$\mathrm{\parallel }(x,p){\mathrm{\parallel }}_{\ast }:=\frac{\mathrm{\mid }x\mathrm{\mid }}{L_{\ast }}+\frac{\mathrm{\mid }p\mathrm{\mid }}{P_{\ast }}\mathrm{.}$

Let ${\pi }_x:\mathrm{\Omega }\times {\mathbb{R}}^3\to \mathrm{\Omega }$ be the spatial projection ${\pi }_x(x,p):=x$, and let $K\mathrm{\ne }\mathrm{\varnothing }$ be a compact convex subset of $\mathrm{\Omega }\times {\mathbb{R}}^3$. With $\mathrm{\mid }\cdot \mathrm{\mid }$ the Euclidean norm, put $P_{\max }:={\sup }_{(x,p)\in K}\mathrm{\mid }p\mathrm{\mid }$, $E_{\max }:={\sup }_{(t,x)\in [0,T]\times {\pi }_{x}K}\mathrm{\mid }\mathbf{E}(t,x)\mathrm{\mid }$, and $B_{\max }:={\sup }_{(t,x)\in [0,T]\times {\pi }_{x}K}\mathrm{\mid }\mathbf{B}(t,x)\mathrm{\mid }$. These field suprema are finite by the $C^1$ hypothesis on a compact cylinder. Set

$B_{\ast }:=\frac{P_{\max }}{m{L}_{\ast }}+\frac{\mathrm{\mid }q\mathrm{\mid }}{P_{\ast }}(E_{\max }+\frac{P_{\max }{B}_{\max }}{m})\mathrm{.}$

For $c>0$ with $c\ge P_{\max }\mathrm{/}(m\delta )$, define the momentum-to-velocity map

$V_c(p):=\frac{p}{m\sqrt{1+\mathrm{\mid }p{\mathrm{\mid }}^2\mathrm{/}(m^2{c}^2)}}\mathrm{.}$

This is the velocity obtained from the massive mass shell and inverse momentum formula of Mass shell inversion, observer work and variable mass. Define the relativistic and Newtonian vector fields on $[0,T]\times K$ by

$G_c(t,x,p):=(V_c(p),q[\mathbf{E}(t,x)+V_c(p)\times \mathbf{B}(t,x)]),G_N(t,x,p):=(p\mathrm{/}m,q[\mathbf{E}(t,x)+(p\mathrm{/}m)\times \mathbf{B}(t,x)])\mathrm{.}$

Suppose there is $L\ge 0$, with units ${\mathrm{s}}^{-1}$, such that for all $t\in [0,T]$ and states $\zeta ,\tilde{\zeta }\in K$,

$\mathrm{\parallel }{G}_N(t,\zeta )-G_N(t,\tilde{\zeta }){\mathrm{\parallel }}_{\ast }\le L\mathrm{\parallel }\zeta -\tilde{\zeta }{\mathrm{\parallel }}_{\ast }\mathrm{.}$

For every $c>0$ in the family, $G_c$ and $G_N$ are bounded by $B_{\ast }$ in $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{\ast }$ on $[0,T]\times K$, since $\mathrm{\mid }{V}_c(p)\mathrm{\mid }\le P_{\max }\mathrm{/}m$. For each $c>0$ with $c\ge P_{\max }\mathrm{/}(m\delta )$, let $J\subseteq I$ be an open interval containing $[0,T]$ and let $x_c\in C^2(J,\mathrm{\Omega })$ satisfy $\mathrm{\mid }{\dot{x}}_c(t)\mathrm{\mid }<c$. Define the coordinate velocity $v_c(t):={\dot{x}}_c(t)$ and Lorentz factor ${\gamma }_c(t):=(1-\mathrm{\mid }{v}_c(t){\mathrm{\mid }}^2\mathrm{/}{c}^2{)}^{-1\mathrm{/}2}$. The curve $X_c(t)=(ct,x_c(t))$ is a future-timelike massive worldline with proper time $d\tau \mathrm{/}dt=\sqrt{1-\mathrm{\mid }{v}_c(t){\mathrm{\mid }}^2\mathrm{/}{c}^2}$ and four-velocity $U_c={\gamma }_c(c,v_c)$; its spatial momentum is $p_c=m{\gamma }_c{v}_c$. Assume $X_c$ satisfies the adopted covariant Lorentz coupling in the prescribed field. Define $z_c(t):=(x_c(t),p_c(t))$. Let $(x_N,p_N)$ be a $C^1$ solution on $[0,T]$ of $\dot{z}=G_N(t,z)$ and set $z_N(t):=(x_N(t),p_N(t))$. Require $z_c(0)=z_N(0)$ and both coordinate-state trajectories to remain in $K$. Then the relativistic coordinates satisfy ${\dot{z}}_c=G_c(t,z_c)$, and, with ${ϵ}_c:=(c_{\ast }\mathrm{/}c{)}^2$ and

$C_{\ast }:=\frac{P_{\max }^{}}{2m^3{c}_{\ast }^2}(\frac{1}{L_{\ast }}+\frac{\mathrm{\mid }q\mathrm{\mid }{B}_{\max }}{P_{\ast }}),$

one has

${\sup }_{0\le t\le T}\mathrm{\parallel }(x_c(t),p_c(t))-(x_N(t),p_N(t)){\mathrm{\parallel }}_{\ast }\le C_{\ast }{ϵ}_{c}T{e}^{LT}\mathrm{.}$

Here $C_{\ast }$ has units ${\mathrm{s}}^{-1}$. Consequently the state difference is $O(c^{-2})$ on this fixed interval as $c\to \mathrm{\infty }$, provided the common containment and Lipschitz hypotheses hold for the family.

For every $t\in [0,T]$, the relativistic speed satisfies $\mathrm{\mid }{v}_c(t)\mathrm{\mid }\mathrm{/}c\le \delta$. If ${\mathcal{E}}_c(t):=\sqrt{m^2{c}^4+c^2\mathrm{\mid }{p}_c(t){\mathrm{\mid }}^2}$ is the particle energy from the same mass shell, then Explicit low-speed momentum and energy remainders also gives

$\mathrm{\mid }{p}_c-m{v}_c\mathrm{\mid }\le \frac{m\mathrm{\mid }{v}_c{\mathrm{\mid }}^3}{2c^2(1-{\delta }^2{)}^{3\mathrm{/}2}},0\le {\mathcal{E}}_c-m{c}^2-\frac{1}{2}m\mathrm{\mid }{v}_c{\mathrm{\mid }}^2\le \frac{3m\mathrm{\mid }{v}_c{\mathrm{\mid }}^4}{8c^2(1-{\delta }^2{)}^{5\mathrm{/}2}}\mathrm{.}$

The limit is a formal comparison between model equations; it does not state that the SI value of $c$ varies in nature.

The field and vector-field estimates are in a named inertial chart and use a formal model parameter $c$. Electric and magnetic fields are prescribed and held fixed, not solved self-consistently as $c$ varies. The common-interval solution and compact-containment assumptions are essential: neither bounded field values alone nor the pointwise low-speed estimates establish finite-time trajectory convergence.