Exact repulsive relativistic Coulomb scattering in the fixed-source test-body model

Use the charged test-body quantities of Charged and spinning test-body quantities on the flat Minkowski specialization, with constant mass $m>0$ in kilograms and constant charge $q$ in coulombs. In a named inertial SI chart write $x^0=ct$, $t\in I:=\mathbb{R}$, and $\mathbf{x}=(x^1,x^2,x^3)\in \mathrm{\Omega }:={\mathbb{R}}^3\setminus \{0\}$, with $r:=\mathrm{\mid }\mathbf{x}\mathrm{\mid }>0$; the fixed point source of charge $Q$ in coulombs is at $\mathbf{x}=0$, which is excluded from the spacetime domain $O:=I\times \mathrm{\Omega }$. The prescribed static scalar potential $\varphi :\mathrm{\Omega }\times I\to \mathbb{R}$, electric field $\mathbf{E}:\mathrm{\Omega }\times I\to {\mathbb{R}}^3$ and magnetic field $\mathbf{B}:\mathrm{\Omega }\times I\to {\mathbb{R}}^3$ are $C^{\mathrm{\infty }}$ on that domain and are

$\varphi (\mathbf{x},t)=\frac{Q}{4\pi {\epsilon }_{0}r},\mathbf{E}(\mathbf{x},t)=\frac{Q\mathbf{x}}{4\pi {\epsilon }_0{r}^3},\mathbf{B}(\mathbf{x},t)=0.$

Here ${\epsilon }_0>0$ is the framework SI parameter of Framework-local SI units, the vacuum constants and the exact/measured status of their values, $\varphi$ has units volts, $\mathbf{E}$ volts per metre, and $\mathbf{B}$ teslas. The external four-potential on $O$ has components $A_0=-\varphi \mathrm{/}c$, $A_i=0$ in the $x^0=ct$ convention, and the electromagnetic two-form convention is that of The electromagnetic two-form, its observer decomposition and the four-current. Thus $E_i=-{\mathrm{\partial }}_{x^i}\varphi$ and $F_{0i}=-E_i\mathrm{/}c$ on $O$; $A$ is $C^{\mathrm{\infty }}$ there and no field value or smoothness across $\mathbf{x}=0$ is asserted. Put $\kappa :=qQ\mathrm{/}(4\pi {\epsilon }_0)>0$, which has units joule metres.

The body's worldline is the $C^2$ future-timelike curve $z:I_z\to O$ of Charged and spinning test-body quantities, with $I_z$ an open proper-time interval and tangent $U=dz\mathrm{/}d\tau$ satisfying $g(U,U)=-c^2$; its four-momentum is $P=mU$.

Under the adopted fixed-endpoint charged-particle action and its proved Lorentz-force consequence, assume ${\mathrm{A}\mathrm{C}}_{\omega }$ for the canonical tangent-bundle smooth structure as required by those suppliers. In the flat specialization also adopt the massive-particle momentum model of Adopted massive-particle momentum, observer measurement and isolated-collision conservation. For $\mathbf{v}:=d\mathbf{x}\mathrm{/}dt$ one has $\mathrm{\mid }\mathbf{v}\mathrm{\mid }<c$ and $\gamma :=(1-\mathrm{\mid }\mathbf{v}{\mathrm{\mid }}^2\mathrm{/}{c}^2{)}^{-1\mathrm{/}2}$. Then the canonical spatial momentum equals the kinetic momentum $\mathbf{p}=m\gamma \mathbf{v}$ (in kilograms metres per second), and the coordinate-time Hamiltonian is

$H(\mathbf{x},\mathbf{p})=\sqrt{m^2{c}^4+c^2\mathrm{\mid }\mathbf{p}{\mathrm{\mid }}^2}+\frac{\kappa }{r}\mathrm{.}$

Thus the trajectories are those of Repulsive relativistic Coulomb Hamiltonian phase system, with $\mathbf{x}\in \mathrm{\Omega }$, $\mathbf{p}\in {\mathbb{R}}^3$, and coordinate time $t\in I$. For asymptotic scattering data, use the unique maximal coordinate-time extension of the given local phase trajectory on the noncolliding scattering branch of that phase system. Incoming and outgoing asymptotic vectors below belong to this extension; a finite interval restriction need not itself contain either asymptotic tail. Let $W:=H(\mathbf{x},\mathbf{p})>m{c}^2$ be its conserved energy in joules and let $J:=\mathrm{\mid }\mathbf{x}\times \mathbf{p}\mathrm{\mid }>\kappa \mathrm{/}c$ be the conserved angular-momentum magnitude in kilograms metres squared per second. The proposition Exact repulsive relativistic Coulomb scattering branch gives the incoming and outgoing asymptotic velocity vectors and defines their principal Euclidean angle $\theta \in (0,\pi )$ as $\pi -2\mathrm{\Delta }$; the limiting momentum vectors have the same directions because $\mathbf{p}=m\gamma \mathbf{v}$ with $\gamma >0$. Define

$\mathcal{A}=\frac{W^2-m^2{c}^4}{c^2},d=\frac{W\kappa }{c^2},C=J^2-\frac{{\kappa }^2}{c^2},$

Here $\mathcal{A}$ is the positive scalar denoted $A$ in Exact repulsive relativistic Coulomb scattering branch. Its exact angle between incoming and outgoing momenta is

$\theta =\pi -\frac{2J}{\sqrt{C}}\arctan \left(\frac{\sqrt{C\mathcal{A}}}{d}\right)\in (0,\pi )\mathrm{.}$

Here $\mathcal{A}$ has units of squared momentum, $d$ has units of $\mathcal{A}$ times length, and $C$ has units of squared angular momentum, so $\sqrt{C\mathcal{A}}\mathrm{/}d$ and $\theta$ are dimensionless. This is a model deduction, not a measured scattering law. The inequality $J>\kappa \mathrm{/}c$ is part of the result's domain; no conclusion is made for $J\le \kappa \mathrm{/}c$.

• The Coulomb source is an external fixed source, not a second dynamical body. The calculation does not conserve source-plus-particle momentum because the source's response is outside the model.
• Landau and Lifshitz §39 gives the exact orbit relation and scattering angle in Gaussian units. The present argument states its own SI conversion through $\kappa =qQ\mathrm{/}(4\pi {\epsilon }_0)$ and does not import a Gaussian-unit formula without conversion.
• Boyer's independent paper treats an attractive $-\alpha \mathrm{/}r$ potential and displays its exact orbit families. The present item uses the opposite, repulsive sign and proves that branch directly; Boyer's attractive trajectories are not imported as a proof of the repulsive formula.