Static Balance of Two Charged Particles in General Relativity

Open Access
Publisher Website
Google Scholar
Abstract
In order to investigate the condition for static balance of two charged particles, the Hamiltonian in the canonical formalism is evaluated. It is shown that the condition for balance is ea = ± (4πG)1/2ma (a = 1, 2) and the alternate conditions ea = ±(4πGm1m2)1/2 given by Barker and O'Connell and e1e2 = 4πGm1m2 conjectured by Bonnor are not valid. To confirm the condition, the Einstein-Maxwell equations with source terms of the two charged particles are solved explicitly up to the post-post-Newtonian (ppN) approximation in axially symmetric coordinates. Since the source terms are included, one new equation is derived in addition to the ordinary equations in empty space. This equation represents the restriction for the metric tensor and the electrostatic potential in order that the solution truly corresponds to the system of the two charged particles. And the restriction equation leads to the same condition obtained in the Hamiltonian method.