Exact Solutions to the Motion of Two Charged Particles in Lineal Gravity

Abstract

We extend the canonical formalism for the motion of $N$-particles in lineal gravity to include charges. Under suitable coordinate conditions and boundary conditions the determining equation of the Hamiltonian (a kind of transcendental equation) is derived from the matching conditions for the dilaton field at the particles' position. The canonical equations of motion are derived from this determining equation. For the equal mass case the canonical equations in terms of the proper time can be exactly solved in terms of hyperbolic and/or trigonometric functions. In electrodynamics with zero cosmological constant the trajectories for repulsive charges exhibit not only bounded motion but also a countably infinite series of unbounded motions for a fixed value of the total energy $H_{0}$, while for attractive charges the trajectories are simply periodic. When the cosmological constant $Λ$ is introduced, the motion for a given $Λ$ and $H_{0}$ is classified in terms of the charge-momentum diagram from which we can predict what type of the motion is realized for a given charge. Since in this theory the charge of each particle appears in the form $e_{1}e_{2}$ in the determining equation, the static balance condition in 1+1 dimensions turns out to be identical with the condition in Newtonian theory. We generalize this condition to non-zero momenta, obtaining the first exact solution to the static balance problem that does not obey the Majumdar-Papapetrou condition.