Exact solutions of relativistic two-body motion in lineal gravity

Abstract

We develop the canonical formalism for a system of N bodies in lineal gravity and obtain exact solutions to the equations of motion for $N=2.$ The determining equation of the Hamiltonian is derived in the form of a transcendental equation, which leads to the exact Hamiltonian to infinite order in the gravitational coupling constant. In the equal mass case, explicit expressions of the trajectories of the particles are given as functions of the proper time, which show characteristic features of the motion depending on the strength of gravity (mass) and the magnitude and sign of the cosmological constant. As expected, we find that a positive cosmological constant has a repulsive effect on the motion, while a negative one has an attractive effect. However, some surprising features emerge that are absent for a vanishing cosmological constant. For a certain range of the negative cosmological constant, the motion shows a double maximum behavior as a combined result of an induced momentum-dependent cosmological potential and the gravitational attraction between the particles. For a positive cosmological constant, not only bounded motions but also unbounded ones are realized. The change of the metric along the movement of the particles is also exactly derived.