Exact Solutions of Relativistic Two-Body Motion in Lineal Gravity

TL;DR

This paper derives exact solutions for the motion of two bodies in lineal gravity, revealing how the cosmological constant influences their trajectories and uncovering novel behaviors absent in zero cosmological constant scenarios.

Contribution

It provides the first exact Hamiltonian solutions for two-body motion in lineal gravity, including effects of the cosmological constant and explicit particle trajectories.

Findings

Positive cosmological constant causes repulsive effects.

Negative cosmological constant can induce double maximum motion.

Both bounded and unbounded motions are possible depending on the cosmological constant.

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 of the gravitational coupling constant. In the equal mass case explicit expressions of the trajectories of the particles are given as the 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 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.