Particles on a Circle in Canonical Lineal Gravity

TL;DR

This paper develops a canonical formulation of lineal gravity with particles on a circle, deriving exact solutions and showing how particles influence the universe's expansion or contraction.

Contribution

It introduces a canonical framework for lineal gravity with point particles on a circular topology and provides exact solutions illustrating particle effects on spacetime evolution.

Findings

Exact solutions for pure gravity and single-particle cases

Particles can slow down or reverse cosmological expansion

Hamiltonian relates to extrinsic curvature and circumference

Abstract

A description of the canonical formulation of lineal gravity minimally coupled to N point particles in a circular topology is given. The Hamiltonian is found to be equal to the time-rate of change of the extrinsic curvature multiplied by the proper circumference of the circle. Exact solutions for pure gravity and for gravity coupled to a single particle are obtained. The presence of a single particle significantly modifies the spacetime evolution by either slowing down or reversing the cosmological expansion of the circle, depending upon the choice of parameters.