Canonical reduction of two-dimensional gravity for Particle Dynamics

TL;DR

This paper develops a canonical reduction formalism for (1+1)-dimensional gravity coupled with point particles, expressing the Hamiltonian explicitly in terms of particle variables and approximations.

Contribution

It introduces a formalism for canonical reduction in (1+1)D gravity with point particles, highlighting the role of an auxiliary scalar field and deriving explicit Hamiltonian forms.

Findings

Reduced Hamiltonian expressed as integral of scalar field derivatives

Explicit Hamiltonian derived in post-linear and post-Newtonian approximations

Formalism analogous to (3+1)-dimensional gravity reduction

Abstract

We develop the formalism for canonical reduction of $(1+1)$--dimensional gravity coupled with a set of point particles by eliminating constraints and imposing coordinate conditions. The formalism itself is quite analogous to the $(3+1)$--dimensional case; however in $(1+1)$ dimensions an auxiliary scalar field is shown to have an important role. The reduced Hamiltonian is expressed as a form of spatial integral of the second derivative of the scalar field. Since in $(1+1)$ dimensions there exists no dynamical degree of freedom of the gravitational field ({\it i.e.} the transverse-traceless part of the metric tensor is zero), the reduced Hamiltonian is completely determined in terms of the particles' canonical variables (coordinates and momenta). The explicit form of the Hamiltonian is calculated both in post-linear and post-Newtonian approximations.