Exact Solution for the Metric and the Motion of Two Bodies in (1+1) Dimensional Gravity

TL;DR

This paper derives an exact solution for the two-body problem in (1+1)-dimensional dilaton gravity, revealing new motion regimes and connecting the Hamiltonian to the Lambert W function, with implications for understanding relativistic effects.

Contribution

It provides the first exact, analytical solution for two-body motion in (1+1)D gravity, including explicit Hamiltonian and metric expressions, and explores novel motion behaviors beyond Newtonian limits.

Findings

Hamiltonian expressed via Lambert W function

Identification of new motion regimes beyond Newtonian physics

Explicit phase space trajectories for various energies

Abstract

We present the exact solution of two-body motion in (1+1) dimensional dilaton gravity by solving the constraint equations in the canonical formalism. The determining equation of the Hamiltonian is derived in a transcendental form and the Hamiltonian is expressed for the system of two identical particles in terms of the Lambert $W$ function. The $W$ function has two real branches which join smoothly onto each other and the Hamiltonian on the principal branch reduces to the Newtonian limit for small coupling constant. On the other branch the Hamiltonian yields a new set of motions which can not be understood as relativistically correcting the Newtonian motion. The explicit trajectory in the phase space $(r, p)$ is illustrated for various values of the energy. The analysis is extended to the case of unequal masses. The full expression of metric tensor is given and the consistency between the solution of the metric and the equations of motion is rigorously proved.