Solution-generating methods of Einstein's equations by Hamiltonian reduction

TL;DR

This paper introduces a novel Hamiltonian reduction method to generate exact solutions of Einstein's equations by identifying privileged coordinates where physical degrees of freedom are explicit, simplifying the equations significantly.

Contribution

The paper presents a new Hamiltonian reduction approach that simplifies Einstein's equations and constructs exact solutions using conformal two-metric data.

Findings

Derived two exact solutions: Einstein-Rosen wave and Schwarzschild.

Showed how to construct spacetime from conformal two-metric data.

Presented gravitational Hamiltonian and momentum densities in privileged coordinates.

Abstract

The purpose of this paper is to demonstrate a new method of generating exact solutions to the Einstein's equations obtained by the Hamiltonian reduction. The key element to the successful Hamiltonian reduction is finding the privileged spacetime coordinates in which physical degrees of freedom manifestly reside in the conformal two-metric, and all the other metric components are determined by the conformal two-metric. In the privileged coordinates the Einstein's constraint equations become trivial; the Hamiltonian and momentum constraints are simply the defining equations of a non-vanishing gravitational Hamiltonian and momentum densities in terms of conformal two-metric and its conjugate momentum, respectively. Thus, given any conformal two-metric, which is a constraint-free data, one can construct the whole 4-dimensional spacetime by integrating the first-order superpotential equations. As the first examples of using Hamiltonian reduction in solving the Einstein's equations, we found two exact solutions to the Einstein's equations in the privileged coordinates. Suitable coordinate transformations from the privileged to the standard coordinates show that they are just the Einstein-Rosen wave and the Schwarzschild solution. The local gravitational Hamiltonian and momentum densities of these spacetimes are also presented in the privileged coordinates.