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

The purpose of this paper is to demonstrate a new method of generating exact solutions to 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, Einstein’s constraint equations become trivial; the Hamiltonian and momentum constraints are simply the defining equations of a nonvanishing 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 four-dimensional spacetime by integrating the first-order superpotential equations. As the first examples of using Hamiltonian reduction in solving Einstein’s equations, we found two exact solutions to 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.

Hamiltonian reduction of Einstein’s equations without isometries

I apply the Hamiltonian reduction procedure to 4-dimensional spacetimes without isometries and find privileged spacetime coordinates in which the physical Hamiltonian is expressed in terms of the conformal two metric and its conjugate momentum. Physical time is the area element of the cross section of null hypersurface, and the physical radial coordinate is defined by equipotential surfaces on a given spacelike hypersurface of constant physical time. The physical Hamiltonian is local and positive in the privileged coordinates. Einstein’s equations in the privileged coordinates are presented as Hamilton’s equations of motions obtained from the physical Hamiltonian.