3+1 Orthogonal and Conformal Decomposition of the Einstein Equation and the ADM Formalism for General Relativity

TL;DR

This paper develops new orthogonal and conformal decompositions of the Einstein equations and ADM formalism, providing a framework that connects conformal solitons, static vacuum solutions, and potential extensions to other gravity theories.

Contribution

It introduces a novel method linking Cotton solitons with static vacuum solutions within the ADM formalism of general relativity.

Findings

Established orthogonal and conformal decompositions of Einstein equations.

Proposed a method relating Cotton solitons to static vacuum solutions.

Suggested extensions to TMG and Ricci solitons.

Abstract

In this work, two particular orthogonal and conformal decompositions of the 3+1 dimensional Einstein equation and Arnowitt-Deser-Misner (ADM) formalism for general relativity are obtained. In order to do these, the 3+1 foliation of the four-dimensional spacetime, the fundamental conformal transformations and the Hamiltonian form of general relativity that leads to the ADM formalism, defined for the conserved quantities of the hypersurfaces of the globally-hyperbolic asymptotically flat spacetimes, are reconstructed. All the calculations up to chapter 7 are just a review. We propose a method in chapter 7 which gives an interesting relation between the Cotton (Conformal) soliton and the static vacuum solutions. The formulation that we introduce can be extended to find the gradient Cotton soliton and the solutions of Topologically Massive Gravity (TMG) as well as the gradient Ricci soliton.