Nonlinear gravitons from the initial value constraints of GR in Ashtekar variables

TL;DR

This paper constructs solutions to Einstein's initial value constraints in Ashtekar variables, realizing Penrose's nonlinear gravitons concept through a formal expansion around Type O spacetimes, highlighting new nonlinear solution methods.

Contribution

It introduces a novel approach to solving Einstein's constraints in Ashtekar variables, enabling the realization of nonlinear gravitons via a formal operatorial expansion.

Findings

Constructed solutions using a spatial SU(2) connection and free functions.

Developed a formal expansion in powers of the cosmological constant.

Provided a simple nonlinear example with a homogeneous connection.

Abstract

In this paper we provide a possible realization of Penrose's idea of nonlinear gravitons by constructing a solution to the initial value constraints in Ashtekar variables. The solution inputs are a spatial SU(2) connection and two free functions of position, and can be constructed as a formal operatorial expansion in powers of the cosmological constant about spacetimes of Petrov Type O. We first present the linear case, and then provide a simple nonlinear example to first order using a spatially homogeneous connection.