Ashtekar variables, self-dual metrics and w-infinity

TL;DR

This paper explores the self-duality equations in general relativity using Ashtekar variables, revealing a connection to the $w_inf$ algebra and constructing explicit self-dual metrics parametrized by this algebra.

Contribution

It demonstrates that the self-duality equations can be expressed as Hamiltonian systems with $w_inf$ algebra-based Hamiltonians, leading to explicit solutions and metrics.

Findings

Self-duality equations can be written as divergence-free vector field equations.

A sector with Hamiltonian form involving $w_inf$ algebra is identified.

Explicit solutions parametrized by $w_inf$ algebra are constructed.

Abstract

The self-duality equations for the Riemann tensor are studied using the Ashtekar Hamiltonian formulation for general relativity. These equations may be written as dynamical equations for three divergence free vector fields on a three dimensional surface in the spacetime. A simplified form of these equations, describing metrics with a one Killing field symmetry are written down, and it shown that a particular sector of these equations has a Hamiltonian form where the Hamiltonian is an arbitrary function on a two-surface. In particular, any element of the $w_\infty$ algebra may be chosen as a the Hamiltonian. For a special choice of this Hamiltonian, an infinite set of solutions of the self-duality equations are given. These solutions are parametrized by elements of the $w_\infty$ algebra, which in turn leads to an explicit form of four dimensional complex self-dual metrics that are in one to one correspondence with elements of this algebra.