Open-Flux Solutions to the Constraints for Plane Gravity Waves

TL;DR

This paper quantizes plane gravitational waves using Ashtekar variables, proposing open flux line solutions that satisfy constraints except at boundaries, with implications for loop quantum gravity.

Contribution

It introduces open-ended flux line solutions for plane gravitational waves in Ashtekar variables, extending spin network concepts to boundary conditions in quantum gravity.

Findings

Solutions are annihilated by constraints except at boundaries.

Flux lines can be terminated on matter, satisfying constraints.

Holonomy matrices can have arbitrary spin, generalizing spin networks.

Abstract

The metric for plane gravitational waves is quantized using the Ashtekar field variables. The z axis (direction of travel of the waves) is taken to be the entire real line. Solutions to the constraints are proposed; they involve open-ended flux lines running along the entire z axis. These solutions are annihilated by the constraints except at the two boundary points, where the Gauss constraint does not annihilate the solutions. This result is in sharp contrast to the situation in the general, 3+1 dimensional case without planar symmetry, where the Gauss constraints do not contribute at boundaries because the Lagrange multipliers for the Gauss constraints vanish there. The constraints annihilate the solutions if classical matter is included (so that flux lines are terminated on the matter). The SU(2) holonomy matrices used in the solutions are (2j+1) dimensional, where j may be any spin, not necessarily j = 1/2. In this respect the solutions resemble the spin network states recently constructed by Rovelli and Smolin in loop space.