Open-Flux Solutions to the Constraints for Plane Gravity Waves

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.