Plane wave holonomies in loop quantum gravity II: sine wave solution

TL;DR

This paper develops an approximate sinusoidal wave packet solution in loop quantum gravity, analyzing how holonomies behave under a passing gravitational wave using a small sine approximation and coherent states.

Contribution

It introduces a novel sinusoidal wave packet solution in LQG, employing a small sine approximation and coherent states tailored to plane wave symmetry.

Findings

Constructed an approximate sinusoidal wave packet in LQG.

Analyzed holonomy behavior under gravitational waves.

Estimated wave energy and coherence lifetime.

Abstract

This paper constructs an approximate sinusoidal wave packet solution to the equations of loop quantum gravity (LQG). There is an SU(2) holonomy on each edge of the LQG simplex, and the goal is to study the behavior of these holonomies under the influence of a passing gravitational wave. The equations are solved in a small sine approximation: holonomies are expanded in powers of sines, and terms beyond $\sin^2$ are dropped; also, fields vary slowly from vertex to vertex. The wave is unidirectional and linearly polarized. The Hilbert space is spanned by a set of coherent states tailored to the symmetry of the plane wave case. Fixing the spatial diffeomorphisms is equivalent to fixing the spatial interval between vertices of the loop quantum gravity lattice. This spacing can be chosen such that the eigenvalues of the triad operators are large, as required in the small sine limit, even though the holonomies are not large. Appendices compute the energy of the wave, estimate the lifetime of the coherent state packet, discuss coarse-graining, and determine the behavior of the spinors used in the U(N) SHO realization of LQG.