The Regularity of Spacetime Perturbations for a Flat Spacetime Under Sobolev Spaces and Spectral Theory

Abstract

This study introduces an analysis concerning spacetime perturbations within the context of quantum foam models. Under the framework of Sobolev spaces, 𝐻�����𝑠�����(ℝ𝑛�����) , we establish the existence and uniqueness of solutions to a linearized wave equation considering the harmonic gauge condition. Energy estimates are derived, demonstrating the conservation of both standard and higher-order energy functionals, which ensures the stability and regularity of metric perturbations over time. In addition, a spectral analysis of the d’Alembertian operator is conducted through Fourier transform techniques. Explicit calculations of Sobolev norms further confirm that these norms remain uniformly bounded, reinforcing the stability of solutions in the Sobolev space framework.