Hyperbolicity analysis of the linearised 3+1 formulation in the Teleparallel Equivalent of General Relativity

TL;DR

This work analyzes the hyperbolicity of the $3+1$ evolution system in the Teleparallel Equivalent of General Relativity using a covariant Hamiltonian formulation with a VAST decomposition. By linearizing around Minkowski spacetime and performing a first order reduction, the authors extract the principal symbol and find sectors with imaginary eigenvalues, indicating a lack of strong hyperbolicity for the chosen gauge and constraint setup. They further show that adding linear combinations of the TEGR constraints can reduce, but not remove, the nonhyperbolic sector, and that extending to full $3$D preserves the issue. The results highlight that hyperbolicity in tetrad-based TEGR is sensitive to gauge and boundary-term choices, and set the stage for exploring constraint damping and gauge strategies to achieve well posedness in contexts like spherical symmetry and numerical relativity.

Abstract

We study the properties of the principal symbol of the 3+1 equations of motion in Teleparallel Equivalent of General Relativity (TEGR) and assess the conditions for hyperbolicity. We use the Hamiltonian formulation based on the vectorial, antisymmetric, symmetric trace-free, and trace (VAST) decomposition of the canonical variables in the Hamiltonian formalism, and the Hamilton's equations previously presented in the literature. We study the system of differential equations at the linear level, and show that the principal symbol has a sector with imaginary eigenvalues, which renders the system not hyperbolic. This situation persists by taking spatial derivatives in either one or three coordinate directions, and it should be interpreted as a problem of the specific gauge choice instead of a general problem with TEGR. The first practical use of Hamilton's equations in this work can be extended for proving well-posedness in spherical symmetry, and establish numerical relativity setups in TEGR.