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

TL;DR

This paper analyzes the hyperbolic properties of the linearised 3+1 TEGR equations, identifying conditions for strong hyperbolicity and proposing gauge fixing to ensure well-posedness.

Contribution

It is the first practical application of Hamilton's equations in TEGR, demonstrating how to achieve strong hyperbolicity through gauge fixing.

Findings

The principal symbol has sectors with imaginary eigenvalues indicating non-hyperbolicity.

Gauge fixing can remove problematic sectors, leading to a strongly hyperbolic system.

The work sets the stage for numerical relativity in TEGR, including spherical symmetry cases.

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 in one dimension, and show that the principal symbol has a sector with imaginary eigenvalues, which renders the system not hyperbolic. This situation is circumvented by identifying the problematic sectors, which are an isolated system and can be removed by a gauge fixing. We prove that the remaining system of equations is strongly hyperbolic. We also present the system in three dimensions. This is the first practical use of Hamilton's equations in TEGR, and our work can be extended for proving well-posedness in spherical symmetry, and establish numerical relativity setups in TEGR.