Discretizing linearized Einstein-Bianchi system by symmetric and traceless tensors

TL;DR

This paper introduces a novel finite element discretization for the linearized Einstein-Bianchi system, leveraging the conformal Hessian complex to preserve symmetry and tracelessness, improving numerical stability and accuracy.

Contribution

It develops a conforming finite element method based on the conformal Hessian complex that maintains algebraic constraints in discretizing the Einstein-Bianchi system.

Findings

Exactness of the finite element complex is proven.

The method preserves symmetry and tracelessness simultaneously.

Applicable to general 3D tetrahedral grids.

Abstract

The Einstein-Bianchi system uses symmetric and traceless tensors to reformulate Einstein's original field equations. However, preserving these algebraic constraints simultaneously remains a challenge for numerical methods. This paper proposes a new formulation that treats the linearized Einstein-Bianchi system (near the trivial Minkowski metric) as the Hodge wave equation associated with the conformal Hessian complex. To discretize this equation, a conforming finite element conformal Hessian complex that preserves symmetry and traceless-ness simultaneously is constructed on general three-dimensional tetrahedral grids, and its exactness is proven.