Consistent and mimetic discretizations in general relativity

TL;DR

This paper demonstrates that a recently introduced discretization technique for linearized general relativity can be mimetic, preserving constraints at the discrete level, and discusses challenges in extending this to full Einstein equations.

Contribution

The paper shows that a new discretization method yields a mimetic scheme for linearized general relativity using metric and Ashtekar variables, and discusses conceptual issues for full Einstein equations.

Findings

Discretization is mimetic for linearized GR around Minkowski space.

The technique works with both metric and Ashtekar variables.

Conceptual difficulties exist in creating a mimetic formulation for full Einstein equations.

Abstract

A discretization of a continuum theory with constraints or conserved quantities is called mimetic if it mirrors the conserved laws or constraints of the continuum theory at the discrete level. Such discretizations have been found useful in continuum mechanics and in electromagnetism. We have recently introduced a new technique for discretizing constrained theories. The technique yields discretizations that are consistent, in the sense that the constraints and evolution equations can be solved simultaneously, but it cannot be considered mimetic since it achieves consistency by determining the Lagrange multipliers. In this paper we would like to show that when applied to general relativity linearized around a Minkowski background the technique yields a discretization that is mimetic in the traditional sense of the word. We show this using the traditional metric variables and also the Ashtekar new variables, but in the latter case we restrict ourselves to the Euclidean case. We also argue that there appear to exist conceptual difficulties to the construction of a mimetic formulation of the full Einstein equations, and suggest that the new discretization scheme can provide an alternative that is nevertheless close in spirit to the traditional mimetic formulations.