# Well-posed Cauchy formulation for Einstein-\ae ther theory

## TL;DR

This paper proves the well-posedness of the initial value problem in Einstein-aether theory by reformulating it into a hyperbolic system, ensuring stability for numerical simulations of this Lorentz-violating gravity model.

## Contribution

It introduces a first-order tetrad-based formulation of Einstein-aether theory and demonstrates conditions under which the equations are strongly or symmetric hyperbolic, establishing well-posedness.

## Key findings

- The formulation yields a well-posed Cauchy problem under certain conditions.
- The approach applies to a Lorentz-violating gravitational theory.
- Ensures stability for numerical evolutions of Einstein-aether equations.

## Abstract

We study the well-posedness of the initial value (Cauchy) problem of vacuum Einstein-aether theory. The latter is a Lorentz-violating gravitational theory consisting of General Relativity with a dynamical timelike 'aether' vector field, which selects a 'preferred time' direction at each spacetime event. The Einstein-aether action is quadratic in the aether, and thus yields second order field equations for the metric and the aether. However, the well-posedness of the Cauchy problem is not easy to prove away from the simple case of perturbations over flat space. This is particularly problematic because well-posedness is a necessary requirement to ensure stability of numerical evolutions of the initial value problem. Here, we employ a first-order formulation of Einstein-aether theory in terms of projections on a tetrad frame. We show that under suitable conditions on the coupling constants of the theory, the resulting evolution equations can be cast into strongly or even symmetric hyperbolic form, and therefore they define a well-posed Cauchy problem.