Well-posed Cauchy formulation for Einstein-æther theory
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.