The Initial Value Formulation of Dynamical Chern-Simons Gravity

TL;DR

This paper develops an initial value formulation for dynamical Chern-Simons gravity, revealing challenges in establishing well-posedness due to potential ill-posedness of the evolution equations.

Contribution

It provides the first detailed analysis of the initial value problem in dynamical Chern-Simons gravity, highlighting issues with hyperbolicity and well-posedness.

Findings

Weak necessary conditions for hyperbolicity are satisfied.

Full field equations are not hyperbolic and may lead to ill-posed evolution.

Certain states cause the evolution to be undetermined.

Abstract

We derive an initial value formulation for dynamical Chern-Simons gravity, a modification of general relativity involving parity-violating higher derivative terms. We investigate the structure of the resulting system of partial differential equations thinking about linearization around arbitrary backgrounds. This type of consideration is necessary if we are to establish well-posedness of the Cauchy problem. Treating the field equations as an effective field theory we find that weak necessary conditions for hyperbolicity are satisfied. For the full field equations we find that there are states from which subsequent evolution is not determined. Generically the evolution system closes, but the full field equations are in no sense hyperbolic. In a cursory mode analysis we find that the equations of motion contain terms that may cause ill-posedness of the initial value problem.