Conditions for constraint preservation and gauge fixing in well posed theories: application to gravitation

ABSTRACT:

We use Partial Differential Equations (PDEs) to describe physical systems; in general, these equations include evolution and constraint equations. One method used to find solutions to these equations is the free-evolution approach, which consists in obtaining solutions of the entire system by solving only the evolution equations.

Certainly, this is valid only when the chosen initial data satisfies the constraints, and the constraints are preserved in the evolution. In this talk, we will establish sufficient conditions for generic first order PDEs to guarantee the constraint preservation. For this purpose, we will discuss the strong hyperbolicity of the PDEs and their connection with the Kronecker decomposition of matrix pencils.

Another method for finding solutions of PDEs with constraints is through extensions; some well-known examples are divergence cleaning for Electrodynamics and Z4 for Einstein equations.
We will discuss how Kronecker decomposition suggests families of strongly hyperbolic extensions.

Finally, we will discuss how this scheme can be extrapolated to PDEs that include Gauge freedom, providing a guide for fixing those freedoms in a well-posed way. We will show how these ideas apply to Einstein equations.