Gravity Seminar: Conditions for preserving constraints in well posed theories: application to gravitation, Julio Fernando Abalos, UIB

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 derive the constraint evolution equations, discuss their strong hyperbolicity and their connection with the Kronecker decomposition of matrix pencils. Finally, we will discuss how this scheme could be extrapolated to PDEs that include Gauge freedom, providing a guide for fixing those freedoms in a well-posed way.