Local well-posedness of the initial value problem in Einstein-Cartan theory

TL;DR

This paper proves the local well-posedness of the initial value problem in Einstein-Cartan theory, extending classical results to include torsion and non-symmetric connections, using advanced hyperbolic PDE techniques.

Contribution

It generalizes the classical well-posedness results of Einstein equations to Einstein-Cartan theory with torsion, employing a harmonic gauge and Leray-Ohya theory for non-strictly hyperbolic systems.

Findings

Established local geometric well-posedness for Einstein-Cartan initial value problem

Derived evolution and constraint equations including torsion

Recovered classical Einstein results when torsion vanishes

Abstract

We study the initial value problem in Einstein-Cartan theory which includes torsion and, therefore, a non-symmetric connection on the spacetime manifold. Generalizing the path of a classical theorem by Choquet-Bruhat and York for the Einstein equations, we use a $n+1$ splitting of the manifold and compute the evolution and constraint equations for the Einstein-Cartan system. In the process, we derive the Gauss-Codazzi-Ricci equations including torsion. We prove that the constraint equations are preserved during evolution. Imposing a generalized harmonic gauge, it is shown that the evolution equations can be cast as a quasilinear system in a Cauchy regular form with a characteristic determinant having a non-trivial multiplicity of characteristics. Using the Leray-Ohya theory for non-strictly hyperbolic systems we then establish the local geometric well-posedness of the Cauchy problem, for sufficiently regular initial data. For vanishing torsion we recover the classical results for the Einstein equations.