The formation of trapped surfaces in spherically-symmetric Einstein–Euler spacetimes with bounded variation

Abstract

We study the evolution of a self-gravitating compressible fluid in spherical symmetry and we prove the existence of weak solutions with bounded variation for the Einstein–Euler equations of general relativity. We formulate the initial value problem in Eddington–Finkelstein coordinates and prescribe spherically symmetric data on a characteristic initial hypersurface. We introduce here a broad class of initial data which contain no trapped surfaces, and we then prove that their Cauchy development contains trapped surfaces. We therefore establish the formation of trapped surfaces in weak solutions to the Einstein equations. This result generalizes a theorem by Christodoulou for regular vacuum spacetimes (but without symmetry restriction). Our method of proof relies on a generalization of the “random choice” method for nonlinear hyperbolic systems and on a detailed analysis of the nonlinear coupling between the Einstein equations and the relativistic Euler equations in spherical symmetry.