  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.
