Formation of Trapped Surfaces for the Einstein-Euler System

Abstract

We prove the formation of trapped surfaces for the Einstein--Euler system in 3+1 dimensions beyond spherical symmetry. We consider smooth characteristic initial data in a short-pulse regime with rapid fluid injection. Starting from initial data containing no trapped surfaces, we obtain two sufficient criteria for trapped surface formation, corresponding to the shear and fluid contributions, respectively. In particular, for the fluid-driven criterion, we impose no positive lower bound on the shear flux. To handle the multiple propagation speeds associated with gravitational waves, sound waves, and fluid transport, our analysis combines three ingredients: a coercive fluid energy current whose null fluxes reflect the unequal velocity components, curvature renormalizations adapted to the matter terms, and weighted geometric estimates controlling derivatives in all spacetime directions. To ensure that fluid injection alone can lead to trapped surface formation, we use particle-number conservation in a designed spacetime region to control the focusing process and obtain the desired sphere-integrated fluxes.