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 detailled analysis of the nonlinear coupling between the Einstein equations and the relativistic Euler equations in spherical symmetry.Keywords: Einstein equations, Euler equations, compressible fluid, trapped surfaces, spherical symmetry, shock wave, bounded variation 2010 MSC: 83C05, 35L60, 76N10 RésuméNousétudions l'évolution d'un fluide compressible auto-gravitant en symétrie radiale et nous démon-trons un résultat d'existence de solutions faiblesà variation bornée pour leséquations d'Einstein-Euler de la relativité générale. Nous formulons le problème de Cauchy en coordonnées d'Eddington-Finkelstein et prescrivons des donnéesà symétrie radiale sur une hypersurface initiale caractéristique. Nous introduisons ici une classe de données initiales qui ne contiennent pas de surfaces piégées, et nous démontrons alors que leur développement de Cauchy contient des surfaces piégées. Nousétablissons ainsi un résultat de formation de surfaces piégées dans les solutions faibles deséquations d'Einstein. Ce résultat généralise un théorème de Christodoulou pour les espaces-temps réguliers sans matière (mais sans restriction de symétrie). Notre méthode de preuve s'appuie sur une généralisation de la méthode "random choice" pour les systèmes hyperboliques nonlinéaires et sur une analyse fine du couplage nonlinéaire entre leséquations d'Einstein et leséquations d'Euler relativistes en symétrie radiale.

We show that spaces of Colombeau generalized functions with smooth parameter dependence are isomorphic to those with continuous parametrization. Based on this result we initiate a systematic study of algebraic properties of the ring Ksm of generalized numbers in this unified setting. In particular, we investigate the ring and order structure of Ksm and establish some properties of its ideals.

The null distance for Lorentzian manifolds was recently introduced by Sormani and Vega. Under mild assumptions on the time function of the spacetime, the null distance gives rise to an intrinsic, conformally invariant metric that induces the manifold topology. We show when warped products of low regularity and globally hyperbolic spacetimes endowed with the null distance are (local) integral current spaces. This metric and integral current structure sets the stage for investigating convergence analogous to Riemannian geometry. Our main theorem is a general convergence result for warped product spacetimes relating uniform, Gromov–Hausdorff, and Sormani–Wenger intrinsic flat convergence of the corresponding null distances. In addition, we show that nonuniform convergence of warping functions in general leads to distinct limiting behavior, such as limits that disagree.

Static spherically symmetric solutions to the Einstein-Euler equations with prescribed central densities are known to exist, be unique and smooth for reasonable equations of state. Some criteria are also available to decide whether solutions have finite extent (stars with a vacuum exterior) or infinite extent. In the latter case, the matter extends globally with the density approaching zero at infinity. The asymptotic behavior largely depends on the equation of state of the fluid and is still poorly understood. While a few such unbounded solutions are known to be asymptotically flat with finite ADM mass, the vast majority are not. We provide a full geometric description of the asymptotic behavior of static spherically symmetric perfect fluid solutions with linear equations of state and polytropic-type equations of state with index n > 5. In order to capture the asymptotic behavior, we introduce a notion of scaled quasi-asymptotic flatness, which encompasses the notion of asymptotic conicality. In particular, these spacetimes are asymptotically simple.