Dissertation Defense: Trapped Surface Formation in General Relativity and Double Null Foliations

Abstract:

This dissertation deals primarily with two main subjects within the field of mathematical general relativity (GR). Mathematical GR is the study of gravitation through analysis of the Einstein equations, which link the matter content of our universe to its geometry. It uses techniques from geometric analysis and nonlinear partial differential equations (PDE) in order to prove rigorous mathematical theorems that answer questions from physics. The first subject in this dissertation is linear first order hyperbolic systems of PDE in a double null foliation. The second is the formation of trapped surfaces in general relativity. Prior to these, we provide a brief introduction to the main ideas of mathematical general relativity which are featured in this thesis.

A double null foliation is a particular type of foliation of spacetime which is well-suited to the study of gravitational radiation. Gravitational waves propagate along the hypersurfaces defined by a double null foliation. The Bianchi identities for the Riemann curvature of a vacuum spacetime exhibit a hyperbolic structure which has been used to obtain detailed nonlinear estimates on the null curvature components. In the first main work of this dissertation, we introduce the notion of a double null hyperbolic system of equations and prove a global existence and uniqueness theorem for these systems. We then discuss the relationship between these systems and the Bianchi equations. We also derive a novel algebraic constraint which must be satisfied, at every point in the spacetime, by tensor fields satisfying the linearized Bianchi equations (the linearized Bianchi equations are obtained from the usual Bianchi equations by replacing the null Weyl tensor components with unknown tensorfields). This work has potential applications in the study of gravitational waves, as well as in any system where the null structure is paramount, such as Einstein-null dust, Einstein-massless Vlasov, or Einstein-Maxwell.

In the second main work of this dissertation, which is based on joint work with Lydia Bieri, Neel Patel, and Pengyu Le, we prove a trapped surface formation result for a certain class of null dust spacetimes, extending the seminal work of Christodoulou. This result extends to past null infinity. We show that if the incoming energy due to gravitation or null dust radiation is sufficiently large, then a trapped surface will form. In particular, we show that a trapped surface can form solely due to the concentration of null dust radiation, in the absence of any gravitational energy. Trapped surfaces are closely related to black holes; a spacetime which contains a closed trapped surface and a complete future null infinity necessarily contains a black hole, with the trapped surface inside. The initial data constructed in this work is in the large data regime, is quite general, and can be arbitrarily far from already containing a trapped surface. New ideas are needed and developed to analyze the matter introduced in this work.