Mass in General Relativity

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The primary topics of this research include three important conjectures concerning mass in General Relativity: the Static Extension Conjecture, the full Penrose Inequality, and the Hoop Conjecture. The first of these arises from a particularly promising definition of quasilocal mass due to Bartnik, which seeks to localize the total (or ADM) mass. Although this definition satisfies most desired properties, and thus has the potential to be very useful, its abstract and nonconstructive nature yield severe limitations on understanding and applicability. In order to rectify this problem the Static Extension Conjecture asserts that the quasilocal mass may be calculated as the ADM mass of a solution to the Static Vacuum Einstein equations which satisfies a certain geometric boundary condition. In recent joint work of the author and M. Anderson, existence for this boundary value problem has been established under a reasonable nondegeneracy assumption on the given boundary data; this development has placed the conjecture within reach. The Penrose Inequality relates the total mass of a spacetime to the area of its event horizons (boundary of black holes) via the inequality: total mass squared is greater than or equal to the total area of the event horizons divided by 16 pi. This may be viewed as a conjecture for an arbitrary spacelike slice of a spacetime, and in this setting an important special case (when the second fundamental form of the slice vanishes) has been confirmed independently by Huisken and Ilmanen (one black hole) and by Bray (finitely many black holes). In recent joint work with Bray, the author has succeeded in reducing this problem to solving a canonical system of partial differential equations, and has confirmed existence in special cases. It is a major goal of this project to complete this program by proving a general existence result, and thus establishing the Penrose Conjecture.Although there are numerous proposed methods for calculating the gravitational plus matter energy content of a bounded domain (the so called quasilocal mass) in General Relativity, all seem to suffer from one affliction or another. In the case of Bartnik's mass, the sole problem is the abstract nature of its definition. If the Static Extension Conjecture were verified, this lone difficulty should be resolved, thus opening the way for several applications. For example, it is generally believed that whenever enough mass is concentrated in a sufficiently small region, gravitational collapse must ensue and result in a black hole - such a statement is often referred to as the Hoop Conjecture. While there are many ways to accurately describe the size of a region, until a proper notion of quasilocal mass is established, a rigorous and complete version of this conjecture will remain elusive. Another promising application of a properly defined notion of quasilocal mass, is to the long time existence problem for the Einstein Equations. As these equations form a hyperbolic system (after fixing the gauge), it is tempting to search for a theory based on an energy method analogous to the classical theory of the scalar wave equation. A major obstacle to realizing such an approach, is the lack of an appropriate notion of energy for the gravitational field (a constituent of quasilocal mass). Moreover the boundary value problem associated with the Static Extension Conjecture, is in fact an elliptic boundary value problem for the Ricci operator. Hence the techniques developed to study this conjecture should prove useful when investigating well-posed boundary value problems in several other settings, such as for instance, the Ricci flow on manifolds with boundary. As for the Penrose Inequality, it was originally put forth by Penrose to study what is perhaps the most important open question in General Relativity today, namely the Cosmic Censorship Conjecture (whether spacetime singularities are always enclosed by black holes), which is related to General Relativity's veracity as a physical theory. Heuristically, the Penrose Inequality is essentially a necessary condition for cosmic censorship to hold. Thus if the Penrose Inequality were to be confirmed it would add significantly to the general belief in the validity of cosmic censorship. Lastly, our methods developed for the full Penrose Inequality are expected to provide a new powerful tool for reducing questions concerning general initial data sets to the case of time symmetry, and therefore will have numerous applications to a wide range of other problems in General Relativity.