Professor Bray studies geometric analysis problems that relate to scalar curvature, many of which are motivated by fundamental questions in General Relativity. Recently, Marcus Khuri and the PI have made important progress on the Penrose Conjecture for asymptotically-flat space-like slices of spacetimes by reducing the conjecture to interesting existence questions for certain naturally motivated systems of p.d.e.'s. One of these existence questions is similar to the one solved by Huisken-Ilmanen to prove the existence of the inverse mean curvature flow with jumps, but for a system of two equations instead of one equation. The physical interpretation of the Penrose Conjecture is the natural idea that the total mass of a spacetime with nonnegative energy density should be at least the mass contributed by the black holes in the spacetime. In 1973, Roger Penrose was able to turn the above statement into a precise geometric conjecture about the Cauchy data of an asymptotically flat space-like slice (itself a Riemannian 3-manifold) of a spacetime. The time-symmetric case, known as the Riemannian Penrose Conjecture, was proved by the PI in 1999, and by Huisken-Ilmanen in 1997 for a single black hole. In this case, the energy density of the spacetime equals the scalar curvature of the slice, the total mass is a parameter describing the rate at which the Riemannian manifold is becoming flat at infinity, and apparent horizons of black holes are area-outerminimizing minimal surfaces. The Riemannian Penrose Inequality is the statement that the total mass is greater than or equal to the square root of the surface area of the apparent horizons of the black holes divided by 16 pi. When we drop the assumption that the Riemannian manifold is time-symmetric in the spacetime and allow the second fundamental form of the slice to be anything, a generalization of this statement is known simply as the Penrose Conjecture. The PI also studies negative point mass singularities in General Relativity and questions relating to quasi-local mass.
As acclaimed a theory as General Relativity is, fundamental aspects of the theory are still not understood. For example, given the state of the universe at one instant of (coordinate) time, it is not currently known if the relevant equations, including the Einstein equation, can be solved forward in time other than for a very short period. Yet the universe, as we observe daily, exists without interruption. Hence, understanding the existence theory of the Cauchy Problem in General Relativity, as this problem is called, is a major question. If General Relativity does not has a physical existence theory, this will be a major hint as to how the theory needs to be modified. If General Relativity does have an existence theory corresponding to the physical universe, then this will be one more piece of evidence supporting the theory. One thing that is clear is that black holes and singularities play an important role in this question. In any case, understanding fundamental theoretical questions about General Relativity will not only deepen our understanding of the theory, but also help current and future researchers develop the next generation of physical theories.
