Scalar Curvature, the Penrose Conjecture, and the Axioms of General Relativity

Duke University, Durham NC

Investigators

Abstract

First, the PI will continue his research on scalar curvature, especially on 3 manifolds. Prior results by the PI in this area include a joint work with Andre Neves in 2002 that classifies prime 3-manifolds with Yamabe invariant greater than RP^3 and a 2008 paper with Pengzi Miao that gives an upper bound on the capacity of surfaces in 3-manifolds with nonnegative scalar curvature. In 2009, the PI's joint paper with Simon Brendle, Michael Eichmair, and AndreNeves proves that A_{min}R_{min} \le 12\pi on compact 3-manifolds which contain embedded incompressible RP^2, where A_{min} is the area of the minimal RP^2 and R_{min} is the minimum value of the scalar curvature. Using Ricci flow, they show that the 3-manifold is a spherical space form in the case of equality. Second, the PI will continue to work toward a proof of the full Penrose conjecture. The PI's 2001 paper proved the Riemannian Penrose conjecture in dimension 3, improving the case of one black hole proved by Huisken and Ilmanen to any number of black holes using a different technique. Since then, the PI proved a similar type of inequality for zero area singularities in 2005 (with some additional hypotheses), the Riemannian Penrose conjecture in dimensions less than 8 in a joint work with Dan Lee in 2007, and showed that the full Penrose conjecture on Cauchy data (M^3,g,k) reduces to the Riemannian case whenever certain systems of p.d.e.s can be solved in a joint work with Marcus Khuri in 2009. These systems of p.d.e.s rely on a new identity that they proved called the Generalized Schoen-Yau identity, which they believe will be a very useful identity for a broad range of problems in mathematical relativity. Third, the PI is opening up a new research direction for himself as he examines the axioms of general relativity to see how they may be modified as little as possible to account for the widely accepted existence of dark matter. Einstein's theory of general relativity was made possible by Gauss and Riemann, both mathematicians, who developed the field of mathematics called differential geometry decades before. Since then, advances in differential geometry have played a crucial role in understanding the implications of Einstein's theory. Einstein used differential geometry to make the qualitative statement ``matter curves spacetime'' precise, thereby showing that gravity results as a consequence of this fundamental idea. By contrast, Newton's inverse square law for gravity has been shown to be false by measuring the precession of the orbit of Mercury. Hence, understanding gravity correctly would appear to require understanding the properties of curvature, currently pursued most directly by mathematicians studying geometric analysis. Black holes, predicted by general relativity and now known to exist, are fundamentally geometric objects, and have been the focus of much of the PI's efforts, resulting in theorems which yield a deeper physical insight into these fascinating phenomena. In light of this rich history of geometric analysis playing a crucial role in understanding the large scale structure of the universe, the PI is now looking to geometric motivations to try to understand the nature of dark matter. While dark matter is known to make up 23% of the mass of the universe and hence has very important gravitational effects, it is otherwise invisible. A geometric idea observed by the PI, as well as other motivations, leads to considering a real-valued scalar field as a model for dark matter, described by the Einstein Klein-Gordon equations. Astrophysicists have already observed that this model for dark matter is consistent with the flat rotation curves of galaxies. The PI is studying the idea that density waves in this scalar field dark matter produce density waves in regular matter, resulting in star formation and both bars and spiral patterns in some galaxies, an exciting possibility supported by preliminary simulations. If correct, this would suggest that while dark matter itself is invisible, its gravitational effects may be quite dramatic.