Proving Numerical Computability of Regularizations of Singular Geometries

The question whether a singularity (points at which the fabric of spacetime breaks down) is removable by coordinate transformation has been of central importance in General Relativity since the time of Einstein. In fact, General Relativity has only been considered a consistent theory when it became clear that the troublesome singularity at the black hole horizon of the Schwarzschild spacetime is removable. Until today, singularity formation is a major challenge in analytical and numerical methods for solving the equations of General Relativity. However, beyond ad-hoc coordinate constructions, General Relativity lacked a unifying principle for identifying when singularities are removable, and it lacked a general procedure for removing them. The P.I., in collaboration with B. Temple, recently discovered a system of non-linear elliptic partial differential equations (the RT-equations) which provides such a general procedure for removing singularities by coordinate transformation, and which establishes boundedness of the spacetime curvature as a definitive criterion for when a singularity is removable. By developing an existence theory for the RT-equations, we gave a mathematical proof that singularities with bounded curvature are removable above a threshold regularity (degree of differentiability). This regularization of the spacetime geometry is general enough to remove spacetime singularities at fluid dynamical relativistic shock waves, (resolving an open problem in Mathematical Physics), but does not yet apply to the more severe singularities of black hole type. The first goal of this project is to develop a mathematical proof that the regularization procedure of singular geometries by the RT-equations is sufficiently stable to be computable by numerical methods. This could be transformative for the theory of Numerical Relativity by proving that the RT-equations provide a computable algorithm for removing singularities in numerical simulations (of, for example, gravitational waves or black hole formation) which does not rely on modifications of the Einstein equations that violate the laws of General Relativity. The second goal of this project is to extend the regularizations by the RT-equations below the threshold regularity they currently require, in order to address more severe singularities, for example, at black hole horizons. This would fully establish the theory of the RT-equations as the unifying principle for identifying removable singularities, together with a computable algorithm for removing them, that has been missing in General Relativity since the days of Einstein.