Uhlenbeck Compactness and Optimal Regularity in Lorentzian Geometry
Abstract
We resolve two problems of Mathematical Physics. First, we prove that any ?? ∞ connection ?? on the tangent bundle of an arbitrary differentiable manifold with ?? ∞ Riemann curvature can be smoothed by coordinate transformation to optimal regularity ?? ∈ ??1,?? , any ?? < ∞, (one derivative smoother than the curvature). This implies in particular that Lorentzian metrics ofshock wave solutions of the Einstein-Euler equations are non-singular---geodesic curves, locally inertial coordinates and the resulting Newtonian limit all exist in a classical sense. This result is based on a system of nonlinear elliptic partial differential equations, the Regularity Transformation equations, and an existence theory for them at the level of ?? ∞ connections. Secondly, we prove that this existence theory suffices to extend Uhlenbeck compactness from the case of connections on vector bundles over Riemannian manifolds, to the case of connections on tangent bundles of arbitrary manifolds, including Lorentzian manifolds of General Relativity.
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