How to smooth a crinkled map of space–time: Uhlenbeck compactness for L ^∞ connections and optimal regularity for general relativistic shock waves by the Reintjes–Temple equations

Abstract

We present the authors’ new theory of the RT-equations (‘regularity transformation’ or ‘Reintjes–Temple’ equations), nonlinear elliptic partial differential equations which determine the coordinate transformations which smooth connections Γ to optimal regularity, one derivative smoother than the Riemann curvature tensor Riem( Γ ). As one application we extend Uhlenbeck compactness from Riemannian to Lorentzian geometry; and as another application we establish that regularity singularities at general relativistic shock waves can always be removed by coordinate transformation. This is based on establishing a general multi-dimensional existence theory for the RT-equations by application of elliptic regularity theory in L p spaces. The theory and results announced in this paper apply to arbitrary L ∞ connections on the tangent bundle T M of arbitrary manifolds M , including Lorentzian manifolds of general relativity.

Optimal regularity and Uhlenbeck compactness for general relativity and Yang–Mills theory

“…Theorem 6.1 extends Uhlenbeck compactness from Uhlenbeck’s setting of connections on vector bundles over Riemannian manifolds with compact gauge groups, to connections on vector bundles over general base manifold, allowing for both compact and non-compact gauge groups, including the Lorentzian geometries of Relativistic Physics. Taken together with our results in [4,22], this establishes Uhlenbeck compactness at every order of regularity at or above Lp. As a first example for how one might apply Uhlenbeck compactness in GR, we give in [1] a new compactness theorem for approximate solutions of the Einstein equations in vacuum space–times, establishing convergence to a solution of the Einstein equations by using that the strong Lp convergence asserted by theorem 6.1 is sufficient to pass limits through nonlinear products.…”

“…As a serendipitous corollary of theorem 5.1, the extra derivative implied by (5.2) of theorem 5.1 directly implies Uhlenbeck compactness for both compact and non-compact gauge groups acting on the fibre, incorporating general affine connections and Lorentzian metrics on the base manifold. The following theorem synthesizes the results in [1,2]. It gives the most general statement of Uhlenbeck compactness for vector bundles at the lowest level of Lp curvature, implied by theorem 2.2 in [1] and theorem 2.4 in [2].…”

“…The following theorem synthesizes the results in [1,2]. It gives the most general statement of Uhlenbeck compactness for vector bundles at the lowest level of Lp curvature, implied by theorem 2.2 in [1] and theorem 2.4 in [2].…”

“…This level of regularity is low enough to include both GR shock waves constructed by the Glimm scheme, as well as shock solutions constructed in multi-d by Israel’s theory of junction conditions, i.e. all of the known examples of GR shock waves [4].…”

“…Remarkably, the elliptic RT-equations determine the optimal regularity of solutions of the hyperbolic Einstein equations. For affine connections, we prove in [1] that any connection Γ defined on the tangent bundle of an n-dimensional differentiable manifold, such that it satisfies the condition that its components Γ∈L2p and the components of its Riemann curvature tensor Riemfalse(Γfalse)∈Lp in some coordinate system, p>n/2, can always be smoothed locally by a W1,2p coordinate transformation to optimal regularity Γ∈W1,p; i.e. Γ one order of differentiability above its curvature Riemfalse(Γfalse).…”