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

Research output: Journal Publications and Reviews (RGC: 21, 22, 62)21_Publication in refereed journalpeer-review

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Original languageEnglish
Article number20200177
Journal / PublicationProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Issue number2241
Online published16 Sept 2020
Publication statusPublished - Sept 2020
Externally publishedYes


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 Lp spaces. The theory and results announced in this paper apply to arbitrary L connections on the tangent bundle TM of arbitrary manifolds M, including Lorentzian manifolds of general relativity.

Research Area(s)

  • general relativity, Lorentzian geometry, optimal metric regularity, regularity singularities, shock waves, Uhlenbeck compactness

Citation Format(s)