"Regularity singularities" and the scattering of gravity waves in approximate locally inertial frames

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

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Original languageEnglish
Pages (from-to)233–258
Number of pages25
Journal / PublicationMethods and Applications of Analysis
Issue number3
Publication statusPublished - Sep 2016
Externally publishedYes


It is an open question whether solutions of the Einstein-Euler equations are smooth enough to admit locally inertial coordinates at points of shock wave interaction, or whether “regularity singularities” can exist at such points. The term regularity singularity was proposed by the authors as a point in spacetime where the gravitational metric tensor is Lipschitz continuous (C0,1), but no smoother, in any coordinate system of the C1,1 atlas. An existence theory for shock wave solutions in C0,1 admitting arbitrary interactions has been proven for the Einstein–Euler equations in spherically symmetric spacetimes, but C1,1 is the requisite smoothness for space-time to be locally flat. Thus the open problem of regularity singularities is the problem as to whether locally inertial coordinate systems exist at shock waves within the larger C1,1 atlas. To clarify this open problem, we identify new “Coriolis type” effects in the geometry of C0,1 shock wave metrics and prove they are essential in the sense that they can never be made to vanish within the atlas of smooth coordinate transformations, the atlas usually assumed in classical differential geometry. Thus the problem of existence of regularity singularities is equivalent to the question as to whether or not these Coriolis type effects are essentially non-removable and ‘real’, or merely coordinate effects that can be removed, (in analogy to classical Coriolis forces), by going to the less regular atlas of C1,1 transformations. If essentially non-removable, it would argue strongly for a ‘real’ new physical effect for General Relativity, providing a physical context to the open problem of regularity singularities.

Research Area(s)

  • Shock waves, regularity singularities, shock wave interaction, gravity waves, linearized Einstein equations, locally inertial frames