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

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 $(C^{0,1})$, but no smoother, in any coordinate system of the $(C^{1,1})$ atlas. An existence theory for shock wave solutions in $(C^{0,1})$ admitting arbitrary interactions has been proven for the Einstein–Euler equations in spherically symmetric spacetimes, but $(C^{1,1})$ 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 $(C^{1,1})$ atlas. To clarify this open problem, we identify new “Coriolis type” effects in the geometry of $(C^{0,1})$ 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 $(C^{1,1})$ 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.

Keywords

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

2010 Mathematics Subject Classification

Primary 83C75. Secondary 76L05.

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Published 9 November 2016