Spacetime is locally inertial at points of general relativistic shock wave interaction between shocks from different characteristic families

Arch Rational Mech Anal

2019

Self Cite

We prove that the essential smoothness of the gravitational metric at shock waves in GR, a PDE regularity issue for weak solutions of the Einstein equations, is equivalent to a geometrical condition which we call the Riemann flat condition. This provides a geometric context for the open problem as to whether regularity singularities (where the essential smoothness of the gravitational metric is Lipschitz continuous) can be created by shock wave interaction in GR, or whether metrics Lipschitz at shocks can always be smoothed one level to C 1,1 by coordinate transformation. As a corollary of the ideas we give a proof that locally inertial frames always exist in a natural sense for shock wave metrics in spherically symmetric spacetimes, independent of whether the metric itself can be smoothed to C 1,1 locally. This final result yields an explicit procedure (analogous to Riemann Normal Coordinates in smooth spacetimes) for constructing inertial coordinates for Lipschitz metrics, and is a new regularity result for GR solutions constructed by the Glimm scheme.

Section: Motivation and Backgroundmentioning

confidence: 99%

Shock Wave Interactions and the Riemann-Flat Condition: The Geometry Behind Metric Smoothing and the Existence of Locally Inertial Frames in General Relativity

Reintjes

Arch Rational Mech Anal

2019

Self Cite

“…Proof. To prove (i), we follow the idea leading to the smoothing condition, first introduced in [12], which lies at the heart of the method in [11,13]: The covariant transformation law of the metric is given by…”

Section: )mentioning

confidence: 99%

“…The first extension of Israel's theorem to the more complicated setting of shock wave interactions was accomplished by the authors in their recent papers [13,11]. The proof demonstrates in the first case ever that the gravitational metric can always be smoothed from C 0,1 to C 1,1 at a point of interacting shock waves in GR, namely the case of regular shock wave interaction in spherical symmetry between shocks from different characteristic families.…”

Section: Introductionmentioning

confidence: 98%

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

Reintjes

Methods and Applications of Analysis

2016

Self Cite

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 "regu-larity 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 required 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.

“…To prove (i), we follow the idea leading to the smoothing condition, first introduced in [12], which lies at the heart of the method in [11,13]: The covariant transformation law of the metric is given by…”

Section: 1mentioning

confidence: 99%

An alternative proposal for the anomalous acceleration

Smoller

Surveys in Differential Geometry

Vogler

2015

Abstract.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 required 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.