On the geometrical structure of shock waves in general relativity

arXiv (Cornell University), 2016

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 determined by a geometrical condition which we introduce and name the Riemann-flat condition. The Riemann-flat condition determines whether or not the essential smoothness of the gravitational metric is two full derivatives more regular than the Riemann curvature tensor. This provides a geometric framework for the open problem as to whether regularity singularities (points where the curvature is in L ∞ but the essential smoothness of the gravitational metric is only 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 latter result yields an explicit procedure (analogous to Riemann Normal Coordinates in smooth spacetimes) for constructing locally inertial coordinates for Lipschitz metrics, and is a new regularity result for GR solutions constructed by the Glimm scheme. 1 The space C 0,1 denotes the space of Lipschitz continuous functions, and C 1,1 the space of functions with Lipschitz continuous derivatives. A function is bounded in C 0,1 if and only if the function and its weak derivatives are bounded in L ∞ , c.f. [7], Chapter 5.8. 2 In geometry, the regularity of a Riemannian or Lorentzian metric is defined to be the regularity of the metric components in each coordinate system of a given atlas. This definition neglects the possibility that the metric might be more regular in particular coordinate systems of this atlas. 3 Since the writing of this paper the Riemann-flat condition has become the starting point for authors' further developments. In fact, we abandoned the Nash embedding idea and used the Riemann-flat condition to derive the Regularity Transformation equations, an elliptic system of PDE's equivalent to the Riemann-flat condition, c.f. .

2010

Abstract. We discuss mathematical issues related to the authors recent paper “Cosmology with a Shock-Wave ” in which we incorporate a shock wave into the standard model of Cosmology. Here we discuss the derivation of the “conservation constraint ” which is used to derive the equation for the shock position in that paper. 1. Introduction. In Einstein’s theory of general relativity, all properties of the gravitational field are determined by the gravitational metric tensor g, a Lorentzian metric of signature (−1,1,1,1), defined at each point of a four dimensional manifold M called “spacetime. ” The equations that describe the time evolution of the metric

Journées équations aux dérivées partielles, 1995

Shock-wave explosions in general relativity Journées Équations aux dérivées partielles (1995), p. 1-20 <http://www.numdam.org/item?id=JEDP_1995____A17_0> © Journées Équations aux dérivées partielles, 1995, tous droits réservés. L'accès aux archives de la revue « Journées Équations aux dérivées partielles » (http://www. math.sciences.univ-nantes.fr/edpa/) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/

General Relativity and Gravitation, 1974

It is shown that the continuity conditions of Lichnerowicz must generally be relaxed in favour of the O'Brien-Synge conditions in the case of shock electromagnetic waves. This is in particular true when two electromagnetic shock waves are in collision. Gravitational impulse waves are produced as a result of the weakened conditions. An exact solution exhibiting this behaviour is derived, and the effect of the impulse waves on a measuring device are compared with experimental results of Weber.

Archive for Rational Mechanics and Analysis, 2019

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 determined by a geometrical condition which we introduce and name the Riemann-flat condition. The Riemann-flat condition determines whether or not the essential smoothness of the gravitational metric is two full derivatives more regular than the Riemann curvature tensor. This provides a geometric framework for the open problem as to whether regularity singularities (points where the curvature is in L ∞ but the essential smoothness of the gravitational metric is only 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 latter result yields an explicit procedure (analogous to Riemann Normal Coordinates in smooth spacetimes) for constructing locally inertial coordinates for Lipschitz metrics, and is a new regularity result for GR solutions constructed by the Glimm scheme.

Elsevier eBooks, 2003

Universe

Both electromagnetic shock-waves and gravitational waves propagate with the speed of light. If they carry significant energy-momentum, this will change the properties of the space-time they propagate through. This can be described in terms of the junction conditions between space-time regions separated by a singular, null hypersurface. We derived generic junction conditions for Brans-Dicke theory in the Jordan frame, exploring a formalism based on a transverse vector, rather than normal, which can be applied to any type of hypersurfaces. In the particular case of a non-null hypersurface we obtain a generalised Lanczos equation, in which the jump of the extrinsic curvature is sourced by both the distributional energy-momentum tensor and by the jump in the transverse derivative of the scalar. In the case of null hypersurfaces, the distributional source is decomposed into surface density, current and pressure. The latter, however, ought to vanish by virtue of the scalar junction condition.

2015

Abstract. We present the analysis of convergence of the locally inertial Godunov method with dynamical time dilation applied to a canonical initial data set which is arguably the simplest initial data that creates a point of shock wave interaction in General Relativity. New applications include the analysis of convergence in the presence of new boundary conditions which enables one to test the validity of the Einstein constraint equations numerically in new Lipschitz continuous space-time metrics. The numerical method, introduced in [14, 15], is an algorithm for simu-lating general relativistic shock-waves in spherically symmetric spacetimes, and the analysis here rigorously establishes claims made in authors ’ PRSA article [15]. 1.

Physical Review D, 1997

I consider the appearance of shocks in hyperbolic formalisms of General Relativity. I study the particular case of the Bona-Massó formalism with zero shift vector and show how shocks associated with two families of characteristic fields can develop. These shocks do not represent discontinuities in the geometry of spacetime, but rather regions where the coordinate system becomes pathological. For this reason I call them 'coordinate shocks'. I show how one family of shocks can be eliminated by restricting the Bona-Massó slicing condition ∂ t α = −α 2 f (α) trK to the case f = 1 + k/α 2 , with k an arbitrary constant. The other family of shocks can not be eliminated even in the case of harmonic slicing (f = 1). I also show the results of the numerical evolution of non-trivial initial slices in the special cases of a flat two-dimensional spacetime, a flat four-dimensional spacetime with a spherically symmetric slicing, and a spherically symmetric black hole spacetime. In all three cases coordinate shocks readily develop, confirming the predictions of the mathematical analysis. Although I concentrate in the Bona-Massó formalism, the phenomena of coordinate shocks should arise in any other hyperbolic formalism. In * Present address:

Classical and Quantum Gravity, 2003

In the present work we approximate an ultrarelativistic jet by a homogeneous beam of null matter with finite width. Then, we study the influence of this beam over the spacetime metric in the framework of higher-derivative gravity. We find an exact shock wave solution of the quadratic gravity field equations and compare it with the solution to Einstein's gravity. We show that the effect of higher-curvature gravity becomes negligible at large distances from the beam axis. We also observe that only the Ricci-squared term contribute to modify the Einstein's gravity prediction. Furthermore, we note that this higher-curvature term contribute to regularize the discontinuities associated to the solution to Einstein's general relativity.