Definition and stability of Lorentzian manifolds with distributional curvature

Following Geroch, Traschen, Mars and Senovilla, we consider Lorentzian manifolds with distributional curvature tensor. Such manifolds represent spacetimes of general relativity that possibly contain gravitational waves, shock waves, and other singular patterns. We aim here at providing a comprehensive and geometric (i.e., coordinate-free) framework. First, we determine the minimal assumptions required on the metric tensor in order to give a rigorous meaning to the spacetime curvature within the framework of distribution theory. This leads us to a direct derivation of the jump relations associated with singular parts of connection and curvature operators. Second, we investigate the induced geometry on a hypersurface with general signature, and we determine the minimal assumptions required to define, in the sense of distributions, the curvature tensors and the second fundamental form of the hypersurface and to establish the Gauss-Codazzi equations.Comment: 28 page

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A basic theorem from differential geometry asserts that, if the Riemann curvature tensor associated with a field C of class C 2 of positive-definite symmetric matrices of order n vanishes in a connected and simply-connected open subset Ω of R n, then there exists an immersion Θ ∈ C 3 (Ω; R n), uniquely determined up to isometries in R n, such that C is the metric tensor field of the manifold Θ(Ω), then isometrically immersed in R n. Let ˙ Θ denote the equivalence class of Θ modulo isometries in R n and let F: C → ˙ Θ denote the mapping determined in this fashion. The first objective of this paper is to show that, if Ω satisfies a certain “geodesic property ” (in effect a mild regularity assumption on the boundary ∂Ω of Ω) and if the field C and its partial derivatives of order ≤ 2 have continuous extensions to Ω, the extension of the field C remaining positive-definite on Ω, then the immersion Θ and its partial derivatives of order ≤ 3 also have continuous extensions to Ω. The second objective is to show that, under a slightly stronger regularity assumption on ∂Ω, the above extension result combined with a fundamental theorem o

Isometric immersions into the Minkowski spacetime for Lorentzian manifolds with limited regularity

Assuming minimal regularity assumptions on the data, we revisit the classical problem of finding isometric immersions into the Minkowski spacetime for hypersurfaces of a Lorentzian manifold. Our approach encompasses metrics having Sobolev regularity and Riemann curvature defined in the distributional sense, only. It applies to timelike, spacelike, or null hypersurfaces with arbitrary signature that possibly changes from point to point

A nonlinear Korn inequality with boundary conditions and its relation to the existence of minimizers in nonlinear elasticity

International audienceWe establish a nonlinear Korn inequality with boundary conditions showing that the H^1-distance between two mappings from Ω ⊂ R^n into R^n preserving orientation is bounded, up to a multiplicative constant, by the L^2-distance between their metrics. This inequality is then used to show the existence of a unique minimizer to the total energy of a hyperelastic body, under the assumptions that the L^p-norm of the density of the applied forces is small enough and the stored energy function is bounded from below by a positive definite quadratic function of the Green-Saint Venant strain tensor

The equations of elastostatics in a Riemannian manifold

To begin with, we identify the equations of elastostatics in a Riemannian manifold, which generalize those of classical elasticity in the three-dimensional Euclidean space. Our approach relies on the principle of least energy, which asserts that the deformation of the elastic body arising in response to given loads minimizes over a specific set of admissible deformations the total energy of the elastic body, defined as the difference between the strain energy and the potential of the loads. Assuming that the strain energy is a function of the metric tensor field induced by the deformation, we first derive the principle of virtual work and the associated nonlinear boundary value problem of nonlinear elasticity from the expression of the total energy of the elastic body. We then show that this boundary value problem possesses a solution if the loads are sufficiently small (in a sense we specify).Comment: 43 page

A nonlinear Korn inequality on a surface with an explicit estimate of the constant

A nonlinear Korn inequality on a surface estimates a distance between a surface $\theta (\omega )$ and another surface $\phi (\omega )$ in terms of distances between their fundamental forms in the space $L^p(\omega )$, $

Expression of Dirichlet boundary conditions in terms of the strain tensor in linearized elasticity

International audienceIn a previous work, it was shown how the linearized strain tensor field e := (∇u^T +∇u)/2 ∈ L^2(Ω) can be considered as the sole unknown in the Neumann problem of linearized elasticity posed over a domain Ω ⊂ R3 , instead of the displacement vector field u ∈ H^1 (Ω ) in the usual approach. The purpose of this Note is to show that the same approach applies as well to the Dirichlet–Neumann problem. To this end, we show how the boundary condition u = 0 on a portion Γ_0 of the boundary of Ω can be recast, again as boundary conditions on Γ_0, but this time expressed only in terms of the new unknown e∈L^2(Ω)

A nonlinear Korn inequality on a surface with an explicit estimate of the constant

A nonlinear Korn inequality on a surface estimates a distance between a surface $\theta (\omega )$ and another surface $\phi (\omega )$ in terms of distances between their fundamental forms in the space $L^p(\omega )$, $