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Relating Material and Space-Time Metrics Within Relativistic Elasticity: A Dynamical Example
Abstract
Given a space-time and a continuous medium with elastic properties described by a 3-dimensional material space, one can ask whether they are compatible in the context of relativistic elasticity. Here a non-static, spherically symmetric spacetime metric is considered and we investigate the conditions for that metric to correspond to different 3-dimensional material metrics.KeywordsKilling VectorParticle Number DensitySymmetric SpacetimeMaterial SpaceProjection TensorThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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The relativistic theory of elasticity is reviewed within the spherically symmetric context with a view towards the modeling of star interiors possessing elastic properties such as theones expected in neutron stars. Emphasis is placed on generality in the main sections of the paper, and the results are then applied to specific examples. Along the way, a few general results for spacetimes admitting isometries are deduced, and their consequences are fully exploited in the case of spherical symmetry relating them next to the the case in which the material content of the spacetime is some elastic material. This paper extends and generalizes the pioneering work by Magli and Kijowski [1], Magli [2] and [3], and complements, in a sense, that by Karlovini and Samuelsson in their interesting series of papers [4], [5] and [6]. Comment: 23 pages
The aim of this paper is twofold. First, we set up the theory of elastic matter sources within the framework of general relativity in a self-contained manner. The discussion is primarily based on the presentation of Carter and Quintana but also includes new methods and results as well as some modifications that in our opinion make the theory more modern and transparent. For instance, the equations of motion for the matter are shown to take a neat form when expressed in terms of the relativistic Hadamard elasticity tensor. Using this formulation we obtain simple formulae for the speeds of elastic wave propagation along eigendirections of the pressure tensor. Secondly, we apply the theory to static spherically symmetric configurations using a specific equation of state and consider models either having an elastic crust or core. Comment: 33 pages, 5 eps-figures, referencenses and comments added
The relativistic interior dynamics of a spherically symmetric, non- rotating star composed of an elastic material is analyzed. A suitable formulation of relativistic elasticity is used, and the corresponding second order Einstein equations are found. The equations arise naturally in a non-comoving frame and reduce, in the static case, to the “generalized” Tolman-Oppenheimer-Volkhoff equations considered by J.Kijowski and the author in a previous paper [J. Geom. Phys. 9, No. 3, 207-223 (1992; Zbl 0771.73004)]. The charged case is also discussed.
The problem of elastomechanical equilibrium for a static, spherically symmetric star composed of an elastic material is analyzed. A suitable formulation of relativistic elasticity theory is used, and the second order equilibrium equations are found. It is shown that the equilibrium conditions with anisotropic pressure introducedad hoc by some authors are in fact the dynamical conditions for a relativistic elastic material. The corresponding first order equations for the components of the metric and of the energy-momentum tensor reduce to the Tolman-Oppenheimer-Volkhoff equations if the material exhibits no shape-rigidity. Two interesting classes of solutions are discussed.
- M Karlovini
- L Samuelsson
- G Magli
- J Kijowski
- Gen
Magli G and Kijowski J Gen. Rel. Grav. 24 139 (1992)
- M Karlovini
- L Samuelsson
Karlovini M and Samuelsson L Class. Quantum Grav. 20 3613 (2003)