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Introduction to the mathematics of general relativity
About: Introduction to the mathematics of general relativity is a research topic. Over the lifetime, 2583 publications have been published within this topic receiving 73295 citations.
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TL;DR: In this paper, an analytical formalism is developed to deal with the occurrence of jump discontinuities in the gmu nu or their derivatives across a hypersurface Sigma, and it is shown that the equations of relativity remain meaningful at Sigma, even when Sigma does not inherit a unique intrinsic geometry, so that the gm nu are discontinuous across Sigma in natural coordinates.
Abstract: An analytical formalism is developed to deal with the occurrence of jump discontinuities in the gmu nu or their derivatives across a hypersurface Sigma . It is shown that the equations of relativity remain meaningful at Sigma , even when Sigma does not inherit a unique intrinsic geometry, so that the gmu nu are discontinuous across Sigma in natural coordinates. The spherically symmetric surface layer at the Schwarzschild-Minkowski junction is used to illustrate these techniques, and to establish rigorously the existence of C0 solutions of the Einstein equations and the conservation equations. The possible validity of relativity at the microscopic level is examined, and it is concluded that, if relativity is valid at the microscopic level, then it is likely that the gmu nu are not globally continuously differentiable.
19 citations
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01 Jan 2011
TL;DR: In this paper, the authors derive the Einstein field and the equations of motion for uncharged and charged self- gravitating fluids from variational principles, and show how singular hyper-surfaces (shock waves) and the equation governing their behavior may be treated by means of these principles.
Abstract: In these lectures we shall derive the Einstein field and the equations of motion for uncharged and charged self- gravitating fluids from variational principles. We shall also see how singular hyper-surfaces (shock waves) and the equations governing their behavior may be treated by means of these principles. In addition we shall show how the “second variation” problem is related to the discussion of the stability of the solutions of the Einstein field equations.
19 citations
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TL;DR: In this paper, the equations for an irrotational perfect fluid with pressure p equal to energy density ρ are studied when the space-time has cylindrical symmetry with no reflection symmetry.
Abstract: The Einstein field equations for an irrotational perfect fluid with pressure p, equal to energy density ρ are studied when the space–time has cylindrical symmetry with no reflection symmetry. The coordinate transformation to comoving coordinates is discussed. The energy and the Hawking–Penrose inequalities are studied. Particular classes of solutions are exhibited.
19 citations
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TL;DR: In this article, the authors show how the use of the normal projection of the Einstein tensor as a set of boundary conditions relates to the propagation of the constraints, for two representations of the equations with vanishing shift vector: the Arnowitt-Deser-Misner formulation and the Einstein-Christoffel formulation.
Abstract: We show how the use of the normal projection of the Einstein tensor as a set of boundary conditions relates to the propagation of the constraints, for two representations of the Einstein equations with vanishing shift vector: the Arnowitt-Deser-Misner formulation, which is ill posed, and the Einstein-Christoffel formulation, which is symmetric hyperbolic. Essentially, the components of the normal projection of the Einstein tensor that act as nontrivial boundary conditions are linear combinations of the evolution equations with the constraints that are not preserved at the boundary, in both cases. In the process, the relationship of the normal projection of the Einstein tensor to the recently introduced ``constraint-preserving'' boundary conditions becomes apparent.
19 citations
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TL;DR: In this article, all space-times admitting a neutrino field having a zero energy-momentum tensor are found, and one of the space times is shown to admit two distinct neutrinos fields.
Abstract: All space-times admitting a neutrino field having a zero energy-momentum tensor are found. One of the space-times is shown to admit two distinct neutrino fields.
19 citations