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Electrogravitic Tensor
In semi-Riemannian geometry, the Bel decomposition, taken with respect to a specific timelike congruence, is a way of breaking up the Riemann tensor of a pseudo-Riemannian manifold into lower order tensors with properties similar to the electric field and magnetic field. Such a decomposition was partially described by Alphonse Matte in 1953 and by Lluis Bel in 1958. This decomposition is particularly important in general relativity. This is the case of four-dimensional Lorentzian manifolds, for which there are only three pieces with simple properties and individual physical interpretations. Decomposition of the Riemann tensor In four dimensions the Bel decomposition of the Riemann tensor, with respect to a timelike unit vector field \vec, not necessarily geodesic or hypersurface orthogonal, consists of three pieces: # the ''electrogravitic tensor'' E vec = R_ \, X^m \, X^n #* Also known as the tidal tensor. It can be physically interpreted as giving the tidal stresses on sma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semi-Riemannian Geometry
In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere non-degenerate bilinear form, nondegenerate. This is a generalization of a Riemannian manifold in which the requirement of Positive-definite bilinear form, positive-definiteness is relaxed. Every tangent space of a pseudo-Riemannian manifold is a pseudo-Euclidean vector space. A special case used in general relativity is a four-dimensional Lorentzian manifold for modeling spacetime, where tangent vectors can be classified as Causal structure, timelike, null, and spacelike. Introduction Manifolds In differential geometry, a differentiable manifold is a space which is locally similar to a Euclidean space. In an ''n''-dimensional Euclidean space any point can be specified by ''n'' real numbers. These are called the coordinates of the point. An ''n''-dimensional differentiable manifold is a generalisation of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |