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Kretschmann Scalar
In the theory of pseudo-Riemannian manifold, Lorentzian manifolds, particularly in the context of applications to general relativity, the Kretschmann scalar is a quadratic curvature invariant (general relativity), scalar invariant. It was introduced by Erich Kretschmann. Definition The Kretschmann invariant is : K = R_ \, R^ where R^_ = \partial_\Gamma^_ - \partial_\Gamma^_ + \Gamma^_\Gamma^_ - \Gamma^_\Gamma^_ is the Riemann curvature tensor and \Gamma is the Christoffel symbol. Because it is a sum of squares of tensor components, this is a ''quadratic'' invariant. Einstein summation convention with Raising_and_lowering_indices, raised and lowered indices is used above and throughout the article. An explicit summation expression is : K = R_ \, R^ =\sum_^\sum_^3 \sum_^3\sum_^3 R_ \, R^ \text R^ =\sum_^3 g^\,\sum_^3 g^\,\sum_^3 g^\,\sum_^3 g^\, R_. \, Examples For a Schwarzschild metric, Schwarzschild black hole of mass M, the Kretschmann scalar is : K = \frac \,. whe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudo-Riemannian Manifold
In mathematical physics, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the requirement of 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 timelike, null, and spacelike. Introduction Manifolds In differential geometry, a differentiable manifold is a space that 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 ''n''-dimensional Euclidean space. In a manifold it may only be possible to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |