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Goldberg–Sachs Theorem
The Goldberg–Sachs theorem is a result in Einstein's theory of general relativity about vacuum solutions of the Einstein field equations relating the existence of a certain type of congruence with algebraic properties of the Weyl tensor. More precisely, the theorem states that ''a vacuum solution of the Einstein field equations will admit a shear-free null geodesic congruence if and only if the Weyl tensor is algebraically special.'' The theorem is often used when searching for algebraically special vacuum solutions. Shear-Free Rays A ray is a family of geodesic light-like curves. That is tangent vector field l^a is null and geodesic: l_a l^a = 0 and l^b \nabla_b l^a = 0. At each point, there is a (nonunique) 2D spatial slice of the tangent space orthogonal to l^a. It is spanned by a complex null vector m^a and its complex conjugate \bar^a. If the metric is time positive, then the metric projected on the slice is \tilde^ = -m^a \bar^b - \bar^a m^b. Goldberg and Sachs conside ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
General Relativity
General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics. General theory of relativity, relativity generalizes special relativity and refines Newton's law of universal gravitation, providing a unified description of gravity as a geometric property of space and time in physics, time, or four-dimensional spacetime. In particular, the ''curvature of spacetime'' is directly related to the energy and momentum of whatever is present, including matter and radiation. The relation is specified by the Einstein field equations, a system of second-order partial differential equations. Newton's law of universal gravitation, which describes gravity in classical mechanics, can be seen as a prediction of general relativity for the almost flat spacetime geometry around stationary mass ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
