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Here, saying that = is irrotational means that the vorticity tensor of the corresponding timelike congruence vanishes; thus, this Killing vector field is hypersurface orthogonal. The fact that the spacetime admits an irrotational timelike Killing vector field is in fact the defining characteristic of a static spacetime .
But another well known generalization of the Schwarzschild vacuum, the Taub–NUT space, is not asymptotically flat. An even simpler generalization, the de Sitter-Schwarzschild metric solution, which models a spherically symmetric massive object immersed in a de Sitter universe , is an example of an asymptotically simple spacetime which is not ...
where is the Ricci curvature tensor and is the Ricci scalar curvature (obtained by taking successive traces of the Riemann tensor). The Ricci tensor vanishes in vacuum spacetimes (such as the Schwarzschild solution mentioned above), and hence there the Riemann tensor and the Weyl tensor coincide, as do their invariants.
In the summer of 2009, the Black Eyed Peas dominated the music charts with their album “The E.N.D.” and went all the way to No. 1 with “I Gotta Feeling,” knocking out their other song ...
This is unfounded because that law has relativistic corrections. For example, the meaning of "r" is physical distance in that classical law, and merely a coordinate in General Relativity.] The Schwarzschild metric can also be derived using the known physics for a circular orbit and a temporarily stationary point mass. [1]
In 1992, Bañados, Teitelboim, and Zanelli discovered the BTZ black hole solution (Bañados, Teitelboim & Zanelli 1992).This came as a surprise, because when the cosmological constant is zero, a vacuum solution of (2+1)-dimensional gravity is necessarily flat (the Weyl tensor vanishes in three dimensions, while the Ricci tensor vanishes due to the Einstein field equations, so the full Riemann ...
A 3-dimensional pseudo-Riemannian manifold is conformally flat if and only if the Cotton tensor vanishes. An n-dimensional pseudo-Riemannian manifold for n ≥ 4 is conformally flat if and only if the Weyl tensor vanishes. Every compact, simply connected, conformally Euclidean Riemannian manifold is conformally equivalent to the round sphere. [4]
In 1916, Karl Schwarzschild found the Schwarzschild metrics, which are Ricci-flat Lorentzian manifolds of nonzero curvature. [3] Roy Kerr later found the Kerr metrics , a two-parameter family containing the Schwarzschild metrics as a special case. [ 4 ]