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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 ...
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.
Schwarzschild solution in Schwarzschild coordinates, with two space dimensions suppressed, leaving just the time t and the distance from the center r. In red the incoming null geodesics. In blue outcoming null geodesics. In green the null light cones on which borders light moves, while massive objects move inside the cones.
In Einstein's theory of general relativity, the Schwarzschild metric (also known as the Schwarzschild solution) is an exact solution to the Einstein field equations that describes the gravitational field outside a spherical mass, on the assumption that the electric charge of the mass, angular momentum of the mass, and universal cosmological constant are all zero.
As noted above, any flat metric is Ricci-flat. However it is nontrivial to identify Ricci-flat manifolds whose full curvature is nonzero. In 1916, Karl Schwarzschild found the Schwarzschild metrics, which are Ricci-flat Lorentzian manifolds of nonzero curvature. [3]
In between these stars and the black hole is a circular band of secondary images of the stars. The duplicate images are instrumental in the identification of the black hole. At r/M = 30, the black hole has become much bigger, spanning a diametrical angle of ~15 degrees in the sky. The band of secondary images has also grown to 10 degrees.
The 50 nakedest red carpet dresses of all time—from Halle Berry's iconic Oscars gown to the time Kendall Jenner wore a casual ... Getty Images. Janelle Monáe at the 2023 Critics Choice Awards. ...
where (,) and (,) are two metric potentials dependent on Weyl's canonical coordinates {,}.The coordinate system {,,,} serves best for symmetries of Weyl's spacetime (with two Killing vector fields being = and =) and often acts like cylindrical coordinates, [2] but is incomplete when describing a black hole as {,} only cover the horizon and its exteriors.
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