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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.
In deriving the Schwarzschild metric, it was assumed that the metric was vacuum, spherically symmetric and static. The static assumption is unneeded, as Birkhoff's theorem states that any spherically symmetric vacuum solution of Einstein's field equations is stationary; the Schwarzschild solution thus follows
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.
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.
The 66-year-old demonstrated a “three-minute” workout to strengthen abs. The fitness star demonstrated a total of five simple moves, including punches, side crunches, and marching twists.
In general relativity, a dust solution is a fluid solution, a type of exact solution of the Einstein field equation, in which the gravitational field is produced entirely by the mass, momentum, and stress density of a perfect fluid that has positive mass density but vanishing pressure.
In isotropic coordinates, the McVittie metric is given by [1] = (() / + / ()) + (+ / ()) () (+), where is the usual line element for the euclidean sphere, M is identified as the mass of the massive object, () is the usual scale factor found in the FLRW metric, which accounts for the expansion of the space-time; and () is a curvature parameter related to the scalar curvature of the 3-space as