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The Schwarzschild metric is a spherically symmetric Lorentzian metric (here, ... The Ricci curvature scalar and the Ricci curvature tensor are both zero.
Like the metric tensor, the Ricci tensor assigns to each tangent space of the manifold a symmetric bilinear form (Besse 1987, ... the trace-free Ricci tensor ...
In general relativity, the de Sitter–Schwarzschild solution describes a black hole in a causal patch of de Sitter space. Unlike a flat-space black hole, there is a largest possible de Sitter black hole, which is the Nariai spacetime .
The Weyl tensor has the same basic symmetries as the Riemann tensor, but its 'analogue' of the Ricci tensor is zero: = = = = The Ricci tensor, the Einstein tensor, and the traceless Ricci tensor are symmetric 2-tensors:
In general relativity, the metric tensor (in this context often abbreviated to simply the metric) is the fundamental object of study. The metric captures all the geometric and causal structure of spacetime , being used to define notions such as time, distance, volume, curvature, angle, and separation of the future and the past.
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] Start with the metric with coefficients that are unknown coefficients of :
The Einstein field equation (EFE) describing the geometry of spacetime is given as = where is the Ricci tensor, is the Ricci scalar, is the energy–momentum tensor, = / is the Einstein gravitational constant, and is the spacetime metric tensor that represents the solutions of the equation.
A pseudo-Riemannian manifold is said to be Ricci-flat if its Ricci curvature is zero. [1] It is direct to verify that, except in dimension two, a metric is Ricci-flat if and only if its Einstein tensor is zero. [2] Ricci-flat manifolds are one of three special types of Einstein manifold, arising as the special case of scalar curvature equaling ...