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Concept
Einstein manifold
Summary
In differential geometry and mathematical physics, an Einstein manifold is a Riemannian or pseudo-Riemannian differentiable manifold whose Ricci tensor is proportional to the metric. They are named after Albert Einstein because this condition is equivalent to saying that the metric is a solution of the vacuum Einstein field equations (with cosmological constant), although both the dimension and the signature of the metric can be arbitrary, thus not being restricted to Lorentzian manifolds (including the four-dimensional Lorentzian manifolds usually studied in general relativity). Einstein manifolds in four Euclidean dimensions are studied as gravitational instantons.
If M is the underlying n-dimensional manifold, and g is its metric tensor, the Einstein condition means that
for some constant k, where Ric denotes the Ricci tensor of g. Einstein manifolds with k = 0 are called Ricci-flat manifolds.
In local coordinates the condition that (M, g) be an Einstein manifold is simply
Taking the trace of both sides reveals that the constant of proportionality k for Einstein manifolds is related to the scalar curvature R by
where n is the dimension of M.
In general relativity, Einstein's equation with a cosmological constant Λ is
where κ is the Einstein gravitational constant. The stress–energy tensor Tab gives the matter and energy content of the underlying spacetime. In vacuum (a region of spacetime devoid of matter) Tab = 0, and Einstein's equation can be rewritten in the form (assuming that n > 2):
Therefore, vacuum solutions of Einstein's equation are (Lorentzian) Einstein manifolds with k proportional to the cosmological constant.
Simple examples of Einstein manifolds include:
Any manifold with constant sectional curvature is an Einstein manifold—in particular:
Euclidean space, which is flat, is a simple example of Ricci-flat, hence Einstein metric.
The n-sphere, , with the round metric is Einstein with .
Hyperbolic space with the canonical metric is Einstein with .
Official source
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