Einstein manifold
In differential geometry and mathematical physics, an Einstein manifold is a Riemannian or pseudo-Riemannian 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 equations (with cosmological constant), although the dimension as signature of the metric can be arbitrary, unlike the four dimensional Lorentzian manifolds usually studied in general relativity.
If M is the underlying m-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.
The Einstein condition and Einstein's equations
If one introduces a coordinate chart, the condition that (M, g) be an Einstein manifold is then simply
By tensor-multiplying both sides of this equation by gab one shows that the constant of proportionality, k for Einstein manifolds is related to the scalar curvature R by
Einstein manifolds with k = 0 are also often referred to as Ricci-flat manifolds.
In general relativity, Einstein's equations relate the curvature of a space-time to the energy-momentum tensor which describes the matter distribution in the space-time according to
where we have used geometrized units G = c = 1.
In a region of spacetime with no matter, Tab = 0. It follows that Lorentzian Einstein manifolds provide vacuum solution to Einstein's equations with a cosmological constant proportional to k. In the four dimensional case, they are therefore mathematical models for regions of spacetime without matter.
Applications
Four dimensional Riemannian Einstein manifolds are also important in mathematical physics as gravitational instantons in quantum theories of gravity. The term "gravitational instanton" is usually used restricted to Einstein 4-manifolds whose Weyl tensor is self-dual, and it is usually assumed that metric is asymptotic to the standard metric of Euclidean 4-space (and are therefore complete but non-compact). In differential geometry, self-dual Einstein 4-manifolds are also known as (4-dimensional) hyperkähler manifolds in the Ricci-flat case, and quaternion Kähler manifolds otherwise.
Higher dimensional Lorentzian Einstein manifolds are used in modern theories of gravity, such as string theory, M-theory and supergravity. Hyperkähler and quaternion Kähler manifolds (which are special kinds of Einstein manifolds) also have applications in physics as target spaces for nonlinear σ-models with supersymmetry.
Compact Einstein manifolds have been much studied in differential geometry, and many examples are known, although constructing them is often challenging. Compact Ricci-flat manifolds are particularly difficult to find: in the monograph on the subject by the pseudonymous author Arthur Besse, readers are offered a meal in a starred restaurant in exchange for a new example.
Examples
Some of the simplest examples of Einstein manifolds are the following.
- Euclidean space, which is flat, is a simple example of Ricci-flat, hence Einstein metric.
- The n-sphere, Sn, with the round metric is Einstein with k = n − 1.
- Hyperbolic space with the canonical metric is Einstein with negative k.
- Complex projective space, CPn, with the Fubini-Study metric.
References
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