Einstein manifold

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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 field equations (with cosmological constant), although the dimension, as well as the signature, of the metric can be arbitrary, unlike the four-dimensional Lorentzian manifolds usually studied in general relativity.

If M is the underlying n-dimensional manifold and g is its metric tensor the Einstein condition means that

\mathrm{Ric} = k\,g,

for some constant k, where Ric denotes the Ricci tensor of g. Einstein manifolds with k = 0 are called Ricci-flat manifolds.

The Einstein condition and Einstein's equation[edit]

In local coordinates the condition that (Mg) be an Einstein manifold is simply

R_{ab} = k\,g_{ab}.

Taking the trace of both sides reveals that the constant of proportionality k for Einstein manifolds is related to the scalar curvature R by

R = nk\,

where n is the dimension of M.

In general relativity, Einstein's equation with a cosmological constant Λ is

R_{ab} - \frac{1}{2}g_{ab}R + g_{ab}\Lambda = 8\pi T_{ab},

written in geometrized units with G = c = 1. The stress–energy tensor Tab gives the matter and energy content of the underlying spacetime. In a vacuum (a region of spacetime with no matter) Tab = 0, and one can rewrite Einstein's equation in the form (assuming n > 2):

R_{ab} = \frac{2\Lambda}{n-2}\,g_{ab}.

Therefore, vacuum solutions of Einstein's equation are (Lorentzian) Einstein manifolds with k proportional to the cosmological constant.

Examples[edit]

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, 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.
  • Calabi Yau manifolds admit an Einstein metric that is also Kähler, with Einstein constant "k"="0". Such metrics are not unique, but rather come in families; there is a Calabi-Yau metric in every Kähler class, and the metric also depends on the choice of complex structure. For example, there is a 60-parameter family of such metrics on K3, 57 parameters of which give rise to Einstein metrics which are not related by isometries or rescalings.

A necessary condition for closed, oriented, 4-manifolds to be Einstein is satisfying the Hitchin–Thorpe inequality.

Applications[edit]

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.

See also[edit]

References[edit]

  • Besse, Arthur L. (1987). Einstein Manifolds. Classics in Mathematics. Berlin: Springer. ISBN 3-540-74120-8.