- Einstein manifold
In
differential geometry andmathematical physics , an Einstein manifold is a Riemannian orpseudo-Riemannian manifold whoseRicci tensor is proportional to the metric. They are named afterAlbert Einstein because this condition is equivalent to saying that the metric is a solution of thevacuum Einstein equations (withcosmological constant ), although the dimension, as well as the signature, of the metric can be arbitrary, unlike the four dimensionalLorentzian manifold s usually studied ingeneral relativity .If "M" is the underlying "n"-dimensional
manifold and "g" is itsmetric 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 manifold s.The Einstein condition and Einstein's equation
In local coordinates the condition that ("M", "g") be an Einstein manifold is simply
:R_{ab} = k,g_{ab}.
Take the trace of both sides one finds 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 acosmological constant Λ is:R_{ab} - frac{1}{2}g_{ab}R + g_{ab}Lambda = 8pi T_{ab},
written in
geometrized units with "G" = "c" = 1. Thestress-energy tensor "T""ab" gives the matter and energy content of the underlying spacetime. In avacuum (a region of spacetime with no matter) "T""ab" = 0, and one can rewrite the Einstein's equation in the form (assuming "n" > 2)::R_{ab} = frac{2Lambda}{n-2},g_{ab}.Therefore, vacuum solutions of Einstein's equation are (Lorentzian) Einstein manifolds with "k" proportional to the cosmological constant.Examples
Some of the simplest examples of Einstein manifolds are the following.
*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, "S""n", with the round metric is Einstein with "k" = "n" − 1.
**Hyperbolic space with the canonical metric is Einstein with negative "k".
*Complex projective space , CP"n", with theFubini-Study metric .
*Calabi Yau manifolds admit a unique Einstein metric which is also KählerApplications
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 whoseWeyl 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 manifold s in the Ricci-flat case, andquaternion Kähler manifold s otherwise.Higher dimensional Lorentzian Einstein manifolds are used in modern theories of gravity, such as
string theory ,M-theory andsupergravity . Hyperkähler and quaternion Kähler manifolds (which are special kinds of Einstein manifolds) also have applications in physics as target spaces fornonlinear σ-model s withsupersymmetry .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.References
*cite book | first = Arthur L. | last = Besse | title = Einstein Manifolds | series = Classics in Mathematics | publisher = Springer | location = Berlin | year = 1987 | isbn = 3-540-74120-8
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