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| In [[differential geometry]] and [[mathematical physics]], an '''Einstein manifold''' is a [[Riemannian manifold|Riemannian]] or [[pseudo-Riemannian manifold]] whose [[Ricci tensor]] is proportional to the [[metric tensor|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 manifold]]s usually studied in [[general relativity]].
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| If ''M'' is the underlying ''n''-dimensional [[manifold]] and ''g'' is its [[metric tensor]] the Einstein condition means that
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| :<math>\mathrm{Ric} = k\,g,</math> | |
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| for some constant ''k'', where Ric denotes the [[Ricci tensor]] of ''g''. Einstein manifolds with ''k'' = 0 are called [[Ricci-flat manifold]]s.
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| ==The Einstein condition and Einstein's equation==
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| In local coordinates the condition that (''M'', ''g'') be an Einstein manifold is simply
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| :<math>R_{ab} = k\,g_{ab}.</math>
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| Taking the trace of both sides reveals that the constant of proportionality ''k'' for Einstein manifolds is related to the [[scalar curvature]] ''R'' by
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| :<math>R = nk\,</math>
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| where ''n'' is the dimension of ''M''.
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| In [[general relativity]], [[Einstein's equation]] with a [[cosmological constant]] Λ is
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| :<math>R_{ab} - \frac{1}{2}g_{ab}R + g_{ab}\Lambda = 8\pi T_{ab}, </math> | |
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| written in [[geometrized units]] with ''G'' = ''c'' = 1. The [[stress-energy tensor]] ''T''<sub>''ab''</sub> gives the matter and energy content of the underlying spacetime. In a [[vacuum]] (a region of spacetime with no matter) ''T''<sub>''ab''</sub> = 0, and one can rewrite Einstein's equation in the form (assuming ''n'' > 2):
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| :<math>R_{ab} = \frac{2\Lambda}{n-2}\,g_{ab}.</math>
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| Therefore, vacuum solutions of Einstein's equation are (Lorentzian) Einstein manifolds with ''k'' proportional to the cosmological constant.
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| == Examples ==
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| Simple examples of Einstein manifolds include:
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| *Any manifold with [[constant sectional curvature]] is an Einstein manifold—in particular:
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| ** [[Euclidean space]], which is flat, is a simple example of Ricci-flat, hence Einstein metric.
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| ** The [[n-sphere|''n''-sphere]], ''S''<sup>''n''</sup>, with the round metric is Einstein with ''k'' = ''n'' − 1.
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| ** [[Hyperbolic space]] with the canonical metric is Einstein with negative ''k''.
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| * [[Complex projective space]], '''CP'''<sup>''n''</sup>, with the [[Fubini-Study metric]].
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| * [[Calabi Yau manifold]]s admit an Einstein metric that is also [[Kähler metric|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.
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| A necessary condition for [[closed manifold|closed]], [[oriented]], [[4-manifold]]s to be Einstein is satisfying the [[Hitchin–Thorpe inequality]].
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| ==Applications==
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| Four dimensional Riemannian Einstein manifolds are also important in mathematical physics as [[gravitational instantons]] in [[quantum gravity|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 metric|complete]] but [[compact space|non-compact]]). In differential geometry, self-dual Einstein 4-manifolds are also known as (4-dimensional) [[hyperkähler manifold]]s in the Ricci-flat case, and [[quaternion Kähler manifold]]s otherwise.
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| 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-model]]s with [[supersymmetry]].
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| 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 [[Michelin star|starred restaurant]] in exchange for a new example.
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| ==See also==
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| *[[Einstein–Hermitian vector bundle]]
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| == References ==
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| *{{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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| [[Category:Riemannian manifolds]]
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| [[Category:Albert Einstein|Manifold]]
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| [[Category:Mathematical physics]]
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