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Einstein manifold

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.

The Einstein condition and Einstein's equation edit

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.[1] 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.

Examples edit

Simple examples of Einstein manifolds include:

  • All 2D manifolds are trivially Einstein manifolds. This is a result of the Riemann tensor having a single degree of freedom.
  • 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  .
  • Complex projective space,  , with the Fubini–Study metric, have  
  • Calabi–Yau manifolds admit an Einstein metric that is also Kähler, with Einstein constant  . 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.
  • Kähler–Einstein metrics exist on a variety of compact complex manifolds due to the existence results of Shing-Tung Yau, and the later study of K-stability especially in the case of Fano manifolds.
  • An Einstein–Weyl geometry is a generalization of an Einstein manifold for a Weyl connection of a conformal class, rather than the Levi-Civita connection of a metric.

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 the 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.[2]

See also edit

Notes and references edit

  1. ^ κ should not be confused with k.
  2. ^ Besse (1987, p. 18)
  • Besse, Arthur L. (1987). Einstein Manifolds. Classics in Mathematics. Berlin: Springer. ISBN 3-540-74120-8.

einstein, manifold, differential, geometry, mathematical, physics, riemannian, pseudo, riemannian, differentiable, manifold, whose, ricci, tensor, proportional, metric, they, named, after, albert, einstein, because, this, condition, equivalent, saying, that, m. 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 R i c k g displaystyle mathrm Ric kg for some constant k where Ric denotes the Ricci tensor of g Einstein manifolds with k 0 are called Ricci flat manifolds Contents 1 The Einstein condition and Einstein s equation 2 Examples 3 Applications 4 See also 5 Notes and referencesThe Einstein condition and Einstein s equation editIn local coordinates the condition that M g be an Einstein manifold is simply R a b k g a b displaystyle R ab kg ab nbsp 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 n k displaystyle R nk nbsp where n is the dimension of M In general relativity Einstein s equation with a cosmological constant L is R a b 1 2 g a b R g a b L k T a b displaystyle R ab frac 1 2 g ab R g ab Lambda kappa T ab nbsp where k is the Einstein gravitational constant 1 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 gt 2 R a b 2 L n 2 g a b displaystyle R ab frac 2 Lambda n 2 g ab nbsp Therefore vacuum solutions of Einstein s equation are Lorentzian Einstein manifolds with k proportional to the cosmological constant Examples editSimple examples of Einstein manifolds include All 2D manifolds are trivially Einstein manifolds This is a result of the Riemann tensor having a single degree of freedom 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 displaystyle S n nbsp with the round metric is Einstein with k n 1 displaystyle k n 1 nbsp Hyperbolic space with the canonical metric is Einstein with k n 1 displaystyle k n 1 nbsp Complex projective space C P n displaystyle mathbf CP n nbsp with the Fubini Study metric have k n 1 displaystyle k n 1 nbsp Calabi Yau manifolds admit an Einstein metric that is also Kahler with Einstein constant k 0 displaystyle k 0 nbsp Such metrics are not unique but rather come in families there is a Calabi Yau metric in every Kahler 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 Kahler Einstein metrics exist on a variety of compact complex manifolds due to the existence results of Shing Tung Yau and the later study of K stability especially in the case of Fano manifolds An Einstein Weyl geometry is a generalization of an Einstein manifold for a Weyl connection of a conformal class rather than the Levi Civita connection of a metric A necessary condition for closed oriented 4 manifolds to be Einstein is satisfying the Hitchin Thorpe inequality Applications editFour 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 the 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 hyperkahler manifolds in the Ricci flat case and quaternion Kahler manifolds otherwise Higher dimensional Lorentzian Einstein manifolds are used in modern theories of gravity such as string theory M theory and supergravity Hyperkahler and quaternion Kahler manifolds which are special kinds of Einstein manifolds also have applications in physics as target spaces for nonlinear s 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 2 See also editEinstein Hermitian vector bundleNotes and references edit k should not be confused with k Besse 1987 p 18 Besse Arthur L 1987 Einstein Manifolds Classics in Mathematics Berlin Springer ISBN 3 540 74120 8 Retrieved from https en wikipedia org w index php title Einstein manifold amp oldid 1214846261, wikipedia, wiki, book, books, library,

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