- Einstein manifold
differential geometryand mathematical physics, an Einstein manifold is a Riemannian or pseudo-Riemannian manifoldwhose Ricci tensoris proportional to the metric. They are named after Albert Einsteinbecause this condition is equivalent to saying that the metric is a solution of the vacuum Einstein 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
manifoldand "g" is its metric tensorthe Einstein condition means that
for some constant "k", where Ric denotes the Ricci tensor of "g". Einstein manifolds with "k" = 0 are called
The Einstein condition and Einstein's equation
In local coordinates the condition that ("M", "g") be an Einstein manifold is simply
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
where "n" is the dimension of "M".
general relativity, Einstein's equationwith a cosmological constantΛ is
geometrized unitswith "G" = "c" = 1. The stress-energy tensor"T""ab" gives the matter and energy content of the underlying spacetime. In a vacuum(a region of spacetime with no matter) "T""ab" = 0, and one can rewrite the Einstein's equation in the form (assuming "n" > 2)::Therefore, vacuum solutions of Einstein's equation are (Lorentzian) Einstein manifolds with "k" proportional to the cosmological constant.
Some of the simplest examples of Einstein manifolds are the following.
*Any manifold with
constant sectional curvatureis 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 spacewith the canonical metric is Einstein with negative "k".
Complex projective space, CP"n", with the Fubini-Study metric.
Calabi Yaumanifolds admit a unique Einstein metric which is also Kähler
Four dimensional Riemannian Einstein manifolds are also important in mathematical physics as
gravitational instantonsin quantum theories of gravity. The term "gravitational instanton" is usually used restricted to Einstein 4-manifolds whose Weyl tensoris 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-theoryand 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.
*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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