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In general relativity, a spacetime is said to be static if it does not change over time and is also irrotational. It is a special case of a
stationary spacetime In general relativity, specifically in the Einstein field equations, a spacetime is said to be stationary if it admits a Killing vector that is asymptotically timelike. Description and analysis In a stationary spacetime, the metric tensor compo ...
, which is the geometry of a stationary spacetime that does not change in time but can rotate. Thus, the
Kerr solution The Kerr metric or Kerr geometry describes the geometry of empty spacetime around a rotating uncharged axially symmetric black hole with a quasispherical event horizon. The Kerr metric tensor, metric is an Exact solutions in general relativity, e ...
provides an example of a stationary spacetime that is not static; the non-rotating
Schwarzschild solution In Einstein's theory of general relativity, the Schwarzschild metric (also known as the Schwarzschild solution) is an exact solution to the Einstein field equations that describes the gravitational field outside a spherical mass, on the assu ...
is an example that is static. Formally, a spacetime is static if it admits a global, non-vanishing,
timelike In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
Killing vector field In mathematics, a Killing vector field (often called a Killing field), named after Wilhelm Killing, is a vector field on a Riemannian manifold (or pseudo-Riemannian manifold) that preserves the metric. Killing fields are the infinitesimal gener ...
K which is irrotational, ''i.e.'', whose orthogonal distribution is involutive. (Note that the leaves of the associated
foliation In mathematics (differential geometry), a foliation is an equivalence relation on an ''n''-manifold, the equivalence classes being connected, injectively immersed submanifolds, all of the same dimension ''p'', modeled on the decomposition of ...
are necessarily space-like
hypersurface In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety of dimension , which is embedded in an ambient space of dimension , generally a Euclidean ...
s.) Thus, a static spacetime is a
stationary spacetime In general relativity, specifically in the Einstein field equations, a spacetime is said to be stationary if it admits a Killing vector that is asymptotically timelike. Description and analysis In a stationary spacetime, the metric tensor compo ...
satisfying this additional integrability condition. These spacetimes form one of the simplest classes of Lorentzian manifolds. Locally, every static spacetime looks like a standard static spacetime which is a Lorentzian warped product ''R'' \times ''S'' with a metric of the form :g t,x)= -\beta(x) dt^ + g_ /math>, where ''R'' is the real line, g_ is a (positive definite) metric and \beta is a positive function on the Riemannian manifold ''S''. In such a local coordinate representation the
Killing field A killing field, in military science, is an area in front of a defensive position that the enemy must cross during an assault and is specifically intended to allow the defending troops to incapacitate a large number of the enemy. Defensive emplacem ...
K may be identified with \partial_t and ''S'', the manifold of K-''trajectories'', may be regarded as the instantaneous 3-space of stationary observers. If \lambda is the square of the norm of the Killing vector field, \lambda = g(K,K), both \lambda and g_S are independent of time (in fact \lambda = - \beta(x)). It is from the latter fact that a static spacetime obtains its name, as the geometry of the space-like slice ''S'' does not change over time.


Examples of static spacetimes

* The (exterior)
Schwarzschild solution In Einstein's theory of general relativity, the Schwarzschild metric (also known as the Schwarzschild solution) is an exact solution to the Einstein field equations that describes the gravitational field outside a spherical mass, on the assu ...
. *
de Sitter space In mathematical physics, ''n''-dimensional de Sitter space (often abbreviated to dS''n'') is a maximally symmetric Lorentzian manifold with constant positive scalar curvature. It is the Lorentzian analogue of an ''n''-sphere (with its canoni ...
(the portion of it covered by the static patch). * Reissner–Nordström space. * The
Weyl solution Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is assoc ...
, a static axisymmetric solution of the Einstein vacuum field equations R_ = 0 discovered by Hermann Weyl.


Examples of non-static spacetimes

In general, "almost all" spacetimes will not be static. Some explicit examples include: * Spherically symmetric spacetimes, which are irrotational, but not static. * The
Kerr solution The Kerr metric or Kerr geometry describes the geometry of empty spacetime around a rotating uncharged axially symmetric black hole with a quasispherical event horizon. The Kerr metric tensor, metric is an Exact solutions in general relativity, e ...
, since it describes a rotating black hole, is a stationary spacetime that is not static. * Spacetimes with gravitational waves in them are not even stationary.


References

* {{relativity-stub Lorentzian manifolds