In
general relativityGeneral relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics. It unifies special relativity and Newton's law of universal gravitation, and describes gravity as a...
, a
spacetimeIn physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space being three-dimensional and time playing the role of a fourth dimension that is of a different sort than the spatial dimensions...
is said to be
static if it admits a global, non-vanishing, timelike
Killing vector fieldIn mathematics, a Killing vector field , named after Wilhelm Killing, is a vector field on a Riemannian manifold that preserves the metric. Killing fields are the infinitesimal generators of isometries; that is, flows generated by Killing fields are continuous isometries of the manifold...
which is
irrotational,
i.e., whose orthogonal distribution is involutive. (Note that the leaves of the associated
foliationIn mathematics, a foliation is a geometric device used to study manifolds, consisting of an integrable subbundle of the tangent bundle. A foliation looks locally like a decomposition of the manifold as a union of parallel submanifolds of smaller dimension....
are necessarily space-like hypersurfaces.) Thus a static spacetime is a
stationary spacetimeIn 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....
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 S with a metric of the form
,
where
R is the real line and is a (positive definite) metric and is a positive function on the Riemannian manifold
S.
In such a local coordinate representation the Killing field may be identified with and
S, the manifold of -
trajectories, may be regarded as the instantaneous 3-space of stationary observers.
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In
general relativityGeneral relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics. It unifies special relativity and Newton's law of universal gravitation, and describes gravity as a...
, a
spacetimeIn physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space being three-dimensional and time playing the role of a fourth dimension that is of a different sort than the spatial dimensions...
is said to be
static if it admits a global, non-vanishing, timelike
Killing vector fieldIn mathematics, a Killing vector field , named after Wilhelm Killing, is a vector field on a Riemannian manifold that preserves the metric. Killing fields are the infinitesimal generators of isometries; that is, flows generated by Killing fields are continuous isometries of the manifold...
which is
irrotational,
i.e., whose orthogonal distribution is involutive. (Note that the leaves of the associated
foliationIn mathematics, a foliation is a geometric device used to study manifolds, consisting of an integrable subbundle of the tangent bundle. A foliation looks locally like a decomposition of the manifold as a union of parallel submanifolds of smaller dimension....
are necessarily space-like hypersurfaces.) Thus a static spacetime is a
stationary spacetimeIn 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....
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 S with a metric of the form
,
where
R is the real line and is a (positive definite) metric and is a positive function on the Riemannian manifold
S.
In such a local coordinate representation the Killing field may be identified with and
S, the manifold of -
trajectories, may be regarded as the instantaneous 3-space of stationary observers. If is the square of the norm of the Killing vector field, , both and are independent of time. 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
- de Sitter space
In mathematics and physics, n-dimensional de Sitter space, denoted , is the Lorentzian analog of an n-sphere...
- Reissner-Nordström space
- The Weyl solution, a static axisymmetric solution of the Einstein vacuum field equations discovered by Hermann Weyl
Hermann Klaus Hugo Weyl was a German mathematician. Although much of his working life was spent in Zürich, Switzerland and then Princeton, he is associated with the University of Göttingen tradition of mathematics, represented by David Hilbert and Hermann Minkowski.His research has had major...
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