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 ...
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''
''S'' with a metric of the form
: