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, which is the geometry of a stationary spacetime that does not change in time but can rotate. Thus, the Kerr solution provides an example of a stationary spacetime that is not static; the non-rotating Schwarzschild solution is an example that is static.
Formally, a spacetime is static if it admits a global, non-vanishing, timelike Killing vector field that is irrotational, i.e., whose orthogonal distribution is involutive. (Note that the leaves of the associated foliation are necessarily space-like hypersurfaces.) Thus, a static spacetime is a stationary spacetime 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 that is a Lorentzian warped product with a metric of the form
where is the real line, is a (positive definite) metric and is a positive function on the Riemannian manifold .
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 (in fact ). It is from the latter fact that a static spacetime obtains its name, as the geometry of the space-like slice does not change over time.
Examples of static spacetimes The (exterior) Schwarzschild solution De Sitter space (the portion of it covered by the static patch) Reissner–Nordström space The Weyl solution, a static axisymmetric solution of the Einstein vacuum field equations 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, a stationary spacetime that is not static Spacetimes with gravitational waves, which are not even stationary.
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