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: the geometry of a stationary spacetime does not change in time; however, it 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 which 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 which is a Lorentzian warped product R S with a metric of the form , where R is the real line, 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 (in fact ). 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 (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:
- The Kerr solution, 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.
- Hawking, S. W.; Ellis, G. F. R. (1973), The large scale structure of space-time, Cambridge Monographs on Mathematical Physics, 1, London-New York: Cambridge University Press, MR 0424186