# Static spacetime

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-NordstrÃ¶m 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.

## References

- 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