# Static spacetime

In general relativity, a spacetime is said to be static if it admits a global, non-vanishing, timelike Killing vector field ${\displaystyle K}$ 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 ${\displaystyle \times }$ S with a metric of the form ${\displaystyle g[(t,x)]=-\beta (x)dt^{2}+g_{S}[x]}$, where R is the real line, ${\displaystyle g_{S}}$ is a (positive definite) metric and ${\displaystyle \beta }$ is a positive function on the Riemannian manifold S.

In such a local coordinate representation the Killing field ${\displaystyle K}$ may be identified with ${\displaystyle \partial _{t}}$ and S, the manifold of ${\displaystyle K}$-trajectories, may be regarded as the instantaneous 3-space of stationary observers. If ${\displaystyle \lambda }$ is the square of the norm of the Killing vector field, ${\displaystyle \lambda =g(K,K)}$, both ${\displaystyle \lambda }$ and ${\displaystyle g_{S}}$ are independent of time (in fact ${\displaystyle \lambda =-\beta (x)}$). 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

1. The (exterior) Schwarzschild solution
2. de Sitter space (the portion of it covered by the static patch).
3. Reissner-Nordström space
4. The Weyl solution, a static axisymmetric solution of the Einstein vacuum field equations ${\displaystyle R_{\mu \nu }=0}$ discovered by Hermann Weyl

## References

{{#invoke:citation/CS1|citation |CitationClass=citation }}