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, 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 ${\displaystyle K}$ 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

• 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 ${\displaystyle R_{\mu \nu }=0}$ 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, 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