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 $K$ 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'' $\times$ ''S'' with a metric of the form

:g t,x)= -\beta(x) dt^ + g_ /math>,

where ''R'' is the real line, $g_$ is a (positive definite) metric and $\beta$ is a positive function on the Riemannian manifold ''S''.

In such a local coordinate representation the Killing field $K$ may be identified with $\partial_t$ and ''S'', the manifold of $K$-''trajectories'', may be regarded as the instantaneous 3-space of stationary observers. If $\lambda$ is the square of the norm of the Killing vector field, $\lambda = g(K,K)$, both $\lambda$ and $g_S$ are independent of time (in fact $\lambda = - \beta(x)$). 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 $R_ = 0$ 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

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Lorentzian manifolds