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


 * $$g[(t,x)] = -\beta(x) dt^{2} + g_{S}[x]$$,

where R is the real line, $$g_{S}$$ 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 $$R_{\mu\nu} = 0$$ 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.