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 <math>K</math> that 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 that is a Lorentzian warped product with a metric of the form
: <math>g[(t,x)] = -\beta(x) dt^{2} + g_{S}[x], </math>
where is the real line, <math>g_{S}</math> is a (positive definite) metric and <math>\beta</math> is a positive function on the Riemannian manifold .
In such a local coordinate representation the Killing field <math>K</math> may be identified with <math>\partial_t</math> and S, the manifold of <math>K</math>-trajectories, may be regarded as the instantaneous 3-space of stationary observers. If <math>\lambda</math> is the square of the norm of the Killing vector field, , both <math>\lambda</math> and <math>g_S</math> 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 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 <math>R_{\mu\nu} = 0</math> 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, a stationary spacetime that is not static
- Spacetimes with gravitational waves, which are not even stationary.