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 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

where is a positive function on the Riemannian manifold .

In such a local coordinate representation the Killing field and S-trajectories is the square of the norm of the Killing vector field, 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 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.

References