Static spacetime explained

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.

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

x S with a metric of the formg[(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.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 λ is the square of the norm of the Killing vector field, λ = g(K,K) , both λ and g _{S} are independent of time (in fact λ = - \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.