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 [math]\displaystyle{ K }[/math] 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 [math]\displaystyle{ \times }[/math] S with a metric of the form

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

where R is the real line, [math]\displaystyle{ g_{S} }[/math] is a (positive definite) metric and [math]\displaystyle{ \beta }[/math] is a positive function on the Riemannian manifold S.

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

• Hawking, S. W.; Ellis, G. F. R. (1973), The large scale structure of space-time, Cambridge Monographs on Mathematical Physics, 1, London-New York: Cambridge University Press