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 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 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

Examples of non-static spacetimes

In general, "almost all" spacetimes will not be static. Some explicit examples include:

References

  • Hawking, S. W. & Ellis, G. F. R. (1973), "The large scale structure of space-time", Vol. 1, Cambridge Monographs on Mathematical Physics, London–New York: Cambridge University Press, MR 0424186