Static spacetime

In general relativity, a spacetime is said to be static if it admits a global, non-vanishing, timelike Killing vector field 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 $\times$ S with a metric of the form g[(t,x)] = − β(x)dt² + g_S[x], where R is the real line, g_S is a (positive definite) metric and β 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 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 = − β(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

1. The (exterior) Schwarzschild solution
2. de Sitter space (the portion of it covered by the static patch).
3. Reissner-Nordström space
4. The Weyl solution, a static axisymmetric solution of the Einstein vacuum field equations R_μν = 0 discovered by Hermann Weyl

External links

• Concepts of General Relativity  - A first principles demonstration of the metric in static spacetime

• M. Sanchez, On the geometry of static space-times, preprint, 2004.