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

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 $R_{\mu \nu }=0$ discovered by Hermann Weyl

References

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