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: the geometry of a stationary spacetime does not change in time; however, it 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 ${\displaystyle K}$ 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 ${\displaystyle \times }$ S with a metric of the form ${\displaystyle g[(t,x)]=-\beta (x)dt^{2}+g_{S}[x]}$, where R is the real line, ${\displaystyle g_{S}}$ is a (positive definite) metric and ${\displaystyle \beta }$ is a positive function on the Riemannian manifold S.

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