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Concept
Static spacetime
Summary
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 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 S with a metric of the form
where R is the real line, is a (positive definite) metric and is a positive function on the Riemannian manifold S.
In such a local coordinate representation the Killing field may be identified with and S, the manifold of -trajectories, may be regarded as the instantaneous 3-space of stationary observers. If is the square of the norm of the Killing vector field, , both and are independent of time (in fact ). 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.
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 discovered by Hermann Weyl.
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.
Official source
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