Static spacetime

Short description: Spacetime that does not change over time and is irrotational

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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 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

g[(t,x)]=−β(x)dt2+gS[x],

where R is the real line, gS 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 ∂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 gS 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

* 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 Rμν=0 discovered by Hermann Weyl.

## Examples of non-static spacetimes

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.
* Spacetimes with gravitational waves in them are not even stationary.

## References

* Hawking, S. W.; Ellis, G. F. R. (1973), The large scale structure of space-time, Cambridge Monographs on Mathematical Physics, 1, London-New York: Cambridge University Press

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