Static spacetime

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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[edit]

  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[edit]

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, MR 0424186