{{short description|Spacetime that does not change over time and is irrotational}} {{more citations needed|date=April 2021}} {{more footnotes|date=April 2021}}
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 <math>K</math> that 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''' that is a Lorentzian warped product {{tmath|R \times S }} with a metric of the form : <math>g[(t,x)] = -\beta(x) dt^{2} + g_{S}[x], </math> where {{tmath| R }} is the real line, <math>g_{S}</math> is a (positive definite) metric and <math>\beta</math> is a positive function on the Riemannian manifold {{tmath| S }}.
In such a local coordinate representation the Killing field <math>K</math> may be identified with <math>\partial_t</math> and ''S'', the manifold of <math>K</math>-''trajectories'', may be regarded as the instantaneous 3-space of stationary observers. If <math>\lambda</math> is the square of the norm of the Killing vector field, {{tmath|1= \lambda = g(K,K) }}, both <math>\lambda</math> and <math>g_S</math> are independent of time (in fact {{tmath|1= \lambda = - \beta(x) }}). It is from the latter fact that a static spacetime obtains its name, as the geometry of the space-like slice {{tmath| 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 <math>R_{\mu\nu} = 0</math> 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, a stationary spacetime that is not static * Spacetimes with gravitational waves, which are not even stationary.
== References == * {{citation |mr=0424186 |last1=Hawking |first1= S. W. |last2= Ellis |first2= G. F. R. |title=The large scale structure of space-time |series=Cambridge Monographs on Mathematical Physics |volume=1 |publisher=Cambridge University Press |place=London–New York |year= 1973 }}
{{relativity-stub}} Category:Lorentzian manifolds