Static spacetime

In <a href="/facts/General_relativity/jVsoIg8X">general relativity</a>, a <a href="/facts/Spacetime/uRIN1x4b">spacetime</a> is said to be static if it does not change over time and is also irrotational. It is a special case of a <a href="/facts/Stationary_spacetime/CXLvlTQ4">stationary spacetime</a>, which is the geometry of a stationary spacetime that does not change in time but can rotate. Thus, the <a href="/facts/Kerr_solution/SjL2K6R0">Kerr solution</a> provides an example of a stationary spacetime that is not static; the non-rotating <a href="/facts/Schwarzschild_solution/gkm25ScM">Schwarzschild solution</a> is an example that is static.
Formally, a spacetime is static if it admits a global, non-vanishing, <a href="/facts/Timelike/uRIN1x4b">timelike</a> <a href="/facts/Killing_vector_field/Ane9xIdV">Killing vector field</a> 
 
 
 
 K
 
 
 {\displaystyle K}
 
 which is irrotational, i.e., whose <a href="/facts/Congruence_(general_relativity)/1Be30Nok">orthogonal distribution</a> is <a href="/facts/Involution_(mathematics)/kKxYrUVa">involutive</a>. (Note that the leaves of the associated <a href="/facts/Foliation/cxml9vyJ">foliation</a> are necessarily space-like <a href="/facts/Hypersurface/cTvtLVsV">hypersurfaces</a>.) Thus, a static spacetime is a <a href="/facts/Stationary_spacetime/CXLvlTQ4">stationary spacetime</a> satisfying this additional integrability condition. These spacetimes form one of the simplest classes of <a href="/facts/Lorentzian_manifold/VFXsVjqb">Lorentzian manifolds</a>. 
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

g
 [
 (
 t
 ,
 x
 )
 ]
 =
 −
 β
 (
 x
 )
 d
 
 t
 
 2
 
 
 +
 
 g
 
 S
 
 
 [
 x
 ]
 
 
 {\displaystyle g[(t,x)]=-\beta (x)dt^{2}+g_{S}[x]}
 
,
where R is the real line, 
 
 
 
 
 g
 
 S
 
 
 
 
 {\displaystyle g_{S}}
 
 is a (positive definite) metric and 
 
 
 
 β
 
 
 {\displaystyle \beta }
 
 is a positive function on the <a href="/facts/Riemannian_manifold/5KYnmEgF">Riemannian manifold</a> S.
In such a local coordinate representation the <a href="/facts/Killing_vector_field/Ane9xIdV">Killing field</a> 
 
 
 
 K
 
 
 {\displaystyle K}
 
 may be identified with 
 
 
 
 
 ∂
 
 t
 
 
 
 
 {\displaystyle \partial _{t}}
 
 and S, the manifold of 
 
 
 
 K
 
 
 {\displaystyle K}
 
-trajectories, may be regarded as the instantaneous 3-space of stationary observers. If 
 
 
 
 λ
 
 
 {\displaystyle \lambda }
 
 is the square of the norm of the Killing vector field, 
 
 
 
 λ
 =
 g
 (
 K
 ,
 K
 )
 
 
 {\displaystyle \lambda =g(K,K)}
 
, both 
 
 
 
 λ
 
 
 {\displaystyle \lambda }
 
 and 
 
 
 
 
 g
 
 S
 
 
 
 
 {\displaystyle g_{S}}
 
 are independent of time (in fact 
 
 
 
 λ
 =
 −
 β
 (
 x
 )
 
 
 {\displaystyle \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.

Static spacetime open-in-new

Static spacetime