Static spacetime

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: the geometry of a stationary spacetime does not change in time; however, it 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.

Contents

• Examples of static spacetimes
• Examples of non static spacetimes
• References

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 ) d t 2 + g S [ x ] , where R is the real line, g S 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 g S 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.