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. 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. Formally, a spacetime is static if it admits a global, non-vanishing, timelike Killing vector field K {displaystyle 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 × {displaystyle imes } S with a metric of the form where R is the real line, g S {displaystyle g_{S}} is a (positive definite) metric and β {displaystyle eta } is a positive function on the Riemannian manifold S. In such a local coordinate representation the Killing field 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 =-eta (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.