Static spacetime

Static spacetime

In general relativity, a spacetime is said to be static if it admits a global, nowhere zero, timelike hypersurface orthogonal Killing vector field. A static spacetime can in effect be split up into time and three dimensional (curved) space. Every static spacetime is stationary but the converse is not true. In a static spacetime, the metric tensor components, g_{mu u} may be chosen so that they are all independent of the time coordinate and the time-space components g_{0i} = 0 , whereas in a stationary spacetime they are in general nonzero. The line element of a static spacetime can be expressed in the form (i,j = 1,2,3)

ds^{2} = lambda dt^{2} - lambda^{-1} h_{ij} dy^{i}dy^{j} where t is the time coordinate, y^{i} are the three spatial coordinates and h_{ij} is metric tensor of 3-dimensional space. As in the more general stationary case, the 3-space can be thought of as the manifold of trajectories of the Killing vector V. But for static spacetimes V can also the regarded as any hypersurface t = const embedded in the spacetime which is now the instantaneous 3-space of stationary observers. lambda is a positive scalar representing the norm of the Killing vector field xi^{mu}, i.e. lambda = g_{mu u}xi^{mu}xi^{ u}. Both lambda and h_{ij} are independent of time. It is in this sense that a static spacetime derives its name, as the geometry of the spacetime does not change. Examples of a static spacetime are the (exterior) Schwarzschild solution and the Weyl solution. The latter are general static axisymmetric solutions of the Einstein vacuum field equations R_{mu u} = 0 discovered by Hermann Weyl.


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