- Static spacetime
general relativity, a spacetimeis 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, may be chosen so that they are all independent of the time coordinate and the time-space components , whereas in a stationary spacetime they are in general nonzero. The line element of a static spacetime can be expressed in the form
where is the time coordinate, are the three spatial coordinates and 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 . But for static spacetimes can also the regarded as any hypersurface = const embedded in the spacetime which is now the instantaneous 3-space of stationary observers. is a positive scalar representing the norm of the Killing vector field , i.e. . Both and 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 solutionand the Weyl solution. The latter are general static axisymmetric solutions of the Einstein vacuum field equations discovered by Hermann Weyl.
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