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.