Definitions

# 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, $g_\left\{munu\right\}$ may be chosen so that they are all independent of the time coordinate and the time-space components $g_\left\{0i\right\} = 0$ , whereas in a stationary spacetime they are in general nonzero. The line element of a static spacetime can be expressed in the form $\left(i,j = 1,2,3\right)$

$ds^\left\{2\right\} = lambda dt^\left\{2\right\} - lambda^\left\{-1\right\} h_\left\{ij\right\} dy^\left\{i\right\}dy^\left\{j\right\}$ where $t$ is the time coordinate, $y^\left\{i\right\}$ are the three spatial coordinates and $h_\left\{ij\right\}$ 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 $xi^\left\{mu\right\}$, i.e. $lambda = g_\left\{munu\right\}xi^\left\{mu\right\}xi^\left\{nu\right\}$. Both $lambda$ and $h_\left\{ij\right\}$ 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 $R_\left\{munu\right\} = 0$ discovered by Hermann Weyl.

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