Stationary spacetime

In general relativity, specifically in the Einstein field equations, a spacetime is said to be stationary if it admits a Killing vector that is asymptotically timelike.

In a stationary spacetime, the metric tensor components, $g_{\mu \nu }$ $g_{\mu \nu }$ , may be chosen so that they are all independent of the time coordinate. The line element of a stationary spacetime has the form $(i,j=1,2,3)$ $(i,j=1,2,3)$

where $t$ $t$ is the time coordinate, $y^{i}$ $y^{{i}}$ are the three spatial coordinates and $h_{ij}$ $h_{ij}$ is the metric tensor of 3-dimensional space. In this coordinate system the Killing vector field $\xi ^{\mu }$ $\xi^{\mu}$ has the components $\xi ^{\mu }=(1,0,0,0)$ $\xi ^{{\mu }}=(1,0,0,0)$ . $\lambda$ $\lambda$ is a positive scalar representing the norm of the Killing vector, i.e., $\lambda =g_{\mu \nu }\xi ^{\mu }\xi ^{\nu }$ $\lambda =g_{{\mu \nu }}\xi ^{{\mu }}\xi ^{{\nu }}$ , and $\omega _{i}$ $\omega _{{i}}$ is a 3-vector, called the twist vector, which vanishes when the Killing vector is hypersurface orthogonal. The latter arises as the spatial components of the twist 4-vector $\omega _{\mu }=e_{\mu \nu \rho \sigma }\xi ^{\nu }\nabla ^{\rho }\xi ^{\sigma }$ $\omega _{{\mu }}=e_{{\mu \nu \rho \sigma }}\xi ^{{\nu }}\nabla ^{{\rho }}\xi ^{{\sigma }}$ (see, for example, p. 163) which is orthogonal to the Killing vector $\xi ^{\mu }$ $\xi^{\mu}$ , i.e., satisfies $\omega _{\mu }\xi ^{\mu }=0$ $\omega _{{\mu }}\xi ^{{\mu }}=0$ . The twist vector measures the extent to which the Killing vector fails to be orthogonal to a family of 3-surfaces. A non-zero twist indicates the presence of rotation in the spacetime geometry.

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