Stationary spacetime

In general relativity , space-time is called stationary if it has a time-like killing vector field . Some authors (e.g. Ludvigsen) use the term “stationary space-time” even for asymptotically flat space-times, which only have a killing vector field that is asymptotically time-like, ie. H. the killing vectors become temporal in the extreme case of great distance. Other authors (Hawking and Ellis) designate a spacetime which only has an asymptotically time-like killing vector field as asymptotically stationary , still others (Carroll) use the term inconsistent.

The metric tensor is invariant along a killing vector field . This clearly means that there are observers for whom the gravitational field does not change over time.

Composition of the line element of a stationary spacetime

An observer field in a spacetime is called stationary if there is a positive function such that it is a killing vector field. If spacetime has a stationary observer field, it is called stationary. In the coordinates of a stationary observer, the components of the metric tensor are independent of the time coordinate. The line element (with signature and ) of a stationary space-time can therefore be applied to the form ${\displaystyle U}$ ${\displaystyle (M,g)}$ ${\displaystyle f\in C^{\infty }(M,\mathbb {R} ^{+})}$ ${\displaystyle fU}$ ${\displaystyle g_{\mu \nu }}$ ${\displaystyle (+,-,-,-)}$ ${\displaystyle i,j=1,2,3}$

${\displaystyle \mathrm {d} s^{2}=\lambda (\mathrm {d} t-\omega _{i}\,\mathrm {d} y^{i})^{2}-\lambda ^{-1}h_{ij}\,\mathrm {d} y^{i}\,\mathrm {d} y^{j}\,,}$

bring, where is the time coordinate, the three spatial coordinates and the metric tensor of three-dimensional space. The Killing vector field has the components in these coordinates . is a positive scalar that determines the norm of the killing vector field, i.e. H. , and a three-way vector that determines the rotation ( English twist ) of spacetime. This vector is calculated from the spatial components of the twist vector (see, for example, p. 163), which is calculated as the antisymmetric product of the Killing vector field and its covariant derivative with the help of the epsilon tensor . ${\displaystyle t}$ ${\displaystyle y^{i}}$ ${\displaystyle h_{ij}}$ ${\displaystyle \xi ^{\mu }}$ ${\displaystyle \xi ^{\mu }=(1,0,0,0)}$ ${\displaystyle \lambda }$ ${\displaystyle \lambda =g_{\mu \nu }\xi ^{\mu }\xi ^{\nu }}$ ${\displaystyle \omega _{i}}$ ${\displaystyle \omega _{\mu }=\varepsilon _{\mu \nu \rho \sigma }\xi ^{\nu }\nabla ^{\rho }\xi ^{\sigma }}$ ${\displaystyle \nabla }$ ${\displaystyle \varepsilon }$

Twist vector and its interpretation

The twist vector describes how much the orientation of the killing vector field deviates from the surface normals of the space-like hypersurfaces. If the Killing vector field is orthogonal to the space-like hypersurfaces, i.e. H. it is true , then the killing vector field is rotation-free and the three vector vanishes. Such spacetime is called static . ${\displaystyle \omega _{\mu }\xi ^{\mu }=0}$ ${\displaystyle \omega _{i}}$

From this definition it follows that a static space-time is always stationary, but a stationary space-time does not have to be static.

Examples

• A flat spacetime is described by a Minkowski metric and is stationary and static.
• The spacetime of a homogeneous, non-charged and non-rotating sphere is described by the Schwarzschild metric and is asymptotically stationary and asymptotically static.
• The gravitational field of a non-charged, rotating black hole is described by the Kerr metric and is asymptotically stationary but not asymptotically static.
• A space-time with gravitational waves is neither stationary nor static.

Individual evidence

1. ↑ Malcolm Ludvigsen: General Relativity: A Geometric Approach . Cambridge University Press, May 28, 1999, ISBN 978-0-521-63976-7 , pp. 123f.
2. ^ Benjamin Crowell: Static and Stationary Spacetimes. April 21, 2019 .  
3. ^ Wald, RM, (1984). General Relativity, (U. Chicago Press)