Stationary spacetime

Description and analysis

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

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

where ${\displaystyle t}$ is the time coordinate, ${\displaystyle y^i}$ are the three spatial coordinates and ${\displaystyle h_{ij}}$ is the metric tensor of 3-dimensional space. In this coordinate system the Killing vector field ${\displaystyle {\xi }^{\mu }}$ has the components ${\displaystyle {\xi }^{\mu }=(1,0,0,0)}$. ${\displaystyle \lambda }$ is a positive scalar representing the norm of the Killing vector, i.e., ${\displaystyle \lambda =g_{\mu \nu }{\xi }^{\mu }{\xi }^{\nu }}$, and ${\displaystyle {\omega }_i}$ 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 ${\displaystyle {\omega }_{\mu }=e_{\mu \nu \rho \sigma }{\xi }^{\nu }{\mathrm{\nabla }}^{\rho }{\xi }^{\sigma }}$(see, for example, p. 163) which is orthogonal to the Killing vector ${\displaystyle {\xi }^{\mu }}$, i.e., satisfies ${\displaystyle {\omega }_{\mu }{\xi }^{\mu }=0}$. 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.

The coordinate representation described above has an interesting geometrical interpretation. The time translation Killing vector generates a one-parameter group of motion ${\displaystyle G}$ in the spacetime ${\displaystyle M}$. By identifying the spacetime points that lie on a particular trajectory (also called orbit) one gets a 3-dimensional space (the manifold of Killing trajectories) ${\displaystyle V=M/G}$, the quotient space. Each point of ${\displaystyle V}$ represents a trajectory in the spacetime ${\displaystyle M}$. This identification, called a canonical projection, ${\displaystyle \pi :M\to V}$ is a mapping that sends each trajectory in ${\displaystyle M}$ onto a point in ${\displaystyle V}$ and induces a metric ${\displaystyle h=-\lambda \pi \ast g}$ on ${\displaystyle V}$ via pullback. The quantities ${\displaystyle \lambda }$, ${\displaystyle {\omega }_i}$ and ${\displaystyle h_{ij}}$ are all fields on ${\displaystyle V}$ and are consequently independent of time. Thus, the geometry of a stationary spacetime does not change in time. In the special case ${\displaystyle {\omega }_i=0}$ the spacetime is said to be static. By definition, every static spacetime is stationary, but the converse is not generally true, as the Kerr metric provides a counterexample.