Stationary spacetime

In general relativity, a spacetime is said to be stationary if it admits a global, nowhere zero timelike Killing vector field. In a stationary spacetime, the metric tensor components, $g\_\{mu\; u\}$, may be chosen so that they are all independent of the time coordinate. The line element of a stationary spacetime has the form $(i,j\; =\; 1,2,3)$::$ds^\{2\}\; =\; lambda\; (dt\; -\; omega\_\{i\}\; dy^\{i\})^\{2\}\; -\; lambda^\{-1\}\; h\_\{ij\}\; dy^\{i\}dy^\{j\}$,where $t$ is the time coordinate, $y^\{i\}$ are the three spatial coordinates and $h\_\{ij\}$ is the metric tensor of 3-dimensional space. In this coordinate system the Killing vector field $xi^\{mu\}$ has the components $xi^\{mu\}\; =\; (1,0,0,0)$. $lambda$ is a positive scalar representing the norm of the Killing vector, i.e., $lambda\; =\; g\_\{mu\; u\}xi^\{mu\}xi^\{\; u\}$, and $omega\_\{i\}$ 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 $omega\_\{mu\}\; =\; e\_\{mu\; u\; hosigma\}xi^\{\; u\}\; abla^\{\; ho\}xi^\{sigma\}$(see, for example, [Wald, R.M., (1984). General Relativity, (U. Chicago Press)] , p. 163) which is orthogonal to the Killing vector $xi^\{mu\}$, i.e., satisfies $omega\_\{mu\}\; xi^\{mu\}\; =\; 0$. 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 [Geroch, R., (1971). J. Math. Phys. 12, 918] . The time translation Killing vector generates a one-parameter group of motion $G$ in the spacetime $M$. 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) $V=\; M/G$, the quotient space. Each point of $V$ represents a trajectory in the spacetime $M$. This identification, called a canonical projection, $pi\; :\; M\; ightarrow\; V$ is a mapping that sends each trajectory in $M$ onto a point in $V$ and induces a metric $h\; =\; -lambda\; pi*g$ on $V$ via pullback. The quantities $lambda$, $omega\_\{i\}$ and $h\_\{ij\}$ are all fields on $V$ and are consequently independent of time. Thus, the geometry of a stationary spacetime does not change in time. In the special case $omega\_\{i\}\; =\; 0$ 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.

In a stationary spacetime satisfying the vacuum Einstein equations $R\_\{mu\; u\}\; =\; 0$ outside the sources, the twist 4-vector $omega\_\{mu\}$ is curl-free, ::$abla\_\{mu\}omega\_\{\; u\}\; -\; abla\_\{\; u\}omega\_\{mu\}\; =\; 0$,and is therefore locally the gradient of a scalar $omega$ (called the twist scalar):::$omega\_\{mu\}\; =\; abla\_\{mu\}omega.$Instead of the scalars $lambda$ and $omega$ it is more convenient to use the two Hansen potentials, the mass and angular momentum potentials, $Phi\_\{M\}$ and $Phi\_\{J\}$, defined asHansen, R.O. (1974). J. Math. Phys. 15, 46.] ::$Phi\_\{M\}\; =\; frac\{1\}\{4\}lambda^\{-1\}(lambda^\{2\}\; +\; omega^\{2\}\; -1)$,::$Phi\_\{J\}\; =\; frac\{1\}\{2\}lambda^\{-1\}omega.$In general relativity the mass potential $Phi\_\{M\}$ plays the role of the Newtonian gravitational potential. A nontrivial angular momentum potential $Phi\_\{J\}$ arises for rotating sources due to the rotational kinetic energy which, because of mass-energy equivalence, can also act as the source of a gravitational field. The situation is analogous to a static electromagnetic field where one has two sets of potentials, electric and magnetic. In general relativity, rotating sources produce a gravitomagnetic field which has no Newtonian analog.

A stationary vacuum metric is thus expressible in terms of the Hansen potentials $Phi\_\{A\}$ ($A=M$, $J$) and the 3-metric $h\_\{ij\}$. In terms of these quantities the Einstein vacuum field equations can be put in the form::$(h^\{ij\}\; abla\_\{i\}\; abla\_\{j\}\; -\; 2R^\{(3)\})Phi\_\{A\}\; =\; 0$,::$R^\{(3)\}\_\{ij\}\; =\; 2\; [\; abla\_\{i\}Phi\_\{A\}\; abla\_\{j\}Phi\_\{A\}\; -\; (1+\; 4\; Phi^\{2\})^\{-1\}\; abla\_\{i\}Phi^\{2\}\; abla\_\{j\}Phi^\{2\}]\; ,$where $Phi^\{2\}\; =\; Phi\_\{A\}Phi\_\{A\}\; =\; (Phi\_\{M\}^\{2\}\; +\; Phi\_\{J\}^\{2\})$, and $R^\{(3)\}\_\{ij\}$ is the Ricci tensor of the spatial metric and $R^\{(3)\}\; =\; h^\{ij\}R^\{(3)\}\_\{ij\}$ the corresponding Ricci scalar. These equations form the starting point for investigating exact stationary vacuum metrics.

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