Stationary spacetime

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, $g_\left\{mu u\right\}$, may be chosen so that they are all independent of the time coordinate. The line element of a stationary spacetime has the form $\left(i,j = 1,2,3\right)$::$ds^\left\{2\right\} = lambda \left(dt - omega_\left\{i\right\} dy^\left\{i\right\}\right)^\left\{2\right\} - lambda^\left\{-1\right\} h_\left\{ij\right\} dy^\left\{i\right\}dy^\left\{j\right\}$,where $t$ is the time coordinate, $y^\left\{i\right\}$ are the three spatial coordinates and $h_\left\{ij\right\}$ is the metric tensor of 3-dimensional space. In this coordinate system the Killing vector field $xi^\left\{mu\right\}$ has the components $xi^\left\{mu\right\} = \left(1,0,0,0\right)$. $lambda$ is a positive scalar representing the norm of the Killing vector, i.e., $lambda = g_\left\{mu u\right\}xi^\left\{mu\right\}xi^\left\{ u\right\}$, and $omega_\left\{i\right\}$ 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 $omega_\left\{mu\right\} = e_\left\{mu u hosigma\right\}xi^\left\{ u\right\} abla^\left\{ ho\right\}xi^\left\{sigma\right\}$(see, for example, [Wald, R.M., (1984). General Relativity, (U. Chicago Press)] , p. 163) which is orthogonal to the Killing vector $xi^\left\{mu\right\}$, i.e., satisfies $omega_\left\{mu\right\} xi^\left\{mu\right\} = 0$. 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 $lambda$, $omega_\left\{i\right\}$ and $h_\left\{ij\right\}$ 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 $omega_\left\{i\right\} = 0$ 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 $R_\left\{mu u\right\} = 0$ outside the sources, the twist 4-vector $omega_\left\{mu\right\}$ is curl-free, ::$abla_\left\{mu\right\}omega_\left\{ u\right\} - abla_\left\{ u\right\}omega_\left\{mu\right\} = 0$,and is therefore locally the gradient of a scalar $omega$ (called the twist scalar):::$omega_\left\{mu\right\} = abla_\left\{mu\right\}omega.$Instead of the scalars $lambda$ and $omega$ it is more convenient to use the two Hansen potentials, the mass and angular momentum potentials, $Phi_\left\{M\right\}$ and $Phi_\left\{J\right\}$, defined asHansen, R.O. (1974). J. Math. Phys. 15, 46.] ::$Phi_\left\{M\right\} = frac\left\{1\right\}\left\{4\right\}lambda^\left\{-1\right\}\left(lambda^\left\{2\right\} + omega^\left\{2\right\} -1\right)$,::$Phi_\left\{J\right\} = frac\left\{1\right\}\left\{2\right\}lambda^\left\{-1\right\}omega.$In general relativity the mass potential $Phi_\left\{M\right\}$ plays the role of the Newtonian gravitational potential. A nontrivial angular momentum potential $Phi_\left\{J\right\}$ 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 $Phi_\left\{A\right\}$ ($A=M$, $J$) and the 3-metric $h_\left\{ij\right\}$. In terms of these quantities the Einstein vacuum field equations can be put in the form::$\left(h^\left\{ij\right\} abla_\left\{i\right\} abla_\left\{j\right\} - 2R^\left\{\left(3\right)\right\}\right)Phi_\left\{A\right\} = 0$,::$R^\left\{\left(3\right)\right\}_\left\{ij\right\} = 2 \left[ abla_\left\{i\right\}Phi_\left\{A\right\} abla_\left\{j\right\}Phi_\left\{A\right\} - \left(1+ 4 Phi^\left\{2\right\}\right)^\left\{-1\right\} abla_\left\{i\right\}Phi^\left\{2\right\} abla_\left\{j\right\}Phi^\left\{2\right\}\right] ,$where $Phi^\left\{2\right\} = Phi_\left\{A\right\}Phi_\left\{A\right\} = \left(Phi_\left\{M\right\}^\left\{2\right\} + Phi_\left\{J\right\}^\left\{2\right\}\right)$, and $R^\left\{\left(3\right)\right\}_\left\{ij\right\}$ is the Ricci tensor of the spatial metric and $R^\left\{\left(3\right)\right\} = h^\left\{ij\right\}R^\left\{\left(3\right)\right\}_\left\{ij\right\}$ the corresponding Ricci scalar. These equations form the starting point for investigating exact stationary vacuum metrics.

References

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