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| {{Expert-subject|Physics|date=November 2008}}
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| In [[general relativity]], specifically in the [[Einstein field equations]], a [[spacetime]] is said to be '''stationary''' if it admits a [[Killing vector]] that is [[Asymptotic curve|asymptotically]] [[timelike]].<ref>http://books.google.com/books?id=YA8rxOn9H1sC&pg=PA123&lpg=PA123&dq=Axisymmetric+space+time+definition&source=bl&ots=EZNOQ4S-bR&sig=FhxfAlI3HBt3U1ZsDErzuc4r_jg&hl=en&ei=9Hs8SpyEH9mntget0IAI&sa=X&oi=book_result&ct=result&resnum=8</ref>
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| In a stationary spacetime, the metric tensor components, <math>g_{\mu\nu}</math>, may be chosen so that they are all independent of the time coordinate. The line element of a stationary spacetime has the form <math>(i,j = 1,2,3)</math>
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| : <math> ds^{2} = \lambda (dt - \omega_{i}\, dy^i)^{2} - \lambda^{-1} h_{ij}\, dy^i\,dy^j,</math>
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| where <math>t</math> is the time coordinate, <math>y^{i}</math> are the three spatial coordinates and <math>h_{ij}</math> is the metric tensor of 3-dimensional space. In this coordinate system the Killing vector field <math>\xi^{\mu}</math> has the components <math>\xi^{\mu} = (1,0,0,0)</math>. <math>\lambda</math> is a positive scalar representing the norm of the Killing vector, i.e., <math>\lambda = g_{\mu\nu}\xi^{\mu}\xi^{\nu}</math>, and <math> \omega_{i} </math> 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 <math> \omega_{\mu} = e_{\mu\nu\rho\sigma}\xi^{\nu}\nabla^{\rho}\xi^{\sigma}</math>(see, for example,<ref>Wald, R.M., (1984). General Relativity, (U. Chicago Press)</ref> p. 163) which is orthogonal to the Killing vector <math>\xi^{\mu}</math>, i.e., satisfies <math>\omega_{\mu} \xi^{\mu} = 0</math>. 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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| The coordinate representation described above has an interesting geometrical interpretation.<ref>Geroch, R., (1971). J. Math. Phys. 12, 918</ref> The time translation Killing vector generates a one-parameter group of motion <math>G</math> in the spacetime <math>M</math>. 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) <math>V= M/G</math>, the quotient space. Each point of <math>V</math> represents a trajectory in the spacetime <math>M</math>. This identification, called a canonical projection, <math> \pi : M \rightarrow V </math> is a mapping that sends each trajectory in <math>M</math> onto a point in <math>V</math> and induces a metric <math>h = -\lambda \pi*g</math> on <math>V</math> via pullback. The quantities <math>\lambda</math>, <math> \omega_{i} </math> and <math>h_{ij}</math> are all fields on <math>V</math> and are consequently independent of time. Thus, the geometry of a stationary spacetime does not change in time. In the special case <math> \omega_{i} = 0 </math> the spacetime is said to be [[static spacetime|static]]. By definition, every [[static spacetime]] is stationary, but the converse is not generally true, as the [[Kerr metric]] provides a counterexample.
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| In a stationary spacetime satisfying the vacuum Einstein equations <math>R_{\mu\nu} = 0</math> outside the sources, the twist 4-vector <math>\omega_{\mu}</math> is curl-free,
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| : <math>\nabla_\mu \omega_\nu - \nabla_\nu \omega_\mu = 0,\,</math>
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| and is therefore locally the gradient of a scalar <math>\omega</math> (called the twist scalar):
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| : <math>\omega_\mu = \nabla_\mu \omega.\,</math>
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| Instead of the scalars <math>\lambda</math> and <math>\omega</math> it is more convenient to use the two Hansen potentials, the mass and angular momentum potentials, <math>\Phi_{M}</math> and <math>\Phi_{J}</math>, defined as<ref name= Hansen>Hansen, R.O. (1974). J. Math. Phys. 15, 46.</ref>
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| : <math>\Phi_{M} = \frac{1}{4}\lambda^{-1}(\lambda^{2} + \omega^{2} -1),</math> | |
| : <math>\Phi_{J} = \frac{1}{2}\lambda^{-1}\omega.</math>
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| In general relativity the mass potential <math>\Phi_{M}</math> plays the role of the Newtonian gravitational potential. A nontrivial angular momentum potential <math>\Phi_{J}</math> 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.
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| A stationary vacuum metric is thus expressible in terms of the Hansen potentials <math>\Phi_{A}</math> (<math>A=M</math>, <math>J</math>) and the 3-metric <math>h_{ij}</math>. In terms of these quantities the Einstein vacuum field equations can be put in the form<ref name=Hansen/>
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| : <math>(h^{ij}\nabla_i \nabla_j - 2R^{(3)})\Phi_A = 0,\,</math>
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| : <math>R^{(3)}_{ij} = 2[\nabla_{i}\Phi_{A}\nabla_{j}\Phi_{A} - (1+ 4 \Phi^{2})^{-1}\nabla_{i}\Phi^{2}\nabla_{j}\Phi^{2}], </math>
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| where <math>\Phi^{2} = \Phi_{A}\Phi_{A} = (\Phi_{M}^{2} + \Phi_{J}^{2})</math>, and <math>R^{(3)}_{ij}</math> is the Ricci tensor of the spatial metric and <math>R^{(3)} = h^{ij}R^{(3)}_{ij}</math> the corresponding Ricci scalar. These equations form the starting point for investigating exact stationary vacuum metrics.
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| ==See also==
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| *[[Axisymmetric spacetime]]
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| *[[Static spacetime]]
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| ==References==
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| <references/>
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| [[Category:Lorentzian manifolds]]
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Hello, I'm Penelope, a 26 year old from Selfoss, Iceland.
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