Principal curvature

Template:Expert-subject In general relativity, specifically in the Einstein field equations, a spacetime is said to be stationary if it admits a Killing vector that is asymptotically timelike.

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

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

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., $\lambda =g_{\mu \nu }{\xi }^{\mu }{\xi }^{\nu }$, 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 ${\omega }_{\mu }=e_{\mu \nu \rho \sigma }{\xi }^{\nu }{\mathrm{\nabla }}^{\rho }{\xi }^{\sigma }$(see, for example, 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. 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, $\pi :M\to V$ 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 $R_{\mu \nu }=0$ outside the sources, the twist 4-vector ${\omega }_{\mu }$ is curl-free,

${\mathrm{\nabla }}_{\mu }{\omega }_{\nu }-{\mathrm{\nabla }}_{\nu }{\omega }_{\mu }=0,$

and is therefore locally the gradient of a scalar $\omega$ (called the twist scalar):

${\omega }_{\mu }={\mathrm{\nabla }}_{\mu }\omega .$

Instead of the scalars $\lambda$ and $\omega$ it is more convenient to use the two Hansen potentials, the mass and angular momentum potentials, ${\mathrm{\Phi }}_M$ and ${\mathrm{\Phi }}_J$, defined as

${\mathrm{\Phi }}_M=\frac{1}{4}{\lambda }^{-1}({\lambda }^2+{\omega }^2-1),$

${\mathrm{\Phi }}_J=\frac{1}{2}{\lambda }^{-1}\omega .$

In general relativity the mass potential ${\mathrm{\Phi }}_M$ plays the role of the Newtonian gravitational potential. A nontrivial angular momentum potential ${\mathrm{\Phi }}_J$ 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 ${\mathrm{\Phi }}_A$ ($A=M$, $J$) and the 3-metric $h_{ij}$. In terms of these quantities the Einstein vacuum field equations can be put in the form

$(h^{ij}{\mathrm{\nabla }}_i{\mathrm{\nabla }}_j-2R^{(3)}){\mathrm{\Phi }}_A=0,$

$R_{ij}^{(3)}=2[{\mathrm{\nabla }}_i{\mathrm{\Phi }}_A{\mathrm{\nabla }}_j{\mathrm{\Phi }}_A-(1+4{\mathrm{\Phi }}^2{)}^{-1}{\mathrm{\nabla }}_i{\mathrm{\Phi }}^2{\mathrm{\nabla }}_j{\mathrm{\Phi }}^2],$

where ${\mathrm{\Phi }}^2={\mathrm{\Phi }}_A{\mathrm{\Phi }}_A=({\mathrm{\Phi }}_M^2+{\mathrm{\Phi }}_J^2)$, and $R_{ij}^{(3)}$ is the Ricci tensor of the spatial metric and $R^{(3)}=h^{ij}{R}_{ij}^{(3)}$ the corresponding Ricci scalar. These equations form the starting point for investigating exact stationary vacuum metrics.

See also

• Axisymmetric spacetime
• Static spacetime