Stationary spacetime
Encyclopedia
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,
, may be chosen so that they are all independent of the time coordinate. The line element of a stationary spacetime has the form 
where
is the time coordinate, 
are the three spatial coordinates and 
is the metric tensor of 3-dimensional space. In this coordinate system the Killing vector field 
has the components 
. 
is a positive scalar representing the norm of the Killing vector, i.e., 
, and 
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 
(see, for example, , p. 163) which is orthogonal to the Killing vector 
, i.e., satisfies 
. 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
in the spacetime 
. 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) 
, the quotient space. Each point of 
represents a trajectory in the spacetime 
. This identification, called a canonical projection, 
is a mapping that sends each trajectory in 
onto a point in 
and induces a metric 
on 
via pullback. The quantities 
, 
and 
are all fields on 
and are consequently independent of time. Thus, the geometry of a stationary spacetime does not change in time. In the special case 
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
outside the sources, the twist 4-vector 
is curl-free,
and is therefore locally the gradient of a scalar
(called the twist scalar):
Instead of the scalars
and 
it is more convenient to use the two Hansen potentials, the mass and angular momentum potentials, 
and 
, defined as
In general relativity the mass potential
plays the role of the Newtonian gravitational potential. A nontrivial angular momentum potential 
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
(
, 
) and the 3-metric 
. In terms of these quantities the Einstein vacuum field equations can be put in the form
where
, and 
is the Ricci tensor of the spatial metric and 
the corresponding Ricci scalar. These equations form the starting point for investigating exact stationary vacuum metrics.
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
, specifically in the Einstein field equations
Einstein field equations
The Einstein field equations or Einstein's equations are a set of ten equations in Albert Einstein's general theory of relativity which describe the fundamental interaction of gravitation as a result of spacetime being curved by matter and energy...
, a spacetime
Spacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
is said to be stationary if it admits a Killing vector that is asymptotically
Asymptotic curve
In the differential geometry of surfaces, an asymptotic curve is a curve always tangent to an asymptotic direction of the surface . It is sometimes called an asymptotic line, although it need not be a line....
timelike.
In a stationary spacetime, the metric tensor components,
, may be chosen so that they are all independent of the time coordinate. The line element of a stationary spacetime has the form 
where
is the time coordinate, are the three spatial coordinates and is the metric tensor of 3-dimensional space. In this coordinate system the Killing vector field has the components . is a positive scalar representing the norm of the Killing vector, i.e., , and 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 (see, for example, , p. 163) which is orthogonal to the Killing vector , i.e., satisfies . 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
in the spacetime . 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) , the quotient space. Each point of represents a trajectory in the spacetime . This identification, called a canonical projection, is a mapping that sends each trajectory in onto a point in and induces a metric on via pullback. The quantities , and are all fields on and are consequently independent of time. Thus, the geometry of a stationary spacetime does not change in time. In the special case the spacetime is said to be staticStatic spacetime
In general relativity, a spacetime is said to be static if it admits a global, non-vanishing, timelike Killing vector field K which is irrotational, i.e., whose orthogonal distribution is involutive...
. By definition, every static spacetime
Static spacetime
In general relativity, a spacetime is said to be static if it admits a global, non-vanishing, timelike Killing vector field K which is irrotational, i.e., whose orthogonal distribution is involutive...
is stationary, but the converse is not generally true, as the Kerr metric
Kerr metric
The Kerr metric describes the geometry of empty spacetime around an uncharged axially-symmetric black-hole with an event horizon which is topologically a sphere. The Kerr metric is an exact solution of the Einstein field equations of general relativity; these equations are highly non-linear, which...
provides a counterexample.
In a stationary spacetime satisfying the vacuum Einstein equations
outside the sources, the twist 4-vector is curl-free,
and is therefore locally the gradient of a scalar
(called the twist scalar):
Instead of the scalars
and it is more convenient to use the two Hansen potentials, the mass and angular momentum potentials, and , defined as
In general relativity the mass potential
plays the role of the Newtonian gravitational potential. A nontrivial angular momentum potential 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
(, ) and the 3-metric . In terms of these quantities the Einstein vacuum field equations can be put in the form
where
, and is the Ricci tensor of the spatial metric and the corresponding Ricci scalar. These equations form the starting point for investigating exact stationary vacuum metrics.