In
general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics. G ...
, specifically in the
Einstein field equations
In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it.
The equations were published by Einstein in 1915 in the form ...
, a
spacetime
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
is said to be stationary if it admits a
Killing vector
In mathematics, a Killing vector field (often called a Killing field), named after Wilhelm Killing, is a vector field on a Riemannian manifold (or pseudo-Riemannian manifold) that preserves the metric. Killing fields are the infinitesimal gener ...
that is
asymptotically
In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related contexts, ...
timelike
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
.
Description and analysis
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.
[Geroch, R., (1971). J. Math. Phys. 12, 918] The
time translation
Time translation symmetry or temporal translation symmetry (TTS) is a mathematical transformation in physics that moves the times of events through a common interval. Time translation symmetry is the law that the laws of physics are unchanged ( ...
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
Static may refer to:
Places
*Static Nunatak, a nunatak in Antarctica
United States
* Static, Kentucky and Tennessee
* Static Peak, a mountain in Wyoming
** Static Peak Divide, a mountain pass near the peak
Science and technology Physics
*Static ...
. By definition, every
static spacetime
In general relativity, a spacetime is said to be static if it does not change over time and is also irrotational. It is a special case of a stationary spacetime, which is the geometry of a stationary spacetime that does not change in time but ca ...
is stationary, but the converse is not generally true, as the
Kerr metric
The Kerr metric or Kerr geometry describes the geometry of empty spacetime around a rotating uncharged axially symmetric black hole with a quasispherical event horizon. The Kerr metric tensor, metric is an Exact solutions in general relativity, e ...
provides a counterexample.
Use as starting point for vacuum field equations
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
[Hansen, R.O. (1974). J. Math. Phys. 15, 46.]
:
:
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
Gravitoelectromagnetism, abbreviated GEM, refers to a set of formal analogies between the equations for electromagnetism and relativistic gravitation; specifically: between Maxwell's field equations and an approximation, valid under certain ...
that 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.
]
See also
* Static spacetime
In general relativity, a spacetime is said to be static if it does not change over time and is also irrotational. It is a special case of a stationary spacetime, which is the geometry of a stationary spacetime that does not change in time but ca ...
* Spherically symmetric spacetime
In physics, spherically symmetric spacetimes are commonly used to obtain analytic and numerical solutions to Einstein's field equations in the presence of radially moving matter or energy. Because spherically symmetric spacetimes are by definition ...
References
Lorentzian manifolds