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
μ
ν
, 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
=
λ
(
d
t
−
ω
i
d
y
i
)
2
−
λ
−
1
h
i
j
d
y
i
d
y
j
,
where
t
is the time coordinate,
y
i
are the three spatial coordinates and
h
i
j
is the metric tensor of 3-dimensional space. In this coordinate system the Killing vector field
ξ
μ
has the components
ξ
μ
=
(
1
,
0
,
0
,
0
)
.
λ
is a positive scalar representing the norm of the Killing vector, i.e.,
λ
=
g
μ
ν
ξ
μ
ξ
ν
, and
ω
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
ω
μ
=
e
μ
ν
ρ
σ
ξ
ν
∇
ρ
ξ
σ
(see, for example, p. 163) which is orthogonal to the Killing vector
ξ
μ
, i.e., satisfies
ω
μ
ξ
μ
=
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,
π
:
M
→
V
is a mapping that sends each trajectory in
M
onto a point in
V
and induces a metric
h
=
−
λ
π
∗
g
on
V
via pullback. The quantities
λ
,
ω
i
and
h
i
j
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
ω
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
μ
ν
=
0
outside the sources, the twist 4-vector
ω
μ
is curl-free,
∇
μ
ω
ν
−
∇
ν
ω
μ
=
0
,
and is therefore locally the gradient of a scalar
ω
(called the twist scalar):
ω
μ
=
∇
μ
ω
.
Instead of the scalars
λ
and
ω
it is more convenient to use the two Hansen potentials, the mass and angular momentum potentials,
Φ
M
and
Φ
J
, defined as
Φ
M
=
1
4
λ
−
1
(
λ
2
+
ω
2
−
1
)
,
Φ
J
=
1
2
λ
−
1
ω
.
In general relativity the mass potential
Φ
M
plays the role of the Newtonian gravitational potential. A nontrivial angular momentum potential
Φ
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
Φ
A
(
A
=
M
,
J
) and the 3-metric
h
i
j
. In terms of these quantities the Einstein vacuum field equations can be put in the form
(
h
i
j
∇
i
∇
j
−
2
R
(
3
)
)
Φ
A
=
0
,
R
i
j
(
3
)
=
2
[
∇
i
Φ
A
∇
j
Φ
A
−
(
1
+
4
Φ
2
)
−
1
∇
i
Φ
2
∇
j
Φ
2
]
,
where
Φ
2
=
Φ
A
Φ
A
=
(
Φ
M
2
+
Φ
J
2
)
, and
R
i
j
(
3
)
is the Ricci tensor of the spatial metric and
R
(
3
)
=
h
i
j
R
i
j
(
3
)
the corresponding Ricci scalar. These equations form the starting point for investigating exact stationary vacuum metrics.
