By definition, a spacetime is conformally
stationary if it admits a timelike conformal Killing vector field
. If
is complete and if there are no closed
timelike curves, the spacetime must be a product,
with a (Hausdorff and paracompact) 3-manifold
and
parallel to the
-lines [148]. If we denote
the projection from
to
by
and choose local coordinates
on
, the metric takes the form
If , where
is a function of
, we
can change the time coordinate according to
, thereby transforming
to zero, i.e., making the surfaces
orthogonal to the
-lines. This is the
conformally static case. Also,
Equation (61
) includes the
stationary case (
independent of
) and the static case (
and
independent of
).
In Section 2.9 we have discussed Kovner’s
version of Fermat’s principle which characterizes the lightlike
geodesics between a point (observation event) and a timelike curve (worldline of light source)
. In a conformally stationary spacetime we may
specialize to the case that
is an integral curve of the
conformal Killing vector field, parametrized by the “conformal
time” coordinate
(in the past-pointing sense, to be in
agreement with Section 2.9). Without loss of generality,
we may assume that the observation event
takes place at
. Then for each
trial path (past-oriented lightlike curve)
from
to
the arrival time is equal to
the travel time in terms of the time function
. By Equation (61
) this puts the arrival
time functional into the following coordinate form
Fermat’s principle in static spacetimes dates
back to Weyl [347] (cf. [206, 319]). The stationary
case was treated by Pham Mau Quan [284], who even took an
isotropic medium into account, and later, in a more elegant
presentation, by Brill [42]. These versions
of Fermat’s principle are discussed in several text-books on
general relativity (see, e.g., [225, 116
, 312
] for the static
and [200] for the stationary
case). A detailed discussion of the conformally stationary case can
be found in [265
]. Fermat’s principle
in conformally stationary spacetimes was used as the starting point
for deriving the lens equation of the quasi-Newtonian
apporoximation formalism by Schneider [297]
(cf. [299]). As an
alternative to the name “Fermat metric” (used, e.g., in [116
, 312, 265
]), the names
“optical metric” (see, e.g., [141, 79]) and “optical reference
geometry” (see, e.g., [4
]) are also used.
In the conformally static case, one can apply the
standard Morse theory for Riemannian geodesics to the Fermat metric
to get results on the number of
-geodesics joining two points in space. This
immediately gives results on the number of lightlike geodesics
joining a point in spacetime to an integral curve of
. Completeness of the Fermat metric corresponds to
global hyperbolicity of the spacetime metric. The relevant
techniques, and their generalization to (conformally) stationary
spacetimes, are detailed in a book by Masiello [218
]. (Note that, in
contrast to standard terminology, Masiello’s definition of a
stationary spacetime includes the assumption that the hypersurfaces
are spacelike.) The resulting Morse
theory is a special case of the Morse theory for Fermat’s principle
in globally hyperbolic spacetimes (see Section 3.3). In addition to Morse theory,
other standard methods from Riemannian geometry have been applied
to the Fermat metric, e.g., convexity techniques [139, 140].
If the metric (61) is conformally
static,
, and if the Fermat metric
is conformal to the Euclidean metric,
,
the arrival time functional (62
) can be written as
Extremizing the functional (67) is formally analogous
to Maupertuis’ principle for a particle in a scalar potential on
flat space, which is discussed in any book on classical mechanics.
Dropping the assumption that the Fermat one-form is a differential,
but still requiring the Fermat metric to be conformal to the
Euclidean metric, corresponds to introducing an additional vector
potential. This form of the optical-mechanical analogy, for light
rays in stationary spacetimes whose Fermat metric is conformal to
the Euclidean metric, is discussed, e.g., in [7
].
The conformal factor in
Equation (61
) does not affect the
paths of light rays. However, it does affect redshifts and distance
measures (recall Section 2.4). If
is of the form (61
), for every lightlike
geodesic
the quantity
is a constant
of motion. This leads to a particularly simple form of the general
redshift formula (36
). We consider an
arbitrary lightlike geodesic
in
terms of its coordinate representation
. If both observer and emitter are at rest in the
sense that their 4-velocities
and
are parallel to
,
Equation (36
) can be rewritten as
Conformally stationary spacetimes can be
characterized by another interesting property. Let be a timelike vector field in a spacetime and fix
three observers whose worldlines are integral curves of
. Then the angle under which two of them are seen by
the third one remains constant in the course of time, for any
choice of the observers, if and only if
is proportional to a
conformal Killing vector field. For a proof see [151].