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 . If we denote the projection from to by and choose local coordinates on , the metric takes the formredshift potential, the Fermat metric and the Fermat one-form. The motivation for these names will become clear from the discussion below.
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 formMaupertuis’ principle. It is often convenient to switch to a parametrization-dependent variational principle which, in the terminology of classical mechanics, is called Hamilton’s principle. The Maupertuis principle with action functional (62) corresponds to Hamilton’s principle with a Lagrangian , Sections 304 – 307). The pertaining Euler–Lagrange equations read Fermat two-form Lorentz force in a magnetostatic field. In this analogy, corresponds to the magnetic (vector) potential. In other words, light rays in a conformally stationary spacetime behave like charged particles, with fixed charge-to-mass ratio, in a magnetostatic field on a Riemannian manifold .
Fermat’s principle in static spacetimes dates back to Weyl  (cf. [207, 318]). The stationary case was treated by Pham Mau Quan , who even took an isotropic medium into account, and later, in a more elegant presentation, by Brill . These versions of Fermat’s principle are discussed in several text-books on general relativity (see, e.g., [226, 115, 311] for the static and  for the stationary case). A detailed discussion of the conformally stationary case can be found in . 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  (cf. ). As an alternative to the name “Fermat metric” (used, e.g., in [115, 311, 265]), the names “optical metric” (see, e.g., [140, 79]) and “optical reference geometry” (see, e.g., ) 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 . (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 [138, 139].
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 asanalogue models for light propagation in general-relativistic spacetimes (see ).
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 .
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 that there is a redshift potential for a congruence of timelike curves in a spacetime if and only if the timelike curves are the integral curves of a conformal Killing vector field. The notion of a redshift potential or redshift function is also discussed in . Note that Equation (68) immediately determines the redshift in conformally stationary spacetimes for any pair of observer and emitter. If the 4-velocity of the observer or of the emitter is not parallel to , one just has to add the usual special-relativistic Doppler factor.
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 .
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