Fermat’s principle for light rays

To set up a variational principle, we have to choose the trial curves among which the solution curves are to be determined and the functional that has to be extremized. Let $ℒpO,γS$ denote the set of all past-pointing lightlike curves from $pO$ to $γS$ . This is the set of trial curves from which the lightlike geodesics are to be singled out by the variational principle. Choose a past-oriented but otherwise arbitrary parametrization for the timelike curve $γS$ and assign to each trial curve the parameter at which it arrives. This gives the arrival time functional $T : ℒpO,γS −→ ℝ$ that is to be extremized. With respect to an appropriate differentiability notion for $T$ , it turns out that the critical points (i.e., the points where the differential of $T$ vanishes) are exactly the geodesics in $ℒ pO,γS$ . This result (or its time-reversed version) can be viewed as a general-relativistic Fermat principle:

Among all ways to move from $pO$ to $γS$ in the past-pointing (or future-pointing) direction at the speed of light, the actual light rays choose those paths that make the arrival time stationary.

This formulation of Fermat’s principle was suggested by Kovner [186

]. The crucial idea is to refer to the arrival time which is given only along the curve $γS$ , and not to some kind of global time which in an arbitrary spacetime does not even exist. The proof that the solution curves of Kovner’s variational principle are, indeed, exactly the lightlike geodesics was given in [264

]. The proof can also be found, with a slight restriction on the spacetime that simplifies matters considerably, in [298

]. An alternative version, based on making $ℒpO,γS$ into a Hilbert manifold, is given in [266

As in ordinary optics, the light rays make the arrival time stationary but not necessarily minimal. A more detailed investigation shows that for a geodesic $λ ∈ ℒp γ O S$ the following holds. (For the notion of conjugate points see Sections 2.2 and 2.7.)

(A1) If along $λ$ there is no point conjugate to $pO$ , $λ$ is a strict local minimum of $T$ .

(A2) If $λ$ passes through a point conjugate to $pO$ before arriving at $γ$ , it is a saddle of $T$ .

(A3) If $λ$ reaches the first point conjugate to $p O$ exactly on its arrival at $γ S$ , it may be a local minimum or a saddle but not a local maximum.

For a proof see [264 ] or [266 ]. The fact that local maxima cannot occur is easily understood from the geometry of the situation: For every trial curve we can find a neighboring trial curve with a larger $T$ by putting “wiggles” into it, preserving the lightlike character of the curve. Also for Fermat’s principle in ordinary optics, the extremum is never a local maximum, as is mentioned, e.g., in Born and Wolf [35 ], p. 137.

The advantage of Kovner’s version of Fermat’s principle is that it works in an arbitrary spacetime. In particular, the spacetime need not be stationary and the light source may arbitrarily move around (at subluminal velocity, of course). This allows applications to dynamical situations, e.g., to lensing by gravitational waves (see Section 5.11). If the spacetime is stationary or conformally stationary, and if the light source is at rest, a purely spatial reformulation of Fermat’s principle is possible. This more specific version of Femat’s principle is known since decades and has found various applications to lensing (see Section 4.2). A more sophisticated application of Fermat’s principle to lensing theory is to put up a Morse theory in order to prove theorems on the possible number of images. In its strongest version, this approach has to presuppose a globally hyperbolic spacetime and will be reviewed in Section 3.3.

For a generalization of Kovner’s version of Fermat’s principle to the case that observer and light source have a spatial extension (see [272 ]).

An alternative variational principle was introduced by Frittelli and Newman [122

] and evaluated in [123

, 121

]. While Kovner’s principle, like the classical Fermat principle, is a varional principle for rays, the Frittelli–Newman principle is a variational principle for wave fronts. (For the definition of wave fronts see Section 2.2.) Although Frittelli and Newman call their variational principle a version of Fermat’s principle, it is actually closer to the classical Huygens principle than to the classical Fermat principle. Again, one fixes $pO$ and $γS$ as above. To define the trial maps, one chooses a set $𝒲 (pO)$ of wave fronts, such that for each lightlike geodesic through $pO$ there is exactly one wave front in $𝒲 (pO)$ that contains this geodesic. Hence, $𝒲 (pO )$ is in one-to-one correspondence to the lightlike directions at $pO$ and thus to the 2-sphere. Now let $𝒲 (pO, γS)$ denote the set of all wave fronts in $𝒲 (pO )$ that meet $γS$ . We can then define the arrival time functional $T : 𝒲 (pO, γS) −→ ℝ$ by assigning to each wave front the parameter value at which it intersects $γS$ . There are some cases to be excluded to make sure that $T$ is defined on an open subset of $𝒲 (pO) ≃ S2$ , single-valued and differentiable. If this is the case, one finds that $T$ is stationary at $W ∈ 𝒲 (pO)$ if and only if $W$ contains a lightlike geodesic from $pO$ to $γS$ . Thus, to each image of $γS$ on the sky of $pO$ there corresponds a critical point of $T$ . The great technical advantage of the Frittelli–Newman principle over the Kovner principle is that $T$ is defined on a finite dimensional manifold, directly to be identified with (part of) the observer’s celestial sphere. The arrival time $T$ in the Frittelli–Newman approach is directly analogous to the “Fermat potential” in the quasi-Newtonian formalism which is discussed, e.g., in [298

]. In view of applications, a crucial point is that the space $𝒲 (pO)$ is a matter of choice; there are many wave fronts which have one light ray in common. There is a natural choice, e.g., in asymptotically simple spacetimes (see Section 3.4).

Frittelli, Newman, and collaborators have used their variational principle in combination with the exact lens map (recall Section 2.1) to discuss thick and thin lens models from a spacetime perspective [123 , 121 ]. Methods from differential topology or global analysis, e.g., Morse theory, have not yet been applied to the Frittelli–Newman principle.

2.9 Fermat’s principle for light rays