Sistema Regional de Derechos Humanos: Belem do Pará

It is often advantageous to characterize light rays by a variational principle, rather than by a differential equation. This is particularly true in view of applications to lensing. If we have chosen a pointpO (observation event) and a timelike curveγS(worldline of light source) in spacetimeM, we want to determine all past-pointing lightlike geodesics from pO to γS. When working with a differential equation for light rays, we have to calculateall light rays issuing frompOinto the past, and to see which of them meetγS. If we work with a variational principle, we can restrict to curves frompO toγSat the outset.

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 frompO 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 thearrival time functional T :LpO,γS−→Rthat is to be extremized. With respect to an appropriate differentiability notion forT, it turns out that the critical points (i.e., the points where the differential ofT vanishes) are exactly the geodesics inLpO,γS. This result (or its time-reversed version) can be viewed as a general-relativistic Fermat principle:

Among all ways to move frompO toγSin the past-pointing (or future-pointing) direc- tion 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 in 1990 by Kovner [240], a local version (restricted to a convex normal neighborhood) can be found already in a 1938 paper by Temple [401]. The crucial idea is to refer to the arrival time which is given only along the curveγS, 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 [332]. The proof can also be found, with a slight restriction on the spacetime that simplifies matters considerably, in [367]. An alternative version, based on making LpO,γS into a Hilbert manifold, is given in [334].

As in ordinary optics, the light rays make the arrival time stationary but not necessarily mini- mal. A more detailed investigation shows that for a geodesicλ∈ LpOγS the following holds. (For the notion of conjugate points see Sections 2.2 and 2.7.)

(A1) If alongλthere is no point conjugate to pO,λis a strict local minimum ofT. (A2) Ifλpasses through a point conjugate topO before arriving atγ, it is a saddle ofT. (A3) If λreaches the first point conjugate topO exactly on its arrival atγS, it may be a local

minimum or a saddle but not a local maximum.

For a proof see [332] or [334]. 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, where light propagation is characterized by a positive index of refraction on Euclidean 3-space, the extremum is never a local maximum, as is mentioned, e.g., in Born and Wolf [46], p. 137. Note, however, that in the quasi-Newtonian lensing approximation with one or more deflector planes, where only broken straight lines are allowed as trial paths, local maximado occur, see, e.g., [343]. Also, in the general formalism of ray optics, where the rays are the solutions of Hamilton’s equations with an unspecified Hamiltonian, local maxima do occur, unless a certain regularity condition is imposed on the Hamiltonian [337].

The advantage of Kovner’s version of Fermat’s principle is that it works in anarbitrary space- time. 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 [340].

An alternative variational principle was introduced by Frittelli and Newman [154] and evaluated in [155, 153]. 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 atpOand thus to the 2-sphere. Now letW(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 thatT is stationary at

W _{∈ W}(pO) if and only ifW contains a lightlike geodesic frompO to γS. Thus, to each image of

γSon the sky ofpO there corresponds a critical point ofT. 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 [367]. 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 [155, 153]. Methods from differential topology or global analysis, e.g., Morse theory, have not yet been applied to the Frittelli–Newman principle.