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 denote the set of all past-pointing lightlike curves
from
to
. 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
and assign to each trial curve the parameter at
which it arrives. This gives the arrival time
functional
that is to be extremized. With respect to an appropriate
differentiability notion for
, it turns out that the
critical points (i.e., the points where the differential of
vanishes) are exactly the geodesics in
. This result (or its time-reversed version) can be
viewed as a general-relativistic Fermat principle:
Among all ways to move from to
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 [187]. The crucial idea
is to refer to the arrival time which is given only along the curve
, 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 [299
]. An alternative
version, based on making
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 the following holds. (For the notion of conjugate
points see Sections 2.2 and 2.7.)
(A1) If along
there is no point conjugate to
,
is a strict local
minimum of
.
(A2) If passes through a
point conjugate to
before arriving at
, it is a saddle of
.
(A3) If reaches the first
point conjugate to
exactly on its arrival at
, 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 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 [123] and evaluated
in [124
, 122
]. 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
and
as above. To define the trial maps, one
chooses a set
of wave fronts, such that for each
lightlike geodesic through
there is exactly one wave
front in
that contains this geodesic. Hence,
is in one-to-one correspondence to the lightlike
directions at
and thus to the 2-sphere. Now let
denote the set of all wave fronts in
that meet
. We can then define the
arrival time functional
by assigning to each wave front the parameter value
at which it intersects
. There are some cases to be
excluded to make sure that
is defined on an open subset
of
, single-valued and differentiable. If
this is the case, one finds that
is stationary at
if and only if
contains a lightlike
geodesic from
to
. Thus, to each image of
on the sky of
there corresponds a critical
point of
. The great technical advantage of the
Frittelli-Newman principle over the Kovner principle is that
is defined on a finite
dimensional manifold, directly to be identified with (part
of) the observer’s celestial sphere. The arrival time
in the Frittelli-Newman approach is directly
analogous to the “Fermat potential” in the quasi-Newtonian
formalism which is discussed, e.g., in [299
]. In view of
applications, a crucial point is that the space
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 [124, 122]. Methods from differential topology or global analysis, e.g., Morse theory, have not yet been applied to the Frittelli-Newman principle.