To briefly review Morse theory, we consider a
differentiable function on a
real manifold
. Points where the differential of
vanishes are called critical
points of
. A critical point is called non-degenerate if the Hessian of
is non-degenerate at this point.
is called a Morse
function if all its critical points are non-degenerate. In
applications to variational problems,
is the space of
trial maps,
is the functional to be varied, and the
critical points of
are the solutions to the variational
problem. The non-degeneracy condition guarantees that the character
of each critical point - local minimum, local maximum, or saddle -
is determined by the Hessian of
at this point. The index of
the Hessian is called the Morse index
of the critical point. It is defined as the maximal dimension of a
subspace on which the Hessian is negative definite. At a local
minimum the Morse index is zero, at a local maximum it is equal to
the dimension of
.
Morse theory was first worked out by
Morse [229] for the case that
is finite-dimensional and compact (see
Milnor [224
] for a detailed
exposition). The main result is the following. On a compact
manifold
, for every Morse function the Morse inequalities
Palais and Smale [251, 252]
realized that the Morse inequalities and the Morse relations are
also true for a Morse function
on a non-compact and
possibly infinite-dimensional Hilbert manifold, provided that
is bounded below and satisfies a technical condition
known as Condition C or Palais-Smale condition. In that case, the
and
need not be finite.
The standard application of Morse theory is the
geodesic problem for Riemannian (i.e., positive definite) metrics:
given two points in a Riemannian manifold, to find the geodesics
that join them. In this case is the “energy functional”
(squared-length functional). Varying the energy functional is
related to varying the length functional like Hamilton’s principle
is related to Maupertuis’ principle in classical mechanics. For the
space
one chooses, in the Palais-Smale approach [251], the
-curves between the given two points. (An
-curve is a curve with locally square-integrable
th derivative). This is an infinite-dimensional
Hilbert manifold. It has the same homotopy type (and thus the same
Betti numbers) as the loop space of
the Riemannian manifold. (The loop space of a connected topological
space is the space of all continuous curves joining any two fixed
points.) On this Hilbert manifold, the energy functional is always
bounded from below, and its critical points are exactly the
geodesics between the given end-points. A critical point (geodesic)
is non-degenerate if the two end-points are not conjugate to each
other, and its Morse index is the number of conjugate points in the
interior, counted with multiplicity (“Morse index theorem”). The
Palais-Smale condition is satisfied if the Riemannian manifold is
complete. So one has the following result: Fix any two points in a
complete Riemannian manifold that are not conjugate to each other
along any geodesic. Then the Morse inequalities (59
) and the Morse
relation (60
) are true, with
denoting the number of geodesics with Morse index
between the two points and
denoting the
th Betti number of the loop
space of the Riemannian manifold. The same result is achieved in
the original version of Morse theory [229] (cf. [224]) by choosing
for
the space of broken geodesics between the two given
points, with
break points, and sending
at the end.
Using this standard example of Morse theory as a pattern, one can prove an analogous result for Kovner’s version of Fermat’s principle. The following hypotheses have to be satisfied:
(M1) is a point and
is a timelike curve in a globally hyperbolic
spacetime
.
(M2) does not meet the
caustic of the past light cone of
.
(M3) Every continuous curve from
to
can be continuously deformed into a
past-oriented lightlike curve, with all intermediary curves
starting at
and terminating on
.
The global hyperbolicity assumption in
Statement (M1) is analogous to the completeness
assumption in the Riemannian case. Statement (M2) is the
direct analogue of the non-conjugacy condition in the Riemmanian
case. Statement (M3) is necessary for relating the space
of trial paths (i.e., of past-oriented lightlike curves from to
) to the loop space of the spacetime
manifold or, equivalently, to the loop space of a Cauchy surface.
If Statements (M1), (M2), and (M3) are valid,
the Morse inequalities (59
) and the Morse
relation (60
) are true, with
denoting the number of past-oriented lightlike
geodesics from
to
that have
conjugate points in its interior, counted with
muliplicity, and
denoting the
th Betti number of the loop space of
or, equivalently, of a Cauchy surface. This result
was proven by Uhlenbeck [327
] à la Morse and
Milnor, and by Giannoni and Masiello [136] in an
infinite-dimensional Hilbert manifold setting à la Palais and
Smale. A more general version, applying to spacetime regions with
boundaries, was worked out by Giannoni, Masiello, and
Piccione [137, 138]. In the work
of Giannoni et al., the proofs are given in greater detail than in
the work of Uhlenbeck.
If Statements (M1), (M2), and (M3) are
satisfied, Morse theory gives us the following results about the
number of images of on the sky of
(cf. [220]):
(R1) If is not contractible
to a point, there are infinitely many images. This follows from
Equation (59
) because for the loop
space of a non-contractible space either
is infinite or almost all
are different from zero [304].
(R2) If is contractible to a
point, the total number of images is infinite or odd. This follows
from Equation (60
) because in this case
the loop space of
is contractible to a point, so all
Betti numbers
vanish with the exception of
. As a consequence, Equation (60
) can be written as
, where
is the number of
images with even parity (geodesics with even Morse index) and
is the number of images with odd parity (geodesics
with odd Morse index), hence
.
These results apply, in particular, to the
following situations of physical interest:
Black hole
spacetimes.
Let be the domain of outer communication of
the Kerr spacetime, i.e., the region between the (outer) horizon
and infinity (see Section 5.8). Then the assumption of global
hyperbolicity is satisfied and
is not contractible to a
point. Statement (M3) is satisfied if
is inextendible and approaches neither the horizon
nor (past lightlike) infinity for
. (This
can be checked with the help of an analytical criterion that is
called the “metric growth condition” in [327].) If, in
addition Statement (M2) is satisfied, the reasoning of
Statement (R1) applies. Hence, a Kerr black hole
produces infinitely many images. The same argument can be applied
to black holes with (electric, magnetic, Yang-Mills,
) charge.
Asymptotically simple and
empty spacetimes.
As discussed in Section 3.4, asymptotically simple and empty
spacetimes are globally hyperbolic and contractible to a point.
They can be viewed as models of isolated transparent gravitational
lenses. Statement (M3) is satisfied if is inextendible and bounded away from past lightlike
infinity
. If, in addition, Statement (M2) is
satisfied, Statement (R2) guarantees that the number of
images is infinite or odd. If it were infinite, we had as the limit
curve a past-inextendible lightlike geodesic that would not go out
to
, in contradiction to the definition of asymptotic
simplicity. So the number of images must be finite and odd. The
same odd-number theorem can also be proven with other methods (see
Section 3.4).
In this way Morse theory provides us with precise mathematical versions of the statements “A black hole produces infinitely many images” and “An isolated transparent gravitational lens produces an odd number of images”. When comparing this theoretical result with observations one has to be aware of the fact that some images might be hidden behind the deflecting mass, some might be too faint for being detected, and some might be too close together for being resolved.
In conformally stationary spacetimes, with being an integral curve of the conformal Killing
vector field, a simpler version of Fermat’s principle and Morse
theory can be used (see Section 4.2).