3.3 Fermat’s principle and Morse theory in globally hyperbolic spacetimes

In an arbitrary spacetime, the past-oriented lightlike geodesics from a point pO (observation event) to a timelike curve γS (worldline of light source) are the solutions of a variational principle (Kovner’s version of Fermat’s principle; see Section 2.9). Every solution of this variational principle corresponds to an image on pO’s sky of γS. Determining the number of images is the same as determining the number of solutions to the variational problem. If the variational functional satisfies some technical conditions, the number of solutions to the variational principle can be related to the topology of the space of trial paths. This is the content of Morse theory. In the case at hand, the “technical conditions” turn out to be satisfied in globally hyperbolic spacetimes.

To briefly review Morse theory, we consider a differentiable function F : 𝒳 −→ ℝ on a real manifold 𝒳. Points where the differential of F vanishes are called critical points of F. A critical point is called non-degenerate if the Hessian of F is non-degenerate at this point. F is called a Morse function if all its critical points are non-degenerate. In applications to variational problems, 𝒳 is the space of trial maps, F is the functional to be varied, and the critical points of F 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 F 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 𝒳.

Morse theory was first worked out by Morse [230Jump To The Next Citation Point] for the case that 𝒳 is finite-dimensional and compact (see Milnor [225Jump To The Next Citation Point] for a detailed exposition). The main result is the following. On a compact manifold 𝒳, for every Morse function the Morse inequalities

Nk ≥ Bk, k = 0,1,2,..., (59 )
and the Morse relation
∞ ∞ ∑ k ∑ k (− 1) Nk = (− 1) Bk (60 ) k=0 k=0
hold true. Here Nk denotes the number of critical points with Morse index k and Bk denotes the kth Betti number of 𝒳. Formally, B k is defined for each topological space 𝒳 in terms of the kth singular homology space Hk (𝒳 ) with coefficients in a field 𝔽 (see, e.g., [78], p. 32). (The results of Morse theory hold for any choice of 𝔽.) Geometrically, B0 counts the connected components of 𝒳 and, for k ≥ 1, Bk counts the “holes” in 𝒳 that prevent a k-cycle with coefficients in 𝔽 from being a boundary. In particular, if 𝒳 is contractible to a point, then B = 0 k for k ≥ 1. The right-hand side of Equation (60View Equation) is, by definition, the Euler characteristic of 𝒳. By compactness of 𝒳, all Nk and Bk are finite and in both sums of Equation (60View Equation) only finitely many summands are different from zero.

Palais and Smale [251Jump To The Next Citation Point252] realized that the Morse inequalities and the Morse relations are also true for a Morse function F on a non-compact and possibly infinite-dimensional Hilbert manifold, provided that F is bounded below and satisfies a technical condition known as Condition C or Palais–Smale condition. In that case, the Nk and Bk 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 F 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 𝒳 one chooses, in the Palais–Smale approach [251], the H1-curves between the given two points. (An Hn-curve is a curve with locally square-integrable nth 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 (59View Equation) and the Morse relation (60View Equation) are true, with Nk denoting the number of geodesics with Morse index k between the two points and Bk denoting the kth Betti number of the loop space of the Riemannian manifold. The same result is achieved in the original version of Morse theory [230] (cf. [225]) by choosing for 𝒳 the space of broken geodesics between the two given points, with N break points, and sending N → ∞ 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)  pO is a point and γS is a timelike curve in a globally hyperbolic spacetime (ℳ, g).

  (M2)  γS does not meet the caustic of the past light cone of pO.

  (M3)  Every continuous curve from pO to γS can be continuously deformed into a past-oriented lightlike curve, with all intermediary curves starting at pO and terminating on γS.

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 pO to γS) 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 (59View Equation) and the Morse relation (60View Equation) are true, with N k denoting the number of past-oriented lightlike geodesics from p O to γS that have k conjugate points in its interior, counted with muliplicity, and Bk denoting the kth Betti number of the loop space of ℳ or, equivalently, of a Cauchy surface. This result was proven by Uhlenbeck [326Jump To The Next Citation Point] à la Morse and Milnor, and by Giannoni and Masiello [135] 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 [136137]. 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 γ S on the sky of p O (cf. [221]):

  (R1)  If ℳ is not contractible to a point, there are infinitely many images. This follows from Equation (59View Equation) because for the loop space of a non-contractible space either B0 is infinite or almost all Bk are different from zero [303].

  (R2)  If ℳ is contractible to a point, the total number of images is infinite or odd. This follows from Equation (60View Equation) because in this case the loop space of ℳ is contractible to a point, so all Betti numbers B k vanish with the exception of B = 1 0. As a consequence, Equation (60View Equation) can be written as N+ − N − = 1, where N+ is the number of images with even parity (geodesics with even Morse index) and N − is the number of images with odd parity (geodesics with odd Morse index), hence N+ + N − = 2N− + 1.

These results apply, in particular, to the following situations of physical interest:

Black hole spacetimes.
Let (ℳ, g) 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 ℳ is not contractible to a point. Statement (M3) is satisfied if γ S is inextendible and approaches neither the horizon nor (past lightlike) infinity for t → − ∞. (This can be checked with the help of an analytical criterion that is called the “metric growth condition” in [326].) 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 γ S is inextendible and bounded away from past lightlike infinity ℐ −. 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 ℐ −, 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 γS 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).

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