A HOMOLOGICAL INVARIANT IN THE RELATIVISTIC RAYS OPTICS

PAOLO PICCIONE AND DANIEL V. TAUSK

Abstract. The optics of light rays in a general-relativistic medium can be described using a Hamiltonian formalism. In this note we discuss some mathematical issues related to the notion of Maslov index for a solution of a Hamiltonian problem, aiming at applications to the study of the gravitational lensing effect.

A few decades ago a new integer valued homological invariant, called the Maslov index, was introduced by the Russian school (see for instance [1] and the references therein) for closed curves in a Lagrangian submanifold M of the space IR^{2n} endowed with its canonical symplectic structure. The notion of Maslov index has been immediately recognized as an important tool in the study of conjugate points, and it has has been thoroughly investigated and extended in several directions by mathematical-physicists, geometers and analysts. There is nowadays a very extensive literature about the subject, and, given the concise nature of this note, it is almost impossible to acknowledge the work of all the many authors who have given significant contributions to the field. For simplicity, we will cite here merely the papers that are more closely related to the brief treatment presented in this paper even though our list of references is far from being exhaustive.

The optics of light rays in a general-relativistic medium, like vacuum, dust, or plasma, magnetized or not, etc., can be described using a Hamiltonian formalism. As the underlying spacetime model one assumes an arbitrary 4dimensional Lorentzian manifold (M, g), and the trajectories of light rays are projections onto M of solutions in the cotangent bundle T M^{∗} of some Hamiltonian function H : T M^{∗} → IR. Typically, the explicit form of the function H involves the spacetime metric and a number of tensor fields by which the medium is characterized. For a comprehensive discussion of this subject we refer to [8]. Associated to each such solution in T M^{∗} (the so-called “lifted light rays”) one can associate an integer number, the Maslov index, which is computed as an intersection number in the Grassmannian of all Lagrangian subspaces of a symplectic space; this homological invariant carries many important geometrical information about the solution in consideration. Roughly speaking, the Maslov index gives a sort of algebraic count of the conjugate points along a solution; here are some of the main properties of this invariant:

• it is always finite (even when the number of conjugate points is infinite);

• it is stable by “small” perturbations of the data;

• it coincides with the geometric index in the case of a causal (timelike or lightlike) Lorentzian geodesic;

• it is related to the analytic index (or, more in general, to the relative index) of the solution, which is the index of the second variation of an associated Lagrangian action functional;

• it is related to the spectral properties of the associated Hamiltonian second order differential operator. The relation between the Maslov index and the analytic index of the solutions is expressed by some generalizations of the classical Morse Index Theorem in Riemannian geometry, that will be touched very briefly in this note; we will not discuss at all the relation between the Maslov index and the so-called spectral index of the Hamiltonian differential operator. A quite extensive treatment of these subjects is found in references [6, 7, 10, 12]. An application of the above properties is presented in [4], where the authors obtain some Morse relations for light rays in vacuum between a pointlike source and a pointlike observer using a limit process in the Morse relations for timelike geodesics.

An excellent reference with the basic properties of the Maslov index for Hamiltonian systems is [3]. In this paper we present an abstract mathematical discussion of the definition and some properties of the Maslov index associated to the solution of a general time-dependent Hamiltonian function on a symplectic manifold; we will use as basic references the articles [7, 10] where the reader will find all the details of the theory exposed.

A natural setup for the definition of the Maslov index is obtained by considering the following objects:

2000 Mathematics Subject Classification. 37J05, 53C22, 53C50, 53D12, 70H05.

The first author is partially sponsored by CNPq, the second author is sponsored by FAPESP.

The participation of the first author to the 9th Marcel Grossmann Meeting was sponsored by CNPq (Brazil), Processo n. 450805/00-0.

• a symplectic manifold (M, ω);

• a smooth (time dependent) Hamiltonian function H : U → IR defined on an open subset U ⊂ IR × M;

• a Lagrangian submanifold P of M;

• a smooth distribution 𝔏 in M of Lagrangian subspaces;

• a smooth curve Γ : [a, b] → M which is a solution of the Hamiltonian H, i.e., an integral curve of the

Hamiltonian vector field H⃗, with Γ(a) ∈ P. We will call a sextuplet (M, ω, H, 𝔏, P, Γ) as above a set of data for the Hamiltonian problem. The basic example to keep in mind of a Hamiltonian problem is the case in which M is the cotangent bundle T M^{∗} of a smooth manifold M, ω is the canonical 2-form of T M^{∗}, P is the annihilator T P^{o} of some smooth submanifold P ⊂ M and 𝔏 is the distribution of vertical subspaces on T M^{∗}, i.e., the subspaces tangent to the fibers of T M^{∗}.

Let F denote the maximal flow of the vector field H⃗, i.e., F is a smooth function defined in an open subset of IR×IR×M taking values in M and such that, for all m ∈ M and all t_{0} ∈ IR, the curve t ↦ F (t, t_{0}, m) is a maximal integral curve of H⃗ with F (t_{0}, t_{0}, m) = m; by F_{t,t_{0}} we will mean the map m ↦ F (t, t_{0}, m). Using a symplectic chart (q, p) in M, the curve Γ(t) = (q(t), p(t)) is a solution of the Hamilton equations \frac{dq}{dt} = \frac{∂H}{∂p} and \frac{dp}{dt} = −\frac{∂H}{∂q}; we will consider the linearized Hamilton equation along the solution Γ, given by:

(1)  ρ(t) = dF_{t,a}(Γ(a))ρ(a),

that can be written in an obvious way using the second derivatives of H with respect to the symplectic coordinates (q, p). A solution ρ of (1) will be called a P-solution if in addition it satisfies ρ(a) ∈ T_{Γ(a)}P. A point Γ(t), t ∈ ]a, b], is said to be a P-focal point along Γ if there exists a non-zero P-solution of (1) such that ρ(t) ∈ 𝔏_{Γ(t)}.

We consider the vector space IR^{n} ⊕ IR^{n}^{∗} (here IR^{n}^{∗} denotes the dual space of IR^{n}) endowed with its canonical symplectic form ω_{0}((v, α), (w, β)) = β(v) − α(w); a symplectic 𝔏-trivialization of T M along Γ is a smooth family of symplectomorphisms ϕ(t) : IR^{n} ⊕ IR^{n}^{∗} → T_{Γ(t)}M such that, for all t ∈ [a, b], ϕ(t)(L_{0}) = 𝔏_{Γ(t)}, where L_{0} = {0} ⊕ IR^{n}^{∗}. The existence of symplectic 𝔏-trivializations of T M along Γ is easily proven. Let Sp(2n, IR) denote the symplectic group of the symplectic space (IR^{n} ⊕ IR^{n}^{∗}, ω_{0}), and let sp(2n, IR) be its Lie algebra. Recall that elements in Sp(2n, IR) are linear automorphisms of IR^{n} ⊕ IR^{n}^{∗} that preserve the canonical symplectic form ω_{0}, and elements X ∈ sp(2n, IR), called also infinitesimally symplectic maps, are 2n × 2n real matrices of the form^{1}:

(2)  X = ( \begin{matrix}A & B \\ C & −A^{∗} \end{matrix}) ,

where A, B, C are real n × n matrices with B and C symmetric. Whenever an infinitesimally symplectic matrix X is given, we will call A, B, C its n × n blocks as in (2).

An 𝔏-symplectic trivialization along Γ produces:

• a trivialization of the quotient bundle T M/𝔏 along Γ given by a family of isomorphisms Z_{t} : IR^{n} → T_{Γ(t)}M/𝔏_{Γ(t)}, t ∈ [a, b], where Z_{t}(x) = ϕ(t)(x, 0) + 𝔏_{Γ(t)};

• a smooth curve Ψ in Sp(2, IR) given by Ψ(t) = ϕ(t)^{−1} ∘ dF_{t,a}(Γ(a)) ∘ ϕ(a), t ∈ [a, b];

• a smooth curve X in sp(2n, IR) given by X(t) = Ψ′(t)Ψ(t)^{−1};

• a Lagrangian subspace ℓ_{0} of IR^{n} ⊕IR^{n}^{∗} corresponding to the submanifold P, given by ℓ_{0} = ϕ(a)^{−1}(T_{Γ(a)}P);

• a family H_{𝔏}(t) of symmetric bilinear forms on (T_{Γ(t)}M/𝔏_{Γ(t)})^{∗} ≃ 𝔏_{Γ(t)}^{o} ≃ 𝔏_{Γ(t)} given by H_{𝔏}(t) = Z_{t} ∘ B(t) ∘ Z_{t}^{∗} (here 𝔏_{Γ(t)}^{o} denotes the annihilator of 𝔏_{Γ(t)} in T M^{∗}).

We remark that, indeed, the bilinear forms H_{𝔏}(t) do not depend on the choice of the symplectic 𝔏-trivialization of T M; we will consider the case of nondegenerate sets of data for the Hamiltonian problem, i.e., those sextuplets (M, ω, H, 𝔏, P, Γ) such that H_{𝔏}(t) is nondegenerate for all t ∈ [a, b] and such that the restriction of H_{𝔏}(a) to the annihilator (T_{Γ(a)}P + 𝔏_{a})^{o} ⊂ 𝔏_{a}^{o} ⊂ T_{Γ(a)}M^{∗} is also nondegenerate.

For all t ∈ [a, b] define

𝔙[t] = {ρ(t) : ρ is a P-solution of (1)} ∩ 𝔏_{Γ(t)}

It is easy to see that 𝔙[a] is identified with the annihilator (T_{Γ(a)}P + 𝔏_{a})^{o}. A point Γ(t) is P-focal if and only if 𝔙[t] is not zero, and the dimension of 𝔙[t] is the multiplicity of Γ(t). If Γ(t) is a P-focal point along Γ, the signature sgn(Γ(t)) is defined to be the signature of the restriction of H_{𝔏}(t) to 𝔙[t] ⊂ 𝔏_{t} ≃ 𝔏_{t}^{o}. Γ(t) is said to be a nondegenerate P-focal point if such restriction is nondegenerate. If Γ has only a finite number of P-focal points, we define the focal index i_{foc}(Γ) as:

1here A^{∗} means the transpose of A

(3)  i_{foc}(Γ) = ∑_{t∈]a,b[} sgn(Γ(t)).

Let H_{Γ} denote the Hilbert space of sections 𝔳 of the quotient bundle T M/𝔏 along Γ satisfying the Sobolev H^{1}- regularity and such that 𝔳(a) ∈ P_{0} = π_{𝔏}(T_{Γ(a)}P) and 𝔳(b) = 0 (here π_{𝔏} : T M → T M/𝔏 denotes the quotient map). We introduce a bounded symmetric bilinear form I_{Γ} on H_{Γ} defined by:

(4)  I_{Γ}(𝔳, 𝔴) = ∫_{a}^{b} [B^{−1}(v′ − Av, w′ − Aw) + C(v, w)] dt − S(v(a), w(a)),

where v = Z^{−1}𝔳, w = Z^{−1}𝔴, A, B, C are the n × n blocks of X as in (2), and S : P × P → IR is a symmetric bilinear form on the subspace P ⊂ IR^{n} such that (P, S) is the unique pair for which ℓ_{0} = {(v, α) ∈ IR^{n} ⊕ IR^{n}^{∗} : α|_{P} + S(v, ·) = 0}.

In the basic example M = T M^{∗}, the Hamiltonian function is called hyper-regular if its fiber derivative 𝔽H : U ⊂ IR × T M^{∗} ↦ IR × T M is a (global) diffeomorphism onto an open subset V ⊂ IR × T M, where 𝔽H(t, p) = (t, d(H|_{T_{({t}×π(p)}M^{∗})∩U}(p)) and π : T M^{∗} → M is the canonical projection. If H is hyper-regular, then it defines a

(time-dependent) Lagrangian function L : V ⊂ IR × T M → IR and an action functional ϑ ↦ ∫_{a}^{b} L(t, ϑ′(t)) dt in the set of curves ϑ : [a, b] → M of class C^{1} such that (t, ϑ′(t)) ∈ V for all t. It is well known that the critical points of this action functional in the set of curves ϑ : [a, b] → M are projections γ = π ∘ Γ on M of the integral curves Γ of the Hamiltonian vector field H⃗. In this situation, the bilinear form I_{Γ} corresponds to the second variation of the Lagrangian action functional at the critical point γ = π ∘ Γ.

Now, we consider the symplectic differential system:

(5)  \{ \begin{matrix}v′ = Av + Bα, \\ α′ = Cv − A^{∗}α, \end{matrix} v ∈ IR^{n}, α ∈ IR^{n}^{∗}, t ∈ [a, b];

with initial conditions (v(a), α(a)) ∈ ℓ_{0} and we denote by Φ : [a, b] → GL(2n, IR) its fundamental matrix, defined by:

Ψ(t)(v(a), α(a)) = (v(t), α(t))

for any solution (v, α) of (5). Since X is in the Lie algebra sp(2n, IR), then Ψ takes values in the symplectic group Sp(2n, IR), and so ℓ(t) = Ψ(t)(ℓ_{0}) is a smooth curve in the Lagrangian Grassmannian Λ of the symplectic space IR^{n} ⊕ IR^{n}^{∗}. Conjugate points along Γ correspond to the intersection of ℓ with the subvariety Λ_{≥1}(L_{0}) of Λ, which is the subset of Λ consisting of those Lagrangians that are not transversal to the Lagrangian L_{0} = {0}^{n} ⊕ IR^{n}^{∗}. The set Λ is a compact, connected real analytic embedded \frac{1}{2}n(n + 1)-dimensional submanifold of the Grassmannian of all n-dimensional subspaces of IR^{n} ⊕ IR^{n}^{∗}; Λ_{≥1}(L_{0}) is not a submanifold of Λ, it is an analytic subset whose regular part is Λ_{1}(L_{0}) = {L ∈ Λ : dim(L ∩ L_{0}) = 1}. Even though Λ and Λ_{≥1}(L_{0}) may fail to be orientable (depending on the parity of n), Λ_{1}(L_{0}) always possesses a natural transverse orientation in Λ; the first relative homology group H_{1}(Λ, Λ \ Λ_{≥1}(L_{0})) is isomorphic to ℤ, and the choice of such isomorphism is related to the choice of a transverse orientation of Λ_{≥1}(L_{0}) in Λ. The curve ℓ in Λ does not in general define a homology class in H_{1}(Λ, Λ \ Λ_{≥1}(L_{0})) because ℓ(a) = ℓ_{0} may fail to be transverse to L_{0} (this happens when the subspace P does not coincide with IR^{n}); also the final endpoint γ(b) may fail to be transverse to L_{0}, and this happens precisely when Γ(b) is a P-focal point along Γ. However, due to the nondegeneracy of H_{𝔏}(a) on (T_{Γ(a)}P + 𝔏_{a})^{o}, ℓ(a) is an isolated intersection of ℓ(t) with Λ_{≥1}(L_{0}), i.e., there exists ε > 0 such that there are no P-focal points along Γ|_{]a,a+ε]} .

1. Definition. Suppose that Γ(b) is not a P-focal point along Γ. The Maslov index i_{Maslov}(Γ) of Γ is the integer number corresponding to the homology class of ℓ|_{[a+ε,b]} in H_{1}(Λ, Λ \ Λ_{≥1}(L_{0})).

The definition of Maslov index can be extended to those solutions Γ for which Γ(b) is a nondegenerate (hence, isolated) P-focal point by considering the homology class of the restriction ℓ|_{[a+ε,b−ε]} for ε > 0 sufficiently small. There is a natural notion of equivalence in the category of symplectic differential systems (5) with initial conditions (v(a), α(a)) ∈ ℓ_{0} (see [10, Section 2.11]), which roughly speaking consists in declaring isomorphic two systems that can be transformed into each other by a symplectic change of coordinates that preserve the subspace L_{0}. Given isomorphic systems, then there is a bijection in the set of curves (v, α) : [a, b] → IR^{n} ⊕ IR^{n}^{∗} that sends the solutions of one into solutions of the other system in a one-to-one way; moreover, such bijection preserves the index forms, conjugate points, their multiplicity and signature, and Maslov index. In the next proposition we collect some results concerning Hamiltonian problems, conjugate points along their solutions, and the Maslov index:

2. Theorem. Let (M, ω, H, 𝔏, P, Γ) be a nondegenerate set of data for the Hamiltonian problem.

(1) Different 𝔏-symplectic trivializations of T M along Γ produce isomorphic symplectic differential problems, hence the notions of index form I_{Γ} and of Maslov index i_{Maslov}(Γ) are independent of the trivialization.

(2) A nondegenerate P-focal point along Γ is isolated; in particular, if all the P-focal points along Γ are nondegenerate, they form a finite set.

(3) If all the P-focal points along Γ are nondegenerate, then i_{foc}(Γ) = i_{Maslov}(Γ).

(4) The index n_{−}(I_{Γ}) of the bilinear form I_{Γ} in the Hilbert space H_{Γ} (i.e., the dimension of a maximal subspace of H_{Γ} on which I_{Γ} is negative definite) is finite if and only if the symmetric bilinear form B(t) is positive definite for all t.

(5) If M = T M^{∗}, where (M, g) is a Riemannian or a Lorentzian manifold, ω is the canonical symplectic form of T M^{∗}, 𝔏 is the distribution of vertical subspaces of T T M^{∗}, P is the annihilator T P^{o} of some smooth (spacelike, in the Lorentzian case) submanifold of M, H(q, p) = \frac{1}{2}g^{−1}(p, p), and Γ is the lift (γ, g(γ̇, ·)) of a (causal, in the Lorentzian case) geodesic in M, then i_{Maslov}(Γ) = i_{foc}(Γ) = n_{−}(I_{Γ}) and these numbers coincide with the number of P -focal points along γ counted with multiplicity.

(6) If Γ(b) is not a P-focal point, then the Maslov index i_{Maslov}(Γ) is stable by “small” perturbations of the sextuplet (M, ω, H, 𝔏, P, Γ). In particular, if Γ_{n} : [a, b] → M is a sequence of solutions of the Hamiltonian problem that converges to a solution Γ_{∞} such that Γ_{∞}(b) is not P-focal, then i_{Maslov}(Γ_{n}) = i_{Maslov}(Γ_{∞}) for n sufficiently large.

Proof. Statement (1) is proven in [10, Subsection 3.1]. Statements (2) and (3) are proven in [10, Theorem 2.3.3] ( see [7, Subsection 7.4] for a counterexample to the equality i_{foc}(Γ) = i_{Maslov}(Γ) in the case of degenerate focal points). Statement (4) is proven in [10, Proposition 2.5.2]. The equality i_{Maslov}(Γ) = i_{foc}(Γ) in (5) is proven in [7, Corollary 2.5.2], the equality i_{foc}(Γ) = n_{−}(I_{Γ}) can then be deduced from the Morse Index Theorem for Riemannian geodesics or for causal Lorentzian geodesics of Beem and Ehrlich (see [2], or [11]); a direct proof is given in [7, Subsection 6.2]. Statement (6) is proven in [7, Subsection 5.2], where it is discussed in what sense the perturbations have to be taken “small” in order to have stability of the Maslov index. □

References

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[8] V. Perlick, Ray Optics, Fermat’s Principle and Applications to General Relativity, Spinger Lecture Notes in Physics, Series m.

[9] V. Perlick, P. Piccione, D. V. Tausk, A Variational Principle for Light Rays in a General Relativistic Plasma, preprint 1999.

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[12] P. Piccione, D. Tausk, The Maslov Index and a Generalized Morse Index Theorem for Non Positive Definite Metrics, preprint 2000, to appear in Comptes Rendus de l’Académie de Sciences, Paris.

Departamento de Matemática, Universidade de São Paulo, Brazil

   E-mail address: piccione@ime.usp.br, tausk@ime.usp.br