arXiv subjects
Publications and source records attributed to Paolo Piccione.
Using the Spectral Theorem for unbounded self-adjoint operators we prove that any countable family of Lagrangian subspaces of a symplectic Hilbert space admits a common complementary Lagrangian.
In this short note we comment about some criticisms - appeared in a recent paper by Iguchi et al - to our previous works on gravitational collapse of perfect fluids. We show that those criticisms are incorrect on their own.
The gravitational collapse of spherical, barotropic perfect fluids is analyzed here. For the first time, the final state of these systems is studied without resorting to simplifying assumptions - such as self-similarity - using a new approach based on non-linear o.d.e. techniques, and formation of naked singularities is shown to occur for solutions such that the mass function is analytic in a neighborhood of the spacetime singularity.
A theorem, giving necessary and sufficient condition for naked singularity formation in spherically symmetric non static spacetimes under hypotheses of physical acceptability, is formulated and proved. The theorem relates existence of singular null geodesics to existence of regular curves which are super-solutions of the radial null geodesic equation, and allows us to treat all the known examples of naked singularities from a unified viewpoint. New examples are also found using this approach, and perspectives are discussed.
A new class of solutions of the Einstein field equations in spherical symmetry is found. The new solutions are mathematically described as the metrics admitting separation of variables in area-radius coordinates. Physically, they describe the gravitational collapse of a class of anisotropic elastic materials. Standard requirements of physical acceptability are satisfied, in particular, existence of an equation of state in closed form, weak energy condition, and existence of a regular Cauchy surface at which the collapse begins. The matter properties are generic in the sense that both the radial and the tangential stresses are non vanishing, and the kinematical properties are generic as well, since shear, expansion, and acceleration are also non-vanishing. As a test-bed for cosmic censorship, the nature of the future singularity forming at the center is analyzed as an existence problem for o.d.e. at a singular point using techniques based on comparison theorems, and the spectrum of endstates - blackholes or naked singularities - is found in full generality. Consequences of these results on the Cosmic Censorship conjecture are discussed.
We give a functional analytical proof of the equality between the Maslov index of a semi-Riemannian geodesic and the spectral flow of the path of self-adjoint Fredholm operators obtained from the index form. This fact, together with recent results on the bifurcation for critical points of strongly indefinite functionals (see P. M. Fitzpatrick, J. Pejsachowicz, L. Recht, Spectral Flow and Bifurcation of Strongly Indefinite Functionals Part I. General Theory, J. Funct. Anal. 162 (1) (1999), 52-95.) imply that each non degenerate and non null conjugate (or $P$-focal) point along a semi-Riemannian geodesic is a bifurcation point.
We consider solutions of Lagrangian variational problems with linear constraints on the derivative. These solutions are given by curves $γ$ in a differentiable manifold $M$ that are everywhere tangent to a smooth distribution $\mathcal D$ on $M$; such curves are called horizontal. We study the manifold structure of the set $Ω_{P,Q}(M,\mathcal D)$ of horizontal curves that join two submanifolds $P$ and $Q$ of $M$. We consider an action functional $\mathcal L$ defined on $Ω_{P,Q}(M,\mathcal D)$ associated to a time-dependent Lagrangian defined on $\mathcal D$. If the Lagrangian satisfies a suitable hyper-regularity assumption, it is shown how to construct an associated degenerate Hamiltonian $H$ on $TM^*$ using a general notion of {\em Legendre transform} for maps on vector bundles. We prove that the solutions of the Hamilton equations of $H$ are precisely the critical points of $\mathcal L$.
We prove a generalized version of the Morse index theorem for geodesics endowed with a non positive definite metric tensor (semi-Riemannian manifolds). We apply the result to obtain lower estimates on the number of geodesics joining two fixed non conjugate points in certain classes of manifolds. More specifically, we consider semi-Riemannian manifolds $(M,\mathfrak g)$ admitting a smooth distribution spanned by commuting Killing vector fields and containing a maximal negative distribution for $\mathfrak$. In particular we obtain Morse relations for stationary semi-Riemannian manifolds and for the {\em Gödel-type} manifolds.
We prove a semi-Riemannian version of the celebrated Morse Index Theorem for geodesics in semi-Riemannian manifolds; we consider the general case of both endpoints variable on two submanifolds. The key role of the theory is played by the notion of the {\em Maslov index} of a semi-Riemannian geodesic, which is a homological invariant and it substitutes the notion of geometric index in Riemannian geometry. Under generic circumstances, the Maslov index of a geodesic is computed as a sort of {\em algebraic count} of the conjugate points along the geodesic. For non positive definite metrics the index of the index form is always infinite; in this paper we prove that the space of all variations of a given geodesic has a {\em natural} splitting into two infinite dimensional subspaces, and the Maslov index is given by the difference of the index and the coindex of the restriction of the index form to these subspaces. In the case of variable endpoints, two suitable correction terms, defined in terms of the endmanifolds, are added to the equality. Using appropriate change of variables, the theory is entirely extended to the more general case of {\em symplectic differential systems}, that can be obtained as linearizations of the Hamilton equations.
Helfer in [Pacific J. Math. 164/2 (1994), p. 321--350] was the first to produce an example of a spacelike Lorentzian geodesic with a continuum of conjugate points. In this paper we show the following result: given an interval $[a,b]$ of $IR$ and any closed subset $F$ of $IR$ contained in $]a,b]$, then there exists a Lorentzian manifold $(M,g)$ and a spacelike geodesic $γ:[a,b]\to M$ such that $γ(t)$ is conjugate to $γ(a)$ along $γ$ iff $t\in F$.
We consider a {\em Hamiltonian setup} $\sextuple$, where $(\mathcal M,ω)$ is a symplectic manifold, $\mathfrak L$ is a distribution of Lagrangian subspaces in $\mathcal M$, $\mathcal P$ a Lagrangian submanifold of $ \mathcal M$, $H$ is a smooth time dependent Hamiltonian function on $\mathcal M$ and $Γ:[a,b]\to\mathcal M$ is an integral curve of the Hamiltonian flow $\Hf$ starting at $\mathcal P$. We do not require any convexity property of the Hamiltonian function $H$. Under the assumption that $Γ(b)$ is not $\mathcal P$-focal it is introduced the Maslov index $\maslov(Γ)$ of $Γ$ given in terms of the first relative homology group of the Lagrangian Grassmannian; under generic circumstances $\maslov(Γ)$ is computed as a sort of {\em algebraic count} of the $\mathcal P$-focal points along $Γ$. We prove the following version of the Index Theorem: under suitable hypotheses, the Morse index of the Lagrangian action functional restricted to suitable variations of $Γ$ is equal to the sum of $\maslov(Γ)$ and a {\em convexity term} of the Hamiltonian $H$ relative to the submanifold $\mathcal P$. When the result is applied to the case of the cotangent bundle $\mathcal M=TM^*$ of a semi-Riemannian manifold $(M,g)$ and to the geodesic Hamiltonian $H(q,p)=\frac12 g^{-1}(p,p)$, we obtain a semi-Riemannian version of the celebrated Morse Index Theorem for geodesics with variable endpoints in Riemannian geometry.
We study the local geometry of the space of horizontal curves with endpoints freely varying in two given submanifolds $\mathcal P$ and $\mathcal Q$ of a manifold $\mathcal M$ endowed with a distribution $\mathcal D\subset T\M$. We give a different proof, that holds in a more general context, of a result by Bismut (Large Deviations and the Malliavin Calculus, Birkhauser, 1984) stating that the normal extremizers that are not abnormal are critical points of the sub-Riemannian action functional. We use the Lagrangian multipliers method in a Hilbert manifold setting, which leads to a characterization of the abnormal extremizers (critical points of the endpoint map) as curves where the linear constraint fails to be regular. Finally, we describe a modification of a result by Liu and Sussmann that shows the global distance minimizing property of sufficiently small portions of normal extremizers between a point and a submanifold.
We investigate the problem of the stability of the number of conjugate or focal points (counted with multiplicity) along a semi-Riemannian geodesic $γ$. For a Riemannian or a non spacelike Lorentzian geodesic, such number is equal to the intersection number (Maslov index) of a continuous curve with a subvariety of codimension one of the Lagrangian Grassmannian of a symplectic space. Such intersection number is proven to be stable in a large variety of circumstances. In the general semi-Riemannian case, under suitable hypotheses this number is equal to an algebraic count of the multiplicities of the conjugate points, and it is related to the spectral properties of a non self-adjoint differential operator. This last relation gives a weak extension of the classical Morse Index Theorem in Riemannian and Lorentzian geometry. In this paper we reprove some results that were incorrectly stated by Helfer in a previous reference; in particular, a counterexample to one of Helfer's results, which is essential for the theory, is given. In the last part of the paper we discuss a general technique for the construction of examples and counterexamples in the index theory for semi-Riemannian metrics, in which some new phenomena appear.
The computation of the index of the Hessian of the action functional in semi-Riemannian geometry at geodesics with two variable endpoints is reduced to the case of a fixed final endpoint. Using this observation, we give an elementary proof of the Morse Index Theorem for Riemannian geodesics with two variable endpoints, in the spirit of the original Morse's proof. This approach reduces substantially the effort required in the proofs of the Theorem given in previous articles on the subject. Exactly the same argument works also in the case of timelike geodesics between two submanifolds of a Lorentzian manifold. For the extension to the lightlike Lorentzian case, just minor changes are required and one obtains easily a proof of the focal index theorems of Beem, Ehrlich and Kim.
In this paper we use a general version of Fermat's principle for light rays in General Relativity and a curve shortening method to write the Morse relations for light rays joining an event with a smooth timelike curve in a Lorentzian manifold with boundary. As a physical meaning, one can apply the Morse relations to have a mathematical description of the "gravitational lens effect" in a very general context.
In this paper we develop a Morse Theory for timelike geodesics parameterized by a constant multiple of proper time. The results are obtained using an extension to the timelike case of the relativistic Fermat Principle, and techniques from Global Analysis on infinite dimensional manifolds. In the second part of the paper we discuss a limit process that allows to obtain also a Morse theory for light rays.
The travel time brachistochrone curves in a general relativistic framework are timelike curves, satisfying a suitable conservation law with respect to a an observer field, that are stationary points of the travel time functional. In this paper we develop a global variational theory for brachistochrones joining an event $p$ and the worldline of an observer $γ$ in a stationary spacetime $M$. More specifically, using the method of Lagrange multipliers, we compute the first and the second variation of the travel time functional, obtaining two variational principles relating the geometry of the brachistochrones with the geometry of geodesics in a suitable Riemannian structure. We present an extension of the classical Morse Theory for Riemannian geodesics to the case of travel time brachistochrones, and we prove a Morse Index Theorem for brachistochrones. Finally, using techniques from Global Analysis, we prove the Morse relations for the travel time functional and we establish some existence and multiplicity results for brachistochrones.