Symmetry Classes of Quasilinear Systems in One Space Variable

Abstract

The family of simple quasilinear systems in one space variable is partitioned into classes of commuting flows, i.e., symmetry classes. The systems in a symmetry class have the same zeroth order conserved densities and the same Hamiltonian structure. The zeroth and first order conservation laws and the Hamiltonian structure of the systems in a complete symmetry class are described. If such a system has a degenerate characteristic speed, then it has conservation laws of arbitrarily high order. Symmetry classes of 2-component hyperbolic systems correspond to coframes on the plane. The invariants of 2-component Hamiltonian hyperbolic symmetry classes are given. An exact symmetry class of 2-component hyperbolic systems is characterized by its canonical representative, and the first order conservation laws of the canonical system correspond to the infinitesimal automorphisms of the coframe. The normal forms of the rank 0 and rank 1 exact classes are listed. A simple symmetry class is tri-Hamiltonian if and only if the metric of its coframe has constant curvature. The normal forms of the tri-Hamiltonian simple classes are listed.     

Letter: On the Global Hyperbolicity of Lorentz 2-Step Nilpotent Lie Groups

This work is concerned with the existence of Lorentz 2-step nilpotent Lie groups having a timelike center and which are not globally hyperbolic. Namely, we prove that any left invariant Lorentz metric with a timelike center on the Heisenberg group H ²ⁿ⁺¹ is not globally hyperbolic.     

Cosmic Censorship of Smooth Structures

It is observed that on many 4-manifolds there is a unique smooth structure underlying a globally hyperbolic Lorentz metric. For instance, every contractible smooth 4-manifold admitting a globally hyperbolic Lorentz metric is diffeomorphic to the standard ${\mathbb{R}^4}$ . Similarly, a smooth 4-manifold homeomorphic to the product of a closed oriented 3-manifold N and ${\mathbb{R}}$ and admitting a globally hyperbolic Lorentz metric is in fact diffeomorphic to ${N\times \mathbb{R}}$ . Thus one may speak of a censorship imposed by the global hyperbolicty assumption on the possible smooth structures on (3 + 1)-dimensional spacetimes.     

Large Deviations for Non-Uniformly Expanding Maps

We obtain large deviation bounds for non-uniformly expanding maps with non-flat singularities or criticalities and for partially hyperbolic non-uniformly expanding attracting sets. That is, given a continuous function we consider its space average with respect to a physical measure and compare this with the time averages along orbits of the map, showing that the Lebesgue measure of the set of points whose time averages stay away from the space average tends to zero exponentially fast with the number of iterates involved. As easy by-products we deduce escape rates from subsets of the basins of physical measures for these types of maps. The rates of decay are naturally related to the metric entropy and pressure function of the system with respect to a family of equilibrium states. 2000 Mathematics Subject Classification: 37D25, 37A50, 37B40, 37C40     

Collisions of Particles in Locally AdS Spacetimes II Moduli of Globally Hyperbolic Spaces

We investigate globally hyperbolic 3-dimensional AdS manifolds containing “particles”, i.e., cone singularities of angles less than 2π along a time-like graph Γ. To each such space (equipped with a time-like vector field satisfying some additional properties) we associate a graph and a finite family of pairs of hyperbolic surfaces with cone singularities. We show that this data is sufficient to recover the space locally (i.e., in the neighborhood of a fixed metric). This is a partial extension of a result of Mess for non-singular globally hyperbolic AdS manifolds.     

A New Parametrization for Tetrad Gravity

A new version of tetrad gravity in globally hyperbolic, asymptotically flat at spatial infinity spacetimes with Cauchy surfaces diffeomorphic to R ³ is obtained by using a new parametrization of arbitrary cotetrads to define a set of configurational variables to be used in the ADM metric action. Seven of the fourteen first class constraints have the form of the vanishing of canonical momenta. A comparison is made with other models of tetrad gravity and with the ADM canonical formalism for metric gravity.     

Local index theorem for orbifold Riemann surfaces

We derive a local index theorem in Quillen’s form for families of Cauchy–Riemann operators on orbifold Riemann surfaces (or Riemann orbisurfaces) that are quotients of the hyperbolic plane by the action of cofinite finitely generated Fuchsian groups. Each conical point (or a conjugacy class of primitive elliptic elements in the Fuchsian group) gives rise to an extra term in the local index theorem that is proportional to the symplectic form of a new Kähler metric on the moduli space of Riemann orbisurfaces. We find a simple formula for a local Kähler potential of the elliptic metric and show that when the order of elliptic element becomes large, the elliptic metric converges to the cuspidal one corresponding to a puncture on the orbisurface (or a conjugacy class of primitive parabolic elements). We also give a simple example of a relation between the elliptic metric and special values of Selberg’s zeta function.

     

From Pythagoras To Einstein: The Hyperbolic Pythagorean Theorem

A new form of the Hyperbolic Pythagorean Theorem, which has a striking intuitive appeal and offers a strong contrast to its standard form, is presented. It expresses the square of the hyperbolic length of the hypotenuse of a hyperbolic right-angled triangle as the Einstein sum of the squares of the hyperbolic lengths of the other two sides, Fig. 1, thus completing the long path from Pythagoras to Einstein. Following the pioneering work of Variak it is well known that relativistic velocities are governed by hyperbolic geometry in the same way that prerelativistic velocities are governed by Euclidean geometry. Unlike prerelativistic velocity composition, given by the ordinary vector addition, the composition of relativistic velocities, given by the Einstein addition, is neither commutative nor associative due to the presence of Thomas precession. Following the discovery of the mathematical regularity that Thomas precession stores, it is now possible to extend Thomas precession by abstraction, (i) allowing hyperbolic geometry to be studied by means of analogies that it shares with Euclidean geometry; and, similarly (ii) allowing velocities and accelerations in relativistic mechanics to be studied by means of analogies that they share with velocities and accelerations in classical mechanics. The abstract Thomas precession, called the Thomas gyration, gives rise to gyrovector space theory in which the prefix gyro is used extensively in terms like gyrogroups and gyrovector spaces, gyroassociative and gyrocommutative laws, gyroautomorphisms, gyrotranslations, etc. We demonstrate the superiority of our gyrovector space formalism in capturing analogies by deriving the Hyperbolic Pythagorean Theorem in a form fully analogous to its Euclidean counterpart, thus contrasting it with the standard form in which the Hyperbolic Pythagorean Theorem is known in the literature. The hyperbolic metric, which supports the Hyperbolic Pythagorean Theorem, has a dual metric. We show that the dual metric does not support a Pythagorean theorem but, instead, it supports the -Theorem according to which the sum of the three dual angles of a hyperbolic triangle is .     

A criterion for global hyperbolicity of left-invariant Lorentz metrics on Lie groups

We show that every left-invariant Lorentz metric on a non-abelian simply connected Lie group is globally hyperbolic whenever its restriction to the commutator ideal of the Lie algebra is positive definite. We also show that a left-invariant Lorentz metric on the three-dimensional Heisenberg group is globally hyperbolic if and only if its restriction to the center of the Lie algebra is positive definite or degenerate.     

Decay of correlations in one and two dimensions

An exposition of some methods of proving exponential (stretched exponential) decay of correlations is given. One-dimensional strictly hyperbolic and quadratic maps and two-dimensional piecewise smooth, uniformly hyperbolic maps are considered. The emphasis is on the fundamental constructions of the Markov sieve method due to Bunimovich-Chernov-Sinai and those of Liverani's Hilbert metric method.     

Ising models on hyperbolic graphs

We consider Ising models on a hyperbolic graph which, loosely speaking, is a discretization of the hyperbolic planeH ² in the same sense asZ ^d is a discretization ofR ^d . We prove that the models exhibit multiple phase transitions. Analogous results for Potts models can be obtained in the same way.     

On nonlocal symmetries of integrable three-field evolutionary systems

Abstract

Nonlocal symmetries for exactly integrable three-field evolutionary systems have been computed. Differentiation the nonlocal symmetries with respect to x gives a few hyperbolic systems for each evolution system. Zero curvature representations for all nonlocal systems and for some of the hyperbolic systems are constructed.     

Etude des Equations des Fluides Chargés Relativistes Inductifs et Conducteurs

In the first part a system of equations for an inductive charged relativistic fluid with finite conductivity is written in a space time with given metric, taking into account thermodynamic phenomena. Speeds of propagation of various types of waves are determined under a restrictive hypothesis concerning the heat currentq: thatq depends only on the thermodynamical quantities and the gradient of one function of these quantities.In the second part it is shown, by a detailed study of the characteristic polynomial and of its irreducible factors, that, whenq is negligible, the proposed system is non-strictly hyperbolic in the sense ofJ. Leray andY. Ohya and existence and uniqueness theorems of a certain Gevrey class are verified; the relativistic causality principle is satisfied under some physically reasonable assumptions on the thermodynamical quantities. The system becomes strictly hyperbolic (existence and uniqueness theorems obtain in classes of functions with a finite number of derivatives) when the fluid is both non inductive and of zero electrical conductivity.In the third part we show briefly, by the methods of the second part, that the equations of relativistic fluids, with an infinite electrical conductivity is also non-strictly hyperbolic. The linearized equations (in the neighborhood of constant values) are strictly hyperbolic.     

Einstein-Maxwell space-times with symmetries and with non-null electromagnetic fields

General properties of solutions (g, F) of the Einstein-Maxwell field equations are discussed, whereg is a metric tensor andF is a non-null Maxwell field. In particular the case is discussed whereg admits a Killing vector fieldv with special emphasis on the case wherev is not admitted byF, i.e., the electromagnetic field does not have a symmetry of the metric tensor. An example is given of a solution (g, F) in whichg admits a hypersurface orthogonal Killing vector not admitted byF.     

Variational equations on mixed Riemannian–Lorentzian metrics

A class of elliptic–hyperbolic equations is placed in the context of a geometric variational theory, in which the change of type is viewed as a change in the character of an underlying metric. A fundamental example of a metric which changes in this way is the extended projective disc, which is Riemannian at ordinary points, Lorentzian at ideal points, and singular on the absolute. Harmonic fields on such a metric can be interpreted as the hodograph image of extremal surfaces in Minkowski 3-space. This suggests an approach to generalized Plateau problems in three-dimensional space-time via Hodge theory on the extended projective disc. Analogous variational problems arise on Riemannian–Lorentzian flow metrics in fiber bundles (twisted nonlinear Hodge equations), and on certain Riemannian–Lorentzian manifolds which occur in relativity and quantum cosmology. The examples surveyed come with natural gauge theories and Hodge dualities. This paper is mainly a review, but some technical extensions are proven.     

Static models of cosmic strings in general relativity

Exact solutions to Einstein’s equations for a cloud of massive strings with a general static metric representing spherical plane and hyperbolic symmetries are derived. Some properties of massive strings for different cases are also discussed.     

The Deformation Quantizations of the Hyperbolic Plane

We describe the space of (all) invariant, both formal and non-formal, deformation quantizations on the hyperbolic plane as solutions of the evolution of a second order hyperbolic differential operator. The construction is entirely explicit and relies on non-commutative harmonic analytical techniques on symplectic symmetric spaces. The present work presents a unified method producing every quantization of , and provides, in the 2-dimensional context, an exact solution to Weinstein’s WKB quantization program within geometric terms. The construction reveals the existence of a metric of Lorentz signature canonically attached (or ‘dual’) to the geometry of the hyperbolic plane through the quantization process.     

Quantum Field Theory on Spacetimes with a Compactly Generated Cauchy Horizon

We prove two theorems which concern difficulties in the formulation of the quantum theory of a linear scalar field on a spacetime, , with a compactly generated Cauchy horizon. These theorems demonstrate the breakdown of the theory at certain base points of the Cauchy horizon, which are defined as ‘past terminal accumulation points’ of the horizon generators. Thus, the theorems may be interpreted as giving support to Hawking's ‘Chronology Protection Conjecture’, according to which the laws of physics prevent one from manufacturing a ’time machine‘. Specifically, we prove: Theorem 1. There is no extension to of the usual field algebra on the initial globally hyperbolic region which satisfies the condition of F-locality at any base point. In other words, any extension of the field algebra must, in any globally hyperbolic neighbourhood of any base point, differ from the algebra one would define on that neighbourhood according to the rules for globally hyperbolic spacetimes. Theorem 2. The two-point distribution for any Hadamard state defined on the initial globally hyperbolic region must (when extended to a distributional bisolution of the covariant Klein-Gordon equation on the full spacetime) be singular at every base point x in the sense that the difference between this two point distribution and a local Hadamard distribution cannot be given by a bounded function in any neighbourhood (in M × M) of (x,x). In consequence of Theorem 2, quantities such as the renormalized expectation value of φ² or of the stress-energy tensor are necessarily ill-defined or singular at any base point. The proof of these theorems relies on the ‘Propagation of Singularities’ theorems of Duistermaat and H?rmander. Received: 14 March 1996/Accepted: 11 June 1996     

Characterization of some causality conditions through the continuity of the Lorentzian distance

A classical result in Lorentzian geometry states that a strongly causal spacetime is globally hyperbolic if and only if the Lorentzian distance is finite valued for every metric choice in the conformal class. It is proved here that a non-total imprisoning spacetime is globally hyperbolic if and only if for every metric choice in the conformal class the Lorentzian distance is continuous. Moreover, it is proved that a non-total imprisoning spacetime is causally simple if and only if for every metric choice in the conformal class the Lorentzian distance is continuous wherever it vanishes. Finally, a strongly causal spacetime is causally continuous if and only if there is at least one metric in the conformal class such that the Lorentzian distance is continuous wherever it vanishes.