Green-Hyperbolic Operators on Globally Hyperbolic Spacetimes

Communications in Mathematical Physics - Green-hyperbolic operators are linear differential operators acting on sections of a vector bundle over a Lorentzian manifold which possess advanced and...     

Special Temporal Functions on Globally Hyperbolic Manifolds

In this article, existence results concerning temporal functions with additional properties on a globally hyperbolic manifold are obtained. These properties are certain bounds on geometric quantities as lapse and shift. The results are linked to completeness properties and the existence of closed isometric embeddings in Minkowski spaces.     

Smoothness of Time Functions and the Metric Splitting of Globally Hyperbolic Spacetimes

The folk questions in Lorentzian Geometry which concerns the smoothness of time functions and slicings by Cauchy hypersurfaces, are solved by giving simple proofs of: (a) any globally hyperbolic spacetime (M, g) admits a smooth time function whose levels are spacelike Cauchy hyperfurfaces and, thus, also a smooth global splitting if a spacetime M admits a (continuous) time function t then it admits a smooth (time) function with timelike gradient on all M.The second-named author has been partially supported by a MCyT-FEDER Grant, MTM2004-04934-C04-01.To Professor P.E. Ehrlich, wishing him a continued recovery and good health     

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.     

The Time Slice Axiom in Perturbative Quantum Field Theory on Globally Hyperbolic Spacetimes

The time slice axiom states that the observables which can be measured within an arbitrarily small time interval suffice to predict all other observables. While well known for free field theories where the validity of the time slice axiom is an immediate consequence of the field equation it was not known whether it also holds in generic interacting theories, the only exception being certain superrenormalizable models in 2 dimensions. In this paper we prove that the time slice axiom holds at least for scalar field theories within formal renormalized perturbation theory.     

Globally Hyperbolic Moment Model of Arbitrary Order for One-Dimensional Special Relativistic Boltzmann Equation

This paper extends the model reduction method by the operator projection to the one-dimensional special relativistic Boltzmann equation. The derivation of arbitrary order globally hyperbolic moment system is built on our careful study of two families of the complicate Grad type orthogonal polynomials depending on a parameter. We derive their recurrence relations, calculate their derivatives with respect to the independent variable and parameter respectively, and study their zeros and coefficient matrices in the recurrence formulas. Some properties of the moment system are also proved. They include the eigenvalues and their bound as well as eigenvectors, hyperbolicity, characteristic fields, linear stability, and Lorentz covariance. A semi-implicit numerical scheme is presented to solve a Cauchy problem of our hyperbolic moment system in order to verify the convergence behavior of the moment method. The results show that the solutions of our hyperbolic moment system converge to the solution of the special relativistic Boltzmann equation as the order of the hyperbolic moment system increases.     

Plane Symmetric Space-times in Bimetric Relativity Theory

It is established that some of the plane symmetric homogeneous models in Rosen's bimetric relativity are singular.     

A C*-Algebra for Quantized Principal U(1)-Connections on Globally Hyperbolic Lorentzian Manifolds

The aim of this work is to complete our program on the quantization of connections on arbitrary principal U(1)-bundles over globally hyperbolic Lorentzian manifolds. In particular, we show that one can assign via a covariant functor to any such bundle an algebra of observables which separates gauge equivalence classes of connections. The C*-algebra we construct generalizes the usual CCR-algebras, since, contrary to the standard field-theoretic models, it is based on a presymplectic Abelian group instead of a symplectic vector space. We prove a no-go theorem according to which neither this functor, nor any of its quotients, satisfies the strict axioms of general local covariance. As a byproduct, we prove that a morphism violates the locality axiom if and only if a certain induced morphism of cohomology groups is non-injective. We show then that, fixing any principal U(1)-bundle, there exists a suitable category of subbundles for which a quotient of our functor yields a quantum field theory in the sense of Haag and Kastler. We shall provide a physical interpretation of this feature and we obtain some new insights concerning electric charges in locally covariant quantum field theory.     

Space-times containing perfect fluids and having a vanishing conformal divergence

The solutions of the Einstein field equations are studied under the assumptions that (1) the source of the gravitational field is a perfect fluid, (2) the divergence of the conformal (Weyl) tensor vanishes, and (3a) either an equation of state exists such thatp=p (w),p being the pressure andw the rest energy density, or (3b) the rest particle density is conserved. Under assumptions (1), (2), and (3a) it is shown that the space-time is conformally flat and the metric is a Robertson-Walker metric. The flow is irrotational, shear-free, and geodesic. Under assumptions (1), (2), and (3b) it is shown that either the line element is static or the fluid has a very special caloric equation of state. Conditions for a static solution to exist are examined, and it is shown that the Schwarzschild interior solution satisfies these conditions as does the Einstein universe. The Schwarzschild interior and the Einstein universe are the only conformally flat, static solutions obeying (1), (2), and (3b).The research reported herein was supported in part by the Atomic Energy Commission under contract number AT (11-1)-34, Project Agreement No. 125.     

Equivalent Forms of Dirac Equations in Curved Space-times and Generalized de Broglie Relations

One may ask whether the relations between energy and frequency and between momentum and wave vector, introduced for matter waves by de Broglie, are rigorously valid in the presence of gravity. In this paper, we show this to be true for Dirac equations in a background of gravitational and electromagnetic fields. We first transform any Dirac equation into an equivalent canonical form, sometimes used in particular cases to solve Dirac equations in a curved space-time. This canonical form is needed to apply Whitham’s Lagrangian method. The latter method, unlike the Wentzel–Kramers–Brillouin method, places no restriction on the magnitude of Planck’s constant to obtain wave packets and furthermore preserves the symmetries of the Dirac Lagrangian. We show by using canonical Dirac fields in a curved space-time that the probability current has a Gordon decomposition into a convection current and a spin current and that the spin current vanishes in the Whitham approximation, which explains the negligible effect of spin on wave packet solutions, independent of the size of Planck’s constant. We further discuss the classical-quantum correspondence in a curved space-time based on both Lagrangian and Hamiltonian formulations of the Whitham equations. We show that the generalized de Broglie relations in a curved space-time are a direct consequence of Whitham’s Lagrangian method and not just a physical hypothesis as introduced by Einstein and de Broglie and by many quantum mechanics textbooks.     

Causality in non-Hausdorff space-times

Some general properties of completely separable, non-Hausdorff manifolds are studied and the notion of a non-Hausdorff space-time is introduced. It is shown that such a space-time must, under very general conditions, display a kind of causal anomaly.     

Causality and quantum mechanics

Statistical causality is recommended as the name of the generalized causality needed in quantum mechanics, instead of statistical correspondence used by Pauli.     

Causality and Cauchy horizons

LetS be a partial Cauchy surface for (M, go) which remains a partial Cauchy surface under small metric perturbations. In general, the Cauchy horizon H⁺(go, S) may be unstable to small changes in the metric. Points of the horizon may move by large amounts and even the topological type of the horizon may change under arbitrarily small changes in the metric tensor. In this paper, we investigate sufficient conditions for existential, locational, and topological stability of Cauchy horizons under metric changes which perturb the light cones by small amounts.     

Composite Structure and Causality

We study the question of whether a composite structure of elementary particles, with a length scale 1/Λ, can leave observable effects of non-locality and causality violation at higher energies (but ≲Λ). We formulate a model-independent approach based on Bogoliubov-Shirkov formulation of causality. We analyze the relation between the fundamental theory (of finer constituents) and the derived theory (of composite particles). We assume that the fundamental theory is causal and formulate a condition which must be fulfilled for the derived theory to be causal. We analyze the condition and exhibit possibilities which fulfil and which violate the condition. We make comments on how causality violating amplitudes can arise.     

Causality in quantum mechanics

We show explicitly how the causal arrow of time that follows from quantum mechanics has already been inserted at a deeper level by the choice of normalisation conditions. This prohibits information being sent backwards in time but does not determine a time direction for state propagation.     

Causality violations and singularities

We announce and justify two theorems (proofs will appear in Refs. 1 and 2): i) A generalization of the singularity theorem of Hawking and Penrose [3] to space-times with chronology violations. Although it is impossible to remove the chronology condition completely the announced theorem is in a well defined sense optimal: the chronology condition is replaced by a strictly weaker condition that cannot be removed because of counter examples, ii) If the chronology violating setV has compact closure and the strong energy and generic conditions hold, thenV is generated by incomplete null geodesics. It follows that if the region of causality violation does not extend to infinity thenV contains singularities.This essay received an honourable mention from the Gravity Research Foundation, 1989     

Faster-than-Light Signals and Causality

The problems of causality are analyzed in terms of space-time models which admit the propagation of signals with superrelativistic velocities. It is shown that there is no violation in causality if the propagation of faster-than-light signals is described by general-covariant equations and occurs along invariant curves, as it is in some well-known models.     

Relativistic Simultaneity and Causality

We analyzed two types of relativistic simultaneity associated to an observer: the spacelike simultaneity, given by Landau submanifolds, and the lightlike simultaneity given by past-pointing horismos submanifolds. We study some geometrical conditions to ensure that Landau submanifolds are spacelike and we prove that horismos submanifolds are always lightlike. Finally, we establish some conditions to guarantee the existence of foliations in the space-time whose leaves are these submanifolds of simultaneity generated by an observer. These foliation structure allows us to incorporate the simultaneity submanifolds for studying some dynamical systems, for instance free elementary massless particles.     

Superluminal Signal Velocity and Causality

A superluminal signal velocity (i.e. faster than light) is said to violate causality. However, superluminal signal velocities have been measured in tunneling experiments recently. The classical dipole interaction approach by Sommerfeld and Brillouin results in a complex refractive index with a finite real part. For the tunneling process with its purely imaginary refractive index this model obtaines a zero-time traversing of tunneling barriers in agreement with wave meechanics. The information of a signal is proportional to the product of its frequency band width and its time duration. The reasons that superluminal signal velocities do not violate causality are: (i) physical signals are frequency band limited and (ii) signals have a finite time duration.