Null Surfaces and Their Singularities in Three-Dimensional Minkowski Space-Time

In this work we integrate the null geodesic equations in three-dimensional Minkowski space-time in order to obtain the light-cone cut function; that is, the function that describes the intersection, C_x ^a, of the light cone from each space-time point, _x ^a, with future null infinity I ⁺. Furthermore, using this result, we locate the singularities of the null surface obtained as the envelope of the past light cones from points on a deformed light-cone cut of I ⁺.     

Horizons and singularities in static,spherically symmetric spacetimes

We make a thorough study of the regions near finite-order metric-singularity boundaries of static, spherically symmetric spacetimes. After distinguishing curvature singularities from other types of metric breakdown, we examine the eigenvalues of the energy tensor near the singularities for positivity and energy dominance, find the causal class of the t-translation (static) Killing field, and ascertain the presence or absence of timelike, null, and spacelike geodesic incompleteness for each spacetime. For a certain subclass of spacetimes, we also show the completeness of all timelike and spacelike curves despite the superficial failure of the metric.     

K-causal structure of space-time in general relativity

Using K-causal relation introduced by Sorkin and Woolgar [1], we generalize results of Garcia-Parrado and Senovilla [2,3] on causal maps. We also introduce causality conditions with respect to K-causality which are analogous to those in classical causality theory and prove their inter-relationships. We introduce a new causality condition following the work of Bombelli and Noldus [4] and show that this condition lies in between global hyperbolicity and causal simplicity. This approach is simpler and more general as compared to traditional causal approach [5,6] and it has been used by Penrose et al [7] in giving a new proof of positivity of mass theorem. C ⁰-space-time structures arise in many mathematical and physical situations like conical singularities, discontinuous matter distributions, phenomena of topology-change in quantum field theory etc.      

Persistently expansive geodesic flows

We prove thatC ¹-persistently expansive geodesic flows of compact, boundaryless Riemannian manifolds have the property that the closure of the set of closed orbits is a hyperbolic set. In the case of compact surfaces we deduce that the geodesic flow isC ¹-persistently expansive if and only if it is an Anosov flow.     

Remarks on J. langer and D. A. singer decomposition theorem for diffeomorphisms of the circle

We give a simple proof that allC ⁴ diffeomorphisms of the torus can be factorized into a finite number of diffeomorphisms commuting with reflection.In one dimension,C ³ suffices and evenC ² can yield that the factors are almost diffeomorphisms. (The derivatives of the function and the inverse are inL ¹ and are positive.)In one dimension underC assumptions, this had been proved by J. Langer and D. A. Singer in their study of geodesic fields by different methods.     

Local Causal Structures,Hadamard States and the Principle of Local Covariance in Quantum Field Theory

In the framework of the algebraic formulation, we discuss and analyse some new features of the local structure of a real scalar quantum field theory in a strongly causal spacetime. In particular, we use the properties of the exponential map to set up a local version of a bulk-to-boundary correspondence. The bulk is a suitable subset of a geodesic neighbourhood of an arbitrary but fixed point p of the underlying background, while the boundary is a part of the future light cone having p as its own tip. In this regime, we provide a novel notion for the extended *-algebra of Wick polynomials on the aforesaid cone and, on the one hand, we prove that it contains the information of the bulk counterpart via an injective *-homomorphism while, on the other hand, we associate to it a distinguished state whose pull-back in the bulk is of Hadamard form. The main advantage of this point of view arises if one uses the universal properties of the exponential map and of the light cone in order to show that, for any two given backgrounds M and M′ and for any two subsets of geodesic neighbourhoods of two arbitrary points, it is possible to engineer the above procedure such that the boundary extended algebras are related via a restriction homomorphism. This allows for the pull-back of boundary states in both spacetimes and, thus, to set up a machinery which permits the comparison of expectation values of local field observables in M and M′.     

Integrable Geodesic Flows on the (Super)extension of the Bott–Virasoro Group

In this Letter, we present an answer to the question posed by Marcel, Ovsienko and Roger in their paper (Lett. Math. Phys. 40 (1997), 31–39). The Itô equation, modified dispersive water wave equation and modified dispersionless long wave equation are shown to be the geodesic flows with respect to an L ² metric on the semidirect product space Diff ^s C (S ¹), where Diff ^s (S ¹) is the group of orientation-preserving Sobolev H ^s diffeomorphisms of the circle. We also study the geodesic flows with respect to H ¹ metric. The geodesic flows in this case yield different integrable systems admitting nonlinear dispersion terms. These systems exhibit more general wave phenomena than usual integrable systems. Finally, we study an integrable geodesic flow on the extended Neveu–Schwarz space.     

Logic of infinite quantum systems

Limits of sequences of finite-dimensional (AF)C ^*-algebras, such as the CAR algebra for the ideal Fermi gas, are a standard mathematical tool to describe quantum statistical systems arising as thermodynamic limits of finite spin systems. Only in the infinite-volume limit one can, for instance, describe phase transitions as singularities in the thermodynamic potentials, and handle the proliferation of physically inequivalent Hilbert space representations of a system with infinitely many degrees of freedom. As is well known, commutative AFC ^*-algebras correspond to countable Boolean algebras, i.e., algebras of propositions in the classical two-valued calculus. We investigate thenoncommutative logic properties of general AFC ^*-algebras, and their corresponding systems. We stress the interplay between Gödel incompleteness and quotient structures in the light of the nature does not have ideals program, stating that there are no quotient structures in physics. We interpret AFC ^*-algebras as algebras of the infinite-valued calculus of Lukasiewicz, i.e., algebras of propositions in Ulam's twenty questions game with lies.     

Stability of geodesic incompleteness for Robertson-Walker space-times

Let (M, g) be a Lorentzian warped product space-timeM=(a, b)×H, g = –dt ² fh, where –a<b+, (H, h) is a Riemannian manifold andf: (a, b)(0, ) is a smooth function. We show that ifa>– and (H, h) is homogeneous, then the past incompleteness of every timelike geodesic of (M,g) is stable under smallC ⁰ perturbations in the space Lor(M) of Lorentzian metrics forM. Also we show that if (H,h) is isotropic and (M,g) contains a past-inextendible, past-incomplete null geodesic, then the past incompleteness of all null geodesics is stable under smallC ¹ perturbations in Lor(M). Given either the isotropy or homogeneity of the Riemannian factor, the background space-time (M,g) is globally hyperbolic. The results of this paper, in particular, answer a question raised by D. Lerner for big bang Robertson-Walker cosmological models affirmatively.Partially supported by a grant from the Research Council of the Graduate School of the University of Missouri-Columbia.Partially supported by a grant from the Research Council of the Graduate School of the University of Missouri-Columbia and NSF grant No. MCS77-18723(02).     

Instability of the boundary in the billiard ball problem

We consider the billiard ball problem in the interior of a plane closed convexC ¹ curve which is piecewiseC ². If the curvature has a discontinuity, then the boundary is unstable, i.e. no caustics exist near the boundary. However, in the interior there can exist caustics, as we show by an example.     

The Numerical Methods for Oscillating Singularities in Elliptic Boundary Value Problems

The singularities near the crack tips of homogeneous materials are monotone of type r^α and r^α log^δr (depending on the boundary conditions along nonsmooth domains). However, the singularities around the interfacial cracks of the heterogeneous bimaterials are oscillatory of type r^α sin( log r). The method of auxiliary mapping (MAM), introduced by Babu ka and Oh, was proven to be successful in dealing with r^α type singularities. However, the effectiveness of MAM is reduced in handling oscillating singularities. This paper deals with oscillating singularities as well as the monotone singularities by extending MAM through introducing the power auxiliary mapping and the exponential auxiliary mapping.     

“Superconducting” causal nets

The world is described as a relativistic quantum neural net with a quantum condensation akin to superconductivity. The sole dynamical variable is an operator representing immediate causal connection. The net enjoys a quantum principle of equivalence implying local LorentzSL(2,C) invariance and causality. The past-future asymmetry of its cell is similar to that of the neutrino. A net phase transition is expected at temperatures on the order of theW mass rather than the Planck mass, and near gravitational singularities.     

A C>1 Completion of the Kerr Space–Time at Spacelike Infinity Including I+ and I−

It is well known that, for asymptotically flat spacetimes, one cannot in general have a smooth differentiable structure at spacelike infinity, i ⁰. Normally, one uses direction dependent structures, whose regularity has to match the regularity of the (rescaled) metric. The standard C ^>1-structure at i ⁰ ensures sufficient regularity in spacelike directions, but examples show very low regularity on I ⁺ and I ^–. The alternative C ¹⁺-structure shows that both null and spacelike directions may be treated on an equal footing, at the expense of some manageable logarithmic singularities at i ⁰. In this paper, we show that the Kerr spacetime may be rescaled and given a structure which is C ^>1 in both null and spacelike directions from i ⁰.     

<Emphasis Type="Italic">J</Emphasis><Subscript><Emphasis Type="Italic">AW,WA</Emphasis></Subscript> functions in Passarino-Veltman reduction

We continue to study a special class of Passarino-Veltman functions J arising at the reduction of infrared divergent box diagrams. We describe a procedure of separation of two types of singularities, infrared and mass singularities, which are absorbed in simple C ₀ functions. The infrared divergences of C ₀’s can be regularized then by any method: photon mass, dimensionally or by the width of an unstable particle. Functions D ₀ are represented as certain linear combinations of the standard C ₀ Passarino-Veltman functions and infrared finite functions J. Then mass singularities are extracted from J to other combinations of C ₀. The rests are free of both types of singularities and are expressed as explicit and compact linear combination of logarithms and dilogarithm functions. The extensive comparison of numerical results with those obtained with the aid of the Loop Tools package is presented.     

Complete affine connection in the causal boundary: static,spherically symmetric spacetimes

The boundary at \(\mathcal {I}^+\), future null infinity, for a standard static, spherically symmetric spactime is examined for possible linear connections. Two independent methods are employed, one for treating \(\mathcal {I}^+\) as the future causal boundary, and one for treating it as a conformal boundary (the latter is subsumed in the former, which is of greater generality). Both methods provide the same result: a constellation of various possible connections, depending on an arbitrary choice of a certain function, a sort of gauge freedom in obtaining a natural connection on \(\mathcal {I}^+\); choosing that function to be constant (for instance) results in a complete connection. Treating \(\mathcal {I}^+\) as part of the future causal boundary, the method is to impute affine connections on null hypersurfaces going out to \(\mathcal {I}^+\), in terms of a transverse vector field on each null hypersurface (there is much gauge freedom on choice of the transverse vector fields). Treating \(\mathcal {I}^+\) as part of a conformal boundary, the method is to make a choice of conformal factor that makes the boundary totally geodesic in the enveloping manifold (there is much gauge freedom in choice of that conformal factor). Similar examination is made of other boundaries, such as timelike infinity and timelike and spacelike singularities. These are much simpler, as they admit a unique connection from a similar limiting process (i.e., no gauge freedom); and that connection is complete.     

On the essential spectrum of the transfer operator for expanding markov maps

The essential spectrum of the transfer operator for expanding markov maps of the interval is studied in detail. To this end we construct explicityly an infinite set of eigenfunctions which allows us to prove that the essential spectrum inC ^k is a disk whose radius is related to the free energy of the Liapunov exponent.     

Vector singularities of superposition of mutually incoherent orthogonally polarized beams

It is shown that, at an incoherent superposition of orthogonally polarized laser beams, a special type of singularities are formed in the cross section of a combined beam in place of the well-known singularities, such as optical vortices (for scalar fields); C points, at which the polarization is circular; and L lines, along which the polarization is linear (for coherent vector fields). These new singularities are U lines, along which the degree of polarization is zero and the state of polarization is undetermined, and P points, at which the degree of polarization is equal to unity and the state of polarization is determined by the nonzero component of the combined beam. Conditions of topological stability of U and P singularities are discussed, as well as peculiarities of the spatial distribution of the degree of polarization of the field in the vicinity of such singularities. First experimental results on the reconstruction of a vector skeleton formed by U and P singularities in combined speckle fields are presented.     

Singularity spectrum of fractal signals from wavelet analysis: Exact results

The multifractal formalism for singular measures is revisited using the wavelet transform. For Bernoulli invariant measures of some expanding Markov maps, the generalized fractal dimensions are proved to be transition points for the scaling exponents of some partition functions defined from the wavelet transform modulus maxima. The generalization of this formalism to fractal signals is established for the class of distribution functions of these singular invariant measures. It is demonstrated that the Hausdorff dimensionD(h) of the set of singularities of Hölder exponenth can be directly determined from the wavelet transform modulus maxima. The singularity spectrum so obtained is shown to be not disturbed by the presence, in the signal, of a superimposed polynomial behavior of ordern, provided one uses an analyzing wavelet that possesses at leastN>n vanishing moments. However, it is shown that aC behavior generally induces a phase transition in theD(h) singularity spectrum that somewhat masks the weakest singularities. This phase transition actually depends on the numberN of vanishing moments of the analyzing wavelet; its observation is emphasized as a reliable experimental test for the existence of nonsingular behavior in the considered signal. These theoretical results are illustrated with numerical examples. They are likely to be valid for a large class of fractal functions as suggested by recent applications to fractional Brownian motions and turbulent velocity signals.     

Geodesic Incompleteness in the ℂP 1 Model on a Compact Riemann Surface

It is proved that the moduli space of static solutions of the P ¹ model on spacetime ×, where is any compact Riemann surface, is geodesically incomplete with respect to the metric induced by the kinetic energy functional. The geodesic approximation predicts, therefore, that lumps can collapse and form singularities in finite time in these models.     

A generalized edge of the wedge theorem

The edge of the wedge theorem is generalized to the case where one only assumes the existence and equality of the distribution boundary values off _±(z) and all their derivatives on some analytic curveC inR ⁿ . Heref _±(z) are holomorphic inR ⁿ ±iC, respectively, whereC is a convex cone, andC has its tangent vector inC at every point. Under these assumptions there exists an analytic continuationf(z) holomorphic in some complex neighbourhood of the double cone generated byC. A proof is also given of the connection between the existence of a distribution boundary value and the growth of the holomorphic function near the boundary.