On the large-N limit of 3D and 4D hermitian matrix models

The large-N limit of the hermitian matrix model in three and four euclidean space-time dimensions is studied with the help of the approximate Renormalization Group recursion formula. The planar graphs contributing to wave-function, mass and coupling-constant renormalization are identified and summed in this approximation. In four dimensions the model fails to have an interacting continuum limit, but in three dimensions there is a non-trivial fixed point for the approximate RG relations. The critical exponents of the three-dimensional model at this fixed point are ν = 0.67 and η = 0.20. The existence (or non-existence) of the fixed point and the critical exponents display a fairly high degree of universality since they do not seem to depend on the specific (non-universal) assumptions made in the approximation.     

The geodesic spray,the vertical projection,and Raychaudhuri's equation

The metric connection on a space-time manifoldM defines on its tangent bundleTM a distribution of subspaces complementary to the vertical subspaces and therefore called horizontal. We give a formula for the Lie derivative with respect to the geodesic spray of the tensor field onTM which defines projection onto the vertical subspace along the horizontal subspace; and we show that this formula is a universal version of the equation, for a geodesic local vector field onM, whose trace is Raychaudhuri's equation.     

Harmonic oscillations of elastic continua and detection of gravitational waves

In this article it is shown that the equation derived by weber from the equation of geodesic deviation — the equation that constitutes the basis for the theoretical studies concerning the detection of gravitational waves — can be obtained as an approximation to an equation deduced from Cauchy's equation which governs an elastic continuum. This is achieved within the frame of the formalism of the theory of relativistic continua developed by A. C. Eringen and his collaborators and the present author. The use of piezoelectric crystals in order to measure the stresses that result from incident gravitational waves is also examined within the framework provided by this theory.     

Complex null geodesics in the extended Schwarzschild universe

The generic null geodesic of the Schwarzschild–Kruskal–Szekeres geometry has a natural complexification, an elliptic curve with a cusp at the singularity. To realize that complexification as a Riemann surface without a cusp, and also to ensure conservation of energy at the singularity, requires a branched cover of the space-time over the singularity, with the geodesic being doubled as well to obtain a genus two hyperelliptic curve with an extra involution. Furthermore, the resulting space-time obtained from this branch cover has a Hamiltonian that is null geodesically complete. The full complex null geodesic can be realized in a natural complexification of the Kruskal–Szekeres metric.     

The correlation of length spectra of two hyperbolic surfaces

We derive an asymptotic formula for the number of free homotopy classes on a closed surface which have (approximately) the same length with respect to two different hyperbolic structures on the surface. The growth rate in the asymptotic formula is described in terms of the thermodynamic formalism for the geodesic flow.     

On the Peierls transition in the presence of disorder

We consider the continuum approximation of a half-filled site-disordered Fröhlich chain in the mean-field approximation. We perform the configurational average over Gaussian-distributed and short-range-correlated potentials by means of the supersymmetry method and obtain an exact analytic formula for the integrated density of states containing many previously known special cases. It enables the minimization of the ground state energy with respect to the Peierls order parameter which is calculated in dependence on two disorder parameters describing forward and umklapp scattering. The Peierls transition proves to be continuous and takes place even if the density of states diverges at the Fermi energy, which eventually happens if only umklapp scattering is present.     

Schwarzschild黑洞整体正规时空中的测地运动

本文研究了Schwarzschild黑洞整体正规时空中的测地运动,求得了无奇性内部时空中检验粒子测地方程的解析解,并与奇性黑洞情况作了比较。 关键词：     

Global energy-momentum conservation in general relativity

It is shown that there exists a family of coordinate systems in which the energy-momentum tensor is globally conserved. Furthermore, this preferred class of frames includes geodesic systems with respect to any arbitrary point or timelike geodesic line. This implies a physically satisfactory conservation law with no need to introduce an extraneous pseudotensor.     

The Continuum Directed Random Polymer

Motivated by discrete directed polymers in one space and one time dimension, we construct a continuum directed random polymer that is modeled by a continuous path interacting with a space-time white noise. The strength of the interaction is determined by an inverse temperature parameter β, and for a given β and realization of the noise the path is a Markov process. The transition probabilities are determined by solutions to the one-dimensional stochastic heat equation. We show that for all β>0 and for almost all realizations of the white noise the path measure has the same Hölder continuity and quadratic variation properties as Brownian motion, but that it is actually singular with respect to the standard Wiener measure on C([0,1]).     

Numerical simulations of two-dimensional QED

We describe the computer simulation of two-dimensional QED on a 64 × 64 Euclidean space-time lattice using the Susskind lattice fermion action. The order parameter for chiral symmetry breaking and the low-lying meson masses are calculated for both the model with two continuum flavours, which arises naturally in this formulation, and the model with one continuum flavour obtained by including a nonsymmetric mass term and setting one fermion mass equal to the cut-off. Results are compared with those obtained using the quenched approximation, and with analytic predictions.     

A two-point boundary value problem on a Lorentz manifold arising in A. Poltorak’s concept of reference frame

Abstract

In A. Poltorak’s concept, the reference frame in General Relativity is a certain manifold equipped with a connection. The question under consideration here is whether it is possible to join two events in the space-time by a time-like geodesic if they are joined by a geodesic of the reference frame connection that has a time-like initial vector. This question is interpreted as whether an event belongs to the proper future of another event in the space-time in case it is so in the reference frame. For reference frames of two special types some geometric conditions are found under which the answer is positive.     

The Stability of Abstract Boundary Essential Singularities

The abstract boundary has, in recent years, proved a general and flexible way to define the singularities of space-time. In this approach an essential singularity is a non-regular boundary point of an embedding which is accessible by a chosen family of curves within finite parameter distance. Ashley and Scott proved the first theorem relating essential singularities in strongly causal space-times to causal geodesic incompleteness. Linking this with the work of Beem on the C ^r -stability of geodesic incompleteness allows proof of the stability of these singularities. Here I present this result stating the conditions under which essential singularities are C ¹-stable against perturbations of the metric.     

Extensions of the Taub and NUT spaces and extensions of their tangent bundles

A system of extensions of the Taub space and the NUT space with the topology due to Misner is constructed having the property: for each incomplete geodesic in these space-times, there is one and only one extension from the system into which the geodesic smoothly continues. Next, the notion of hypermanifold is introduced which is a generalization of tangent bundle of a space-time, and an untrivial hypermanifold is constructed that contains the tangent bundles of the Taub and NUT spaces as proper submanifolds, and within which almost all geodesics are complete. Locally, the hypermanifolds do not yield anything new, but they provide much broader choice of global properties than any four-dimensional space-time manifold.     

The Lepton Brusselator: Creation of structure in the early universe

The creation of structure in the Lepton Era of the early universe is discussed in terms of a suitably chosen dynamical system of Brusselator type. A bifurcation analysis is performed and a criterion is extracted for the onset of instabilities in the system which in turn lead to self-organizing processes in the associated conglomerate of elementary particles which may be visualized as an imperfect fluid in the hydrodynamic approximation chosen here. This may be interpreted as a symmetry-breaking mechanism which might be responsible for the creation of protostructures on space-time due to density fluctuations, the latter being thought of as controlling the further agglomeration of matter once matter and radiation have decoupled at a temperature level of some 4000 K. This may eventually lead to the formation of galaxies as are observed in the present stage of the universe. The influence of curvature is discussed in considering a metric background of Bianchi type I with respect to the onset of instability. The notion of a local concept on space-time is reviewed from an interactive point of view.     

Real miniheaven

For the class of axisymmetric gravitational fields originally treated by Bondi, the existence of a two-dimensional family of asymptotically shear-free slices (good slices) of null infinity is exhibited. They form a real two-dimensional submanifold (miniheaven) of the four-dimensional complex manifold of good slices (heaven) constructed by Newman. The real good slices can be described as surfaces of revolution of geodesic curves. Geodesic deviation due to the Bondi news function gives rise to the geometrical properties of miniheaven. These properties are examined for the special case of axisymmetric Robinson-Trautman solutions. The Robinson-Trautman family of shear-free null hypersurfaces corresponds to a geodesic in miniheaven possessing an infinite number of angular momentum and supermomentum constants of the motion. The global properties of the Robinson-Trautman miniheaven are examined in the linear approximation in which a remarkably simple expression for the curvature of miniheaven is found. The asymptotic properties of miniheaven turn out to be identical to those of the original Robinson-Trautman space-time.Research supported by grant No. MPS74-18020 from the National Science Foundation. The work constitutes part of a doctoral dissertation submitted by R. Dubisch to the University of Pittsburgh.     

Singularity of Projections of 2-Dimensional Measures Invariant Under the Geodesic Flow

We show that on any compact Riemann surface with variable negative curvature there exists a measure which is invariant and ergodic under the geodesic flow and whose projection to the base manifold is 2-dimensional and singular with respect to the 2-dimensional Lebesgue measure.     

Mappings of empty space-times leaving the curvature tensor invariant

It is shown that if in some local coordinate system the componentsR ⁱ _jkl of the curvature tensor of an empty space-time are known, then, provided the space-time is not of Petrov typeN with hypersurface orthogonal geodesic rays, the components of the metric tensor are uniquely determined up to a trivial constant scaling factor. The Petrov type-N empty space-times with hypersurface orthogonal geodesic rays are investigated. The most general mappings leaving the curvature tensorR ⁱ _jkl invariant are found for each class of these space-times.     

Orthonormal tetrads and the charged fluid in general relativity

A space-time filled with the self-gravitating charged fluid with constant electric permitivity and constant magnetic permeability is investigated. On expressing the stress-energy tensor in terms of an orthonormal tetrad, the equations of motion and Maxwell equations are formulated. In case of the geodesic flow, the conditions for divergence-free electric and magnetic fields are obtained. It is shown that the space-time permeated by the charged fluid with the electric field orthogonal to the magnetic one is embedded in 5-dimensional flat class-one space-time if and only if the electromagnetic energy flux vector vanishes.     

The computer simulation and classical interpretation of fast ion emission profiles near planes

Abstract

The techniques of computer simulation are used to generate the velocity distribution of ions emerging near close packed planes from a crystal surface. We concern ourselves in particular with protons and α-particles, 5 to 100 keV, near (100) planes in copper. The ions are initially emitted from either a lattice site-the ‘blocking case ‘or any other given point within a unit cell’ intermediate emission’. The resultant profiles are compared with an analytic model due to Andersen⁽⁶⁾ based on the continuum approximation, and the validity of this model is thereby assessed. A simpler analytic model in which the interplanar continuum potential is approximated by a pure harmonic is also developed and compared with the previous two. Using the harmonic approximation we obtain a relatively simple relationship between the yield measured exactly parallel to a plane and the position of the original emission site. It is suggested that such a relationship could be of importance for experiments that can measure nuclear lifetimes of the order of 10^?18 to 10^?16 seconds.