Propagation of density perturbations in the deDonder gauge on a gravitational background field II

For the problem of propagation of density waves in a preexisting gravitational field, the advantages of the deDonder gauge over the commonly used synchronous gauge are outlined. In a background matter substratum withp as equation of state there are in the deDonder gauge only decaying modes of the perturbation density contrast with arbitrary large spatial extension, whereas in synchronous gauge there is one growing mode (calculated for vanishing spatial divergency of the perturbation in the 4-velocity, i.e.,usk(1),j/j0). The calculations are extended to the case of finite spatial extensions of the density perturbations. This is done by expanding all perturbations in a power series of the inverse square of the speed of light with the result of getting a recursive set of differential equations in both gauges for the equation of motion of the density perturbations. The lowest orders of this equation are the same in both gauges, but only in the deDonder gauge is the correct Newtonian limit of propagation of density waves in an expanding universe obtained. The correction by the next higher orders in the deDonder gauge are dependent explicitly on the spatial extension of the perturbations; whereas in synchronous gauge this is not the case. For attaining the Newtonian limit this dependence is a necessary condition. At appropriately large spatial extensions, however exact, this dependence in deDonder gauge leads ultimately to a decaying of density contrast modes growing in zeroth order (at least forp=0 andp/3 as equations of state for the background matter substratum). Hence, there are upper boundaries in the spatial extensions of instable growing modes of density contrast.     

Propagation of self-gravitating density waves in the deDonder gauge on a gravitational background field

Arguments are given for using the deDonder instead of the synchronous gauge in describing the propagation of density perturbations in a preexisting gravitational field. Since in the deDonder gauge the corresponding reference frame is fixed on the background, the physical interpretation of results is obvious, while in the synchronous gauge it is at least very difficult to extract the physical consequences from the results. For the propagation of density perturbations, with large spatial extension, a decisive difference is found between the two gauges. While in the synchronous gauge there is a growing mode in the density contrast (at least for adiabatic perturbations on a background matter substratum withp as equation of state), in the deDonder gauge there is not. The calculation in deDonder gauge leads to upper boundaries for the spatial extension of unstable density perturbations, and thus may give a hint for upper boundaries of galaxy masses.     

Propagation of self-gravitating density waves in the deDonder gauge: the physical importance of gauge solutions and the problem of initial conditions

The propagation of perturbations on a spatially flat Robertson-Walker background is studied within linear perturbation theory in deDonder gauge and for comparison in synchronous gauge. The metric perturbations should be determined uniquely by the density/pressure perturbations, therefore only two initial conditions, namely for the density contrast and its time derivative, should be needed. Since the number of fundamental solutions for the density perturbations is higher than 2 in both gauges (6 resp. 3) an additional reduction of possible initial conditions, resp. a physically motivated exclusion of solutions, is needed. It is shown that the common treatment of excluding the so-called gauge solutions (solutions which can be gauged to zero in an already chosen gauge) leads to unphysical results. If gauge solutions are excluded the density perturbation solutions are the same in both gauges. But the correct Newtonian limit — which is present in deDonder gauge but not in synchronous gauge — is bound to the differences in the two gauges for large spatial scales of perturbations. Furthermore, compressional wave solutions should vanish for infinite spatial scales of perturbations (isotropy), but this is guaranteed in deDonder gauge by gauge solutions again. Gauge solutions should therefore not be taken as unphysical.     

Global C*-Dynamics and Its KMS States of Weakly Inhomogeneous Bipolaronic Superconductors

We establish the limiting dynamics of a class of inhomogeneous bipolaronic models for superconductivity which incorporate deviations from the homogeneous models strong enough to require disjoint representations. The models are of the Hubbard type and the thermodynamics of their homogeneous part has been already elaborated by the authors. Now the dynamics of the systems is evaluated in terms of a generalized perturbation theory and leads to a C*-dynamical system over a classically extended algebra of observables. The classical part of the dynamical system, expressed by a set of 15 nonlinear differential equations, is observed to be independent from the perturbations. The KMS states of the C*-dynamical system are determined on the state space of the extended algebra of observables. The subsimplices of KMS states with unbroken symmetries are investigated and used to define the type of a phase. The KMS phase diagrams are worked out explicitly and compared with the thermodynamic phase structures obtained in the preceding works.     

Linear and nonlinear response of discrete dynamical systems II: Chaotic attractors

We investigate the average response to small external perturbations for discrete dynamical systems with chaotic attractors. The average linear response satisfies a fluctuation theorem, and in general diverges exponentially in the long-time limitt. It vanishes identically for allt>0 only in a number of special cases including the logistic model with bifurcation parameter =4. The nonlinear response turns out to be crucial. Its average is analyzed for a time-localized (pulse) perturbation. Near the onset of chaos it exhibits universal scaling behaviour expressed by two critical exponents. For static perturbations the resulting dynamics is extremely sensitive to the perturbation strength.Work supported by the Swiss National Science Foundation     

Analytic relativistic model for a superdense star

A new analytic relativistic model has been obtained for superdense stars by solving the Einstein field equations for the spherically symmetric and static case. The model stands all the tests of physical reality. The density,, remains positive under all conditions and decreases smoothly from the center to the surface of the structure. The pressure,P, the ratioP/ anddP/d decrease with decreasing density. For all the finite values of pressure, the configurations are stable under radial perturbation. FordP/d 1, the maximum mass of neutron star model is 4.17, and the surface and central red shifts are 0.63 and 1.60, respectively. For an infinite central pressure, the surface red shift is 1.61. The structures are bound and the binding coefficients increase with the increasing mass.     

Asymptotic regularizations and shocks in noninductive heat-current-free relativistic fluids

We have developed elsewhere (cf. [1]) a method, which we call asymptotic regularizations, designed to give shock and infinitesimal shock wave equations for systems of partial differential equations. What we intend to do here is to illustrate its usefulness when applied to a specific system of partial differential equations, namely, that of noninductive, heatcurrent-free perfect relativistic fluids.     

Generic instability of rotating relativistic stars

All rotating perfect fluid configurations having two-parameter equations of state are shown to be dynamically unstable to nonaxisymmetric perturbations in the framework of general relativity. Perturbations of an equilibrium fluid are described by means of a Lagrangian displacement, and an action for the linearized field equations is obtained, in terms of which the symplectic product and canonical energy of the system can be expressed. Previous criteria governing stability were based on the sign of the canonical energy, but this functional fails to be invariant under the gauge freedom associated with a class of trivial Lagrangian displacements, whose existence was first pointed out by Schutz and Sorkin [12]. In order to regain a stability criterion, one must eliminate the trivials, and this is accomplished by restricting consideration to a class of canonical displacements, orthogonal to the trivials with respect to the symplectic product. There nevertheless remain perturbations having angular dependencee ^im ( the azimuthal angle) which, for sufficiently largem, make the canonical energy negative; consequently, even slowly rotating stars are unstable to short wavelength perturbations. To show strict instability, it is necessary to assume that time-dependent nonaxisymmetric perturbations radiate energy to null infinity. As a byproduct of the work, the relativistic generalization of Ertel's theorem (conservation of vorticity in constant entropy surfaces) is obtained and shown to be Noetherrelated to the symmetry associated with the trivial displacements.Supported in part by the National Science Foundation under grant number MPS 74-17456     

Perturbations of a universe filled with dust and radiation

A first-order perturbation approach tok=0 Friedmann cosmologies filled with dust and radiation is developed. Adopting the coordinate gauge comoving with the perturbed matter, and neglecting the vorticity of the radiation, a pair of coupled equations is obtained for the traceh of the metric perturbations and for the velocity potentialv. A power series solution with upward cutoff exists such that the leading terms for large values of the dimensionless time agree with the relatively growing terms of the dust solution of Sachs and Wolfe.     

Unified fields of analytic space-time-energy

An analytic gravitational fieldZ (Z ^y ) is shown to include electromagnetic phenomena. In an almost flat and almost static complex geometryds ² =zdzdz of four complex variables z=t, x, y, x the field equationsR Rz = –(U U – Z ) imply the conventional equations of motion and the conventional electromagnetic field equations to first order if =(Z ^v) and =(z ) are expressed in terms of the conventional mass density function , the conventional charge density function , and a pressurep as follows: v=const=p/c ²–10^–29 gm/cm³.     

Relativistic Hydrodynamic Cosmological Perturbations

Relativistic cosmological perturbation analysescan be made based on several different fundamental gaugeconditions. In the pressureless limit the variables incertain gauge conditions show the correct Newtonian behaviors. Considering the generalcurvature (K) and the cosmological constant ()in the background medium, the perturbed density in thecomoving gauge, and the perturbed velocity and the perturbed potential in the zero-shear gaugeshow the same behavior as the Newtonian ones in generalscales. In the first part, we elaborate these Newtoniancorrespondences. In the second part, using the identified gauge-in variant variables withcorrect Newtonian correspondences, we present therelativistic results with general pressures in thebackground and perturbation. We present the generalsuper-sound-horizon scale solutions of the above mentionedvariables valid for general K, Lambda, and generallyevolving equation of state. We show that, for vanishingK, the super-sound-horizon scale evolution ischaracterised by a conserved variable which is the perturbedthree-space curvature in the comoving gauge. We alsopresent equations for the multi-component hydrodynamicsituation and for the rotation and gravitational wave.     

Kinetics of small perturbations in an isotropic world

Long-wavelength gravitational perturbations are studied in an isotropic expanding universe filled with an ultrarelativistic gas. A kinetic study in the collisionless approximation shows that scalar and vector perturbations which appear at a time ₀ 1/n, where N is the wave vector and is the time coordinate x⁴, grow if the perturbation of the macroscopic momentum density of the gas at time ₀ is nonvanishing. The growth continues until the time ₁=27₀, at which the perturbation of the macroscopic momentum density of the gas vanishes. A solution is also derived for tensor perturbations in the limit n 1.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 37–45, April, 1978.     

A functional integral approach to shock wave solutions of Euler equations with spherical symmetry

Forn×n systems of conservation laws in one dimension without source terms, the existence of global weak solutions was proved by Glimm [1]. Glimm constructed approximate solutions using a difference scheme by solving a class of Riemann problems.In this paper, we consider the Cauchy problem for the Euler equations in the spherically symmetric case when the initial data are small perturbations of the trivial solution, i.e.,u0 and constant, whereu is velocity and is density. We show that this Cauchy problem can be reduced to an ideal nonlinear problem approximately. If we assume all the waves move at constant speeds in the ideal problem, by using Glimm's scheme and an integral approach to sum the contributions of the reflected waves that correspond to each path through the solution, we get uniform bounds on theL norm and total variational norm of the solutions for all time. The geometric effects of spherical symmetry leads to a non-integrable source term in the Euler equations. Correspondingly, we consider an infinite reflection problem and solve it by considering the cancellations between reflections of different orders in our ideal problem. Thus we view this as an analysis of the interaction effects at the quadratic level in a nonlinear model problem for the Euler equations. Although it is far more difficult to obtain estimates in the exact solutions of the Euler equations due to the problem of controlling the time at which the cancellations occur, we believe that this analysis of the wave behaviour will be the first step in solving the problem of existence of global weak solutions for the spherically symmetric Euler equations outside of fixed ball.     

A topological characterization of classical BRST cohomology

The recent identification of classical BRST cohomology with the vertical cohomology of a certain fibration is used to compute it in terms of the classical observables and the topology of the gauge orbits. When the gauge orbits are compact and orientable, a duality theorem is exhibited.     

Geometro-stochastic quantization of a theory for extended elementary objects

The geometro-stochastic quantization of a gauge theory based on the (4,1)-de Sitter group is presented. The theory contains an intrinsic elementary length parameter R of geometric origin taken to be of a size typical for hadron physics. Use is made of a soldered Hilbert bundle over curved spacetime carrying a phase space representation of SO(4, 1) with the Lorentz subgroup related to a vierbein formulation of gravitation. The typical fiber of is a resolution kernel Hilbert space constructed in terms of generalized coherent states related to the principal series of unitary irreducible representations of SO(4, 1), namely de Sitter horospherical waves for spinless particles characterized by the parameter . The framework is, finally, extended to a quantum field-theoretical formalism by using bundles with Fock space fibers constructed from .Supported in part by NSERC Research Grant No. A5206.     

Point-defect-driven dislocation patterning during deformation under irradiation

Dislocation patterning driven by interactions of dislocations with deformation-induced point defects is considered. The effect of concurrent irradiation-induced production of point defects is also included. The uniform time-dependent solution of the set of equations describing the evolution of the system is probed by small periodic perturbations. A linear stability condition obtained in this way as well as the preferred wavelength of the emerging pattern depend on the values of the parameters reflecting biases in the production and annihilation of vacancies and interstitial atoms. It is proposed that by studying the effect of different types of radiation and different irradiation intensities on the occurrence and the wavelength of the dislocation pattern information about the deformation-induced point defect production bias may be obtained.     

Gauge theories on four dimensional Riemannian manifolds

This paper develops the Riemannian geometry of classical gauge theories — Yang-Mills fields coupled with scalar and spinor fields — on compact four-dimensional manifolds. Some important properties of these fields are derived from elliptic theory: regularity, an energy gap theorem, the manifold structure of the configuration space, and a bound for the supremum of the field in terms of the energy. It is then shown that finite energy solutions of the coupled field equations cannot have isolated singularities (this extends a theorem of K. Uhlenbeck).The author holds an A.M.S. Postdoctoral Fellowship     

Proof of chiral symmetry breaking in lattice gauge theory

Using the method of infrared bounds and partial-integration formulas, we prove that there is a chiral phase transition in four-dimensional strongly coupled lattice gauge theory with gauge group U(N) and staggered fermions for all N5.     

Yang-Mills fields which are not self-dual

The purpose of this paper is to prove the existence of a new family of non-self-dual finite-energy solutions to the Yang-Mills equations on Euclidean four-space, withSU(2) as a gauge group. The approach is that of equivariant geometry: attention is restricted to a special class of fields, those that satisfy a certain kind of rotational symmetry, for which it is proved that (1) a solution to the Yang-Mills equations exists among them; and (2) no solution to the self-duality equations exists among them. The first assertion is proved by an application of the direct method of the calculus of variations (existence and regularity of minimizers), and the second assertion by studying the symmetry properties of the linearized self-duality equations. The same technique yields a new family of non-self-dual solutions on the complex projective plane.