Conformal cosmological model and SNe Ia data

Now there is a huge scientific activity in astrophysical studies and cosmological ones in particular. Cosmology transforms from a pure theoretical branch of science into an observational one. All the cosmological models have to pass observational tests. The supernovae type Ia (SNe Ia) test is among the most important ones. If one applies the test to determine parameters of the standard Friedmann-Robertson-Walker cosmological model one can conclude that observations lead to the discovery of the dominance of the ?? term and as a result to an acceleration of the Universe. However, there are big mysteries connected with an origin and an essence of dark matter (DM) and the ?? term or dark energy (DE). Alternative theories of gravitation are treated as a possible solution of DM and DE puzzles. The conformal cosmological approach is one of possible alternatives to the standard ??CDM model. As it was noted several years ago, in the framework of the conformal cosmological approach an introduction of a rigid matter can explain observational data without ?? term (or dark energy). We confirm the claim with much larger set of observational data.     

Conformally rescaled spacetimes and Hawking radiation

We study various derivations of Hawking radiation in conformally rescaled metrics. We focus on two important properties, the location of the horizon under a conformal transformation and its associated temperature. We find that the production of Hawking radiation cannot be associated in all cases to a trapping horizon because its location is not invariant under a conformal transformation. We also find evidence that the temperature of the Hawking radiation should transform simply under a conformal transformation, being invariant for asymptotic observers in the limit that the conformal transformation factor is unity at their location.     

共形整流罩像差特性分析及校正方法

共形结构不仅具有良好的空气动力学性能,且不存在由于表面不连续造成的热梯度等问题,因而采用该结构的导弹整流罩更有利于导弹系统作战性能的提高。但共形结构中采用的非球面罩曲面使整流罩表现出许多不同于球形结构的动态特性,这给导引头中光学系统的设计带来很多困难。在分析共形结构一阶特性的基础上,利用矢量像差理论详细分析该结构中初级像差产生的原因及特点,并提出通过控制元件的倾斜和偏心来平衡所有观察视场中像差的方法。软件分析结果表明：加入倾斜偏心元件后可适当放大小观察视场中的像差,共形整流罩在各观察视场中具有较为稳定的像差特性,有效地改善了该结构的成像质量。     

Path-integral derivation of quantum duality in nonlinear sigma-models

We show that a previously derived shift in the dilaton field, which necessarily augments the classical effects of duality transformation on the geometry of a nonlinear sigma-model if conformal invariance is to be preserved at the one-loop level, can be extended without change to the case of sigma-models with Wess-Zumino-Witten term (torsion) before and after duality. We also construct a path-integral implementation of the duality transformation, and discover the origin of the dilaton shift in a functional determinant resulting from the elimination of the first-order field. The path-integral formulation in principle allows a derivation of “quantum” duality transformations which preserve conformal invariance to all orders in α', the string tension parameter.     

Two discretisations of the Ermakov-Pinney equation

We propose two candidates for discrete analogues to the nonlinear Ermakov-Pinney equation. The first one based on an association with a two-dimensional conformal mapping defines a second-degree difference scheme. It possesses the same features as in the continuum: a nonlinear superposition principle relating its general solution to a second-order linear difference equation and by direct linearisation a relationship with a third-order difference equation. The second form, which is new, is obtained from a slight improvement of the superposition principle. It has the advantage of leading to a first degree difference scheme and preserves all the nice properties of its linearisation.     

Causal boundary entropy from horizon conformal field theory

The effective quantum theory of near horizon regions of classical four-dimensional spatially flat, Friedman-Robertson-Walker spacetimes is shown to be approximately a two-dimensional conformal field theory. The central charge and expectation value of the Hamiltonian of this theory, and the statistical entropy of horizon states which can be calculated using Cardy's formula, are all proportional to the horizon area in units of Newton's constant. The proportionality constant which is determined by Planck scale physics can be fixed such that the entropy is equal to a quarter of the horizon area in units of Newton's constant, in agreement with thermodynamic considerations.     

TFT construction of RCFT correlators IV:: Structure constants and correlation functions

We compute the fundamental correlation functions in two-dimensional rational conformal field theory, from which all other correlators can be obtained by sewing: the correlators of three bulk fields on the sphere, one bulk and one boundary field on the disk, three boundary fields on the disk, and one bulk field on the cross cap. We also consider conformal defects and calculate the correlators of three defect fields on the sphere and of one defect field on the cross cap.Each of these correlators is presented as the product of a structure constant and the appropriate conformal two- or three-point block. The structure constants are expressed as invariants of ribbon graphs in three-manifolds.     

Anti-self-dual Conformal Structures with Null Killing Vectors from Projective Structures

Using twistor methods, we explicitly construct all local forms of four–dimensional real analytic neutral signature anti–self–dual conformal structures (M, [g]) with a null conformal Killing vector. We show that M is foliated by anti-self-dual null surfaces, and the two-dimensional leaf space inherits a natural projective structure. The twistor space of this projective structure is the quotient of the twistor space of (M, [g]) by the group action induced by the conformal Killing vector. We obtain a local classification which branches according to whether or not the conformal Killing vector is hyper-surface orthogonal in (M, [g]). We give examples of conformal classes which contain Ricci–flat metrics on compact complex surfaces and discuss other conformal classes with no Ricci–flat metrics. Dedicated to the memory of Jerzy Plebański     

Theory of the conformally invariant mass

By interpreting the conformal transformations as space-time-dependent change of units and introducing the concept of the conformally invariant mass and charge, we develop new conformally invariant Maxwell equations with source terms and equations of motion for massive particles. Although the usual equations of motion with mass terms break the conformal symmetry, it is shown that the Minkowski space is not the most general framework to describe physical processes and there exists a wider consistent dynamics in which conformal invariance is exact. New results also include the general transformation laws of the electromagnetic fields, of currents and force densities. The theory leads naturally to an affine connection and to the 21-parameter inhomogeneous conformal group, ISO(4, 2).     

Higher spin fields and the Gelfand-Dickey algebra

We show that in 2-dimensional field theory, higher spin algebras are contained in the algebra of formal pseudodifferential operators introduced by Gelfand and Dickey to describe integrable nonlinear differential equations in Lax form. The spin 2 and 3 algebras are discussed in detail and the generalization to all higher spins is outlined. This provides a conformal field theory approach to the representation theory of Gelfand—Dickey algebras.Supported in part by the NSF Grant PHY-84-04931     

Complete Classification of Conformal Killing Symmetries of Plane-Symmetric Static Spacetimes in Teleparallel Theory

We have studied the conformal, homothetic and Killing vectors in the context of teleparallel theory of gravitation for plane-symmetric static spacetimes. We have solved completely the non-linear coupled teleparallel conformal Killing equations. This yields the general form of teleparallel conformal vectors along with the conformal factor for all possible cases of metric functions. We have found four solutions which are divided into one Killing symmetries and three conformal Killing symmetries. One of these teleparalel conformal vectors depends on x only and other is a function of all spacetime coordinates. The three conformal Killing symmetries contain three proper homothetic symmetries where the conformal factor is an arbitrary non-zero constant.     

Multi-cut solutions of Laplacian growth

A new class of solutions to Laplacian growth (LG) with zero surface tension is presented and shown to contain all other known solutions as special or limiting cases. These solutions, which are time-dependent conformal maps with branch cuts inside the unit circle, are governed by a nonlinear integral equation and describe oil fjords with non-parallel walls in viscous fingering experiments in Hele-Shaw cells. Integrals of motion for the multi-cut LG solutions in terms of singularities of the Schwarz function are found, and the dynamics of densities (jumps) on the cuts are derived. The subclass of these solutions with linear Cauchy densities on the cuts of the Schwarz function is of particular interest, because in this case the integral equation for the conformal map becomes linear. These solutions can also be of physical importance by representing oil/air interfaces, which form oil fjords with a constant opening angle, in accordance with recent experiments in a Hele-shaw cell.     

Conformal invariance and conserved quantity of third-order Lagrange equations for non-conserved mechanical systems

This paper studies conformal invariance and conserved quantity of third-order Lagrange equations for non-conserved mechanical systems. Third-order Lagrange equations, the definition and a determining equation of conformal invariance of the system are presented. The conformal factor expression is deduced from conformal invariance and Lie symmetry. The necessary and sufficient condition that conformal invariance of the system would have Lie symmetry under single-parameter infinitesimal transformations is obtained. The corresponding conserved quantity of conformal invariance is derived with the aid of a structure equation. Lastly, an example is given to illustrate the application of the results.     

Gravitational coupling,dynamical changes of units and the cosmological constant problem

A spacetime interval connecting two neighbouring points can be measured in different unit systems.For instance,it can be measured in atomic unit defined in terms of fundamental constants existing in quantum theories.It is also possible to use a gravitational unit which is defined by the use of properties of macroscopic objects.These two unit systems are usually regarded as indistinguishable up to a constant conversion factor.Here we consider the possibility that these two units are related by an epoch-dependent conversion factor.This is a dynamical changes of units.Regarding a conformal transformation as a local unit transformation,we use a gravitational model in which the gravitational and the matter sectors are given in different conformal frames(or unit systems).It is relevant to the cosmological constant problem,namely the huge discrepancy between the estimated and the observational values of the cosmological constant in particle physics and cosmology,respectively.We argue that the problem arises when one ignores evolution of the conversion factor relating the two units during expansion of the Universe.Connection of the model with violation of equivalence principle and possible variation of fundamental constants are also discussed.     

Conformal couplings and “azimuthal matching” of QCD pomerons

Using the asymptotic conformal invariance of perturbative QCD we derive the expression of the coupling of external states to all conformal spin p components of the forward elastic amplitude. Using the wave function formalism for structure functions at small x, we derive the perturbative coupling of the virtual photon for , which is maximal for linear transverse polarization. The non-perturbative coupling to the proton is discussed in terms of “azimuthal matching” between the proton color dipoles and the configurations of the photon. As an application, the recent conjecture of a second QCD pomeron related to the conformal spin-1 component is shown to rely upon a strong azimuthal matching of the component in –proton scattering. Received: 25 October 1999 / Revised version: 19 January 2000 / Published online: 14 April 2000     

Space-times admitting a three-dimensional conformal group

Perfect fluid space-times admitting a three-dimensional Lie group of conformal motions containing a two-dimensional Abelian Lie subgroup of isometries are studied. Demanding that the conformal Killing vector be proper (i.e., not homothetic nor Killing), all such space-times are classified according to the structure of their corresponding three-dimensional conformal Lie group and the nature of their corresponding orbits (that are assumed to be non-null). Each metric is then explicitly displayed in coordinates adapted to the symmetry vectors. Attention is then restricted to the diagonal case, and exact perfect fluid solutions are obtained in both the cases in which the fluid four-velocity is tangential or orthogonal to the conformal orbits, as well as in the more general tilting case.     

Twistor Geometry of a Pair of Second Order ODEs

We discuss the twistor correspondence between path geometries in three dimensions with vanishing Wilczynski invariants and anti-self-dual conformal structures of signature (2, 2). We show how to reconstruct a system of ODEs with vanishing invariants for a given conformal structure, highlighting the Ricci-flat case in particular. Using this framework, we give a new derivation of the Wilczynski invariants for a system of ODEs whose solution space is endowed with a conformal structure. We explain how to reconstruct the conformal structure directly from the integral curves, and present new examples of systems of ODEs with point symmetry algebra of dimension four and greater which give rise to anti–self–dual structures with conformal symmetry algebra of the same dimension. Some of these examples are (2, 2) analogues of plane wave space–times in General Relativity. Finally we discuss a variational principle for twistor curves arising from the Finsler structures with scalar flag curvature.