On Level Sets of Lorentzian Distance Function

Level sets of Lorentzian distance functions with respect to a point and with respect to an achronal spacelike hypersurface, are analyzed. Some bounds for the Laplacian of such Lorentzian distance functions are obtained and, in relation to them, some spacetime singularity theorems are given.Supported by project BFM2001-3778-C03-01 (Spain).     

Killing spinors on Lorentzian manifolds

The aim of this paper is to describe some results concerning the geometry of Lorentzian manifolds admitting Killing spinors. We prove that there are imaginary Killing spinors on simply connected Lorentzian Einstein–Sasaki manifolds. In the Riemannian case, an odd-dimensional complete simply connected manifold (of dimension n≠7) is Einstein–Sasaki if and only if it admits a non-trivial Killing spinor to . The analogous result does not hold in the Lorentzian case. We give an example of a non-Einstein Lorentzian manifold admitting an imaginary Killing spinor. A Lorentzian manifold admitting a real Killing spinor is at least locally a codimension one warped product with a special warping function. The fiber of the warped product is either a Riemannian manifold with a real or imaginary Killing spinor or with a parallel spinor, or it again is a Lorentzian manifold with a real Killing spinor. Conversely, all warped products of that form admit real Killing spinors.     

A formula for the contribution of a resonance to the canonical partition function

An explicit expression for the Laplace transform of a Lorentzian function is applied to evaluate the contribution to the canonical partition function of ideal-gas molecules stemming from resonances. This formula can be used instead of numerical integration. A discussion of potential errors in replacing this contribution by the conventional formula for the Boltzmann weight is given. When generalised to the complex plane, the known expression is shown to be discontinuous and a new formulation is proposed that removes the discontinuity. As a result, an explicit expression is obtained for the Fourier transform of a Lorentzian function that is truncated at a lower energy bound.     

Analytical solution of Everett integral using Lorentzian Preisach function approximation

In this paper, we take in to account the Lorentzian function, already proposed as analytical function approximation in scalar Preisach-type modeling of hysteresis of soft magnetic materials. Preliminarily, we point out the properties of the Lorentzian function and the physical and mathematical meaning of its parameters. Successively, we show how the use of the Lorentzian function approximation allows to solve in complete analytical way the Everett's integral. In particular, we present in the paper the analytical expression in closed form of the first magnetization curve, of the symmetric and non-symmetric minor loops, and of the first- and second-order reversal curves. In addition, we show the use of the complete analytical formulas of the symmetric magnetic loops above-mentioned, applied to a simple identification procedure of the Lorentzian function parameters, by the knowledge of the measured major loop. Finally, in order to show the practical use of the analytical expression found, some computation examples and comparisons with experimental data are shown.     

Locally inertial reference frames in Lorentzian and Riemann‐Cartan spacetimes

In this paper the concept of locally inertial reference frames ( LIRF s) in Lorentzian and Riemann‐Cartan spacetime structures is scrutinized. A rigorous mathematical definition of a LIRF in both structures is given, something that needs preliminary a clear mathematical distinction between the concepts of observers, reference frames, naturally adapted coordinate functions to a given reference frame and which properties may characterize an inertial reference frame (if any) in the Lorentzian and Riemann‐Cartan structures. Hopefully, the paper clarifies some obscure issues associated to the concept of a LIRF appearing in the literature, in particular the relationship between LIRF s in Lorentzian and Riemann‐Cartan spacetimes and Einstein's most happy thought, i.e., the equivalence principle.     

Homogeneous structures on three-dimensional Lorentzian manifolds

We prove that any non-symmetric three-dimensional homogeneous Lorentzian manifold is isometric to a Lie group equipped with a left-invariant Lorentzian metric. We then classify all three-dimensional homogeneous Lorentzian manifolds.     

A new procedure for obtaining the Voigt function dependent upon the complex error function

This work proposes a new method for obtaining the differential equation of the Voigt function and, from this equation, expressing the Voigt function as dependent upon the complex error function. In addition, the integral expression of the successive derivatives of the Voigt function is given, and from this a method is generalized which permits the representation, also, of other functions depending on the complex error function. This enables us to simplify other functions which are the convolution of a Gaussian function with rational polynomial functions. Moreover, the relationship between the Lorentzian (w_L), Gaussian (w_G) and Voigt (w_V) widths at half maximum for the function is given, which is of great interest in diverse branches of physics, such as plasma spectroscopy, astrophysics, nuclear magnetic resonance, etc.     

Twistor spinors on Lorentzian symmetric spaces

An indecomposable Riemannian symmetric space which admits non-trivial twistor spinors has constant sectional curvature. Furthermore, each homogeneous Riemannian manifold with parallel spinors is flat. In the present paper we solve the twistor equation on all indecomposable Lorentzian symmetric spaces explicitly. In particular, we show that there are — in contrast to the Riemannian case — indecomposable Lorentzian symmetric spaces with twistor spinors, which have non-constant sectional curvature and non-flat and non-Ricci flat homogeneous Lorentzian manifolds with parallel spinors.     

Semi-symmetric Lorentzian hypersurfaces in Lorentzian space forms

In the present paper, we shall investigate and classify Lorentzian hypersurfaces in Lorentzian space forms satisfying some curvature conditions. We shall focus especially on semi-symmetric Lorentzian hypersurfaces. Those of constant curvature and those so-called “good” are explicitly described and classified. We shall also classify Einstein–Lorentzian hypersurfaces.     

Holomorphic 4-Metrics and Lorentzian Structures

Methods of constructing Lorentzian metrics on real four dimensional manifolds from complex and holomorphic 4-metrics are presented. In particular half-flat holomorphic 4-metrics are used to construct real Lorentzian 4-metrics. Holomorphic and real solutions of Einstein's equations, and relations between them, are discussed.     

Comparison theorems in Lorentzian geometry and applications to spacelike hypersurfaces

In this paper we prove Hessian and Laplacian comparison theorems for the Lorentzian distance function in a spacetime with sectional (or Ricci) curvature bounded by a certain function by means of a comparison criterion for Riccati equations. Using these results, under suitable conditions, we are able to obtain some estimates on the higher order mean curvatures of spacelike hypersurfaces satisfying a Omori-Yau maximum principle for certain elliptic operators.     

基于洛伦兹函数微扰的磁光光纤布喇格光栅线性透射谱和非线性双稳特性研究

基于非线性耦合模理论,利用逆向递推龙格-库塔法,数值研究了在磁光耦合系数中引入洛伦兹函数微扰后对磁光光纤布喇格光栅线性透射谱和非线性双稳特性的影响.研究结果表明:给磁光耦合系数引入洛伦兹微扰能在线性透射谱阻带中打开线宽极窄的一个透射窗口,微扰宽度和微扰中心位置可以影响透射窗口的位置、宽度以及峰值大小;当微扰宽度和微扰中心位置发生变化时,光栅的双稳特性表现出明显的差异,合理地选择微扰参量可以实现对其双稳特性的优化.     

洛伦兹光谱线型的高层大气风场被动探测原理分析

通过探测高层大气中气辉(极光)辐射线的多普勒频移,可以反演出高层大气中的速度、温度和压强等物理量。以高层大气中的极光(原子氧跃迁所辐射的两条主要谱线)为被探测源,对于洛伦兹光谱线型高层大气风场的探测原理和方法进行了研究;给出了基于洛伦兹光谱线型高层大气的速度场、温度场和压力场的分布规律和理论计算公式;采用计算机模拟,描绘了洛伦兹光谱线型风场的误差曲线,表明了洛伦兹光谱线型在高层大气风场探测中占有相当重要的地位。     

Study of a lidar for measurement of thermospheric He I 1083 nm for Lorentzian profile

Using the lidar backscatter cross-section and the normalized Lorentzian profile, we investigate the backscatter cross-section and the returned signal of He I 1083 nm for the Lorentzian profile theoretically. Taking the average temperature and pressure, we find that the backscattering cross-section for Lorentzian profile is larger than that for Gaussian profile near the line center, while is smaller than that for Gaussian profile far away from the line center. Through simulation, we find that the returned signal is strong enough for measurement of thermospheric metastable helium. This investigation is significant in the thermospheric metastable helium by lidar.     

Optical geometries and related structures

Two natural optical geometries on the space of all null directions over a four-dimensional Lorentzian manifold are defined and studied. One of this geometries is never integrable and the other is integrable iff the metric of is conformally flat. Sections of forming a zero set of integrability conditions for the latter optical geometry are interpreted as principal null directions on .

Certain well-defined conditions on are shown to be equivalent to the vanishing of the traceless part of the Ricci tensor of . Sections of forming a zero set for these new conditions correspond to the eigendirections of the Ricci tensor of .

An analogy between optical and Hermitian geometries is discussed. Existing (or possible to exist) mutual counterparts between facts from optical and Hermitian geometries are listed. In this analogy, construction of the optical geometries on constitutes a Lorentzian counterpart of the Atiyah-Hitchin-Singer construction of two natural almost Hermitian structures on the twistor space of four-dimensional Euclidean manifold.     

Vibrational relaxation study in carbonyl containing molecule: role of hydrodynamic and dispersion forces

The isotropic Raman component of the C = O stretching mode of ethyl acetate (EA) was analyzed using various polar and nonpolar solvents. It was found that the bandshape approaches towards Lorentzian at high dilution using the curve‐fitting method. The isotropic Raman band was also analyzed by estimating the correlation coefficient with reference to the Lorentzian lineshape using a simple method of linear‐curve fitting. The effects of dispersion and hydrodynamic forces on bandwidth were studied in details. The vibrational relaxation rate was studied using certain parameters and it was found that the microscopic based parameter can explain the complexities occurring in solute–solvent interactions. Copyright © 2007 John Wiley & Sons, Ltd.     

Automatic optimization design of Gaussian beam shaping system by using ZEMAX software

This paper studies the method of automatic optimization design of Gaussian beam shaping. Firstly, the principle of Gaussian beam shaping was studied theoretically and the flattened Lorentzian function (FL) is chosen as the distribution of flattened beam. The mapping function of arbitrary ray in incident plane and image plane was deduced based on the law of conservation of energy. And then, according to the characteristics of this system, Zemax programming language (ZPL) was used to compile ZPL macro orders to optimize the Gaussian beam shaping automatically. Finally, optical analyzing software was used to test the aspheric lens system. The method is not only simple but also practical with high efficient energy conversion and significant engineering application value.     

Causality in noncommutative two-sheeted space-times

We investigate the causal structure of two-sheeted space-times using the tools of Lorentzian spectral triples. We show that the noncommutative geometry of these spaces allows for causal relations between the two sheets. The computation is given in detail when the sheet is a 2- or 4-dimensional globally hyperbolic spin manifold. The conclusions are then generalised to a point-dependent distance between the two sheets resulting from the fluctuations of the Dirac operator.     

On curvature homogeneous three-dimensional Lorentzian manifolds

The aim of this paper is the study of three-dimensional Lorentzian manifolds whose Ricci tensor has three equal constant eigenvalues, whose associated eigenspace is two-dimensional. A complete local classification of this class of curvature homogeneous manifolds is presented. It turns out that, if the eigenvalue is zero, these are exactly the curvature homogeneous manifolds modelled on an indecomposable, non-irreducible Lorentzian symmetric space, which were first studied in Cahen etaal. (1990), and the techniques presented in this paper can therefore be applied to obtain a complete (local) classification of these manifolds, and to construct a number of new examples of such manifolds.     

On the isotropy subalgebras of Lie algebras of conformal vector fields

The conformal isotropy algebra of a point m in an n-manifold with a metric of arbitrary signature is shown to be locally reducible, by a conformal change of the metric, to a homothetic algebra near m iff, by choice of a chart, its constituent vector fields are simultaneously linearisable at m and, for n≥3, a necessary and sufficient condition for this in terms of the first and second derivatives of these fields at m is given. The implications for the Riemannian case and the Lorentzian case are investigated. In contrast to the former, a Lorentzian manifold admitting a conformal vector field that is not linearisable at some point need not be conformally flat. Relevant four-dimensional examples are provided.