Einstein equations and the joining of discontinuous metrics

We consider the problem of joining of metrics when these are not continuous across the joining (hyper-) surface. We confine ourselves to static, spherically symmetric metrics which join without requiring gradients of a δ-function in the energy-momentum tensor. It is found that a surface tension is always associated in cases where the metrics are discontinuous. In some cases, the joined metrics satisfy Einstein equations (in the sense of distributions) while, in others, the surface tension associated with the limiting discontinuous metric depends on the interpolating functions used to produce it suggesting sensitivity to short distance effects beyond general relativity.     

Classification of Nilsoliton metrics in dimension seven

The aim of this paper is to classify Ricci soliton metrics on 7-dimensional nilpotent Lie groups. It can be considered as a continuation of our paper in Fernández Culma (2012). To this end, we use the classification of 7-dimensional real nilpotent Lie algebras given by Ming-Peng Gong in his Ph.D thesis and some techniques from the results of Michael Jablonski (2010, 2012) and of Yuri Nikolayevsky (2011). Of the 9 one-parameter families and 140 isolated 7-dimensional indecomposable real nilpotent Lie algebras, we have 99 nilsoliton metrics given in an explicit form and 7 one-parameter families admitting nilsoliton metrics.Our classification is the result of a case-by-case analysis, so many illustrative examples are carefully developed to explain how to obtain the main result.     

On a class of Ricci-flat Finsler metrics in Finsler geometry

In this paper, we study a class of Finsler metrics defined by a Riemannian metric and a one-form on a manifold. We completely determine the local structure of Ricci-flat metrics in this class which are also of Douglas type.     

Comments on the unification of electromagnetism and gravitation through “Generalized Einstein manifolds”

Akbar-Zadeh [J. Geom. Phys. 17 (1995) 342] has recently proposed a new geometric formulation of Einstein-Maxwell system with source in terms of what are called “Generalized Einstein manifolds”. We show that, contrary to the claim, Maxwell equations have not been derived in this formulation, and that the assumed equations can be identified only as source-free Maxwell equations in the proposed geometric setup. A genuine derivation of source-free Maxwell equations is presented within the same framework. We draw a conclusion that the proposed unification scheme can pertain only to source-free situations.     

Letter: Spatial Metrics Which are Static in Many Ways

We consider Riemannian 3-metrics which can form the spatial part of vacuum solutions of the Einstein equations, possibly with a cosmological constant, in more than one way (in a sense made precise). The locally rotationally symmetric (LRS) Kasner metric gives the simplest example, and we find that the resulting space-time metrics are always of Petrov type D.     

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.     

Spin Zero Quantum Relativistic Particles in Einstein Universe

We find exact eigenvalues and eigenfunctions of relativistic massless scalar particle conformally coupled to a background Einstein universe.     

Conformally Steckel metrics in Einstein spaces

Anisotropic conformally Steckel metrics that satisfy Einstein vacuum equations containing a Λ term are examined in the paper. The metrics allow the integration of the isotropic geodetic Hamilton-Jacobi equations by complete variable separation. Tomsk State University. Translated from Izeestiya Vysshikn Uchebnykh Zavedenii, Fizika, No. 10, pp. 34–43, October, 1999.     

Algebraically special hypersurface-homogeneous Einstein spaces in general relativity

This paper illustrates the value of the Newman—Penrose complex null tetrad formalism by using it to obtain all algebraically special Einstein spaces admitting three-parameter groups of motions acting on timelike surfaces containing the repeated principal null direction. Taken together with earlier work, this enables us to give a complete list of Einstein spaces which are both algebraically special and hypersurface-homogeneous or homogeneous.     

关于爱因斯坦的若干资料

列出了有关爱因斯坦生平的一些有趣问题,包括他的学习、科学研究,以及他对教育的批评等.     

Einstein-like Lorentzian Lie groups of dimension four

Einstein-like examples of four-dimensional Lorentzian Lie groups are listed and geometric properties of each class have been investigated.     

Odd-Soliton Solutions to the Einstein Equations

Using the Inverse Scattering Method (ISM) of Belinskii and Zakharov a new odd-soliton solutions to the Einstein's field equations for an axially symmetric space-time in general relativity are obtained in the determinant form and shown to include Weyl's half-integral delta static solutions in a special case.     

Flat connections and Wigner-Yanase-Dyson metrics

On the manifold of positive definite matrices, we investigate the existence of pairs of flat affine connections, dual with respect to a given monotone metric. The connections are defined either using the α-embeddings and finding the duals with respect to the metric, or by means of contrast functionals. We show that in both cases, the existence of such a pair of connections is possible if and only if the metric is given by the Wigner-Yanase-Dyson skew information.     

Radiating Kerr particle in Einstein universe

A generalized Kerr-NUT type metric is considered in connection with Einstein field equations corresponding to perfect fluid plus a pure radiation field. A general scheme for obtaining the exact solutions of these field equations is developed. Two physically meaningful particular cases are investigated in detail. One gives the field of a radiating Kerr particle embedded in the Einstein universe. The other solution may probably represent a deSitter-like universe pervaded by a pure radiation field.     

Minimal orbits of metrics

The group of diffeomorphisms of a compact manifold acts isometrically on the space of Riemannian metrics with its L² metric. Following Arnaudon and Paycha (1995) and Maeda, Rosenberg and Tondeur (1993), we define minimal orbits for this action by a zeta function regularization. We show that odd dimensional isotropy irreducible homogeneous spaces give rise to minimal orbits, the first known examples of minimal submanifolds of infinite dimension and codimension. We also find a flat 2-torus giving a stable minimal orbit. We prove that isolated orbits are minimal, as in finite dimensions.     

Schlesinger System,Einstein Equations and Hyperelliptic Curves

We review recent developments in the method of algebro-geometric integration of integrable systems related to deformations of algebraic curves. In particular, we discuss the theta-functional solutions of the Schlesinger system, Ernst equation, and self-dual SU(2)-invariant Einstein equations.