Some New Solutions Derived From the 5-Dimensional Theory—The New Methods of Creating Solutions

Application of the 5-dimensional coordinate transformations in the 5-dimensional theory lead us to some new solutions for the 4-dimensional Einstein–Maxwell equations and the relevant scaler equation. From the Kerr solution we derive the corresponding solution. And we propose a new method to solve the usual 4-dimensional Einstein–Maxwell equations and the scalar equation, illustrating by three examples.     

一个4+1维宇宙模型

我们讨论的4+1维宇宙模型是通常的四维时空和一个紧致的一维内禀空间的直积空间。我们假定四维时空的能量密度是以辐射为主的,而内禀子空间的能动张量是一个阶跃函数。通过求解五维的Einstein场方程得到前四维时空由de-Sitter解过渡到标准模型的辐射为主解,与此同时内禀子空间的尺度由减幅振荡过渡到为按t的负幂次收缩而趋于一常量。 关键词：     

The embedding of General Relativity in five dimensions

We argue that General Relativistic solutions can always be locally embedded in Ricci-flat 5-dimensional spaces. This is a direct consequence of a theorem of Campbell (given here for both a timelike and spacelike extra dimension, together with a special case of this theorem) which guarantees that anyn-dimensional Riemannian manifold can be locally embedded in an (n+1)-dimensional Ricci-flat Riemannian manifold. This is of great importance in establishing local generality for a proposal recently put forward and developed by Wesson and others, whereby vacuum (4+1)-dimensional field equations give rise to (3+1)-dimensional equations with sources. An important feature of Campbell's procedure is that it automatically guarantees the compatibility of Gauss-Codazzi equations and therefore allows the construction of embeddings to be in principle always possible. We employ this procedure to construct such embeddings in a number of simple cases.     

Minimal immersions,Einstein's equations and Mach's principle

A geometrical stress energy tensor for semi-Riemannian manifolds is described and a Mach's principle is formulated. It is shown that vacuum occurs if and only if the manifold is a totally geodesic submanifold of a flat semi-Euclidean space. Furthermore the Einstein equations are attained with the cosmological constant appearing as the mean curvature of an isometric immersion. A minimal submanifold of a semi-Euclidean space can thereby be regarded as a solution to Einsteins equations without a cosmological constant. Intrinsic conditions that will allow a 4-dimensional semi-Riemannian manifold to be immersed isometrically into 5-dimensional semi-Euclidean space as a minimal hypersurface are found. From this result it is possible to find explicit minimal hypersurfaces of Robertson-Walker type in a 5-dimensional Minkowski space and it is observed that they all contain an initial singularity.     

A Five-Dimensional Model of the Gravitational Interaction of an Electromagnetic Field in an Affine-Metric Space

A geometric model for the gravitational interaction of an electromagnetic field in an affine-metric space with torsion and nonmetricity is proposed which describes the dynamics of an empty 5-dimensional affine-metric space. The gravitational and the electromagnetic field are presented in terms of the metric tensor of a 5-dimensional space-time. The equations of the theory are deduced from the variation principle with the use of the (4 + 1)-splitting formalism. Exact spherically symmetrical solutions have been obtained for the system of equations of the presented theory, and their possible astrophysical consequences have been investigated.     

Dynamics in Kaluza-Klein Gravity and a Fifth Force

Using a novel coordinate system, we rederive the field equations and equa tions of motion for 5-dimensional relativity in the general case where the metric can depend on all 5 coordinates (i.e., Kaluza-Klein theory without the cylinder restriction). We show that in general the fifth dimension produces a new dynamical force in 4-dimensional spacetime. This timelike fifth force is proportional to the 4-velocity of a particle and is thus unlike any known 4-force. We briefly examine the properties of some simple models, and suggest that the detection of the fifth force is the most promising way to investigate the existence of an extra dimension in nature.     

Particle Production in 5-Dimensional Cosmological Relativity

The 5-dimensional extension of cosmological special and general relativity is considered. In this framework it is possible to define a 5-dimensional perfect fluid stress-energy tensor and to unify the equations of perfect hydrodynamics in a single 5-dimensional tensor conservation law. This picture in principle permits to interpret particle production phenomena as cosmological effects, in the spirit of open system cosmology.     

Spinor matter in a particular 5-dimensional projective unified field theory

After presenting the foundation and the basic equations of a new 5-dimensional projective unified field theory, the problem of incorporating spinor fields into this framework is investigated. Apart from Pauli's method, we propose a new approach which leads to a consistent 5-dimensional spinor theory with a series of physical consequences (variability of the 4-dimensional rest mass, instability of 4-dimensional stationary states, etc.).Dedicated to Prof. Peter Bergmann on the occasion of this 70th birthday.     

Dynamical Noncommutative Spheres

We introduce a family of noncommutative 4-spheres, such that the instanton projector has its first Chern class trivial: ch ₁(e)=B+b. We construct for them a 4-dimensional cycle and calculate explicitly the Chern-Connes pairing for the instanton projector.Supported by Marie Curie Fellowship HPMF-CT-1999-00053, at Laboratoire de Physique Theórique, Université Paris-Sud, Bat. 210, 91405 Orsay, Cedex, France     

Cosmological application on five-dimensional teleparallel theory equivalent to general relativity

A theory of(4+1)-dimensional gravity has been developed on the basis of which equivalent to the theory of general relativity by teleparallel.The fundamental gravitational field variables are the 5-dimensional(5D) vector fields(pentad),defined globally on a manifold M,and gravity is attributed to the torsion.The Lagrangian density is quadratic in the torsion tensor.We then apply the field equations to two different homogenous and isotropic geometric structures which give the same line element,i.e.,FRW in five dimensions.The cosmological parameters are calculated and some cosmological problems are discussed.     

用光电混合系统解线性方程组

本文提出一种用于迭代法求解线性方程组的光电混合系统。该系统的光学部分主要由单个全息透镜组成,它执行矩阵与矢量的乘法运算;系统的其余部分执行矢量的测量与求和,它由CCD探测器件和一台微机组成。使用这个光电混合系统,用迭代法对一个4元线性方程组求解,实验结果与理论解比较,误差约为5％。     

Noncommutative integration of the Dirac equation in Riemann spaces possessing a group of automorphisms

Applying the method of noncommutative integration for linear differential equations, we build exact solutions for the Dirac equation in 4-dimensional Riemann spaces, which have a 5-parameter group of automorphisms and where the Klein-Gordon and the Dirac equations are nonintegrable using the technique of complete separation of variables.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 43–46, September, 1991.     

（2＋2）-Dimensional Discrete Soliton Equations and Integrable Coupling System

In this paper, we extend a （2＋2）-dimensional continuous zero curvature equation to （2＋2）-dimensional discrete zero curvature equation, then a new （2＋2）-dimensional cubic Volterra lattice hierarchy is obtained. Fhrthermore, the integrable coupling systems of the （2＋2）-dimensional cubic Volterra lattice hierarchy and the generalized Toda lattice soliton equations are presented by using a Lie algebraic system sl（4）.     

<Emphasis Type="Italic">D</Emphasis>-Dimensional Metrics with <Emphasis Type="Italic">D</Emphasis>−3 Symmetries

Symmetry transformations in a space of D-dimensional vacuum metrics with D?3 commuting Killing vectors are studied. We solve directly the Einstein equations in the Maison formulation under additional assumptions. We show that the Reissner-Nordström solution is related by the symmetry transformation to a particular case of the 5-dimensional Gross-Perry metric and the 5-dimensional plane wave solution is related to the Gross-Perry-Sorkin metric.     

Curved Space-Times by Crystallization of Liquid Fiber Bundles

Motivated by the search for a Hamiltonian formulation of Einstein equations of gravity which depends in a minimal way on choices of coordinates, nor on a choice of gauge, we develop a multisymplectic formulation on the total space of the principal bundle of orthonormal frames on the 4-dimensional space-time. This leads quite naturally to a new theory which takes place on 10-dimensional manifolds. The fields are pairs of \(((\alpha ,\omega ),\varpi )\), where \((\alpha ,\omega )\) is a 1-form with coefficients in the Lie algebra of the Poincaré group and \(\varpi \) is an 8-form with coefficients in the dual of this Lie algebra. The dynamical equations derive from a simple variational principle and imply that the 10-dimensional manifold looks locally like the total space of a fiber bundle over a 4-dimensional base manifold. Moreover this base manifold inherits a metric and a connection which are solutions of a system of Einstein–Cartan equations.     

Cohomogeneity One Einstein-Sasaki 5-Manifolds

We consider hypersurfaces in Einstein-Sasaki 5-manifolds which are tangent to the characteristic vector field. We introduce evolution equations that can be used to reconstruct the 5-dimensional metric from such a hypersurface, analogous to the (nearly) hypo and half-flat evolution equations in higher dimensions. We use these equations to classify Einstein-Sasaki 5-manifolds of cohomogeneity one.     

Wesson’s Induced Matter Theory with a Weylian Bulk

The foundations of Wesson’s induced matter theory are analyzed. It is shown that the empty—without matter—5-dimensional bulk must be regarded as a Weylian space rather than as a Riemannian one. Revising the geometry of the bulk, we have assumed that a Weylian connection vector and a gauge function exist in addition to the metric tensor. The framework of a Weyl–Dirac version of Wesson’s theory is elaborated and discussed. In the 4-dimensional hypersurface (brane), one obtains equations describing both fields, the gravitational and the electromagnetic. The result is a geometrically based unified theory of gravitation and electromagnetism with mass and current induced by the bulk. In special cases on obtains on the brane the equations of Einstein–Maxwell, or these of the original induced matter theory.     

Three-variable solution in the $$$$-dimensional null-surface formulation

The null-surface formulation of general relativity (NSF) describes gravity by using families of null surfaces instead of a spacetime metric. Despite the fact that the NSF is (to within a conformal factor) equivalent to general relativity, the equations of the NSF are exceptionally difficult to solve, even in 2+1 dimensions. The present paper gives the first exact \((2+1)\)-dimensional solution that depends nontrivially upon all three of the NSF’s intrinsic spacetime variables. The metric derived from this solution is shown to represent a spacetime whose source is a massless scalar field that satisfies the general relativistic wave equation and the Einstein equations with minimal coupling. The spacetime is identified as one of a family of \((2+1)\)-dimensional general relativistic spacetimes discovered by Cavaglià.