Class two analogue of T. Y. Thomas's theorem and different types of embeddings of static spherically symmetric space-times

A Riemannian space of embedding class two is characterised by two symmetric tensors a _ij , b _ij and a vector s_i, satisfying the equations of Gauss, Codazzi and Ricci. It is proved that the Gauss equations together with one set of Codazzi equations imply the other set of Codazzi equations and the Ricci equations, provided that the matrix of the tensor b _ij (or a _ij ) is nonsingular. (The class m generalisation of the result has also been suggested). The result so proved has further been utilized in finding explicitly the a _ij 's and b _ij 's in the case of the static spherically symmetric line element. It is further indicated that the a _ij 's and b _ij 's so obtained are responsible for the different types of embeddings of the spacetime considered.     

Quantum cosmology inR 1×S 1×S n space time

Classical and quantum cosmological aspects for (n + 2) dimensional anisotropic spherically symmetric space-time with topology of (n + 1) spaceS ¹×S ⁿ have been studied. The Lorentzian field equations are reduced to an autonomous system by a change of field variables and are discussed near the critical points. The path integral expression for propagation amplitude is converted to a single ordinary integration over the lapse function by the usual technique and is evaluated in terms of Bessel functions.     

The R.I. Pimenov unified gravitation and electromagnetism field theory as semi-Riemannian geometry

More than forty years ago R.I. Pimenov introduced a new geometry—semi-Riemannian one—as a set of geometrical objects consistent with a fibering pr: M _n → M _m . He suggested the heuristic principle according to which the physically different quantities (meter, second, Coulomb, etc.) are geometrically modelled as space coordinates that are not superposed by automorphisms. As there is only one type of coordinates in Riemannian geometry and only three types of coordinates in pseudo-Riemannian one, a multiple-fibered semi-Riemannian geometry is the most appropriate one for the treatment of more than three different physical quantities as unified geometrical field theory. Semi-Euclidean geometry ³ R ₅⁴ with 1-dimensional fiber x ⁵ and 4-dimensional Minkowski space-time as a base is naturally interpreted as classical electrodynamics. Semi-Riemannian geometry ³ V ₅⁴ with the general relativity pseudo-Riemannian space-time ³ V ₄, and 1-dimensional fiber x ⁵, responsible for the electromagnetism, provides the unified field theory of gravitation and electromagnetism. Unlike Kaluza-Klein theories, where the fifth coordinate appears in nondegenerate Riemannian or pseudo-Riemannian geometry, the theory based on semi-Riemannian geometry is free from defects of the former. In particular, scalar field does not arise. The text was submitted by the author in English.     

On “Conformal spinor geometry”: An attempt to “Understand” internal symmetry

The natural homomorphism of pure spinors corresponding to a given Clifford algebraC _2n to polarized isotropicn-planes of complex Euclidean spaceE _2n ^c is taken as a starting point for the construction of a geometry called spinor geometry where pure spinors are the only elements out of which all tensors have to be constructed (analytically as bilinear polynomials of the components of a pure spinor).C ₄ andC ₆ spinor geometry are analyzed, but it seems that C₈ spinor geometry is necessary to construct Minkowski spaceM ^3,1.C ₆ spinor field equations give rise in Minkowski space to a pair of Dirac equations (for conformal semispinors) presenting ansu(2) internal symmetry algebra. Mass is generated by breaking spontaneously the originalO(4,2) symmetry of the spinor equation.Invited talk presented at the International Symposium Selected Topics in Quantum Field Theory and Mathematical Physics, Bechyn, Czechoslovakia, June 14–19, 1981.     

Unified gravitational and Yang-Mills fields

We unify the gravitational and Yang-Mills fields by extending the diffeomorphisms in (N=4+n)-dimensional space-time to a larger group, called the conservation group. This is the largest group of coordinate transformations under which conservation laws are covariant statements. We present two theories that are invariant under the conservation group. Both theories have field equations that imply the validity of Einstein's equations for general relativity with the stress-energy tensor of a non-Abelian Yang-Mills field (with massive quanta) and associated currents. Both provide a geometrical foundation for string theory and admit solutions that describe the direct product of a compactn-dimensional space and flat four-dimensional space-time. One of the theories requires that the cosmological constant shall vanish. The conservation group symmetry is so large that there is reason to believe the theories are finite or renormalizable.     

Relativistic geometrical optics

We investigate the gravitational and electromagnetic fields on the generalized Lagrange space endowed with the metricg _ij(x, y) = _ij(x) + {1 + 1/n ² (x, y)}y _iy_j. The generalized Lagrange spacesM ^m do not reduce to Lagrange spaces. Consequently, they cannot be studied by methods of symplectic geometry. The restriction of the spacesM ^m to a sectionS (M) leads to the Maxwell equations and Einstein equations for the electromagnetic and gravitational fields in dispersive media with the refractive indexn(x, V) endowed with the Synge metric. Whenn(x, V) = 1 we have the classical Einstein equations. If 1/n ²=1–1/c ² (c being the light velocity), we get results given previously by the authors. The present paper is a detailed version of a work in preparation.     

Symmetries and the Einstein-Maxwell field equations the null field case

This paper deals with space-times that satisfy the Einstein-Maxwell field equations in the presence of a perfect fluid, which may be charged. The electromagnetic field is assumed to be null. It is proved that if the space-time admits a group of isometrics then the fluid velocityu ⁱ, energy density, pressurep, and charge density are invariant under the group. In addition, if the charge density is nonzero, the electromagnetic field tensorf _ij is also invariant. On the other hand, examples of exact solutions are given which establish that if = 0, thenF _ij is not necessarily invariant under the group. In the case of spherically symmetric space-times, however, in which the group of isometries acting isSO (3),f _ij is invariant, independently of whether or not is nonzero. This result leads to the conclusion that in a spherically symmetric space-time the field equations in question admit no solutions with non-trivial null electromagnetic field.     

A simple Fourier photopolarimeter with rotating polarizer and analyzer for measuring Jones and Mueller matrices

The classic and simplest polarimetric scheme of examining a “sample” by placing it between a pair of linear polarizers and observing the intensity of the transmitted light can be transformed into a powerful photopolarimeter if the two polarizers are synchronously rotated at different speeds and the transmitted flux is linearly detected and its periodic waveform Fourier analyzed. In particular, if the angular speed of rotation of one polarizer is ω and that of the other is 3ω, the detected signal has the waveform, i=a₀ + Σ⁴_n=1a_n cos nω_ft + b_n sin nω_ft, where ω_f=2ω is the fundamental frequency. The nine Fourier amplitudes a₀ and (a_n, b_n), n=1,2,3,4, to be derived by performing a discrete Fourier transform (DFT) of the signal i, determine all nine elements of the 3×3 submatrix M_3×3 obtained by deleting the fourth row and fourth column of the Mueller matrix M. If the sample is nondepolarizing, the absolute values of all the four elements of the equivalent Jones matrix $\text{J}\text{=(J}{}_{\mathrm{ij}}\text{)}$, i,j=1,2 and their angle differences θ_ij?θ₂₂ (where θ_ij=argJ_ij) can be determined.     

On the instability of the vacuum in multidimensional scalar theories

The one-loop effective potential in a scalar theory with quartic interaction on the spaceM ⁴×T ⁿ forn=2 is calculated and is shown to be unbounded from below. This is an indication of a possible instability of the vacuum of theλψ ⁴ model onM ⁴, when it is regarded as a low-energy sector of the theory obtained by dimensional reduction of the original six-dimensional one. It is argued that, in some range of the values of the parameters of the theory, higher-loop corrections do not change this result qualitatively. The issue of stability for other values of the numbern of extra dimensions is also discussed.     

Some jacobi matrices with decaying potential and dense point spectrum

We discuss doubly infinite matrices of the formM _ij= _i,j+1+ _i,j–1+V _{i ij} as operators on ². We present for each >0, examples of potentialsV _n with |V _n|=O(n ^–1/2+) and whereM has only point spectrum. Our discussion should be viewed as a remark on the recent work of Delyon, Kunz, and Souillard.Research partially supported by USNSF under grant MCS 81-20833     

Topological Geometrodynamics

The identification of spacetime as a 4-surface in the space H =M⁴×CP₂ (product of Minkowski space and complex projective space of complex dimension two) as means of obtaining Poincare invariant theory of gravitation was the triggering idea of topological geometrodynamics (TGD), which can be regarded as an attempt to unify basic interactions in terms of submanifold geometry instead of abstract manifold geometry as in case of General Relativity. One can however regard TGD also as a generalization of string model: instead of strings free particles are regarded as 3-surfaces. In this article I want to describe these two approaches and to show how they merge into a single coherent scheme provided macroscopic 3-space with matter is identified as a 3-surface containing particles as topological inhomogenities. Also the quantization program of TGD based on the idea that interacting field theory can be regarded as a classical, free field theory for Grassmann algebra valued Schrödinger amplitude in the space of all possible 3-surfaces of H, is described.     

Exact solutions in gravity with a sigma model source

We consider a D-dimensional model of gravity with non-linear “scalar fields” as a matter source. The model is defined on the product manifold M, which contains n Einstein factor spaces. General cosmological type solutions to the field equations are obtained when n − 1 factor spaces are Ricci-flat, e.g. when one space M ₁ of dimension d ₁ > 1 has nonzero scalar curvature. The solutions are defined up to solutions to geodesic equations corresponding to a sigma model target space. Several examples of sigma models are presented. A subclass of spherically symmetric solutions is studied and a restricted version of “no-hair theorem” for black holes is proved. For the case d ₁ = 2 a subclass of latent soliton solutions is singled out.     

Tensor Invariants of the Poisson Brackets of Hydrodynamic Type

Form-invariant solutions for the Poisson brackets of hydrodynamic type on a manifold M ⁿ with (2,0)-tensor g ^ij (u) of rank m ≤ n are derived. Tensor invariants of the Poisson brackets are introduced that include a vector field V (or dynamical system V) on M ⁿ , the Lie derivative L _V g ^ij and symmetric (k, 0)-tensors . Several scalar invariants of the Poisson brackets are defined. A nilpotent Lie algebra structure is disclosed in the space of 1-forms that annihilate the (2,0)-tensor g ^ij (u). Applications to the one-dimensional gas dynamics are presented.     

Left-flat spaces and their associated space-times

It is already known that for an asymptotically flat space-time the metric coefficients and the other Newman-Penrose variables (in a suitable frame) can be constructed, in principle, by specifying certain initial data at conformal null infinity (and one further function on another null hypersurface), integrating the Newman-Penrose equations in the conformally rescaled “unphysical” space, and then transforming the results back to the physical space-time. If this is done approximately near ?⁺, for vacuum, the well-known Newman-Unti expansion is obtained. In this paper, after complexifying null infinity ?⁺ we generate, in a similar fashion, a left-flat spaceH using as much of the initial data of a given asymptotically flat space-timeM as possible, and show that the left-flat spaceH thus constructed is, in fact, the H-space corresponding toM. The advantage of our method is that it allows a reversal of procedure. Under suitable conditions we can generate from a given left-flat spaceH a class of physical space-times whose H-space is precisely the given left-flat spaceH. We shall see that the formal procedure requires only the local but not the global properties of ?⁺.     

An identity for 4-spacetimes embedded intoE 5

We show that if a 4-spacetimeV ₄ can be embedded intoE ₅ then, ifb _ijis the second fundamental form tensor associated withV ₄, the quantity (traceb)·b _ij ^/−1 depends only on intrinsic geometric properties of the spacetime. Such fact is used to obtain a necessary condition for the embedding of aV ₄ intoE ₅.     

Multidimensional Unified Theories

An extended spacetime, M^4+N, is a Riemannian (4 + N)-dimensional manifold which admits an N-parameter group G of (spacelike) isometries and is such that ordinary spacetime M⁴ is the space M^4+N/G of the equivalence classes under G-transformations of M^4+N. A multidimensional unified theory (MUT) is a dynamical theory of the metric tensor on M^4+N, the metric being determined from the Einstein-Hilbert action principle: in absence of matter, the Lagrangian is (essentially) the total curvature scalar of M^4+N. A MUT is an extension of the Cho-Freund generalization of Jordan's five-dimensional theory. A MUT can be faithfully translated in four-dimensional language: as a theory on M⁴, a MUT is a gauge field theory with gauge group G. A unifying aspect of MUT's is that all fields occur as elements of the metric tensor on M^4+N. When the isometry generators are subjected to strongest constraints, a MUT becomes the De Witt-Trautman generalization of Kaluza's five-dimensional theory; in four-dimensional language, this is the theory of Yang-Mills gauge fields coupled to gravity. With weaker constraints, a MUT appears to be more natural than a Yang-Mills theory as a physical realization of the gauge principle for an exact symmetry of gauged confined color. Such weakly-constrained MUT leads to bag-type models without the need for ad hoc surgery on the basic. Lagrangian. The present paper provides a detailed introduction to the formalism of multidimensional unified gauge field theory.