Spherically symmetric solution in higher-dimensional teleparallel equivalent of general relativity

A theory of(N+1)-dimensional gravity is developed on the basis of the teleparallel equivalent of general relativity(TEGR).The fundamental gravitational field variables are the(N+1)-dimensional vector fields,defined globally on a manifold M,and the gravitational field is attributed to the torsion.The form of Lagrangian density is quadratic in torsion tensor.We then give an exact five-dimensional spherically symmetric solution(Schwarzschild(4+1)-dimensions).Finally,we calculate energy and spatial momentum using gravitational energy-momentum tensor and superpotential 2-form.     

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.     

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.     

Quantization of antisymmetric tensors and ray-singer torsion

The quantization of antisymmetric tensor fields on an n-dimensional riemannian manifold is studied. The connection between quantized antisymmetric fields of ranks k?1 and n?k?1 is analysed. It is shown that the quotient of the corresponding partition functions can be expressed through so-called Ray-Singer torsion. Ray-Singer torsion is calculated for a three-dimensional manifold with a boundary. Some general results on the quantization of degenerate functionals are obtained.     

Application of the slaving principle to quenching problems

The slaving principle theory is applied to quenching problems. It is shown that a system, which starts from the close vicinity of an unstable point and the evolution of which is described by a set ofn-dimensional equations, will approach and then follow the unstable manifold. In the presence of fluctuations, it is even possible that the system approaches the most unstable manifold. Thus, the actual dimension related to the problem can be considerably reduced.     

Scalar fields in a seven-dimensional manifold behaving as Lorentz-covariant spinor fields in space-time

A Lorentz-invariant model of vacuum is given in the form of a 7-dimensional manifold endowed with a statistical metrical tensor. Certain scalar fields on this manifold behave then as spinor fields when viewed from their space-time projection. This paper generalizes previous work fromSO(3)-covariance to Lorentz-covariance.     

Physical symmetries in a theory of several scalar real fields

Let us consider a theory ofn scalar, real, local, Poincaré covariant quantum fields forming an irreducible set and giving rise to one particle states belonging to the same mass different from zero. The vacuum is unique. It is shown under fairly weak assumptions that every Poincaré and TCP invariant symmetry of the theory, implemented unitarily, which mapps localized elements of the field algebra into operators almost local with respect to the former (such a symmetry we call a physical one) can be defined uniquely in terms of the incoming or outgoing fields and ann-dimensional (real) orthogonal matrix. The symmetry commutes with the scattering matrix. Incidentally we show also that the symmetry groups are compact. A special case of these symmetries are the internal symmetries and symmetries induced by locally conserved currents local with respect to the basic fields and transforming under the same representation of the Poincaré group. We may make linear combinations out the original fields resulting in complex fields and its complex conjugate in a suitable way. The inspection of the representations of the groupsSO(n) and their subgroups sheds some light on the s.c. generalized Carruthers Theorem concerning the self- and pair-conjugate multiplets.     

Geometry of the submanifolds of SEXn. I. The C-nonholonomic frame of reference

A connection which is both Einstein and semisymmetric is called an SE connection. A generalizedn-dimensional Riemannian manifold on which the differential geometric structure is imposed byg through an SE connection is called ann-dimensional SE manifold and denoted by SEX_n. This paper is the introductory part of a systematic study of the submanifolds of SEX_n. It introduces a new concept of the C-nonholonomic frame of references in SEX_n at points of its submanifold and deals with its consequences. The second part will deal with the generalized fundamental equations on an SE hypersubmanifold of SEX_n. The third part will be devoted to the theory of parallelism in SEX_n and in its submanifold. Finally, the last part will study the curvature theory in a submanifold of SEX_n.     

A bosonic model for relativistic spinors

In an 11-dimensional manifold with topology ⁷ ×S ⁷ (Minkowski-space times 7-dimensional sphere) we construct a completely antisymmetrical 8-rank tensor field (equivalent to the Pauli matrices) performing a symmetry breaking so that a Lorentz-transformation must be linked with a rotation of the internal spaceS ⁷. Certain scalar fields are then described as and have the same transformation properties as relativistic 4-dimensional spinor fields.Research supported by the Commission of the European Communities (Directorate General for Science, Research and Development-Joint Research Center).     

Curvature tensors unified field equations on SEXn

We study the curvature tensors and field equations in then-dimensional SE manifold SEX_n. We obtain several basic properties of the vectorsS andU and then of the SE curvature tensor and its contractions, such as a generalized Ricci identity, a generalized Bianchi identity, and two variations of the Bianchi identity satisfied by the SE Einstein tensor. Finally, a system of field equations is discussed in SEX_n and one of its particular solutions is constructed and displayed.     

A theory of generally invariant lagrangians for the metric fields. II

The second-order generally invariant Lagrangians for the metric fields on anyn-dimensional manifold are studied as certain special coordinate functions on a space of jets. The number of independent Lagrangians of this type is determined. The dimensionsn=3 andn=4 are examined in detail with the help of a computer.     

A connection between the Einstein and Yang-Mills equations

It is our purpose here to show an unusual relationship between the Einstein equations and the Yang-Mills equations. We give a correspondence between solutions of the self-dual Einstein vacuum equations and the self-dual Yang-Mills equations with a special choice of gauge group. The extension of the argument to the full Yang-Mills equations yields Einstein's unifield equations. We try to incorporate the full Einstein vacuum equations, but the approach is incomplete. We first consider Yang-Mills theory for an arbitrary Lie-algebra with the condition that the connection 1-form and curvature are constant on Minkowski space. This leads to a set of algebraic equations on the connection components. We then specialize the Lie-algebra to be the (infinite dimensional) Lie-algebra of a group of diffeomorphisms of some manifold. The algebraic equations then become differential equations for four vector fields on the manifold on which the diffeomorphisms act. In the self-dual case, if we choose the connection components from the Lie-algebra of the volume preserving 4-dimensional diffeomorphism group, the resulting equations are the same as those obtained by Ashtekar, Jacobsen and Smolin, in their remarkable simplification of the self-dual Einstein vacuum equations. (An alternative derivation of the same equations begins with the self-dual Yang-Mills connection now depending only on the time, then choosing the Lie algebra as that of the volume preserving 3-dimensional diffeomorphisms.) When the reduced full Yang-Mills equations are used in the same context, we get Einstein's equations for his unified theory based on absolute parallelism. To incorporate the full Einsteinvacuum equations we use as the Lie group the semi-direct product of the diffeomorphism group of a 4-dimensional manifold with the group of frame rotations of anSO(1, 3) bundle over the 4-manifold. This last approach, however, yields equations more general than the vacuum equations.Andrew Mellon Postdoctoral fellow and Fulbright ScholarSupported in part by NSF grant no. PHY 80023     

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.     

On conformal Killing 2-form of the electromagnetic field

Let D:C^∞Λ^pM→C^∞(T^*MΛ^pM) be the first order linear differential operator on an n-dimensional (1≤p≤n−1) pseudo-Riemannian manifold (M,g). We have by the representation theory of orthogonal group, that the tangent bundle of this operation decomposes into the orthogonal and irreducible sum of forms of degree p+1 (which gives the exterior differential d), the forms of degree p−1 (defining the codifferential d^*) and the trace-free part of the partial symmetrization (the corresponding first order operator is denoted by D). The general forms in the kernel of D are closely related to conformal Killing vector fields, called conformal Killing p-forms, while those in kernel of d are called closed conformal Killing p-forms or, according to another terminology, planar p-forms. In particular an arbitrary planar 1-form ω is dual (by g) to the special concircular vector field ξ. We consider some local properties for the closed conformal Killing p-forms. As an application we present examples of decomposition into irreducible components for the electromagnetic field 2-form ω and its covariant derivative in four-dimensional space–time. In particular, we prove that the energy–momentum tensor T of the electromagnetic field is a symmetric conformal Killing tensor if the electromagnetic field 2-form ω is a conformal Killing form.     

The goldstone fields of interacting higher spin field theory in AdS(4)

A higher spin field theory on AdS(4) possesses a conformal theory on the boundary R(3) which can be identified with the critical O(N) sigma model of O(N) invariant fields only. The notions of quasiprimary and secondary fields can be carried over to the AdS theory. If de Donder’s gauge is applied, the traceless part of the higher spin field on AdS(4) is quasiprimary and the Goldstone fields are quasiprimary fields to leading order too. Those fields corresponding to the Goldstone fields in the critical O(N) sigma model are odd-rank symmetric tensor currents which vanish in the free-field limit. The text was submitted by the author in English.     

Conformal invariance and scalar-tensor theories

A new generalisation of Einstein’s theory is proposed which is invariant under conformal mappings. Two scalar fields are introduced in addition to the metric tensor field, so that two special choices of gauge are available for physical interpretation, the ‘Einstein gauge’ and the ‘atomic gauge’. The theory is not unique but contains two adjustable parameters ζ anda. Witha=1 the theory viewed from the atomic gauge is Brans-Dicke theory (ω=−3/2+ζ/4). Any other choice ofa leads to a creation-field theory. In particular the theory given by the choicea=−3 possesses a cosmological solution satisfying Dirac’s ‘large numbers’ hypothesis.     

Nijenhuis tensors and lie algebras

Let M be a 2n-dimensional almost complex manifold: we construct a local almost complex structure starting from a Nijenhuis tensor given at a point. Moreover, we determine, on a 6-manifold, the conditions ensuring that its Nijenhuis tensor induces a Lie algebra structure on the tangent space. We give a class of examples for every kind.     

Fields,statistics and non-Abelian gauge groups

We examine field theories with a compact groupG of exact internal gauge symmetries so that the superselection sectors are labelled by the inequivalent irreducible representations ofG. A particle in one of these sectors obeys a parastatistics of orderd if and only if the corresponding representation ofG isd-dimensional. The correspondence between representations of the observable algebra and representations ofG extends to a mapping of the intertwining operators for these representations preserving linearity, tensor products and conjugation. Although we assume no explicit commutation property between fields, the commutation relations of fields of the same irreducible tensor character underG at spacelike separations are largely determined by the statistics parameter of the corresponding sector. For fields of conjugate irreducible tensor character the observable part of the commutator (anticommutator) vanishes at spacelike separations if the corresponding sector has para-Bose (para-Fermi) statistics.     

Brane cosmology and motion of test particles in five-dimensional warped product spacetimes

In the “braneworld scenario” ordinary standard model matter and non-gravitational fields are confined by some trapping mechanism to the 4-dimensional universe constituting the D3-branes which are embedded in a (4 + n)-dimensional manifold referred to as the ‘bulk’ (n being the number of extra dimensions). The notion of particle confinement is necessary for theories with non-compact extra dimensions, otherwise, the particles would escape from our 4-dimensional world along unseen directions. In this paper, we have considered a five-dimensional warped product space-time having an exponential warping function which depends both on time as well as on the extra coordinates and a non-compact fifth dimension. Assuming that the lapse function may either be a constant or a function of both time and of the extra coordinates, we have studied the nature of the geodesics of test particles and photons and have analyzed the conditions of stability in this geometrical framework. We have also discussed the possible cosmology of the corresponding (3 + 1)-dimensional hypersurfaces.