Covariant construction of the symmetric energy-momentum tensor of vector and spinor fields in an orthogonal frame of reference

A covariant method is devised to construct the symmetric energy-momentum tensor for vector fields in an orthogonal frame of reference. The method is then used to construct the symmetric energy-momentum tensor for spinor fields. Kazan’ University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 27–30, March, 1997.     

Symmetric hyperbolic systems for a large class of fields in arbitrary dimension

Symmetric hyperbolic systems of equations are explicitly constructed for a general class of tensor fields by considering their structure as r-fold forms. The hyperbolizations depend on 2r−1 arbitrary timelike vectors. The importance of the so-called “superenergy” tensors, which provide the necessary symmetric positive matrices, is emphasized and made explicit. Thereby, a unified treatment of many physical systems is achieved, as well as of the sometimes called “higher order” systems. The characteristics of these symmetric hyperbolic systems are always physical, and directly related to the null directions of the superenergy tensor, which are in particular principal null directions of the tensor field solutions. Generic energy estimates and inequalities are presented too. Examples are included, in particular a mixed gravitational-scalar field system at the level of the Bianchi equations.     

Symmetries at stationary Killing horizons

It has often been suggested (especially by S. Carlip) that spacetime symmetries in the neighborhood of a black hole horizon may be relevant to a statistical understanding of the Bekenstein–Hawking entropy. A prime candidate for this type of symmetry is that one which is exhibited by the Einstein tensor. More precisely, it is now known that this tensor takes on a strongly constrained (block-diagonal) form as it approaches any stationary, non-extremal Killing horizon. Presently, exploiting the geometrical properties of such horizons, we provide a particularly elegant argument that substantiates this highly symmetric form for the Einstein tensor. It is, however, duly noted that, on account of a “loophole,” the argument does fall just short of attaining the status of a rigorous proof.     

On the algebraic types of the Bel–Robinson tensor

The Bel–Robinson tensor is analyzed as a linear map on the space of the traceless symmetric tensors. This study leads to an algebraic classification that refines the usual Petrov–Bel classification of the Weyl tensor. The new classes correspond to degenerate type I space-times which have already been introduced in literature from another point of view. The Petrov–Bel types and the additional ones are intrinsically characterized in terms of the sole Bel–Robinson tensor, and an algorithm is proposed that enables the different classes to be distinguished. Results are presented that solve the problem of obtaining the Weyl tensor from the Bel–Robinson tensor in regular cases.     

Conformal invariant two and three-point functions for fields with arbitrary spin

The conformal invariant two and three-point functions for any “fundamental” fields with an arbitrary spin and scale dimensions are found in the Minkowsky x-space. The two-point functions for Dirac, symmetric and antisymmetric tensor fields are given. The three-point functions for two Dirac fields and one symmetrical tensor field, as well as any other field for which this function is nonvanishing, are given. In the case of conserved currents the Ward identities are considered.     

Decomposition of symmetric tensor fields in the presence of a flat contact projective structure

Abstract

Let M be an odd-dimensional Euclidean space endowed with a contact 1-form α. We investigate the space of symmetric contravariant tensor fields over M as a module over the Lie algebra of contact vector fields, i.e. over the Lie subalgebra made up of those vector fields that preserve the contact structure defined by a. If we consider symmetric tensor fields with coefficients in tensor densities (also called symbols), the vertical cotangent lift of the contact form a defines a contact invariant operator. We also extend the classical contact Hamiltonian to the space of symbols. This generalized Hamiltonian operator on the space of symbols is invariant with respect to the action of the projective contact algebra sp(2n+2) the algebra of vector fields which preserve both the contact structure and the projective structure of the Euclidean space. These two operators lead to a decomposition of the space of symbols, except for some critical density weights, which generalizes a splitting proposed by V. Ovsienko in [18].     

A spectral method for the wave equation of divergence-free vectors and symmetric tensors inside a sphere

The wave equation for vectors and symmetric tensors in spherical coordinates is studied under the divergence-free constraint. We describe a numerical method, based on the spectral decomposition of vector/tensor components onto spherical harmonics, that allows for the evolution of only those scalar fields which correspond to the divergence-free degrees of freedom of the vector/tensor. The full vector/tensor field is recovered at each time-step from these two (in the vector case), or three (symmetric tensor case) scalar fields, through the solution of a first-order system of ordinary differential equations (ODE) for each spherical harmonic. The correspondence with the poloidal–toroidal decomposition is shown for the vector case. Numerical tests are presented using an explicit Chebyshev-tau method for the radial coordinate.     

A natural one-form for the Schouten concomitant

The Poisson bracket in classical mechanics arises from the existence of a natural one-form on a cotangent bundle. The Schouten concomitant of two symmetric contravariant tensor fields is closely related to the Poisson bracket. We show that it arises in an analogous way from a natural onecochain, where the chains are chains of derivations from the module of symmetric contravariant tensor fields into itself.     

Casimir Effect for Moving Branes in Static dS4+1 Bulk

In this paper we study the Casimir effect for conformally coupled massless scalar fields on background of Static dS₄₊₁ spacetime. We will consider the general plane–symmetric solutions of the gravitational field equations and boundary conditions of the Dirichlet type on the branes. Then we calculate the vacuum energy-momentum tensor in a configuration in which the boundary branes are moving by uniform proper acceleration in static de Sitter background. Static de Sitter space is conformally related to the Rindler space, as a result we can obtain vacuum expectation values of energy-momentum tensor for conformally invariant field in static de Sitter space from the corresponding Rindler counterpart by the conformal transformation.     

Proof of a decomposition theorem for symmetric tensors on spaces with constant curvature

In cosmological perturbation theory a first major step consists in the decomposition of the various perturbation amplitudes into scalar, vector and tensor perturbations, which mutually decouple. In performing this decomposition one uses – beside the Hodge decomposition for one‐forms – an analogous decomposition of symmetric tensor fields of second rank on Riemannian manifolds with constant curvature. While the uniqueness of such a decomposition follows from Gauss' theorem, a rigorous existence proof is not obvious. In this note we establish this for smooth tensor fields, by making use of some important results for linear elliptic differential equations.     

Reconsiderations on the formulation of general relativity based on Riemannian structures

We prove that some basic aspects of gravity commonly attributed to the modern connection-based approaches, can be reached naturally within the usual Riemannian geometry-based approach, by assuming the independence between the metric and the connection of the background manifold. These aspects are: 1) the BF-like field theory structure of the Einstein–Hilbert action, of the cosmological term, and of the corresponding equations of motion; 2) the formulation of Maxwellian field theories using only the Riemannian connection and its corresponding curvature tensor, and the subsequent unification of gravity and gauge interactions in a four dimensional field theory; 3) the construction of four and three dimensional geometrical invariants in terms of the Riemann tensor and its traces, particularly the formulation of an anomalous Chern–Simons topological model where the action of diffeomorphisms is identified with the action of a gauge symmetry, close to Witten’s formulation of three-dimensional gravity as a Chern–Simon gauge theory. 4) Tordions as propagating and non-propagating fields are also formulated in this approach. This new formulation collapses to the usual one when the metric connection is invoked, and certain geometrical structures very known in the traditional literature can be identified as remanent structures in this collapse.     

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.     

Extended string field theory for massless higher-spin fields

We propose a new gauge field theory which is an extension of ordinary string field theory by assembling multiple state spaces of the bosonic string. The theory includes higher-spin fields in its massless spectrum together with the infinite tower of massive fields. From the theory, we can easily extract the minimal gauge-invariant quadratic action for tensor fields with any symmetry. As examples, we explicitly derive the gauge-invariant actions for some simple mixed symmetric tensor fields. We also construct covariantly gauge-fixed action by extending the method developed for string field theory.     

Bargmann–Wigner方程与高自旋场方程

在坐标表象中由Bargmann-Wigner方程导出了便于求解的高自旋场方程,并给出了相应的拉氏函数.     

Paths of spinning particles in general relativity as geodesics of an Einstein connection

This paper deals with the Papapetrou-Pirani equations of motion for a spinning test particle in general relativity. The motion of the center of mass can be represented by the geodesic equation of an affine connection that is the sum of the Christoffel connection and a tensor that depends on the Riemann-Christoffel curvature tensor, the mass of the particle, its 4-velocity, and its spin tensor. The connection is not unique, and here it is chosen to satisfy one of the basic geometrical principles of Einstein's unified field theory: The symmetric part of the fundamental tensor of the geometry is specified to be the metric tensor of general relativity. The special case of conformally flat space-times is discussed.     

Letter: Non Conducting Spherically Symmetric Fluids

A class of spherically symmetric spacetimes invariantly defined by a zero flux condition is examined first from a purely geometrical point of view and then physically by way of Einstein's equations for a general fluid decomposition of the energy-momentum tensor. The approach, which allows a formal inversion of Einstein's equations, explains, for example, why spherically symmetric perfect fluids with spatially homogeneous energy density must be shearfree.     

Unified Field Theory From Enlarged Transformation Group. The Covariant Derivative for Conservative Coordinate Transformations and Local Frame Transformations

Pandres has developed a theory in which the geometrical structure of a real four-dimensional space-time is expressed by a real orthonormal tetrad, and the group of diffeomorphisms is replaced by a larger group called the conservation group. This paper extends the geometrical foundation for Pandres’ theory by developing an appropriate covariant derivative which is covariant under all local Lorentz (frame) transformations, including complex Lorentz transformations, as well as conservative transformations. After defining this extended covariant derivative, an appropriate Lagrangian and its resulting field equations are derived. As in Pandres’ theory, these field equations result in a stress-energy tensor that has terms which may automatically represent the electroweak field. Finally, the theory is extended to include 2-spinors and 4-spinors.     

Standard model and gravity from spinors

It is shown how the Lorentz and standard-model gauge groups can be unified by using algebraic spinors of the standard four-dimensional Clifford algebra, in left–right symmetric fashion. This defines a framework of unification with gravity and generates exactly a standard-model family of fermions, while a Pati–Salam unification group emerges, at the Planck scale, where (chiral) gravity decouples. We show that this low-energy broken phase emerges from the VEV of extended vierbein fields, which at this stage are assumed to be dynamically generated from a theory in the fully symmetric phase valid beyond the Planck scale (and whose consistency and dynamics is thus yet to be assessed) providing thus a geometrical and group-theoretical framework for the unification and breaking. At low energy, on the other hand, it is intriguing to find, as a remnant of this unification, new isospin-triplet spin-two particles that may naturally lie at the weak scale, providing a striking signal at the LHC.     

RI′/SMOM scheme amplitudes for quark currents at two loops

We determine the two loop corrections to the Green’s function of a quark current inserted in a quark 2-point function at the symmetric subtraction point. The amplitudes for the scalar, vector and tensor currents are presented in both the [`(MS)]\overline {\mathrm {MS}} and RI′/SMOM renormalization schemes. The RI′/SMOM scheme two loop renormalization for the scalar and tensor cases agree with previous work. The vector current renormalization requires special treatment as it must be consistent with the Slavnov–Taylor identity which we demonstrate. We also discuss the possibility of an alternative definition of the RI′/SMOM scheme in the case of the tensor current.     

Complexified Gravity in Noncommutative Spaces

The presence of a constant background antisymmetric tensor for open strings or D-branes forces the space-time coordinates to be noncommutative. This effect is equivalent to replacing ordinary products in the effective theory by the deformed star product. An immediate consequence of this is that all fields get complexified. The only possible noncommutative Yang–Mills theory is the one with U(N) gauge symmetry. By applying this idea to gravity one discovers that the metric becomes complex. We show in this article that this procedure is completely consistent and one can obtain complexified gravity by gauging the symmetry U(1,D−1) instead of the usual SO(1,D−1). The final theory depends on a Hermitian tensor containing both the symmetric metric and antisymmetric tensor. In contrast to other theories of nonsymmetric gravity the action is both unique and gauge invariant. The results are then generalized to noncommutative spaces. Received: 1 June 2000 / Accepted: 27 November 2000