Fermion Fields in Theories of Gravitation

After an introduction to the formalism used throughout the paper there follows a concise presentation of the theory of fermion fields in one-tetrad gravitational theories. That presentation gives a hint to the construction of a bi-tetrad theory, the two tetrad fields being denoted by h^A_k and h?^A_k. The tetrad field h^A_k. gives the Riemannian metric g_kl while the tetrad field h?h^A_k is orthonormalized with respect to the flat metric a_kl. Specializing h?^A_k in such a way that they have the form δ^A_k in the preferred coordinates of Minkowski space and using a matter Lagrangian which contains these h?^A_k only by the anholonomic components of the metric Christoffel symbols, we obtain a dynamical energy momentum tensor which is equal to the canonical one. Then we consider the relations of the bi-tetrad theory to other theories which are only covariant with respect to global Lorentz transformations from the beginning. As an example we formulate the main relations of the two-component neutrino theory.     

Teleparallel equivalent theory of (1+1)-dimensional gravity

A theory of (1+1)-dimensional gravity is constructed on the basis of the teleparallel equivalent of general relativity.The fundamental field variables are the tetrad fields e i μ and the gravity is attributed to the torsion.A dilatonic spherically symmetric exact solution of the gravitational field equations characterized by two parameters M and Q is derived.The energy associated with this solution is calculated using the two-dimensional gravitational energy-momentum formula.     

A new class of interior solutions for a stress-energy tensor of Segré characteristic [111, 1]

The interior solutions of (the tetrad versions of) Einstein's field equations withT _AB having Segré characteristic [111, 1] (which has all four eigenvalues distinct), are investigated. For this purpose amixed method, which combines Synge'sg method andT method, is introduced. Some of the tetrad equations are solved for the metric functions while the remaining equations are used to define the corresponding components ofT _AB . As necessary conditions of the consistency of the mixed method the conservation equationsT ^AB _B =0 are explicitly verified. Several simplifications and analysis of some differential inequalities show the existence of a new class of solutions which, in addition to having Segré characteristic [111, 1], also satisfy the strong energy conditions of Hawking and Ellis.This Author is a member of the Theoretical Sciences Institute, Simon Fraser University, Burnaby, B.C., V5A 1S6, Canada.     

Interacting Gravitational and Dirac Fields with Torsion

A general interaction scheme is formulated in a general space–time with torsion from the action principle by considering the gravitational, the Dirac, and the torsion field as independent fields. Some components of the torsion field come out to be automatically zero. Both the resulting Einstein-like and the Dirac-like fields equations contain nonlinear terms given by a self-interaction of the Dirac spinor and originally produced by torsion. The theory is specialized to the Robertson–Walker space–time without torsion. To solve he corresponding equations, that still have a complex structure, the spin coefficients have to be calculated explicitly from the tetrad employed. A solution, even if simple and elementary, is then determined.     

Integration in the GHP Formalism IV: A New Lie Derivative Operator Leading to an Efficient Treatment of Killing Vectors

In order to achieve efficient calculations and easy interpretations of symmetries, a strategy for investigations in tetrad formalisms is outlined: work in an intrinsic tetrad using intrinsic coordinates. The key result is that a vector field is a Killing vector field if and only if there exists a tetrad which is Lie derived with respect to ; this result is translated into the GHP formalism using a new generalised Lie derivative operator with respect to a vector field . We identify a class of it intrinsic GHP tetrads, which belongs to the class of GHP tetrads which is generalised Lie derived by this new generalised Lie derivative operator in the presence of a Killing vector field . This new operator also has the important property that, with respect to an intrinsic GHP tetrad, it commutes with the usual GHP operators if and only if is a Killing vector field. Practically, this means, for any spacetime obtained by integration in the GHP formalism using an intrinsic GHP tetrad, that the Killing vector properties can be deduced from the tetrad or metric using the Lie-GHP commutator equations, without a detailed additional analysis. Killing vectors are found in this manner for a number of special spaces.     

One-Parameter Continuous Fields of Kirchberg Algebras

We prove that all unital separable continuous fields of C*-algebras over [0,1] with fibers isomorphic to the Cuntz algebra are trivial. More generally, we show that if A is a separable, unital or stable, continuous field over [0,1] of Kirchberg C*-algebras satisfying the UCT and having finitely generated K-theory groups, then A is isomorphic to a trivial field if and only if the associated K-theory presheaf is trivial. For fixed we also show that, under the additional assumption that the fibers have torsion free K _d -group and trivial K _d+1-group, the K _d -sheaf is a complete invariant for separable stable continuous fields of Kirchberg algebras. M.D. was supported in part by NSF Grant #DMS-0500693. G.A.E. held a Discovery Grant from NSERC Canada.     

A solution of the non‐uniqueness problem of the Dirac Hamiltonian and energy operators

In a general spacetime, the possible choices for the field of orthonormal tetrads lead (in standard conditions) to equivalent Dirac equations. However, the Hamiltonian operator is got from rewriting the Dirac equation in a form adapted to a particular reference frame, or class of coordinate systems. That rewriting does not commute with changing the tetrad field (u_α ). The data of a reference frame F fixes a four‐velocity field v, and also fixes a rotation‐rate field Ω . It is natural to impose that u₀ = v. We show that then the spatial triad (u_p) can only be rotating w.r.t. F, and that the title problem is solved if one imposes that the corresponding rotation rate Ξ be equal to Ω – or also, if one imposes that Ξ = 0 . We also analyze other proposals which were aimed at solving the problem of the non‐uniqueness of the Dirac Hamiltonian.     

All Stable Characteristic Classes of Homological Vector Fields

An odd vector field Q on a supermanifold M is called homological, if Q ² = 0. The operator of Lie derivative L _Q makes the algebra of smooth tensor fields on M into a differential tensor algebra. In this paper, we give a complete classification of certain invariants of homological vector fields called characteristic classes. These take values in the cohomology of the operator L _Q and are represented by Q-invariant tensors made up of the homological vector field and a symmetric connection on M by means of the algebraic tensor operations and covariant differentiation.     

Electromagnetic perturbations of charged Kerr geometry

The freedom associated with infinitesimal co-ordinate transformations and infinitesimal tetrad rotations within the Newman Penrose formalism is used when considering electromagnetic perturbations of type D geometries. In particular, it is shown that for the charged Kerr geometry, a suitable choice of tetrad rotation leads to equations for the perturbed Newman Penrose components ₀ and ₂ which are closely related to those for the uncharged case.     

Integrability and Explicit Solutions in Some Bianchi Cosmological Dynamical Systems

Abstract

The Einstein field equations for several cosmological models reduce to polynomial systems of ordinary differential equations. In this paper we shall concentrate our attention to the spatially homogeneous diagonal G ₂ cosmologies. By using Darboux’s theory in order to study ordinary differential equations in the complex projective plane ??² we solve the Bianchi V models totally. Moreover, we carry out a study of Bianchi VI models and first integrals are given in particular cases.     

Double-horizon limit, AdS geometry and entropy function

We start from a generic metric which describes four-dimensional stationary black holes in an arbitrary theory of gravity and show that the AdS₂ part of the near-horizon geometry is a consequence of the double-horizon limit and finiteness. We also show that the field configurations of the near horizon are determined if the same conditions are applied to the equations of motion. This is done by showing that in the double-horizon limit field equations at the horizon decouple from the bulk of the space. Solving these equations gives the near-horizon field configurations. It is shown that these decoupled equations can be obtained from an action derived from the original action by applying the double-horizon condition. Our results agree with the entropy function method.     

Hestenes' Tetrad and Spin Connections

Defining a spin connection is necessary for formulating Dirac's bispinor equation in a curved space-time. Hestenes has shown that a bispinor field is equivalent to an orthonormal tetrad of vector fields together with a complex scalar field. In this paper, we show that using Hestenes' tetrad for the spin connection in a Riemannian space-time leads to a Yang-Mills formulation of the Dirac Lagrangian in which the bispinor field Ψ is mapped to a set of SL(2,R)× U(1) gauge potentials F_α^K and a complex scalar field ρ. This result was previously proved for a Minkowski space-time using Fierz identities. As an application we derive several different non-Riemannian spin connections found in the literature directly from an arbitrary linear connection acting on the tensor fields (F_α^K, ρ). We also derive spin connections for which Dirac's bispinor equation is form invariant. Previous work has not considered form invariance of the Dirac equation as a criterion for defining a general spin connection.     

Xanthopoulos Theorem in the Kaluza–Klein Theory

It is shown that if _AB is an exact solution of the Einstein vacuum field equations in 4 + 1 dimensions, R^ _AB = 0, and l _A is a null vector field, then _AB + l _A l _B is also an exact solution of the Einstein equations R^ _AB = 0 if and only if the perturbation l _A l _B satisfies the Einstein equations linearized about _AB. Then, making use of the Kaluza–Klein approach, it is shown that this result allows us to obtain exact solutions of the Einstein–Maxwell equations (possibly coupled to a scalar field) by solving a system of linear equations.     

Direct gauging of the Poincaré group V. Group scaling,classical gauge theory,and gravitational corrections

Homogeneous scaling of the group space of the Poincaré group,P ₁₀, is shown to induce scalings of all geometric quantities associated with the local action ofP ₁₀. The field equations for both the translation and the Lorentz rotation compensating fields reduce toO(1) equations if the scaling parameter is set equal to the general relativistic gravitational coupling constant 8Gc ^–4. Standard expansions of all field variables in power series in the scaling parameter give the following results. The zeroth-order field equations are exactly the classical field equations for matter fields on Minkowski space subject to local action of an internal symmetry group (classical gauge theory). The expansion process is shown to breakP ₁₀-gauge covariance of the theory, and hence solving the zeroth-order field equations imposes an implicit system ofP ₁₀-gauge conditions. Explicit systems of field equations are obtained for the first- and higher-order approximations. The first-order translation field equations are driven by the momentum-energy tensor of the matter and internal compensating fields in the zeroth order (classical gauge theory), while the first-order Lorentz rotation field equations are driven by the spin currents of the same classical gauge theory. Field equations for the first-order gravitational corrections to the matter fields and the gauge fields for the internal symmetry group are obtained. Direct Poincaré gauge theory is thus shown to satisfy the first two of the three-part acid test of any unified field theory. Satisfaction of the third part of the test, at least for finite neighborhoods, seems probable.     

On Treder-Type Tetrad Theories of Gravitation II. Treder's Tetrad Theories in Linear Approximation

We consider some properties of TREDER'S tetrad theories, derived in I, using the field equations proposed by KASPER and LIEBSCHER . The linearized theory is considered, because the field energy becomes positive, if the energy of the weak field is a positive one. Using the dynamical equations, the field equations lead for the symmetric part of the field to the gauge invariant field equations in Hilbert gauge and to corresponding equations for the antisymmetric part. This means that in this approximation the dynamical equations replace the gauge invariance and the tetrad field corresponds to a mixture of tensor and scalar gravitons. We discuss possible experiments for showing the existence of scalar gravitons and limiting the free parameter of the theory.     

Teleparallelism as a universal connection on null hypersurfaces in general relativity

We show that there exists a close relationship between inner geometry of a null hypersurfaceN ₃ and the Newman-Penrose (NP) spin coefficient formalism. Projecting the null complexNP tetrad ontoN ₃ we get two triads of basis vectors inN ₃. Inner geometry ofN ₃ is based on the assumption that these vectors are parallelly transported along the surface; this gives rise to the teleparallel connection as a metric nonsymmetric affine connection. The gauge freedom for the choice of the basis triads is given by the isotropy subgroup of the local Lorentz group leaving invariant the direction of the null generators ofN ₃, and teleparallelism is determined by the equivalence class of the basis triads with respect to the global gauge group. Nine of the twelve NP coefficients are identified as the triad components of the torsion and the second fundamental form ofN ₃. The resulting generalized Gauss-Codazzi equations are identical to 9 of the NP equations, i.e., to the half of the Ricci identities. This result gives a geometrical meaning to the entire formalism. Finally we present a general proof of Penrose's theorem that the shear of the null generators ofN ₃ is the only initial null datum for a gravitational field onN ₃.     

Lanczos Spintensor and GHP Formalism

This work shows how the equations which relate the Lanczos potential K _ijr to the conformal tensor C _abpq, can be structured in a simple form when written in terms of the formalism proposed by Geroch-Held-Penrose. As working examples, we present the immediate construction of K _ijk for any spacetime with Petrov type O, N or III. Our findings are in good agreement with already published results, which indicates a relationship between the spin coefficients and the Lanczos spintensor onto the canonical tetrad.     

Remarks on Singularities, Dimension and Energy Dissipation for Ideal Hydrodynamics and MHD

For weak solutions of the incompressible Euler equations, there is energy conservation if the velocity is in the Besov space B ³ _s with s greater than 1/3. B ³ _s consists of functions that are Lip(s) (i.e., H?lder continuous with exponent s) measured in the L ^p norm. Here this result is applied to a velocity field that is Lip(α₀) except on a set of co-dimension on which it is Lip($agr;₁), with uniformity that will be made precise below. We show that the Frisch-Parisi multifractal formalism is valid (at least in one direction) for such a function, and that there is energy conservation if . Analogous conservation results are derived for the equations of incompressible ideal MHD (i.e., zero viscosity and resistivity) for both energy and helicity . In addition, a necessary condition is derived for singularity development in ideal MHD generalizing the Beale-Kato-Majda condition for ideal hydrodynamics. Received: 21 March 1995 / Accepted: 6 August 1996     

A Lagrangian stochastic model for tetrad dispersion

We present results for tetrad (four-particle) dispersion in homogeneous isotropic turbulence by means of a simple Lagrangian stochastic model with a focus on the inertial subrange. We show that for appropriate values of C ₀, the constant of proportionality in the second-order Lagrangian velocity structure function, the shape statistics agree well with equivalent results from a direct numerical simulation (DNS) of turbulence. Moreover, we show that the shape statistics are independent of C ₀ for a wide range of C ₀-values. We also show that the parameters which characterise the shape of the tetrads can be approximately related to appropriate ratios of the growth rates of the mean square separation, the mean square area and the mean square volume of the tetrads. By means of exit times, we are able to estimate the equivalent values for the DNS data. We also consider the statistics of four-point velocity differences (via a diffusion tensor) which agree well with the DNS. We show that the nature of the velocity field experienced by the tetrad varies significantly with C ₀.     

Exact self-consistent plane-symmetric solutions of the equations of two interacting fields (spinor field and scalar field)

A spinor field interacting with a zero-mass neutral scalar field is considered for the case of the simplest type of direct interaction, where the interaction Lagrangian has the formL _int =1/2 ϕ_αϕ_,α F(S) whereF(S) is an arbitrary function of the spinor field invariantS=ψψ. Exact solutions of the corresponding systems of equations that take into account the natural gravitational field in a plane-symmetric metric are obtained. It is proved that the initial system of equations has regular localized soliton-type solutions only if the energy density of the zero-mass scalar field is negative as it “disengages” from interaction with the spinor field. In two-dimensional space-time the system of field equations we are studying describes the configuration of fields with constant energy densityT ₀₀ , i.e., no soliton-like solutions exist in this case. Russian People’s Friendship University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 69–75, July, 1998.