REVIEW: Spin Gravity

A theory of gravitation with torsion that isderived from an antisymmetric second rank tensor isreviewed. A non-symmetric energy momentum tensor isdeveloped and the explicit material action is presented. This is done from a phenomenological point ofview and using the Dirac Lagrangian. The equations ofmotion are derived and it is shown that the source oftorsion is the intrinsic spin of elementary particles. The torsion sector is reduced to a low energy3-vector formulation and the interaction energies arederived. The theory is reformulated in terms of theDirac field, and it is shown that precisely the same interaction energy is predicted. The theory iscompared to the low energy string theory effective fieldlimit and the scalar field is introduced. It is shownhow this field is a mandatory requirement of the theory, and a particular limit is derivedin which the scalar field acts as a strong non-minimalcoupling of the torsion field. Physical predictions arecompared to experiment, including effects in hydrogen and paramagnetic salts. Other physicalmanifestations that are discussed include spin flippingof neutrinos, torsion waves and their power, how thenon-linear Dirac and Schrodinger equations arise from torsion, and the physical origin andcorrect prediction of the magnetic dipole moment ofelementary particles.     

An Entropy Functional for Riemann-Cartan Space-Times

By viewing space-time as a continuum elastic medium and introducing an entropy functional for its elastic deformations, T. Padmanabhan has shown that general relativity emerges from varying the functional and that the latter suggests holography for gravity and yields the Bekenstein-Hawking entropy formula. In this paper we extend this idea to Riemann-Cartan space-times by constructing an entropy functional for the elastic deformations of space-times with torsion. We show that varying this generalized entropy functional permits to recover the full set of field equations of the Cartan-Sciama-Kibble theory. Our generalized functional shows that the contributions to the on-shell entropy of a bulk region in Riemann-Cartan space-times come from the boundary as well as the bulk and hence does not suggest that holography would also apply for gravity with spin in space-times with torsion. It is nevertheless shown that for the specific cases of Dirac fields and spin fluids the system does become holographic. The entropy of a black hole with spin is evaluated and found to be in agreement with Bekenstein-Hawking formula.     

Classical unified fields: Physical space

We develop a general unified theory of classical mechanics and classical electromagnetism in a gravitational field on Friedman-Schöuten space-time (FSS). In this formalism (i) local equations of a charged fluid in an electromagnetic field are the same as in classical mechanics, (ii) local equations for a moving charged fluid are the same as in electromagnetism, (iii) the path of a charged particle under gravity and electromagnetism is a geodesic of the four-dimensional FSS, and (iv) the strong equivalence principle and a nonzero torsion coexist without conflict.     

Exact solutions inU 4 gravity. I. The ansatz for self double dual curvature

We investigate the energy-momentum and spin field equations of gravity theory on a Riemann-Cartan space-time (including metric and torsion,U ₄-manifold). The structure of the rather complicated nonlinear differential equations of second order is made considerably easier to survey by decomposing curvature into its self and anti-self double dual parts. This leads to an obvious ansatz for the self double dual curvature, whereby the field equations are reduced to Einstein's equations with cosmological term. To solve the double dual ansatz, we choose proper variables adopted to its double duality, and perform a (3+1)-decomposition of exterior calculus. We examine these equations further on a Kerr background with cosmological constant for the Riemannian geometry.     

Finding Dirac Spin Effect in NUT Spacetime

In the present study, we are interested in finding the spin precession of a Dirac particle in expanding and rotating NUT spaeetime. A tetrad with two functions to be determined is applied to the field equation of the teleparallel theory of gravity via a coordinate transformation. The vector, the axial-vector and the tensor parts of the torsion tensor are obtained. We found that the vector parts are in the radial and Ф-directions. The axial-vector torsion is along r-direction while its other components along θ and oh-directions vanish everywhere. The vector connected with Dirac spin has been evaluated as well.     

On the weak field approximation of the de Sitter gauge theory of gravity

The weak field approximation of a model of de Sitter gauge theory of gravity is studied in two cases. Without torsion and spin current, the model cannot give the right non-relativistic approximation unless the density is a constant. With small torsion, a satisfactory Newtonian approximation can be obtained.     

Geometrization of the physics with teleparallelism. I. The classical interactions

A connection viewed from the perspective of integration has the Bianchi identities as constraints. It is shown that the removal of these constraints admits a natural solution on manifolds endowed with a metric and teleparallelism. In the process, the equations of structure and the Bianchi identities take standard forms of field equations and conservation laws.The Levi-Civita (part of the) connection ends up as the potential for the gravity sector, where the source is geometric and tensorial and contains an explicit gravitational contribution.Nonlinear field equations for the torsion result. In a low-energy approximation (linearity andlow energy-momentumtransfer), the postulate that only charge and velocities contribute to the source transforms these equations into the Maxwell system. Moreover, the affine geodesics become the equations of motion of special relativity with Lorentz force in the same approximation [J. G. Vargas,Found. Phys. 21, 379 (1991)]. The field equations for the torsion must then be viewed as applying to an electromagnetic/strong interaction.A classical unified theory thus arises where the underlying geometry confers their contrasting characters to Maxwell-Lorentz electrodynamics and to an Einstein's-like theory of gravity. The highly compact field equations must, however, be developed in phase-spacetime, since the connection is velocity-dependent, i.e., Finsler-like.Further opportunities for similarities with present-day physics are discussed: (a) teleparallelism allows for the formulation of the torsion sector of the theory as a flat space theory with concomitant point-dependent transformations; (b) spinors should replace Lorentz frames in their role as the subjects to which the connection refers; (c) the Dirac equation consistent with the frame bundle for a velocity-dependent metric with Lorentz signature generates a weak-like interaction in the torsion sector.Work done at the Department of Mathematics and Physics of the Interamerican University of Puerto Rico, San German, Puerto Rico 00683.     

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.     

The Brans-Dicke scalar field in Einstein-Cartan theory

In the framework of Einstein-Cartan (EC) theory, the Brans-Dicke (BD) theory is considered and it is found that a scalar field nonminimally coupled to the gravitational field gives rise to torsion, even though the scalar field has zero spin. The metric equations stay the same if the coupling constant is rescaled, but the equations of motion of a test particle, derived from the conservation equations, differ from those of the usual BD theory without torsion. The gravitational red-shift value differs considerably from the usual prediction of general theory of relativity (GTR), and rules out the possibility of a torsion version of BD theory for<6.     

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.     

Extension of the York field decomposition to general gravitationally coupled fields

We extend the York decomposition analysis of the initial value constraints to general gravitationally coupled classical field theories. The decomposition is found to be particularly useful in solving the constraint equations for all theories of current physical interest. These include Einstein gravity or Einstein-Cartan (torsion) gravity coupled to the massive or massless version of the following: general scalar (including Klein-Gordon, Brans-Dicke, and Higgs), Dirac spin 1/2, Maxwell (Proca) and Yang-Mills (any gauge group). We show in detail how the program works for the general Yang-Mills field and for the Einstein-Cartan-Proca field.     

Massive, Non-ghost Solutions for the Dirac Field Coupled Self-consistently to Gravity

We present new, massive, non-ghost solutions for the Dirac field coupled self-consistently to gravity. We employ a gauge-theoretic formulation of gravity which automatically identifies the spin of the Dirac field with the torsion of the gauge fields. Homogeneity of the field observables requires that the spatial sections be flat. Expanding and collapsing singular solutions are given, as well as a solution which expands from a singularity before recollapsing. Torsion effects are only important while the Compton wavelength of the Dirac field is larger than the Hubble radius. We study the motion of spinning point-particles in the background of the expanding solution. The anisotropy due to the torsion is manifest in the particle trajectories.     

“Bouncing” of simple cosmological models with torsion

The known cosmological solutions of the Einstein-Cartan-Sciama-Kibble (ECSK) field equations are reviewed. The prevention of singularities is explained by means of the extension of the Hawking-Penrose singularity theorems to the ECSK theory. Singularity prevention in semiclassical “spinning dust” models derives from the postulated form of the canonical energy-momentum and spin angular momentum tensors for the matter distribution. The effects of shear, vorticity, and pressure are examined. The singularity behavior of cosmological models incorporating the Dirac field as the source of the metric and torsion is discussed. In these models one finds an enhancement, rather than a prevention of singularity formation. Finally, the consequences of spin and torsion for observational cosmology and for particle creation in the early universe are noted.     

Field Equations of Arbitrary Spin in Space-time with Torsion

A two spinor lagrangian formulation of field equations for massive particle of arbitrary spin is proposed in a curved space-time with torsion. The interaction between fields and torsion is expressed by generalizing the situation of the Dirac equation. The resulting field equations are different (except for the spin-1/2 case) from those obtained by promoting the covariant derivatives of the torsion free equations to include torsion. The non linearity of the equations, that is induced by torsion, can be interpreted as a self-interaction of the particle. The spin-1 and spin-3/2 cases are studied with some details by translating into tensor form. There result the Proca and Rarita-Schwinger field equations with torsion, respectively. PACS numbers: 03.65.Pm; 04.20.Cv; 04.20.Fy.     

Motion of test particles in a torsion field

The solution of the Mathisson-Papapetrou equations generalized to the case of the Einstein-Cartan theory, when they describe the motion of a test particle in an external torsion field, is considered. It is shown that a particle of nonzero rest mass moves inertially in a constant polarized torsion field, but its spin precesses around the direction of polarization of the spin of the torsion source. It is also found that the motion of a massless particle in a variable torsion field leads to a torsion frequency displacement effect of a photon under the assumption that the photon spin interacts with the torsion.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 84–87, August, 1980.     

On Non-Riemannian Domain Walls

Two classes of non-Riemannian domain walls are obtained as distributional planar sources for the linearized Einstein-Cartan (EC) field equations of gravity. The first class represents spin polarized particles distributed on the non-Minkowskian side of the wall and an analogy with the ferromagnetic domains is displayed since the spin distributions are different on both sides of the walls. The other class represents a gravitational analog of Type I superconductor where Cartan torsion plays the role of the magnetic field. The interior solution is obtained by using matching conditions in EC-gravity and is matched to a torsionless vacuum in the last case.     

A new class of exact solutions of the vacuum quadratic Poincaré gauge field theory

A class of algebraically special exact solutions of the vacuum quadratic Poincaré gauge field theory is presented. These solutions are of type III and type N and have a nonexpanding, shear-free and twist-free geodesic repeated principal null congruence. The metric is of Kundt's class, and the torsion components are solutions of certain differential equations. The solutions have been obtained using a generalised spin coefficient formalism.     

Stability ofR + T 2 theories of gravitation

In a framework with metric and spin affine connection as independent field variables, we show that the total energy of a genericR + T ²theory of gravitation is positive definite for an asymptotically flat space-time. This suggests that a more thorough treatment of the perturbative quantization of quadratic theories of gravity (including curvature and torsion squared terms) does not yield violation of unitarity.