The mathematical and geometrical structure of the spacetime and the concept of unification, matter and energy

Geometrical analysis of a new type of Unified Field Theoretical models follow the guidelines of previous works of the authors is presented. These new unified theoretical models are characterized by an underlying hypercomplex structure, zero non-metricity and the geometrical action is determined fundamentally by the curvature provenient of the breaking of symmetry of a group manifold in higher dimensions. This mechanism of Cartan-MacDowell-Mansouri type, permits us to construct geometrical actions of determinantal type leading a non topological physical Lagrangian due the splitting of a reductive geometry. Our goal is to take advantage of the geometrical and topological properties of this theory in order to determine the minimal group structure of the resultant spacetime Manifold able to support a fermionic structure. From this fact, the relation between antisymmetric torsion and Dirac structure of the spacetime is determined and the existence of an important contribution of the torsion to the giromagnetic factor of the fermions, shown. Also we resume and analyze previous cosmological solutions in this new UFT where, as in our work [Class. Quantum Grav. 22 (2005) 4987–5004] for the non abelian Born-Infeld model, the Hosoya and Ogura ansatz is introduced for the important cases of tratorial, totally antisymmetric and general torsion fields. In the case of spacetimes with torsion the real meaning of the spin-frame alignment is find and the question of the minimal coupling is discussed.     

Spinning test particle in Kalb-Ramond background

In this work we explore the geodesic deviations of spinning test particles in a string inspired Einstein-Kalb-Ramond background. Such a background is known to be equivalent to a spacetime geometry with torsion. We have shown here that the antisymmetric Kalb-Ramond field has a significant effect on the geodesic deviation of a spinning test particle. A search for observational evidence of such an effect in astrophysical experiments may lead to a better understanding of the geometry of the background spacetime.Received: 5 April 2005, Revised: 19 May 2005, Published online: 8 July 2005     

Nonholonomic Mapping Principle for Classical and Quantum Mechanics in Spaces with Curvature and Torsion

I explain the geometric basis for the recently-discovered nonholonomic mapping principle which permits deriving laws of nature in spacetimes with curvature and torsion from those in flat spacetime, thus replacing and extending Einstein's equivalence principle. As an important consequence, it yields a new action principle for determining the equation of motion of a free spinless point particle in such spacetimes. Surprisingly, this equation contains a torsion force, although the action involves only the metric. This force makes trajectories autoparallel rather than geodesic, as a manifestation of inertia. A generalization of the mapping principle transforms path integrals from flat spacetimes to those with curvature and torsion, thus playing the role of a quantum equivalence principle. This generalization yields consistent results only for completely antisymmetric or for gradient torsion.     

The geometrical form for the string space-time action

In the present article, we derive the space-time action of the bosonic string in terms of geometrical quantities. First, we study the space-time geometry felt by a probe bosonic string moving in antisymmetric and dilaton background fields. We show that the presence of the antisymmetric field leads to space-time torsion, and the presence of the dilaton field leads to space-time non-metricity. Using these results we obtain the integration measure for space-time with stringy non-metricity, requiring its preservation under parallel transport. We derive the Lagrangian depending on stringy curvature, torsion and non-metricity.     

Uniqueness of rotating charged black holes in five-dimensional minimal gauged supergravity

We study a five-dimensional spacetime admitting, in the presence of torsion, a non-degenerate conformal Killing–Yano 2-form which is closed with respect to both the usual exterior differentiation and the exterior differentiation with torsion. Furthermore, assuming that the torsion is closed and co-closed with respect to the exterior differentiation with torsion, we prove that such a spacetime is the only spacetime given by the Chong–Cvetič–Lü–Pope solution for stationary, rotating charged black holes with two independent angular momenta in five-dimensional minimal gauged supergravity.     

Parity violation in four and higher dimensional spacetime with torsion

The possibility of parity violation in a gravitational theory with torsion is extensively explored in four and higher dimensions. In the former case, we have listed our conclusions on when and whether parity ceases to be conserved, with both two- and three-index antisymmetry of the torsion field. In the latter, the bulk spacetime is assumed to have torsion, and the survival of parity violating terms in the four dimensional effective action is studied, using the compactification schemes proposed by Arkani-Hamed-Dimopoulos-Dvali and Randall-Sundrum. An interesting conclusion is that the torsion-axion duality arising in a stringy scenario via the second rank antisymmetric Kalb-Ramond field leads to conservation of parity in the gravity sector in any dimension. However, parity violating interactions do appear for spin-1/2 fermions in such theories, which can have crucial phenomenological implications.Received: 17 July 2003, Revised: 12 February 2004, Published online: 23 April 2004     

Propagating Torsion from First Principles

A propagating torsion model is derived from the requirement of compatibility between minimal action principle and minimal coupling procedure in Riemann-Cartan spacetimes. In the proposed model, the trace of the torsion tensor is derived from a scalar potential that determines the volume element of the spacetime. The equations of the model are written down for the vacuum and for various types of matter fields. Some of their properties are discussed. In particular, we show that gauge fields can interact minimally with the torsion without the breaking of gauge symmetry.     

A minimal time and time-temperature uncertainty principle

We show that introducing torsion in general relativity, that is, physically, considering the effect of the spin and linking the torsion to defects in spacetime topology, we can have a minimal unit of time. Also an uncertainty relation between time and temperature is suggested. The interesting thing is that with this minimal time we can eliminate the divergence of the self-energy integral without introducing any ad hoc cut-off, and it is also possible to understand black-hole evaporation as a process of quantum diffusion which leads directly to the Hawking formula. A minimal operationally definable temperature in a cosmological context is discussed.     

TheB (3) field as a link between gravitation and electromagnetism in the vacuum

The emergence of theB ⁽³⁾ field in vacuo has shown that electromagnetism is non-Abelian and similar in structure to gravitation. In this paper the Christoffel symbol used in general relativity is developed for electromagnetism in curvilinear coordinates: The former becomes describable as the antisymmetric part of the gravitational Ricci tensor. Therefore gravitation and electromagnetism are respectively the symmetric and antisymmetric parts of thesame Ricci tensor within a proportionality factor. Both fields are obtained from the Riemann curvature tensor, both are expressions of curvature in spacetime.     

Gravitational energy of a magnetized Schwarzschild black hole: a teleparallel approach

We investigate the distribution of gravitational energy in the spacetime of a Schwarzschild black hole immersed in a cosmic magnetic field. This is done in the context of the teleparallel equivalent of general relativity, which is an alternative geometrical formulation of general relativity, where gravity is described by a spacetime endowed with torsion rather than curvature, whose fundamental field variables are tetrad fields. We calculate the energy enclosed by a two-surface of constant radius—in particular, the energy enclosed by the event horizon of the black hole. In this case we find that the magnetic field has the effect of increasing the gravitational energy as compared to the vacuum Schwarzschild case. We also compute the energy (i) in the weak magnetic field limit, (ii) in the limit of vanishing magnetic field, and (iii) in the absence of the black hole. In all cases our results are consistent with what should be expected on physical grounds.     

On the axioms of topological electromagnetism

The axioms of topological electromagnetism that were given by Hehl, Obukhov, and Rubilar are refined by the use of geometrical and topological notions that are found on orientable manifolds. The central problem of defining the spacetime electromagnetic constitutive law in terms of the geometrical and topological structure of the spacetime manifold is elaborated upon in the linear and nonlinear cases. The manner by which the spacetime metric might follow from the electromagnetic constitutive law is examined in the linear case. The possibility that the intersection form of the spacetime manifold might play a role in defining a topological basis for a nonlinear electromagnetic constitutive law is explored. The manner by which electromagnetic wave motion relates to the geometric structure is also discussed.     

Curvature from spin

We show that, considering the dislocation defect induced by torsion in spacetime, which behaves like a string with tension, we are lead also to defect angle and then to curvature of spacetime. The space with torsion and curvature is then equivalent to an elastic continuum which has undergone plastic deformations and, following Sakharov idea of the spacetime as a elastic continuum, we are lead to a gravitational constant, which occurs in the Einstein action, as the metrical elasticity of spacetime with the exact value without introducing any arbitrary cutoff, when also torsion is considered.     

Superstrings with tensor degrees of freedom

We add antisymmetric tensor degrees of freedom to the usual superstring coordinates. We show that super and kappa symmetries are only achieved for the spacetime dimensionD=4. We also address problems related to the quantization of the model and discuss the influences of this extended spacetime in the usual quantum field theory.     

Q gravity

The geometrical structure of an extended theory of gravitation incorporating the quaternion field is investigated. A lagrangian is constructed and shown to be pure real as a consequence of the geometrical structure. The structure of spacetime becomes nonabelian; however, the base manifold in which physics takes place, remains real and n-dimensional.     

Formulation of 11-dimensional supergravity in superspace

We formulate on-shell 11-dimensional supergravity in superspace and express its equations of motion in terms of purely geometrical quantities. All torsion and curvature components are solved in terms of a single superfield W_rstu, totally antisymmetric in its (flat vector) indices. The dimensional reduction of this formulation is expected to be related to the superspace formulation of N=8 extended supergravity and might explain the origin of the hidden (local) SU(8) and (global) E₇ symmetries present in this theory.     

Enlarged Geometries of Gauge Bundles

The geometrical picture of gauge theories must be enlarged when a gauge potentialceases to behave like a connection, as it does in electroweak interactions. Whenthe gauge group has dimension four, the vector space isomorphism betweenspacetime and the gauge algebra is realized by a tetrad-like field. The objectmeasuring the deviation from a strict bundle structure has the formal behaviorof a spacetime connection, of which the deformed gauge field strength is thetorsion. A generalized derivative emerges in terms of which the two Bianchiidentities are formally recovered. Effects of gravitational type turn up. Thedynamical equations obtained correspond to a broken gauge model on acurved spacetime.     

The cornerstone role of the torsion in Finslerian physical worlds

It is shown that there are no metric-compatible connections with zero torsion onproperly Finslerian, i.e. post-Riemannian, metrics. Since Finslerian connections exist on Riemannian metrics, the torsion rather than the metric becomes the object which determines whether the geometry is properly Finslerian or not. On the other hand, the solder forms and connection are determined by the torsion if the affine curvature is zero, the torsion then containing all the information about the geometric reality of spacetime. Since the metric curvature may still be Riemannian, the question arises of whether its present central role in spacetime physics is but a consequence of requiring that all the geometric content of spacetime be contained in the metric.     

Recurrent Riemann-Cartan spacetimes

An example of a Riemann recurrent spacetime with torsion is given. The implications for the Segré classification are investigated. It is also shown that the recurrent vector coincides with the torsion vector in the case where the Ricci recurrent spacetime is an Einstein space.     

Dirac particle in Riemann–Cartan spacetimes

Dynamics of a Dirac particle in general Riemann–Cartan spacetimes is considered. The Hermitian Dirac Hamiltonian is derived and is transformed to the Foldy–Wouthuysen representation for an arbitrary spacetime geometry. The contribution of the torsion field to the Foldy–Wouthuysen Hamiltonian is found. The new bounds on Cartan’s spacetime torsion are obtained.     

Strings Have Spin

If a string source is used to replace the conventional point-like source of gravitation in spacetime with torsion, then intrinsic spin arises naturally as an attribute of the string. This spin stems from the structure of the string, and not from internal motions.