External fields as intrinsic geometry

There is an interesting dichotomy between a space-time metric considered as external field in a flat background and the same considered as an intrinsic part of the geometry of space-time. We shall describe and compare two other external fields which can be absorbed into an appropriate redefinition of the geometry, this time a noncommutative one. We shall also recall some previous incidences of the same phenomena involving bosonic field theories. It is known that some such theories on the commutative geometry of space-time can be re-expressed as abelian-gauge theory in an appropriate noncommutative geometry. The noncommutative structure can be considered as containing extra modes all of whose dynamics are given by the one abelian action. Received: 1 December 2000 / Published online: 23 January 2001     

On the Higgs feature of gravity

The gauge gravitation theory, based on the equivalence principle besides the gauge principle, is formulated in the fibre bundle terms. The correlation between gauge geometry on spinor bundles describing Dirac fermion fields and space-time geometry on a tangent bundle is investigated. We show that field functions of fermion fields in presence of different gravitational fields are always written with respect to different reference frames. Therefore, the conventional quantization procedure is applicable to fermion fields only if gravitational field is fixed. Quantum gravitational fields violate the above mentioned correlation between two geometries.     

A self-consistent approach to quantum field theory for extended particles

A notion of quantum space-time is introduced, physically defined as the totality of all flows of quantum test particles in free fall. In quantum space-time the classical notion of deterministic inertial frames is replaced by that of stochastic frames marked by extended particles. The same particles are used both as markers of quantum space-time points as well as natural clocks, each species of quantum test particle thus providing a standard for space-time measurements. In the considered flat-space case, the fluctuations in coordinate values with respect to stochastic frames are described by coordinate probability amplitudes related to irreducible stochastic phase space representations of the Poincaré group. Lagrangian field theory on quantum space-time is formulated. The ensuing equations of motion for interacting fields contain no singularities in their nonlinear terms, and therefore can be handled by methods borrowed from classical nonlinear analysis.Supported in part by an NSERC grant.     

Parity violation in a gravitational theory with torsion: A geometrical interpretation

In a space-time with torsion, the action for the gravitational field can be extended with a parity-violating piece. We show how to obtain such a piece from geometry itself, by suitably modifying the affine connection so as to include a pseudo-tensorial part. A consistent method is thus suggested for incorporating parity-violation in the Lagrangians of all matter fields with spin in a space-time background with torsion.     

Wave fields in Weyl spaces and conditions for the existence of a preferred pseudo-Riemannian structure

Previous axiomatic approaches to general relativity which led to a Weylian structure of space-time are supplemented by a physical condition which implies the existence of a preferred pseudo-Riemannian structure. It is stipulated that the trajectories of the short wave limit of classical massive fields agree with the geodesics of the Weyl connection and it is shown that this is equivalent to the vanishing of the covariant derivative of a mass function of nontrivial Weyl type. This in turn is proven to be equivalent to the existence of a preferred metric of the conformal structure such that the Weyl connection is reducible to a connection of the bundle of orthonormal frames belonging to this distinguished metric.     

A geometrical interpretation of parity violation in gravity with torsion

In a space-time with torsion, the action for the gravitational field can be extended with a parity-violating piece. We show how to obtain such a piece from geometry itself, by suitably modifying the affine connection so as to include a pseudo-tensorial part. The merit of such an approach is that it provides one with a consistent method for incorporating parity-violation in the Lagrangians for matter fields with arbitrary spin in a space-time background with torsion.     

On a new variational principle in general relativity and the energy of the gravitational field

A new variational principle based on the affine connection in space-time is proposed. This leads to a new formulation of general relativity. The gravitational field is a field of inertial frames in space-time. The metricg appears as a momentum canonically conjugate to the gravitational field. In the case of simple matter fields, e.g., scalar fields, electromagnetic fields, Proca fields, or hydrodynamical matter, the new formulation is equivalent to the traditional one. A new formulation of conservation laws is proposed.     

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.     

Spinor fields at spacelike infinity

The concept of a spinor structure at spacelike infinity is introduced for space-times which are asymptotically flat. It is shown how zero-rest-mass fields on space-time acquire smooth limits on this structure and that these limits satisfy certain differential equations characterized by the helicity and regularity of the field. The geometry of the limits of twistor fields is also discussed, and it seems possible that one can define the momentum and angular momentum of an asymptotically flat space-time in terms of a twistor space at spacelike infinity.     

Finsler formulation of the equations of physical fields with internal symmetries

The Finsler formulation of the space-time geometry incorporates such internal symmetries of the physical fields as isotopic invariance of strongly interacting fields. The explicit dependence of a Yang-Mills field on the elements of the Finsler structure is found. The corresponding equations for scalar and spinor fields with internal symmetries are constructed.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 37–41, August, 1984.     

On unified field theories,dynamical torsion and geometrical models: II

We analyze in this letter the same space-time structure as that presented in our previous reference (Part. Nucl., Lett. 2010. V. 7, No. 5, P. 299–307), but relaxing now the condition a priori of the existence of a potential for the torsion. We show through exact cosmological solutions from this model, where the geometry is Euclidean R ⊗ O ₃ ∼ R ⊗ SU(2), the relation between the space-time geometry and the structure of the gauge group. Precisely this relation is directly connected with the relation between the spin and torsion fields. The solution of this model is explicitly compared with our previous ones and we find that: (i) the torsion is not identified directly with the Yang-Mills type strength field, (ii) there exists a compatibility condition connected with the identification of the gauge group with the geometric structure of the space-time: this fact leads to the identification between derivatives of the scale factor with the components of the torsion in order to allow the Hosoya-Ogura ansatz (namely, the alignment of the isospin with the frame geometry of the space-time), and (iii) of two possible structures of the torsion the “tratorial” form (the only one studied here) forbids wormhole configurations, leading only to cosmological space-time solution in eternal expansion.     

Critical comments on the discussion about tachyonic causal paradoxes and on the concept of superluminal reference frame

The discussions of the tachyonic causal paradoxes and the concept of superluminal reference frame are criticized. The essence of the construction of the known paradoxes is revealed. Some possibilities of eliminating these paradoxes without contradicting the theory of relativity, are discussed. The tachyonic causal loop in an arbitrarily dimensional flat space-time is formally defined. The logical relations between assumptions on existence (or nonexistence) of the tachyonic causal loops and of inertial reference frames preferred in the tachyon kinematics are given. Such frames are not preferred in relation to bradyons and luxons, and maybe are not preferred in the dynamics of the tachyons. The theorem is proved which shows that the discussion on the tachyonic causal loops concerns also the preferred frames. The operational definitions of spacelike, timelike, and null vectors are given. It is shown that superluminal transformations and reference frames do not exist inside the theory of relativity. It is also shown that the so-called superluminal Lorentz transformations are not in fact transformations but mappings. It is concluded that the existence of tachyonic phenomena is not contradictory to the theory of relativity, while the concept of usual superluminal reference frame is contradictory to that theory.     

The Embedding of Schwarzschild in Braneworld

The braneworlds models were inspired partly by Kaluza-Klein’s theory, where both the gravitational and the gauge fields are obtained from the geometry of a higher dimensional space. The positive aspects of these models consist in perspectives of modifications it could bring in to particle physics, such as: unification in a TeV scale, quantum gravity in this scale and deviation of Newton’s law for small distances. One of the principles of these models is to suppose that all space-times can be embedded in a bulk of higher dimension. The main result in these notes is a theorem showing a mathematical inconsistency of the Randall-Sundrum braneworld model, namely that the Schwarzschild space-time cannot be embedded locally and isometrically in a five dimensional bulk with constant curvature (for example AdS-5). From the point of view of semi-Riemannian geometry this last result represents a serious restriction to the Randall-Sundrum’s braneworld model.     

The conformal structure of space-time in a constructive axiomatics based on elements of quantum mechanics

A constructive axiomatic approach to the geometry of space-time in the domain of classical and quantum physics can be founded on basic experiences made with quantum objects. The corresponding matter fields are described by linear systems of partial differential equations. Their characteristics are studied. Postulates concerning the related jump cones and jump amplitudes lead to a unique conformal structure attributed to the space-time geometry.     

Gravitation and Electrodynamics over SO(3,3)

In a series of papers, an approach to field theory is developed in which matter appears by interpreting source-free (homogeneous) fields over a 6-dimensional space of signature (3,3), as interacting (inhomogeneous) fields in space-time. The extra dimensions are given a physical meaning as coordinatized matter. The inhomogeneous energy-momentum relations for the interacting fields in space-time are automatically generated by the simple homogeneous relations in 6-d. We then develop a Weyl geometry over SO(3,3) as base, under which gravity and electromagnetism are essentially unified via an irreducible 6-calibration invariant Lagrange density and corresponding variational principle. The Einstein–Maxwell equations are shown to represent a low-order approximation, and the cosmological constant must vanish in order that this limit exist.     

Friedman-like Robertson–Walker model in generalized metric space-time with weak anisotropy

A generalized FRW model of space-time is studied, taking into consideration the anisotropic structure of fields which are depended on the position and the direction (velocity). The Raychaudhouri and Friedman-like equations are investigated assuming the Finslerian character of space-time. A long range vector field of cosmological origin is considered in relation to a physical geometry where the Cartan connection has a fundamental role. The Friedman equations are produced including extra anisotropic terms. The variation of anisotropy z _t is expressed in terms of the Cartan torsion tensor of the Finslerian manifold. A physical generalization of the Hubble and other cosmological parameters arises as a direct consequence of the equations of motion.     

Deformed special relativity with an invariant minimum speed and its cosmological implications

The paper aims to introduce a new symmetry principle in the space-time geometry through the elimination of the classical idea of rest and by including a universal minimum speed limit in the subatomic world. Such a limit, unattainable by particles, represents the preferred reference frame associated with a universal background field that breaks Lorentz symmetry. Thus, the structure of space-time is extended due to the presence of a vacuum energy density, which leads to a negative pressure at cosmological scales. The tiny values of the cosmological constant and the vacuum energy density shall be successfully obtained, which are in good agreement with current observational results.      

Group deformations and relativity

A theory of deformation (homcomorphism but non-isomorphism) of topological groups is developed. In particular, a theory of deformation of subgroups structure is considered. The whole formalism is based on conceptions of holonomicity and relative geometry.A field is postulated to deform the symmetry group of a free physical system. It is shown that the classical fields deform the Poincare group. Thanks to this fact, gravitation appears as space-time curvature (non-holonomicity of the Lorentz subgroup mapping); and electromagnetism reveals itself by space-time torsion (non-holonomicity of the translation subgroup mapping). From physically evident premises it follows that space-time also has a torsion in the rotating and accelerated systems of reference.