Ricci Rotation Coefficients Research Articles

The Ricci Rotation Coefficients in the Description of Trajectories of Spinning Test Particles Off-equatorial Plane in the Gravitational Field of a Rotating Source

We describe the trajectories of circular orbits of spinless and spinning test particles around a rotating body in equatorial and non-equatorial planes via the Mathisson-Papapetrou-Dixon equations. In this paper, these equations include the Ricci rotation coefficients with the purpose of describing not only the curvature of space time, but also the rotation of the spinning test particles that orbit around a rotating massive body. We found a numerical difference between the trajectories of spinless test particles and spinning test particles in the order of 10-7\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\varvec{10}}^{\\varvec{-7}}$$\\end{document}. We take as parameters: radius, energy, Carter’s constant and angular momentum.

A Newman Penrose approach to Bertrand spacetime in general relativity

Newman Penrose (NP) formalism is a new approach to general relativity was presented. In various geometrical groups are estimate on a chosen null tetrad basis for tetrad formalism. This approach widely used in analytical and numerical studies of Einstein’s equations. The reasonable utilization of explicit complex straight emulsions of Ricci rotation coefficients which present, in the product, the spinor relative association. And we construct a null tetrad for general case and special attension for Bertrand’s space-time type-1 metric for special case k=0.

An attempt to geometrize electromagnetism

AbstractThis study investigates the curved worldline of a charged particle accelerated by an electromagnetic field in flat spacetime. A new metric, which dependes on the charge-to-mass ratio and electromagnetic potential, is proposed to describe the curve characteristic of the world-line. The main result of this paper is that an equivalent equation of the Lorentz equation of motion is put forward based on a 4-dimensional Riemannian manifold defined by the metric. Using the Ricci rotation coefficients, the equivalent equation is self-consistently constructed. Additionally, the Lorentz equation of motion in the non-inertial reference frames is studied with the local Lorentz covariance of the equivalent equation. The model attempts to geometrize classical electromagnetism in the absence of the other interactions, and it is conducive to the establishment of the unified theory between electromagnetism and gravitation.

Spacetimes of Weyl and Ricci type N in higher dimensions

We study the geometrical properties of null congruences generated by an aligned null direction of the Weyl tensor (WAND) in spacetimes of Weyl and Ricci type N (possibly with a non-vanishing cosmological constant) in an arbitrary dimension. We prove that a type N Ricci tensor and a type III or N Weyl tensor have to be aligned. In such spacetimes, the multiple WAND has to be geodetic. For spacetimes with type N aligned Weyl and Ricci tensors, the canonical form of the optical matrix in the twisting and non-twisting cases is derived and the dependence of the Weyl and the Ricci tensors and Ricci rotation coefficients on the affine parameter of the geodetic null congruence generated by the WAND is obtained.

Torsion in extra-dimensions

We consider a variant of the 5 dimensional Kaluza-Klein theory within the framework of Einstein-Cartan formalism. By imposing a set of constraints on torsion and Ricci rotation coefficients, we show that the torsion components are completely expressed in terms of the metric. and the Ricci tensor in 5D corresponds exactly to what one would obtain from torsion-free general relativity on a 4D hypersurface. The contributions of the scalar and vector fields of the standard K-K theory to the Ricci tensor and the affine connections are completely nullified by the contributions from the torsion. As a consequence, geodesic motions do not distinguish the torsion free 4D space-time from a hypersurface of 5D space-time with torsion satisfying the constraints. Since torsion is not an independent dynamical variable in this formalism, the modified Einstein equations are different from those in the general Einstein-Cartan theory. This leads to important cosmological consequences such as the emergence of cosmic acceleration.

KALUZA–KLEIN THEORY WITH TORSION CONFINED TO THE EXTRA DIMENSION

Here we consider a variant of the five-dimensional Kaluza–Klein (KK) theory within the framework of Einstein–Cartan formalism that includes torsion. By imposing a set of constraints on torsion and Ricci rotation coefficients, we show that the torsion components are completely expressed in terms of the metric. Moreover, the Ricci tensor in 5D corresponds exactly to what one would obtain from torsion-free general relativity on a 4D hypersurface. The contributions of the scalar and vector fields of the standard KK theory to the Ricci tensor and the affine connections are completely nullified by the contributions from torsion. As a consequence, geodesic motions do not distinguish the torsion free 4D spacetime from a hypersurface of 5D spacetime with torsion satisfying the constraints. Since torsion is not an independent dynamical variable in this formalism, the modified Einstein equations are different from those in the general Einstein–Cartan theory. This leads to important cosmological consequences such as the emergence of cosmic acceleration.

Tilted cosmological models of Bianchi type V

Cosmological models of Bianchi type V and I containing a perfect fluid with a linear equation of state plus cosmological constant are investigated using a tetrad approach where our variables are the Riemann tensor, the Ricci rotation coefficients and a subset of the tetrad vector components. This set, in the following called S, describes a spacetime when its elements are constrained by certain integrability conditions and due to a theorem by Cartan this set gives a complete local description of the spacetime. The system obtained by imposing the integrability conditions and Einstein's equations can be reduced to an integrable system of five coupled first order ordinary differential equations. The general solution is tilted and describes a fluid with expansion, shear and vorticity. With the help of standard bases for Bianchi V and I the full line element is found in terms of the elements in S. We then construct the solutions to the linearized equations around the open Friedmann model. The full system is also studied numerically and the perturbative solutions agree well with the numerical ones in the appropriate domains. We also give some examples of numerical solutions in the non-perturbative regime.

Erratum: Hyperbolic tetrad formulation of the Einstein equations for numerical relativity[Phys. Rev. D 67, 084017 (2003)

The tetrad-based equations for vacuum gravity published by Estabrook, Robinson, and Wahlquist are simplified and adapted for numerical relativity. We show that the evolution equations as partial differential equations for the Ricci rotation coefficients constitute a rather simple first-order symmetrizable hyperbolic system, not only for the Nester gauge condition on the acceleration and angular velocity of the tetrad frames considered by Estabrook et al., but also for the Lorentz gauge condition of van Putten and Eardley, and for a fixed gauge condition. We introduce a lapse function and a shift vector to allow general coordinate evolution relative to the timelike congruence defined by the tetrad vector field.

Second-order properties of a generalized frame of reference with null congruence

We extend to the isotropic case, by adapted non-holonomic techniques 2nd-order properties of a relativistic frame of reference, generalized in the polar sense: Riemann and gravitational tensors decomposition, Lie derivative of the Ricci rotation coefficients, commutation formulas and finally Bianchi identity. All the decompositions tensors are formally invariant (as regards the standard case), by means of the longitudinal derivative extension.

Hyperbolic tetrad formulation of the Einstein equations for numerical relativity

The tetrad-based equations for vacuum gravity published by Estabrook, Robinson, and Wahlquist are simplified and adapted for numerical relativity. We show that the evolution equations as partial differential equations for the Ricci rotation coefficients constitute a rather simple first-order symmetrizable hyperbolic system, not only for the Nester gauge condition on the acceleration and angular velocity of the tetrad frames considered by Estabrook et al., but also for the Lorentz gauge condition of van Putten and Eardley and for a fixed gauge condition. We introduce a lapse function and a shift vector to allow general coordinate evolution relative to the timelike congruence defined by the tetrad vector field.

Symmetric hyperbolic form of central-moment relativistic gas dynamics with gravitation

Recently, Banach and Larecki (2002 Rev. Math. Phys. 14 469) systematically derived the evolution equations for a gas obtained by taking moments of the relativistic Boltzmann equation, truncating and closing the resulting system of moment equations by a maximum-entropy ansatz. The moments just mentioned (so-called central moments) are defined with respect to a network of fiducial observers whose 4-velocities are at first arbitrary (though fixed ab initio) and later specialized to be nonrotating (hypersurface-orthogonal). The basic advantage of the latter choice is that if one expresses the central moments and the collision operator moments in terms of Lagrange multipliers, the evolution equations for these multipliers are then automatically symmetric hyperbolic and causal at every order of truncation. In this paper, the formulation of central-moment relativistic gas dynamics goes further by combining it with the dynamics of gravitational fields. Explicitly, we employ the 1 + 3 orthonormal frame formalism for spacetime, which contains the equations for the tetrad basis and the Ricci rotation coefficients, to derive a full causal evolution system of first-order symmetric hyperbolic form for a set of Lagrange multipliers and gravitational fields. An important issue arising in our studies is the question of consistency of equations of two different kinds: constraint equations and evolution equations. Thus we show that these two sets of equations are consistent with each other, in the sense that the constraint equations are preserved by the evolution equations.

Riferimenti isotropi in relatività generale

We consider anew approach for a relativistic frame of reference in the polar sense [1] (two-congruences) and the related non-holonomic techniques for anull congruence (light flux); 1° order characteristic tensors, Ricci rotation coefficients, longitudinal and transversal covariant derivatives are deduced in general form. Thus, the method of a standard frame of reference is extended to thenull case.

Proprietà di 20 ordine di un riferimento isotropo

We extend to the isotropic case, by adapted non-holonomic techniques [1], 2° order properties of a relativistic frame of reference, generalized in the polar sense [2,3]: Riemann and gravitational tensors decomposition, Lie derivative of the Ricci rotation coefficients, commutation formulae and finally Bianchi identity. All the decomposition tensors are formally invariant (as regards the standard case), by means of the longitudinal derivative extension.

Axistationary perfect fluids - a tetrad approach

Stationary axisymmetric perfect fluid spacetimes are investigated using the curvature description of geometries. We formulate the equations in terms of components of the Riemann tensor and the Ricci rotation coefficients in a comoving Lorentz tetrad. It is shown that the only incompressible axistationary magnetic perfect fluid is the interior Schwarzschild solution. Further, we find that all rigidly rotating axistationary fluids with magnetic Weyl tensor have local rotational symmetry. Rigidly rotating fluid spacetimes with purely electric or purely magnetic Weyl tensor are shown to be of Petrov type D.

Invariant construction ofsolutions to Einstein'sfield equations - LRS perfect fluids II

The properties of LRS class II perfect fluid spacetimes are analysed using the description of geometries in terms of the Riemann tensor and a finite number of its covariant derivatives. In this manner it is straightforward to obtain the plane and hyperbolic analogues to the spherical symmetric case. For spherically symmetric static models the set of equations is reduced to the Tolman-Oppenheimer-Volkoff equation only. Some new non-stationary and inhomogeneous solutions with shear, expansion and acceleration of the fluid are presented. Among these are some of temporally self-similar solutions with equation of state given by p = ( - 1)µ, 1 < < 2 and a class of solutions characterized by = -(1/6). We give an example of a geometry where the Riemann tensor and the Ricci rotation coefficients are not sufficient to give a complete description of the geometry. Using an extension of the method, we find the full metric in terms of curvature quantities.

Gravitational and Electroweak Unification

Einstein suggested that a unified field theorybe constructed by replacing the diffeomorphisms (thecoordinate transformations of general relativity) withsome larger group. We have constructed a theory that unifies the gravitational and electroweakfields by replacing the diffeomorphisms with the largestgroup of coordinate transformations under whichconservation laws are covariant statements. Thisreplacement leads to a theory with field equations whichimply the validity of the Einstein equations of generalrelativity, with a stress-energy tensor that is justwhat one expects for the electroweak field andassociated currents. The electroweak field appears as aconsequence of the field equations (rather than as a"compensating field" introduced to secure gaugeinvariance). There is no need for symmetry breaking toaccommodate mass, because the U(1) × SU(2) gaugesymmetry is approximate from the outset. Thegravitational field is described by the space-timemetric, as in general relativity. The electroweak fieldis described by the "mixed symmetry" part of the Riccirotation coefficients. The gauge symmetry-breakingquantity is a vector formed by contracting theLevi-Civita symbol with the totally antisymmetric partof the Ricci rotation coefficients.

Proprietà di secondo ordine di un riferimento generalizzato

We consider a relativistic frame of reference, generalized in the polar sense [1]–[2],and the adapted non-holonomic techniques;then, we extend to non-orthogonal case second order propertiesof a standard frame of reference: Riemann tensor decomposition, Lie derivatives of the Ricci rotation coefficients, commutation formulae and spatial Bianchi identity.

A spacetime calculus based on a single null direction

A spacetime calculus based on a single null direction is developed. The fundamental objects are totally symmetric spinors formed from the Ricci rotation coefficients together with four new differential operators which only depend upon the choice of up to a complex scaling. The formalism combines features of the GHP formalism with those of one which is invariant under null rotations. Einstein's equations and the full Bianchi identities are given in this new formalism.

A spacetime calculus invariant under null rotations

A spacetime calculus which is invariant under null rotations is presented. The fundamental objects are totally symmetric spinors formed from the Ricci rotation coefficients whose components transform covariantly under null rotations, and four new differential operators which are formed from the directional derivatives and the remaining Ricci rotation coefficients which transform ‘badly’ under null rotations. Einstein’s equations and the full Bianchi identities are translated into this new formalism. Because only totally symmetric spinors are used there is no need to explicitly use indices in any of the equations. Several applications of the formalism are mentioned and the geometrical interpretation is briefley discussed.

Classical Electromagnetic Radiation in Multidimensional Space-Times

The dependence of various features of electromagnetic radiation on the space-time dimension is investigated. The dimension is d ≥ 4, the lower dimensional theory is not considered. Some features known in four dimensions are carried over (possibly with slight modifications) to extra dimensions, whereas other ones exist only for d = 4. The first group contains, e.g., the notion of null characteristic surfaces and null electromagnetic fields, the geometrical optics approximation ( nearly plane waves), and Fermat′s principle. It turns out that the null rays of the null field are geodesic lines for any dimension (generalization of Mariot′s theorem). The independent functional (algebraic) invariants for a generic electromagnetic field are explicitly constructed, there are [ d/2] of such invariants for d dimensions. A generalized Newman-Penrose formalism (complex Ricci rotation coefficients) is developed in d = 6 and it is used to show that four dimensions are exceptional: only for d = 4 are the rays of a null field shear-free (Robinson′s theorem); in extra dimensions Maxwell′s equations are less restrictive. This formalism is also applied for proving a generalization of Sachs′ optical theorem: for any d, the shadow cast by a small object in a light beam is rotated, expanded, and sheared, but it is not Lorentz contracted.