Spacetimes Admitting Concircular Curvaure Tensor in f(R) Gravity

The main object of this paper is to investigate spacetimes admitting concircular curvature tensor in f(R) gravity theory. At first, concircularly flat and concircularly flat perfect fluid spacetimes in fR gravity are studied. In this case, the forms of the isotropic pressure p and the energy density σ are obtained. Next, some energy conditions are considered. Finally, perfect fluid spacetimes with divergence free concircular curvature tensor in f(R) gravity are studied; amongst many results, it is proved that if the energy-momentum tensor of such spacetimes is recurrent or bi-recurrent, then the Ricci tensor is semi-symmetric and hence these spacetimes either represent inflation or their isotropic pressure and energy density are constants.

Curvature Tensor of the Stationary Accelerated Frame in Gravity Field

We define an accelerated frame that moves along rˆ -axis in the general relativistic curved space-time. We then calculate the curvature tensor of this accelerated frame in the stationary gravity field. The curvature tensor is divided into two parts: the curvature tensor as observed by the observer and the curvature tensor of the observer’s own planet in the gravity field.

A Study of Generalized Projective P − Curvature Tensor on Warped Product Manifolds

The main aim of this study is to investigate the effects of the P − curvature flatness, P − divergence-free characteristic, and P − symmetry of a warped product manifold on its base and fiber (factor) manifolds. It is proved that the base and the fiber manifolds of the P − curvature flat warped manifold are Einstein manifold. Besides that, the forms of the P − curvature tensor on the base and the fiber manifolds are obtained. The warped product manifold with P − divergence-free characteristic is investigated, and amongst many results, it is proved that the factor manifolds are of constant scalar curvature. Finally, P − symmetric warped product manifold is considered.

The Geometrization of Maxwell’s Equations and the Emergence of Gravity

A recently proposed classical field theory in which Maxwell’s equations are replaced by an equation that couples the Maxwell tensor to the Riemann-Christoffel curvature tensor in a fundamentally new way is reviewed and extended. This proposed geometrization of the Maxwell tensor provides a succinct framework for the classical Maxwell equations which are left intact as a derivable consequence. Beyond providing a basis for the classical Maxwell equations, the coupling of the Riemann-Christoffel curvature tensor to the Maxwell tensor leads to the emergence of gravity, with all solutions of the proposed theory also being solutions of Einstein’s equation of General Relativity augmented by a term that can mimic the properties of dark matter and/or dark energy. Both electromagnetic and gravitational phenomena are put an equal footing with both being tied to the curvature of space-time. Using specific solutions to the proposed theory, the unification brought to electromagnetic and gravitational phenomena as well as the consistency of solutions with those of the classical Maxwell and Einstein field equations is emphasized throughout. Unique to the proposed theory and based on specific solutions are the emergence of antimatter and its behavior in gravitational fields, the emergence of dark matter and dark energy mimicking terms in the context of General Relativity, an underlying relationship between electromagnetic and gravitational radiation, and the impossibility of negative mass solutions that would generate repulsive gravitational fields or antigravity. Finally, a method for quantizing the charge, mass, and angular momentum of particle-like solutions as well as the possibility of superluminal transport of specific curvature related quantities is conjectured.

Discrimination of Surface Topographies Created by Two-Stage Process by Means of Multiscale Analysis

The fundamental issue in surface metrology is to provide methods that can allow the establishment of correlations between measured topographies and performance or processes, or that can discriminate confidently topographies that are processed or performed differently. This article presents a set of topographies from two-staged processed steel rings, measured with a 3D contact profilometer. Data were captured individually from four different regions, namely the top, bottom, inner, and outer surfaces. The rings were manufactured by drop forging and hot rolling. Final surface texture was achieved by mass finishing with spherical ceramic media or cut wire. In this study, we compared four different multiscale methods: sliding bandpass filtering, three geometric length- and area-scale analyses, and the multiscale curvature tensor approach. In the first method, ISO standard parameters were evaluated as a function of the central wavelength and bandwidth for measured textures. In the second and third method, complexity and relative length and area were utilized. In the last, multiscale curvature tensor statistics were calculated for a range of scales from the original sampling interval to its forty-five times multiplication. These characterization parameters were then utilized to determine how confident we can discriminate (through F-test) topographies between regions of the same specimen and between topographies resulting from processing with various technological parameters. Characterization methods that focus on the geometrical properties of topographic features allowed for discrimination at the finest scales only. Bandpass filtration and basic height parameters Sa and Sq proved to confidently discriminate against all factors at all three considered bandwidths.

Renormalizable and Unitary Lorentz Invariant Model of Quantum Gravity

We analyze the R+R2 model of quantum gravity where terms quadratic in the curvature tensor are added to the General Relativity action. This model was recently proved to be a self-consistent quantum theory of gravitation, being both renormalizable and unitary. The model can be made practically indistinguishable from General Relativity at astrophysical and cosmological scales by the proper choice of parameters.

Curvature

The mathematics of Riemannian curvature is presented. The Riemann curvature tensor and its role in parallel transport, in the metric, and in geodesic deviation are expounded at length. We begin by defining the curvature tensor and the torsion tensor by relating them to covariant derivatives. Then the local metric is obtained up to second order in terms of Minkowski metric and curvature tensor. Geometric issues such as the closure or non-closure of parallelograms are discussed. Next, the relation between curvature and parallel transport around a loop is derived. Then we proceed to geodesic deviation. The influence of global properties of the manifold on parallel transport is briefly expounded. The Lie derivative is then defined, and isometries of spacetime are discussed. Killing’s equation and properties of Killing vectors are obtained. Finally, the Weyl tensor (conformal tensor) is introduced.

Variational theory of the Ricci curvature tensor dynamics

In this letter a new Lagrangian variational principle is proved to hold for the Einstein field equations, in which the independent variational tensor field is identified with the Ricci curvature tensor $$R^{\mu \nu }$$ R μ ν rather than the metric tensor $$g_{\mu \nu }$$ g μ ν . The corresponding Lagrangian function, denoted as $$L_{R}$$ L R , is realized by a polynomial expression of the Ricci 4-scalar $$R\equiv g_{\mu \nu }R^{\mu \nu }$$ R ≡ g μ ν R μ ν and of the quadratic curvature 4-scalar $$\rho \equiv R^{\mu \nu }R_{\mu \nu }$$ ρ ≡ R μ ν R μ ν . The Lagrangian variational principle applies both to vacuum and non-vacuum cases and for its validity it demands a non-vanishing, and actually also positive, cosmological constant $$\Lambda >0$$ Λ > 0 . Then, by implementing the deDonder–Weyl formalism, the physical conditions for the existence of a manifestly-covariant Hamiltonian structure associated with such a Lagrangian formulation are investigated. As a consequence, it is proved that the Ricci tensor can obey a Hamiltonian dynamics which is consistent with the solutions predicted by the Einstein field equations.

The structure of the curvature tensor of plane gravitational waves

Plane gravitational waves in the Riemann space of General Relativity is considered. The criterion of plane gravitational waves is used based on the analogy between plane gravitational and electromagnetic waves. The Theorem is proved that the action of the Lie derivative on the plane wave curvature 2-form in the direction of the vector generating the invariance group of this wave in the Riemann space is equal to zero. It is justified that the gravitational waves can be used to transmit information in the Riemann space.

On W0 and W2 ø-Symmetric Contact Manifold Admitting Quarter-Symmetric Metric Connection

The paper deals locally W0 and W2 curvature tensor of ø-symmetric K-contact manifolds with quarter-symmetric metric connection and some results are obtained.