Narrow Results

The field equations following from a Lagrangian L(R) will be deduced and solved for special cases. If L is a non-linear function of the curvature scalar, then these equations are of fourth order in the metric. In the introduction, we present the history of these equations beginning with the paper of H. Weyl from 1918, who first discussed them as alternative to Einstein's theory. In the third part, we give details about the cosmic no hair theorem, i.e. the details of how within fourth order gravity with L= R + R², the inflationary phase of cosmic evolution turns out to be a transient attractor. Finally, the Bicknell theorem, i.e. the conformal relation from fourth order gravity to scalar-tensor theory, will be shortly presented.

We calculate the relative conserved currents, superpotentials and conserved quantities between two homogeneous and isotropic universes. In particular, we prove that their relative "energy" (defined as the conserved quantity associated to cosmic time coordinate translations for a comoving observer) is vanishing and so are the other conserved quantities related to a Lie subalgebra of vector fields isomorphic to the Poincaré algebra. These quantities are also conserved in time. We also find a relative conserved quantity for such a kind of solution which is conserved in time though non-vanishing. This example provides at least two insights in the theory of conserved quantities in General Relativity. First, the contribution of the cosmological matter fluid to the conserved quantities is carefully studied and proved to be vanishing. Second, we explicitly show that our superpotential (that happens to coincide with the so-called KBL potential although it is generated differently) provides strong conservation laws under much weaker hypotheses than the ones usually required. In particular, the symmetry generator is not needed to be Killing (nor Killing of the background, nor asymptotically Killing), the prescription is quasi-local and it works fine in a finite region too and no matching condition on the boundary is required.

In this study, we investigate the special type of magnetic trajectories associated with a magnetic field ${ℬ}$ defined on a 3D Riemannian manifold. First, we consider a moving charged particle under the action of a frictional force, $f$, in the magnetic field ${ℬ}$. Then, we assume that trajectories of the particle associated with the magnetic field ${ℬ}$ correspond to frictional magnetic curves ($f$-magnetic curves$)$ of magnetic vector field ${ℬ}$ on the 3D Riemannian manifold. Thus, we are able to investigate some geometrical properties and physical consequences of the particle under the action of frictional force in the magnetic field ${ℬ}$ on the 3D Riemannian manifold.

In this paper, we study a special type of magnetic trajectories associated with a magnetic field ${ℬ}$ defined on a 3D Riemannian manifold. First, we assume that we have a moving charged particle which is supposed to be under the action of a gravitational force $G$ in the magnetic field ${ℬ}$ on the 3D Riemannian manifold. Then, we determine trajectories of the charged particle associated with the magnetic field ${ℬ}$ and we define gravitational magnetic curves ($G$-magnetic curves) of the magnetic vector field ${ℬ}$ on the 3D Riemannian manifold. Finally, we investigate some geometrical and physical features of the moving charged particle corresponding to the $G$-magnetic curve. Namely, we compute the energy, magnetic force, and uniformity of the $G$-magnetic curve.

In this paper, we show that the velocity vector field of classical Bernoulli–Euler elastic curve is harmonic in $R^n$ space. We propose a new characterization for classical Bernoulli–Euler elastic curves and plot graphs of examples that satisfy this characterization.

In this study, we firstly characterize modified magnetic particles and associated magnetic fields by considering the evolution of the charged particle together with its modified frame construction in 3D space. Then, we compute the energy flows of each modified magnetic particles in terms of its curvature and torsion in the space. Finally, we obtain energy flux density, the intensity of the light, power of modified magnetic particles in the space.

The more precise definition and the more fundamental understanding of the concepts of time, energy, entropy and information are building upon the new, relativistic foundation of gravity. This lecture is an attempt to explain the basic principles that underpin this progress, by focusing on the simple but subtle universal definition of energy. The principles are unearthed from Einstein’s theory and Noether’s theorems, beneath a century of misconceptions.

In this work, we calculate new concept for their Fermi–Walker conformable derivatives with spherical timelike magnetic fiber in spherical de-Sitter space ${𝕊}_1^2$. The description is advanced for timelike magnetic trajectory. First, we find conformable fractional derivatives of spherical magnetic Lorentz fields. Moreover, we compute the Fermi–Walker conformable derivatives of the normalization and recursion operators for these spherical vector fields. Furthermore, we give some characterizations of Lorentz fields of electromagnetic fields of associated with spherical magnetic conformable particle. Finally, we obtain the energy of the conformable derivative of the normalization function of the Lorentz forces and we have energy relations of some vector fields associated with spherical frame in the de-Sitter space ${𝕊}_1^2$.