Related papers: On equation of geodesic deviation and its solution…
In this thesis we consider several aspects of general relativity relating to exact solutions of the Einstein equations. In the first part gravitational plane waves in the Rosen form are investigated, and we develop a formalism for writing…
This article explores solutions to a generalised form of the Seiberg--Witten equations in higher dimensions, first introduced by Fine and the author. Starting with an oriented $n$ dimensional Riemannian manifold with a…
We present a method to compute exact expressions for optical observables for static spherically symmetric spacetimes in the framework of the bilocal geodesic operator formalism. The expressions are obtained by solving the linear geodesic…
The geodesic deviation equation is generalized to worldline deviation equations describing the relative accelerations of charged spinning particles in the framework of Dixon-Souriau equations of motion.
Geodesic equations are solved when at least two of $\theta$, $\phi$ and $\psi$ are constant, or $r$ is constant, on scalar flat metrics of Eguchi-Hanson type. They can also be solved also on Eguchi-Hanson metrics which are Ricci flat if…
A three-dimensional Riccati differential equation of complex quaternion-valued functions is studied. Many properties similar to those of the ordinary differential Riccati equation such that linearization and Picard theorem are obtained. Lie…
We present a study of the geodesic equations of a black hole space-time which is a solution of the three-dimensional NMG theory and is asymptotically Lifshitz with $z=3$ and $d=1$ as found in [Ayon-Beato E., Garbarz A., Giribet G. and…
Through this article, I will overview the use of Noether symmetry approach in discussing the integration of geodesic equations for the geodesic Lagrangians of spacetimes. I will also give some examples to reveal the efficiency of Noether…
In this paper, the Riemannian gradient algorithm and the natural gradient algorithm are applied to solve descent direction problems on the manifold of positive definite Hermitian matrices, where the geodesic distance is considered as the…
This paper investigates tidal forces in multidimensional spherically symmetric spacetimes. We consider geodesic deviation equation in Schwarzschild-Tangherlini metric and its electrically charged analog. It was shown that for radial…
An exact 2-dimensional conical Riemannian defect solution of 3-dimensional Euclidean Einstein equations of stresses and defects representing a shear-free Heisenberg ferromagnet is given.The system is equivalent to the Einstein equations in…
We consider the geodesic deviation equation, describing the relative accelerations of nearby particles, and the Raychaudhuri equation, giving the evolution of the kinematical quantities associated with deformations (expansion, shear and…
The three-body general problem is formulated as a problem of geodesic trajectories flows on the Riemannian manifold. It is proved that a curved space with local coordinate system allows to detect new hidden symmetries of the internal motion…
The quantum fluctuations of the geodesic deviation equation in a flat background spacetime are discussed. We calculate the resulting mean squared fluctuations in the relative velocity and separation of test particles. The effect of these…
We consider spaces of smooth immersed plane curves (modulo translations and/or rotations), equipped with reparameterization invariant weak Riemannian metrics involving second derivatives. This includes the full $H^2$-metric without zero…
A local classification of locally conformal flat Riemannian Einstein-like four-manifolds as well as a local classification of all locally conformal flat Riemannian four-manifolds for which all Jacobi operators have parallel eigenspaces…
For the first time, Schr\"odinger equations with cubic and more complex nonlinearities containing the unknown function with constant delay are analyzed. The physical considerations that can lead to the appearance of a delay in such…
Recently, Riemannian Gaussian distributions were defined on spaces of positive-definite real and complex matrices. The present paper extends this definition to the space of positive-definite quaternion matrices. In order to do so, it…
The method of Hamilton-Jacobi is used to obtain geodesics around non- Riemannian planar torsional defects.It is shown that by perturbation expansion in the Cartan torsion the geodesics obtained are parabolic curves along the plane x-z when…
Hilbert's fourth problem asks for the construction and the study of metrics on subsets of projective space for which the projective line segments are geodesics. Several solutions of the problem were given so far, depending on more precise…