Christoffel Symbols Research Articles

Exploring Torus Black-Holes In (1+3)-Dimensions: A Novel Approach to Higher Genus Solution

A torus black-hole solution of the vacuum gravitational field equation of general relativity in (1 + 3)-dimensions is obtained. Starting with a metricansatz associated with the torus, our method is based on straight forward computations the usual geometric mathematical tools of the Christoffel symbols and the Riemann tensor. Specifically, after deriving such mathematical tools the field equations of general relativity are considered. The resultatning equations are properly combained to find the solution. Moreover, the novelty and potential implications of this solution emerges from the fact that is based on a coordinate transformation metric ansatz. This provides with broad implications and future research directions. In particular we argue that our formalism can properly be used for a search of higher genus black-hole solution.

An application of recurrence relations to central force fields

By generalizing the relationships between auxiliary variables that appear in the works of A.E. Roy, we investigate the dynamics of non-interacting particles under the action of a central force field whose force function is a solution to a certain differential equation. The method used by A.E. Roy, R. Broucke, and later by G. Sitarski, was utilized in such a way that the obtained recursive equations can be used to describe the motion of a particle under such a field. The class of such fields includes both gravitational and non-gravitational force fields. Several numerical and historically essential examples and detailed discussions of various cases are given.

Kinetic and hydrodynamic post-Newtonian equations in the Brans–Dicke theory

Abstract A kinetic theory for the post-Newtonian Brans–Dicke theory is developed. The Boltzmann equation and the equilibrium Maxwell-Jüttner distribution function are determined from the knowledge of the components of the metric tensor and Christoffel symbols in the post-Newtonian Brans–Dicke theory. The hydrodynamic equations for the mass density, momentum density and mass-energy density are obtained from a transfer equation derived from the Boltzmann equation. The problem of self-gravitating fluid instabilities in the post-Newtonian Brans–Dicke theory is investigated.

Relativistic periastron advance beyond Einstein theory: analytical solution with applications

We find a new solution to calculate the orbital periastron advance of a test body subject to a central gravitational force field for relativistic theories and models beyond Einstein. This analitycal formula has general validity that includes all the post-Newtonian (PN) contributions to the dynamics and is useful for high-precision gravitational tests. The solution is directly applicable to corrective potentials of various forms, without the need for numerical integration. Later, we apply it to the scalar tensor fourth order gravity (STFOG) and noncommutative geometry, providing corrections to the Newtonian potential of Yukawa-like form V(r)=αe-βrr\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$V(r)=\\alpha \\frac{e^{-\\beta r}}{r}$$\\end{document}, and we conduct the first analysis involving all the PN terms for these theories. The same work is performed with a Schwarzschild geometry perturbed by a Quintessence Field, leading to a power-law potential V(r)=αqrq\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$V(r)=\\alpha _q {r}^q$$\\end{document}. Finally, by using astrometric data of the Solar System planetary precessions and those of the S2 star around Sgr A*, we infer new theoretical constraints and improvements in the bounds for β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta $$\\end{document}. The resulting simulated orbits turn out to be compatible with general relativity.

Three-dimensional quasi-umbilical locally strongly convex homogeneous affine hypersurfaces

The study of strongly convex, locally homogeneous hypersurfaces in the affine space ℝn+1 has a rich history. In dimension 2, these hypersurfaces were first examined in Guggenheimer's 1977 book “Differential Geometry” [4], and the classification was completed by Nomizu and Sasaki in their 1991 paper “A new model of unimodular-affinely homogeneous surfaces” [6]. Apart from the work of Magid and Vrancken in their 1995 paper [5] and Ooguri’s 2013 paper [9], most studies have focused on locally strongly convex affine hypersurfaces. These are characterized by an induced affine metric that is positive definite, and have been extensively studied in Sasaki’s 1980 paper “Hyperbolic affine hyperspheres” [10]. When the hypersurface is an affine sphere, where the shape operator has all eigenvalues equal to zero, Dillen and Vrancken provided a complete classification in a series of papers. In dimension 3, their 1993 paper “The classification of 3-dimensional locally strongly convex homogeneous affine hypersurfaces” [3] offers a comprehensive classification for any dimensions of eigenvalues of the shape operator. In dimension 4, the quasi-umbilical case was explored by Dillen and Vrancken in their 1994 paper “Quasi-umbilical, locally strongly convex homogeneous affine hypersurfaces” [2]. The case where the affine shape operator has two distinct eigenvalues, each with a multiplicity of two, was investigated by Chikh-Salah and Vrancken in their 2017 paper “Four-dimensional locally strongly convex homogeneous affine hypersurfaces” [1]. In this paper, we provide a different proof of the final theorem in Dillen and Vrancken’s [3], utilizing Mathematica software to calculate all Christoffel symbol functions.

To geometric aspects of in nite-dimensional dynamical systems

The main goal of the work is to construct analogues of Christoffel symbols for infinitedimensional systems and on this basis to obtain geodesic equations for such systems. These analogies are of particular interest in terms of identifying the relationship between the dynamics of systems with an infinite number of degrees of freedom and Riemannian geometry, as well as geometry defined by the pseudo-Riemannian metric.

Connection Between Gravity and Quantum World

Gravity is the most important force in cosmology. A lot of research deals with it, but we only know its perceptible properties. Two basic descriptions of gravity are accepted today. One is based on classical physics based on Newton\’s theory. The other is the theory of gravity described by Einstein. However, another description of gravity is also possible. Gravitation can also be described in other ways using the equations of Kepler, Newton, Coulomb, and Maxwell. Using their equations, it can be deduced that: The magnitude of the gravitational force field is not determined only by mass. It depends on the speed of the mass. It depends on the size of the gravitational field in which the mass moves Newton\’s and Coulomb\’s laws are equivalent. This has already been determined by several researchers. It was also conducted by Constantin Meis. Newton\’s, Coulomb\’s, Maxwell\’s law can be used to determine a new cosmic constant. It can be counted on in the Solar System, but also in the Milky Way. Denote by μ_2 and √ (μ_2) This constant connects the macro world and the micro or quantum world. This constant can also be calculated using Coulomb\’s and Maxwell\’s equations Maxwell\’s and Coulomb\’s equation can be used to determine the resultant of the associated vibrating charges for each mass and celestial body. Coulomb\’s law can also be used to accurately determine the force effect between masses. Even the force effect between celestial bodies can be determined with it. This is exactly the same as the force effect calculated by Newton\’s law. For a mass of 1 kg, we get exactly Cavendish\’s measurement result with Coulomb\’s law. Already in the 20th century, it was suggested that there is a connection between gravity and the electromagnetic field. This has already been conducted by Constantin Meis and Takaaki Musha and B. Ivanov. Gravitational constant can not only be measured, but also calculated in several ways. This has already been determined by several researchers. The gravitational constant can also be calculated using the mass and its speed. But it can also be calculated using the Coulomb constant and μ_2 The gravitational force field for a given mass or celestial body can be calculated using √(μ_2) and the charge of the mass. The speed of a given planet can be calculated using its charge and √ (μ_2) Based on the derivations, gravity is a force field that has an electric force field component. Therefore, it can be the carrier medium of electromagnetic waves. And so is the carrier of light If we assume that the gravitational force field is a very high frequency electromagnetic wave, then we also understand that gravity is unipolar.

Invariants of Immersions on n-Dimensional Affine Manifold

Main results: The system of Christoffel symbols of the connection of an immersion ξ:J→R^n of an n-dimensional manifold J in the n-dimensional linear space R^n is a system of generators of the differential field of all Aff(n)-invariant differential rational functions of ξ, where Aff(n) is the group of all affine transformations of R^n. A similar result have obtained for the subgroup SAff(n) of ⁡Aff(n) generated by all unimodular linear transformations and parallel translations of R^n. Rigidity and uniqueness theorems for immersions ξ:J→R^n in geometries of groups Aff(n) and SAff(n) were obtained. These theorems are given in terms of the affine connection and the volume form of immersions.

Static spherically symmetric black holes of Brans–Dicke theory of gravity

This paper investigates the static spherically symmetric vacuum solutions of Brans–Dicke Theory of Gravity. In this paper, we systematically derive the static spherically symmetric solutions by starting with the general static spherically symmetric anzats, discovering the nonzero Christoffel symbols, nonzero Ricci tensor components and the Ricci Scalar. These results allow us to rewrite the four Brans–Dicke gravitational field equations explicitly in terms of the metric components, the scalar field and its scalar potential. This paper systematically explains and shows how to modify these linear second-order ordinary differential equations in order to be able to find the solution of the scalar field, the metric components and the scalar potential analytically. We successfully find the most general solution of the static spherically symmetric vacuum solutions of Brans–Dicke gravity and the well-known Schwarzschild solution is recovered as the simplest solution.

The gravitational force field of proto-pancakes

It is well known that the first structures that form from small fluctuations in a self-gravitating, collisionless, and initially smooth cold dark matter (CDM) fluid are pancakes. We studied the gravitational force generated by such pancakes just after shell crossing and have found a simple analytical formula for the force along the collapse direction, which can be applied to both the single- and multi-stream regimes. We tested the formula on the early growth of CDM proto-haloes seeded by two or three crossed sine waves. Adopting the high-order Lagrangian perturbation theory (LPT) solution as a proxy for the dynamics, we confirm that our analytical prediction agrees well with the exact solution computed via a direct resolution of the Poisson equation, as long as the local caustic structure remains sufficiently one-dimensional. These results are further confirmed by comparisons of the LPT predictions performed this way to measurements in Vlasov simulations performed with the public code ColDICE. We also show that the component of the force orthogonal to the collapse direction preserves its single-stream nature – it does not change qualitatively before or after the collapse – allowing sufficiently high-order LPT acceleration to be used to approximate it accurately as long as the LPT series converges. As expected, solving the Poisson equation on the density field generated with LPT displacement provides a more accurate force than the LPT acceleration itself, as a direct consequence of the faster convergence of the LPT series for the positions than for the accelerations. This may provide a clue as to how we can improve standard LPT predictions. Our investigations represent a very needed first step in the study of gravitational dynamics in the multi-stream regime analytically: we estimate, at the leading order in time and space, the proper backreaction on the gravitational field inside the pancakes.

Some Aspects of Rectifying Curves on Regular Surfaces Under Different Transformations

An essential space curve in the study of differential geometry is the rectifying curve. In this paper, we studied the adequate requirement for a rectifying curve under the isometry of the surfaces. The normal components of the rectifying curves are also studied, and it is investigated that for rectifying curves, the Christoffel symbols and the normal components along the surface normal are invariant under the isometric transformation. Moreover, we also studied some properties for the first fundamental form of the surfaces.

Design and Analysis of a Four DoF Robotic Arm with Two Grippers Used in Agricultural Operations

Both academic and commercial interest in agricultural robots has increased recently. This is due to the fact that agricultural robots address significant issues such as seasonal labor shortages during harvest and the rising concern over environmentally friendly practices. Because of these, several distinct agricultural robots have already been created for a variety of purposes, with varying degrees of success, including monitoring, spraying, harvesting, transport, etc. As a result, agriculture automation became unfeasible and unprofitable. The purpose of this study is to provide a new methodology for multitasking in performing agriculture operations by designing a 4 Degrees of freedom (4DoF) robotic arm with a new mechanism that has a different configuration with 2 grippers. We did kinematics and kinetics calculations using Denavit-Hardenberg (D-H) method with Lagrangian mechanics, and with the help of Christoffel Symbols of the First Kind, also We used Robot Operating System (ROS) to provide potential solutions using it. It is easy to be paired with other open-source technologies, such as android or IoT technologies. The robot arm can work synchronously with other hardware, sensors, cameras, and agricultural machines concerning farming operations. In conclusion, we believe that this new configuration will open a door for agricultural tasks to be easily automated and achieved using robotic technologies.

Quantum Riemannian geometry of the discrete interval and q-deformation

We solve for quantum Riemannian geometries on the finite lattice interval • – • –⋯– • with n nodes (the Dynkin graph of type An) and find that they are necessarily q-deformed with q=eıπn+1. This comes out of the intrinsic geometry and not by assuming any quantum group in the picture. Specifically, we discover a novel “boundary effect” whereby, in order to admit a quantum Levi-Cività connection, the “metric weight” at any edge is forced to be greater pointing toward the bulk compared to toward the boundary, with the ratio given by (i + 1)q/(i)q at node i, where (i)q is a q-integer. The Christoffel symbols are also q-deformed. The limit q → 1 likewise forces the quantum Riemannian geometry of the natural numbers N to have rational metric multiples (i + 1)/i in the direction of increasing i. In both cases, there is a unique Ricci-scalar flat metric up to normalization. Elements of quantum field theory and quantum gravity are exhibited for n = 3 and for the continuum limit of the geometry of N. The Laplacian for the scalar-flat metric becomes the Airy equation operator 1xd2dx2 in so far as a limit exists. Scaling this metric by a conformal factor eψ(i) gives a limiting Ricci scalar curvature proportional to e−ψxd2ψdx2.

Possible relationship between initial conditions for inflation and past geodesic incompleteness of the inflationary spacetime

According to the Borde-Guth-Vilenkin (BGV) theorem an expanding region of spacetime cannot be extended to the past beyond some boundary ℬ. Therefore, the inflationary universe must have had some kind of beginning. However, the BGW theorem says nothing about the boundary conditions on ℬ, or even about its location. Here we present a single-scalar field model of the Two-Measure Theory, where the non-Riemannian volume element Υ d 4 x is present in the action. As a result of the model dynamics, an upper bound φ 0 of admissible values of the scalar field φ appears, which sets the position of ℬ in the form of a spacelike hypersurface Υ(x) = 0 with a boundary condition: Υ → 0+ as φ → φ 0 -. A detailed study has established that if the initial kinetic energy density ρ kin (in) prevails over initial gradient energy density ρ grad (in) then there is an interval of initial values φ in (min) ≤ φ in < φ 0, where ρ kin (in) and ρ grad (in) cannot exceed the potential energy density and hence the initial conditions necessary for the onset of inflation are satisfied. It is shown that under almost all possible left-handed boundary conditions on ℬ, that is where Υ → 0-, the metric tensor in the Einstein frame has a jump discontinuity on ℬ, so the Christoffel connection coefficients are not defined on the spacelike hypersurface Υ = 0. Thus, if φ in (min) ≤ φ in < φ 0 and ρ kin (in) > ρ grad (in), then there was an inflationary stage in the history of our Universe and the congruence of timelike geodesics cannot be extended to the past beyond the hypersurface Υ = 0.

Nonlinear photoconductivities and quantum geometry of chiral multifold fermions

Chiral multifold fermions are quasiparticles that appear only in chiral crystals such as transition metal silicides in the cubic B20 structure (i.e., the CoSi family), and they may show exotic physical properties. Here we study the injection and shift photoconductivities and also the related geometrical quantities for several types of chiral multifold fermions, including spin-1/2 as well as pseudospin-1 and -3/2 fermions, dubbed as Kramers Weyl, triple-point, and Rarita-Schwinger-Weyl (RSW) fermions, respectively. We utilize the minimal symmorphic model to describe the triple-point fermions (TPF). We also consider the more realistic model Hamiltonian for the CoSi family including both linear and quadratic terms. We find that injection currents due to circularly polarized light are quantized as a result of the Chern numbers carried by the multifold fermions within the linear models. Surprisingly, we discover that in the TPF model, the linear shift conductivities, responsible for the shift current generation by linearly polarized light, are proportional to the pseudo spin-orbit coupling and independent of photon frequency. In contrast, for the RSW and Kramer Weyl fermions, the linear shift conductivity is linearly proportional to photon frequency. The numerical results agree with the power-counting analysis for quadratic Hamiltonians. The frequency independence of the linear shift conductivity could be attributed to the strong resonant symplectic Christoffel symbols of the flat bands. Moreover, the calculated symplectic Christoffel symbols show significant peaks at the nodes, suggesting that the shift currents are due to the strong geometrical response near the topological nodes.

On Equivalence between Kinetic Equations and Geodesic Equations in Spaces with Affine Connection

Discrete kinetic equations describing binary processes of agglomeration and fragmentation are considered using formal equivalence between the kinetic equations and the geodesic equations of some affinely connected space A associated with the kinetic equation and called the kinetic space of affine connection. The geometric properties of equations are treated locally in some coordinate chart (x;U). The peculiarity of the space A is that in the coordinates (x) of some selected local chart, the Christoffel symbols defining the affine connection of the space A are constant. Examples of the Smoluchowski equation for agglomeration processes without fragmentation and the exchange-driven growth equation are considered for small dimensions in terms of geodesic equations. When fragmentation is taken into account, the kinetic equations can be written as equations of quasigeodesics. Particular cases of spaces with symmetries are discussed.

Analysis of the Conditions of Mechanical Equilibrium on the Curved Surface of a Vapor–Liquid System in a Gravitational Field

A thermodynamic analysis of mechanical equilibrium conditions is performed for the curved interface of a vapor–liquid system in the gravitational field potential with regard to ratios of experimental times of relaxation of thermodynamic parameters (momentum, energy, and mass). In a kinetic analysis of the relaxation stage of reaching the equilibrium state, it is revealed that it is important to consider the physical nature of the boundary of separating phases. The mechanical model of the boundary is one with an intermediate foreign film preventing the attainment of chemical equilibrium between the neighboring phases. The real boundary of coexisting phases has a variable density profile corresponding to the condition of the constant chemical potential throughout the region of transition and in the neighboring coexisting phases. The absence of an intermediate film excludes the priority of the mechanical equilibrium over the chemical equilibrium and results in pressure being the function of local temperature and chemical potential values (excluding the application of the Laplace equation). Considering times of relaxation is found to change the familiar Gibbs expression for the pressure jump between coexisting vapor and fluid as functions of the boundary in the gravitational field potential. A consequence of considering the correct ratio of momentum and mass times of relaxation is discussed: the phase equilibrium conditions of a separating vapor–liquid system in combined gravitational and surface force fields must be analyzed in order to solve problems of the capillary theory.

On the second Piola-Kirchhoff and Cauchy stress tensors in nonlinear shells subjected to displacement-dependent loads

In this article, a nonlinear geometrically exact (GeX) hybrid-mixed four-node solid-shell element under non-conservative loading using the sampling surfaces (SaS) method is developed. The term “geometrically exact” means that the parametrization of the middle surface is known and, therefore, the coefficients of the first and second fundamental forms and the Christoffel symbols are taken exactly at element nodes. The SaS formulation is based on the choice of N SaS parallel to the middle surface and located at Chebyshev polynomial nodes in order to introduce the displacements of these surfaces as fundamental shell unknowns. Such a choice of unknowns with the use of Lagrange polynomials of degree N–1 in the through-thickness distributions of displacements, strains, and stresses leads to an efficient higher-order shell formulation. To develop the hybrid-mixed four-node solid-shell element, which is free of shear and membrane locking, the Hu–Washizu variational principle is utilized. The proposed GeX hybrid-mixed solid-shell element subjected to follower loads shows superior performance in the case of coarse meshes and allows the use of only one load step in most of the considered numerical examples. It is established that the difference between the second Piola-Kirchhoff and Cauchy stresses in some non-conservative problems for isotropic and composite shells undergoing finite rotations can be significant.

Space-Time Properties of Extreme RN Black Holes in Static Triangular Distribution

We studied the space-time properties of the triangular symmetric black hole in the case of extreme RN black hole. Because the neutral test particle is only affected by space-time in the curved space-time, we chose the triangular symmetric black hole as the model with which to study the motion of the test particle in this case. The curvature tensor and curvature scalar were calculated by giving the metric and the Christoffel Symbol, and then the kinematics equation of the test particle was obtained and analyzed by using these quantities. Then we analyzed the relationship between the coordinate distance and the inherent distance, the relationship between the coordinate time and the inherent time, the inherent velocity and the coordinate velocity of light, and then verified the correctness of general relativity. Next, the one-dimensional effective potential and two-dimensional effective potential of the system under different separation distances were analyzed. Finally, we analyzed and explored the innermost stable circular orbit, calculated all the Lagrange points under this model, and expounded some applications of circular orbit in astrophysics.

Effects of Variations in Earth’s Gravitational Force Fields on Climate Change

World climate continues to deteriorate at varying rates in spite of different anthropogenic mitigative and adaptive interventions. Conversely, the gravitational force fields of the sun and moon hold Earth in orbit, amongst others. Herein, we explore the impacts of changes in these gravitational fields on global climate and examine their influences on natural events, such as ocean tides, volcanoes, geomagnetic storms and movement of tectonic plates. Beyond reports on greenhouse effect and mass transport, we discuss the influence of gravity on tidal bulges, melting glaciers, condensable atmosphere and other events, such as El Ni&amp;ntilde;o and La Ni&amp;ntilde;a, and correlate their typically subtle impacts on climate worldwide with variations in Earth&amp;rsquo;s gravity. The intention is to highlight other causative factors implicated in climatic change without diminishing the contributions of greenhouse gasses and other factors, which are currently regarded, by specialists and lay-public alike, as the major culprits of climate change. We submit that the shifts in Earth&amp;rsquo;s gravitational fields to sustain equilibrium and remain in orbit can manifest as perturbations of atmospheric temperatures, pressures and air concentrations as well as volume changes and ion effects; in hydrological bodies, and are some of the non-anthropogenic agents driving changes in global climate. We surmise, therefore, that the influence of changes in gravity, albeit subtle, on climate change is significant. It is envisaged that highlighting these subtle agents of change would intensify efforts toward ameliorating and/or eliminating drastic and deleterious changes in climate as well as at embracing adaptive measures at local and international levels.