Magnetic trajectories corresponding to Killing magnetic fields in a three-dimensional warped product

2020 ◽  
Vol 17 (14) ◽  
pp. 2050212
Author(s):  
Zafar Iqbal ◽  
Joydeep Sengupta ◽  
Subenoy Chakraborty
Keyword(s):  
Magnetic Fields ◽  
Warped Product ◽  
Vector Fields ◽  
Killing Vector ◽  
Three Dimensional ◽  
Charged Particles ◽  
Open Interval ◽  
Product Formula ◽  
Warping Function ◽  
Killing Vector Fields

The aim of this paper is to investigate Killing magnetic trajectories of varying electrically charged particles in a three-dimensional warped product [Formula: see text] with positive warping function [Formula: see text], where [Formula: see text] is an open interval in [Formula: see text] equipped with an induced semi-Euclidean metric on [Formula: see text]. First, Killing vector fields on [Formula: see text] are characterized and it is observed that lifts to [Formula: see text] of Killing vector fields tangent to [Formula: see text] are also Killing on [Formula: see text]. Now, any Killing vector field on [Formula: see text] corresponds to a Killing magnetic field on [Formula: see text]. Magnetic trajectories (also known as magnetic curves) of charged particles which move under the influence of Lorentz force generated by Killing magnetic fields on [Formula: see text] are obtained in both Riemannian and Lorentzian cases. Moreover, some examples are exhibited with pictures determining Killing magnetic trajectories in hyperbolic [Formula: see text]-space [Formula: see text] modeled by the Riemannian warped product [Formula: see text]. Furthermore, some examples of spacelike, timelike and lightlike Killing magnetic trajectories are given with their possible graphs in the Lorentzian warped product [Formula: see text].

scholarly journals On the symmetries of three-dimensional generalized symmetric spaces

2021 ◽  
Vol 13(62) (2) ◽  
pp. 451-462
Author(s):  
Lakehal Belarbi
Keyword(s):  
Symmetric Spaces ◽  
Vector Fields ◽  
Killing Vector ◽  
Three Dimensional ◽  
Riemannian Metric ◽  
Killing Vector Fields ◽  
Generalized Symmetric Space ◽  
Generalized Symmetric Spaces ◽  
Full Classification

In this work we consider the three-dimensional generalized symmetric space, equipped with the left-invariant pseudo-Riemannian metric. We determine Killing vector fields and affine vectors fields. Also we obtain a full classification of Ricci, curvature and matter collineations

scholarly journals Conformal and Disformal Structure of 3D Circularly Symmetric Static Metric in f(R) Theory of Gravity

2020 ◽  
Vol 39 (1) ◽  
pp. 111-116
Author(s):  
Muhammad Ramzan ◽  
Murtaza Ali ◽  
Fiaz Hussain
Keyword(s):  
Vector Fields ◽  
Killing Vector ◽  
Three Dimensional ◽  
Conformal Killing ◽  
Field Equations ◽  
Killing Vector Fields ◽  
Homothetic Vector ◽  
First Case ◽  
Conformal Vector Fields ◽  
Theory Of Gravity

Conformal vector fields are treated as generalization of homothetic vector fields while disformal vector fields are defined through disformal transformations which are generalization of conformal transformations, therefore it is important to study conformal and disformal vector fields. In this paper, conformal and disformal structure of 3D (Three Dimensional) circularly symmetric static metric is discussed in the framework of f(R) theory of gravity. The purpose of this paper is twofold. Firstly, we have found some dust matter solutions of EFEs (Einstein Field Equations) by considering 3D circularly symmetric static metric in the f(R) theory of gravity. Secondly, we have found CKVFs (Conformal Killing Vector Fields) and DKVFs (Disformal Killing Vector Fields) of the obtained solutions by means of some algebraic and direct integration techniques. A metric version of f(R) theory of gravity is used to explore the solutions and dust matter as a source of energy momentum tensor. This study reveals that no proper DVFs exists. Here, DVFs for the solutions under consideration are either HVFs (Homothetic Vector Fields) or KVFs (Killing Vector Fields) in the f(R) theory of gravity. In this study, two cases have been discussed. In the first case, both CKVFs and DKVFs become HVFs with dimension three. In the second case, there exists two subcases. In the first subcase, DKVFs become HVFs with dimension seven. In the second subcase, CKVFs and DKVFs become KVFs having dimension four.

scholarly journals 2-Killing vector fields on warped product manifolds

2015 ◽  
Vol 26 (09) ◽  
pp. 1550065 ◽  
Author(s):  
Sameh Shenawy ◽  
Bülent Ünal
Keyword(s):  
Vector Field ◽  
Product Space ◽  
Warped Product ◽  
Vector Fields ◽  
Killing Vector ◽  
Killing Vector Field ◽  
Product Manifold ◽  
Killing Vector Fields ◽  
Warped Product Manifold

This paper provides a study of 2-Killing vector fields on warped product manifolds as well as characterization of this structure on standard static and generalized Robertson–Walker space-times. Some conditions for a 2-Killing vector field on a warped product manifold to be parallel are obtained. Moreover, some results on the curvature of a warped product manifolds in terms of 2-Killing vector fields are derived. Finally, we apply our results to describe 2-Killing vector fields of some well-known warped product space-time models.

scholarly journals Solutions to the affine quasi-Einstein equation for homogeneous surfaces

2020 ◽  
Vol 20 (3) ◽  
pp. 413-432
Author(s):  
M. Brozos-Vázquez ◽  
E. García-Río ◽  
P. Gilkey ◽  
X. Valle-Regueiro
Keyword(s):  
Einstein Equation ◽  
Warped Product ◽  
Vector Fields ◽  
Einstein Metrics ◽  
Killing Vector ◽  
Einstein Manifolds ◽  
Killing Vector Fields ◽  
Organizational Framework ◽  
Yamabe Solitons

AbstractWe examine the space of solutions to the affine quasi–Einstein equation in the context of homogeneous surfaces. As these spaces can be used to create gradient Yamabe solitons, conformally Einstein metrics, and warped product Einstein manifolds using the modified Riemannian extension, we provide very explicit descriptions of these solution spaces. We use the dimension of the space of affine Killing vector fields to structure our discussion as this provides a convenient organizational framework.

Spacetime symmetries

2018 ◽  
Author(s):  
Michael Kachelriess
Keyword(s):  
Riemannian Manifold ◽  
Conservation Laws ◽  
Vector Fields ◽  
Killing Vector ◽  
Geodesic Equation ◽  
Spacetime Symmetries ◽  
Tensor Fields ◽  
Killing Vector Fields ◽  
Covariant Derivatives ◽  
Killing Equation

This chapter introduces tensor fields, covariant derivatives and the geodesic equation on a (pseudo-) Riemannian manifold. It discusses how symmetries of a general space-time can be found from the Killing equation, and how the existence of Killing vector fields is connected to global conservation laws.

scholarly journals Off-Shell Noether Currents and Potentials for First-Order General Relativity

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 348
Author(s):  
Merced Montesinos ◽  
Diego Gonzalez ◽  
Rodrigo Romero ◽  
Mariano Celada
Keyword(s):  
General Relativity ◽  
Vector Fields ◽  
Killing Vector ◽  
Spherically Symmetric ◽  
Killing Vector Fields ◽  
Arbitrary Vector ◽  
Four Dimensions ◽  
First Order ◽  
Direct Form ◽  
Definition Of

We report off-shell Noether currents obtained from off-shell Noether potentials for first-order general relativity described by n-dimensional Palatini and Holst Lagrangians including the cosmological constant. These off-shell currents and potentials are achieved by using the corresponding Lagrangian and the off-shell Noether identities satisfied by diffeomorphisms generated by arbitrary vector fields, local SO(n) or SO(n−1,1) transformations, ‘improved diffeomorphisms’, and the ‘generalization of local translations’ of the orthonormal frame and the connection. A remarkable aspect of our approach is that we do not use Noether’s theorem in its direct form. By construction, the currents are off-shell conserved and lead naturally to the definition of off-shell Noether charges. We also study what we call the ‘half off-shell’ case for both Palatini and Holst Lagrangians. In particular, we find that the resulting diffeomorphism and local SO(3,1) or SO(4) off-shell Noether currents and potentials for the Holst Lagrangian generically depend on the Immirzi parameter, which holds even in the ‘half off-shell’ and on-shell cases. We also study Killing vector fields in the ‘half off-shell’ and on-shell cases. The current theoretical framework is illustrated for the ‘half off-shell’ case in static spherically symmetric and Friedmann–Lemaitre–Robertson–Walker spacetimes in four dimensions.

scholarly journals The geometry of null-like disformal transformations

2019 ◽  
Vol 16 (11) ◽  
pp. 1950180 ◽  
Author(s):  
I. P. Lobo ◽  
G. G. Carvalho
Keyword(s):  
Vector Field ◽  
Conformal Transformation ◽  
Vector Fields ◽  
Killing Vector ◽  
Killing Vector Fields ◽  
Spinor Structure ◽  
Schwarzschild Metric ◽  
Killing Vectors ◽  
Symmetry Properties ◽  
Geometric Objects

Motivated by the hindrance of defining metric tensors compatible with the underlying spinor structure, other than the ones obtained via a conformal transformation, we study how some geometric objects are affected by the action of a disformal transformation in the closest scenario possible: the disformal transformation in the direction of a null-like vector field. Subsequently, we analyze symmetry properties such as mutual geodesics and mutual Killing vectors, generalized Weyl transformations that leave the disformal relation invariant, and introduce the concept of disformal Killing vector fields. In most cases, we use the Schwarzschild metric, in the Kerr–Schild formulation, to verify our calculations and results. We also revisit the disformal operator using a Newman–Penrose basis to show that, in the null-like case, this operator is not diagonalizable.

scholarly journals PROPER CURVATURE COLLINEATIONS IN NONSTATIC SPHERICALLY SYMMETRIC SPACE–TIMES

2008 ◽  
Vol 23 (05) ◽  
pp. 749-759 ◽  
Author(s):  
GHULAM SHABBIR ◽  
M. RAMZAN
Keyword(s):  
Symmetric Space ◽  
Vector Fields ◽  
Killing Vector ◽  
Spherically Symmetric ◽  
Killing Vector Fields ◽  
Infinite Dimensional ◽  
Riemann Matrix ◽  
Integration Techniques ◽  
Curvature Collineations ◽  
Proper Curvature

A study of nonstatic spherically symmetric space–times according to their proper curvature collineations is given by using the rank of the 6×6 Riemann matrix and direct integration techniques. Studying proper curvature collineations in each case of the above space–times it is shown that when the above space–times admit proper curvature collineations, they turn out to be static spherically symmetric and form an infinite dimensional vector space. In the nonstatic cases curvature collineations are just Killing vector fields.

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