Homothetic Vector Field Research Articles

Some remarks on wave solutions in general relativity theory

This paper reviews the concept of pp-waves and plane waves in classical general relativity theory. The first four sections give discussions of some algebraic constructions and symmetry concepts which will be needed in what is to follow. The final sections deal with the definitions of such wave solutions, their associated geometrical tensors in space–time and their Killing, homothetic, conformal and wave surface symmetries. Some unusual geometrical features of these solutions (compared with standard positive definite geometry) are described.

Classification of static spherically symmetric space-times in f(R) theory of gravity according to their conformal vector fields

A classification of static spherically symmetric space-times in [Formula: see text] theory of gravity according to their conformal vector fields (CVFs) is presented. For this analysis, a direct integration technique is used. This study reveals that for static spherically symmetric space-times in [Formula: see text] theory of gravity, CVFs are just Killing vector fields (KVFs) or homothetic vector fields (HVFs). For this classification, six cases have been discussed out of which there exists only one case for which CVFs become HVFs while in the rest of the cases CVFs become KVFs.

On two topologies that were suggested by Zeeman

The class of Zeeman topologies on spacetimes in the frame of relativity theory is considered to be of powerful intuitive justification, satisfying a sequence of properties with physical meaning, such as the group of homeomorphisms under such a topology is isomorphic to the Lorentz group and dilatations, in Minkowski spacetime, and to the group of homothetic symmetries in any curved spacetime. In this article, we focus on two distinct topologies that were suggested by Zeeman as alternatives to his fine topology, showing their connection with two orders, ie, a timelike and a (noncausal) spacelike one. For the (noncausal) spacelike order, we introduce a partition of the null cone, which gives the desired topology invariantly from the choice of the hyperplane of partition. In particular, we observe that these two orders induce topologies within the class of Zeeman topologies, while the two suggested topologies by Zeeman himself are intersection topologies of these two order topologies (respectively) with the manifold topology. We end up with a list of open questions and a discussion, comparing the topologies with bounded against those with unbounded open sets and their possible physical interpretation.

Kaluza–Klein type Ricci solitons on unit tangent sphere bundles

We consider g-natural pseudo-Riemannian metrics of Kaluza–Klein type on the unit tangent sphere bundle of a Riemannian manifold of constant sectional curvature and give necessary and sufficient conditions for these metrics to give rise to a Ricci soliton.On the one hand, we obtain a rigidity result in dimension three, showing that there are no nontrivial Ricci solitons among g-natural metrics of Kaluza–Klein type on the unit tangent sphere bundle of any Riemannian surface. On the other hand, while Ricci solitons determined by tangential lifts remain trivial in arbitrary dimension, horizontal lifts of vector fields related to the geometry of the base manifold (namely, homothetic vector fields) produce nontrivial Ricci solitons metrics of Kaluza–Klein type. Gradient Ricci solitons of Kaluza–Klein type are also completely characterized.

On twisting type [N] ⊗ [N] Ricci flat complex spacetimes with two homothetic symmetries

In this article, HH spaces of type [N] ⊗ [N] with twisting congruence of null geodesics defined by the 4-fold undotted and dotted Penrose spinors are investigated. It is assumed that these spaces admit two homothetic symmetries. The general form of the homothetic vector fields is found. New coordinates are introduced, which enable us to reduce the HH system of partial differential equations to one ordinary differential equation (ODE) on one holomorphic function. In a special case, this is a second-order ODE and its general solution is explicitly given. In the generic case, one gets rather involved fifth-order ODE.

Homothetic Motion in a Bianchi Type –V Model in Lyra Geometry

In this paper we study a homothetic vector field of a Bianchi type-V model based on Lyra geometry. The aim of this paper is to get the components of homothetic vector field in Lyra geometry for Bianchi type-V, compare between it and the case of Bianchi type-I. Using ordinary method and Computer program to get the components of the vector . The results are compatible in the two methods and get the condition that the results tends to the case of Bianchi type-I. The cases when a displacement vector is a function of t and when it is constant are considered. In both two cases we investigate the equation of state.

Proper Teleparallel Homothetic Vector Fields in General Cylindrically Symmetric Space-Times in Teleparallel Theory of Gravitation Using Diagonal Tetrads

In this paper we consider the most general form of non-static cylindrically symmetric space-times in order to study proper teleparallel homothetic vector fields using the direct integration technique and diagonal tetrads. This study also covers static cylindrically symmetric, Bianchi type I, non-static and static plane symmetric space-times as well. Here, we will only discuss the cases which do not fall in the category of static cylindrically symmetric, Bianchi type I, non-static and static plane symmetric space-times. From the above study we show that very special classes of the above space-times yield 6, 7 and 8 teleparallel homothetic vector fields with non-zero torsion.

Vacuum self similar anisotropic cosmologies in F(R)-gravity

The implications from the existence of a proper Homothetic Vector Field (HVF) on the dynamics of vacuum anisotropic models in $F(R)$ gravitational theory are studied. The fact that \emph{every} Spatially Homogeneous vacuum model is equivalent, formally, with a "flux" -free anisotropic fluid model in standard gravity and the induced power-law form of the functional $F(R)$ due to self-similarity enable us to close the system of equations. We found some new exact anisotropic solutions that arise as fixed points in the associated dynamical system. The non-existence of Kasner-like (Bianchi type I) solutions in proper $F(R)-$gravity (i.e. $R\neq 0$) strengthens the belief that curvature corrections will prevent the shear influence into the past thus permitting an isotropic singularity. We also discuss certain issues regarding the lack of vacuum models of type III, IV, VII$_{h}$ in comparison with the corresponding results in standard gravity.

Some results on harmonic and bi-harmonic maps

In this paper, we prove that any bi-harmonic map from a compact orientable Riemannian manifold without boundary [Formula: see text] to Riemannian manifold [Formula: see text] is necessarily constant with [Formula: see text] admitting a strongly convex function [Formula: see text] such that [Formula: see text] is a Jacobi-type vector field (or [Formula: see text] admitting a proper homothetic vector field). We also prove that every harmonic map from a complete Riemannian manifold into a Riemannian manifold admitting a proper homothetic vector field, satisfying some condition, is constant. We present an open problem.

Teleparallel conformal Killing vector fields of LRS Bianchi type V spacetimes in teleparallel gravity

The aim of this paper is to explore teleparallel conformal Killing vector fields (CKVFs) of locally rotationally symmetric (LRS) Bianchi type V spacetimes in the context of teleparallel gravity and compare the obtained results with those of general relativity (GR). The general solution of teleparallel conformal Killing's equations is found in terms of some unknown functions of [Formula: see text] and [Formula: see text], along with a set of integrability conditions. The integrability conditions are solved in some particular cases to get the final form of teleparallel CKVFs. It is observed that the LRS Bianchi type V spacetimes admit proper teleparallel CKVF in only one case, while in remaining cases the teleparallel CKVFs reduce to teleparallel Killing vector fields (KVFs). Moreover, it is shown that the LRS Bianchi type V spacetimes do not admit any proper teleparallel homothetic vector field (HVF).

Some uniqueness results for Ricci solitons

We investigate the relationship between Ricci soliton structures and homothetic vector fields, especially Killing vector fields. More precisely, we present some characterizations for Ricci solitons endowed with Killing vector fields of constant norm as well as homothetic vector fields. In particular, we relate different Ricci soliton structures on a Riemannian manifolds in order to deduce some uniqueness results.

On axially symmetric space–times admitting homothetic vector fields in Lyra’s geometry

This paper investigates axially symmetric space–times that admit a homothetic vector field based on Lyra’s geometry. The cases when the displacement vector is a function of t and when it is constant are studied. In the context of this geometry, we find and classify the solutions of the Einstein’s field equations for the space–time under consideration, which display a homothetic symmetry.

Noether symmetries of vacuum classes of pp-waves and the wave equation

The Noether symmetry algebras admitted by wave equations on plane-fronted gravitational waves with parallel rays are determined. We apply the classification of different metric functions to determine generators for the wave equation, and also adopt Noether's theorem to derive conserved forms. For the possible cases considered, there exist symmetry groups with dimensions two, three, five, six and eight. These symmetry groups contain the homothetic symmetries of the spacetime.

Self-similar motion of a Nambu-Goto string

We study the self-similar motion of a string in a self-similar spacetime by introducing the concept of a self-similar string, which is defined as the world sheet to which a homothetic vector field is tangent. It is shown that in Nambu-Goto theory, the equations of motion for a self-similar string reduce to those for a particle. Moreover, under certain conditions such as the hypersurface orthogonality of the homothetic vector field, the equations of motion for a self-similar string simplify to the geodesic equations on a (pseudo) Riemannian space. As a concrete example, we investigate a self-similar Nambu-Goto string in a spatially flat Friedmann-Lema\^itre-Robertson-Walker expanding universe with self-similarity and obtain solutions of open and closed strings, which have various nontrivial configurations depending on the rate of the cosmic expansion. For instance, we obtain a circular solution that evolves linearly in the cosmic time while keeping its configuration by the balance between the effects of the cosmic expansion and string tension. We also show the instability for linear radial perturbation of the circular solutions.

A note on teleparallel conformal Killing vector fields in plane symmetric non-static spacetimes

The aim of this paper is to investigate teleparallel conformal Killing vector fields (CKVFs) in plane symmetric non-static spacetimes. Ten teleparallel conformal Killing’s equations are obtained which are linear in the components of the teleparallel CKVF [Formula: see text]. A general solution of these equations comprising the components of CKVF and conformal factor is presented, which subject to some integrability conditions. For seven particular choices of the metric functions, the integrability conditions are completely solved to get the final form of teleparallel CKVFs and conformal factor. In four different cases we get proper CKVFs, while in the remaining three cases it is shown that teleparallel CKVFs reduce to teleparallel homothetic or teleparallel Killing vector fields.

Complete Classification of Conformal Killing Symmetries of Plane-Symmetric Static Spacetimes in Teleparallel Theory

We have studied the conformal, homothetic and Killing vectors in the context of teleparallel theory of gravitation for plane-symmetric static spacetimes. We have solved completely the non-linear coupled teleparallel conformal Killing equations. This yields the general form of teleparallel conformal vectors along with the conformal factor for all possible cases of metric functions. We have found four solutions which are divided into one Killing symmetries and three conformal Killing symmetries. One of these teleparalel conformal vectors depends on x only and other is a function of all spacetime coordinates. The three conformal Killing symmetries contain three proper homothetic symmetries where the conformal factor is an arbitrary non-zero constant.

Homothetic vector fields in a specially homogenous Bianchi type-I cosmological model in Lyra geometry

In this paper, homothetic vector fields of a spatially homogenous Bianchi type-I cosmological model have been evaluated based on Lyra geometry. Further, we investigate the equation of state in cases when a displacement vector [Formula: see text] is a function of t and when it is constant. We give a comparison between the obtained results, using Lyra geometry, and those obtained previously in the context of general relativity, based on Riemannian geometry.

A note on Classification of Proper Teleparallel Homothetic Vector Field in Spherically Symmetric Static Space-times in the Teleparallel Theory of Gravitation using Diagonal Tetrads

In this paper a classification of proper teleparallel homothetic vector field in spherically symmetric static space-times is given using the direct integration technique. From the above study we show that the above space-times do not admit proper teleparallel homothetic vector fields.

Homothetic Motion in a Bianchi Type-I Model in Lyra Geometry

In this paper we study a homothetic vector field of a Bianchi type-I model based on Lyra geometry. The cases when a displacement vector is function of $t$ and when it is constant are considered. In both two cases we investigate the equation of state. A comparison between the obtained results, using Lyra geometry, and that have obtained previously in the context of General Relativity, based on Riemannian geometry, will be given.

Teleparallel Homothetic Symmetry of Special Axially Symmetric Static Spacetime

In this paper we are interested to find teleparallel proper homothetic vector fields over the Lorentzian manifold of special axially symmetric static spacetime. For the purpose we applied teleparallel Lie derivative to the metric for homothetic equations and obtained ten coupled non linear differential equations. These equations are then solved for each possibility of the metric functions and it comes out that only in one case teleparallel proper homothetic vector fields exist for the special choice of metric functions. The spacetime admit eight dimensional teleparallel homothetic vector fields in which one is proper teleparallel homothetic vector and remaining seven are teleparallel Killing vector fields