Corrigendum: Killing vector fields in three dimensions: a method to solve massive gravity field equations

2012 ◽  
Vol 29 (5) ◽  
pp. 059501 ◽  
Author(s):  
Metin Gürses
Keyword(s):  
Gravity Field ◽  
Vector Fields ◽  
Killing Vector ◽  
Massive Gravity ◽  
Three Dimensions ◽  
Field Equations ◽  
Killing Vector Fields

Conformal vector fields of some vacuum classes of static spherically symmetric space-times in f(T,B) gravity

2020 ◽  
Vol 17 (10) ◽  
pp. 2050149 ◽  
Author(s):  
Ghulam Shabbir ◽  
Fiaz Hussain ◽  
Shabeela Malik ◽  
Muhammad Ramzan
Keyword(s):  
Symmetric Space ◽  
Vector Fields ◽  
Killing Vector ◽  
Spherically Symmetric ◽  
Einstein Field Equations ◽  
Field Equations ◽  
Killing Vector Fields ◽  
Integration Approach ◽  
Conformal Vector ◽  
Conformal Vector Fields

The aim of this paper is to investigate the conformal vector fields (CVFs) for some vacuum classes of static spherically symmetric space-times in [Formula: see text] gravity. First, we have explored the space-times by solving the Einstein field equations in [Formula: see text] gravity. These solutions have been obtained by imposing various conditions on the space-time components and selecting separable form of the bivariate function [Formula: see text]. Second, we find the CVFs of the obtained space-times via direct integration approach. The overall study reveals that there exist 17 cases. From these 17 cases, the space-times in five cases admit proper CVFs whereas in rest of the 12 cases, CVFs become Killing vector fields (KVFs). We have also calculated the torsion scalar and boundary term for each of the obtained solutions.

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.

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.

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