Killing vector fields and the Einstein-Maxwell field equations in general relativity

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.

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On Almost Contingent Manifolds of Second Class with Applications in Relativity

Canadian Mathematical Bulletin ◽

10.4153/cmb-1978-051-4 ◽

1978 ◽

Vol 21 (3) ◽

pp. 289-295 ◽

Cited By ~ 4

Author(s):

K. L. Duggal

Keyword(s):

General Relativity ◽

Electromagnetic Fields ◽

Vector Fields ◽

Killing Vector ◽

Positive Definite ◽

Geometric Proof ◽

Sasakian Manifolds ◽

Killing Vector Fields ◽

Degenerate Metric ◽

Frame Work

D. E. Blair [1] has introduced the notion of K-manifolds as an analogue of the even dimensional Kähler manifolds and of the odd dimensional quasi-Sasakian manifolds. These manifolds have been studied with respect to a positive definite metric. In this paper, we study a more general case of if-manifolds carrying an arbitrary non-degenerate metric, in particular, a metric of Lorentz signature. This theory is then applied within the frame-work of general relativity. Using the Ruse-Synge classification [8, 9] of non-null electromagnetic fields with source, we develop a geometric proof for the existence of either two space like or one space like and one time like Killing vector fields on the space-time manifold.

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CLASSIFICATION OF CYLINDRICALLY SYMMETRIC STATIC SPACETIMES ACCORDING TO THEIR KILLING VECTOR FIELDS IN TELEPARALLEL THEORY OF GRAVITATION

Modern Physics Letters A ◽

10.1142/s0217732310032007 ◽

2010 ◽

Vol 25 (07) ◽

pp. 525-533 ◽

Cited By ~ 17

Author(s):

GHULAM SHABBIR ◽

SUHAIL KHAN

Keyword(s):

General Relativity ◽

Vector Fields ◽

Killing Vector ◽

Minkowski Spacetime ◽

Killing Vector Fields ◽

Plane Symmetric ◽

Cylindrically Symmetric ◽

Teleparallel Theory ◽

Theory Of Gravitation

In this paper we classify cylindrically symmetric static spacetimes according to their teleparallel Killing vector fields using direct integration technique. It turns out that the dimension of the teleparallel Killing vector fields are 3, 4, 6 or 10 which are the same in numbers as in general relativity. In case of 3, 4 or 6 the teleparallel Killing vector fields are multiple of the corresponding Killing vector fields in general relativity by some function of r. In the case of 10 Killing vector fields the spacetime becomes Minkowski spacetime and all the torsion components are zero. The Killing vector fields in this case are exactly the same as in general relativity. Here we also discuss the Lie algebra in each case. It is important to note that this classification also covers the plane symmetric static spacetimes.

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f-SYMBOLS, KILLING TENSORS AND CONSERVED BEL-TYPE CURRENTS

Modern Physics Letters A ◽

10.1142/s0217732311034888 ◽

2011 ◽

Vol 26 (05) ◽

pp. 337-349

Author(s):

OVIDIU TINTAREANU-MIRCEA

Keyword(s):

General Relativity ◽

Vector Fields ◽

Killing Vector ◽

Killing Vector Fields ◽

Killing Tensors ◽

Einstein Spaces ◽

Conserved Currents

In the framework of the General Relativity we show that from three generalizations of Killing vector fields, namely f-symbols, symmetric Stäckel–Killing and antisymmetric Killing–Yano tensors, some conserved currents can be obtained through adequate contractions of the above-mentioned objects with rank-four tensors having the properties of Bel or Bel–Robinson tensors in Einstein spaces.

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

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887820501492 ◽

2020 ◽

Vol 17 (10) ◽

pp. 2050149 ◽

Cited By ~ 1

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.

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A NOTE ON CLASSIFICATION OF BIANCHI TYPE I SPACETIMES ACCORDING TO THEIR TELEPARALLEL KILLING VECTOR FIELDS

Modern Physics Letters A ◽

10.1142/s0217732310031403 ◽

2010 ◽

Vol 25 (01) ◽

pp. 55-61 ◽

Cited By ~ 14

Author(s):

GHULAM SHABBIR ◽

SUHAIL KHAN

Keyword(s):

General Relativity ◽

Vector Fields ◽

Bianchi Type ◽

Killing Vector ◽

Type I ◽

Direct Integration ◽

Integration Technique ◽

Killing Vector Fields ◽

Bianchi Type I

In this paper we classify Bianchi type I spacetimes according to their teleparallel Killing vector fields using direct integration technique. It turns out that the dimension of the teleparallel Killing vector fields is 3, 4, 6 or 10 which are the same in numbers as in general relativity. In case of 3, 4 or 6 the teleparallel Killing vector fields are multiple of the corresponding Killing vector fields in general relativity by some function of t. In the case of 10 Killing vector fields, the spacetime becomes Minkowski and all the torsion components are zero. The Killing vector fields in this case are exactly the same as in the general relativity.

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