Classification of Teleparallel Homothetic Vector Fields in Cylindrically Symmetric Static Space-Times in Teleparallel Theory of Gravitation

Abstract

In this paper we classify cylindrically symmetric static space-times according to their teleparallel homothetic vector fields using direct integration technique. It turns out that the dimensions of the teleparallel homothetic vector fields are 4, 5, 7 or 11, which are the same in numbers as in general relativity. In case of 4, 5 or 7 proper teleparallel homothetic vector fields exist for the special choice to the space-times. In the case of 11 teleparallel homothetic vector fields the space-time becomes Minkowski with all the zero torsion components. Teleparallel homothetic vector fields in this case are exactly the same as in general relativity. It is important to note that this classification also covers the plane symmetric static space-times.

Killing Symmetry in Special Axially Symmetric Static Spacetimes in Teleparalel Theory of Gravitation

“…0 3 3 ,  X (20) To find the Killing vector fields in teleparallel theory of gravitation, we proceed as follows: The following system is formed when we solve equations (11), (12), (19) and (20). are the functions of integration.…”

“…In general relativity Killing vector fields are considered as an essential symmetry. In the background of teleparallel theory, a lot of work has been finished in obtaining Killing vector fields for different spacetimes [14][15][16][17][18][19][20][21][22]. To understand the physical and geometrical feature of our unexplained universe, much work is required to be done through symmetries in the presence of curvature or torsion only.…”

Teleparallel Homothetic Symmetry of Special Axially Symmetric Static Spacetime

“…One can write (6) explicitly using (7) and (11) (12) 0, e X θ) (r, A 2 (14) 0, e (20) . 3 , 3   X (21) In order to get teleparallel homothetic vector fields from the above equations we proceed as follows: Solving equations (12), (13), (20) and (21)  are functions of integration.…”

“…Homothetic vector fields are considered as an important symmetry in general relativity, as it preserve the metric up to a constant factor and give an extra symmetry ignored by the Killing symmetry. Homothetic vector fields in the presence of torsion in teleparallel theory are obtained in [19][20][21][22]. In order to understand the physical and geometrical aspects of our mysterious universe much work is needed to be done through different symmetries in the presence of curvature or torsion only.…”

Classification of static cylindrically symmetric spacetimes in f(R) theory of gravity by conformal motions with perfect fluid matter

“…Moreover, by classifying spacetimes, according to some symmetry, restriction makes easier to find solutions of EFEs. A reasonable amount of work has been done on the classification of cylindrically symmetric spacetimes according to different symmetries in GR and teleparallel theory of gravitation [2,3,13,[15][16][17][18][19]. Over the past few decades, finding exact solutions in f (R) theory of gravity has remained a key subject for researchers using constant and non-constant curvature condition.…”