Classification of Cylindrically Symmetric Static Spacetimes According to Their Killing Vector Fields in Teleparallel Theory of Gravitation

Abstract

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.

A Note on Classification of Cylindrically Symmetric Non-static Space–Times According to Their Teleparallel Killing Vector Fields in the Teleparallel Theory of Gravitation

“…It is clear from the teleparallel Killing equation that both the theories will produce the same Killing vector fields only if the torsion components become zero. This is only possible when the space-time becomes Minkowski [8,12]. One can easily see that in Minkowski space-times [8,12] Killing vector fields in both the theories are same.…”

“…In this paper, we will only present the results of non-static cylindrically symmetric and non-static plane symmetric space-times. The results of static cylindrically symmetric space-times and plane static space-times can be found in [12]. The current study will not only help to understand the geometrical and physical properties of the space-time but the effect of torsion on the gravitational laws can also be deduced.…”

“…Keeping this point in mind, Sharif and Amir [7] introduced the teleparallel version of the Lie derivative for Killing vector fields and used those equations to find the teleparallel Killing vector fields in Einstein universe. Using the teleparallel Lie derivative proposed in [7], we have explored teleparallel Killing vector fields in Bianchi type I space-times [8], Bianchi type II space-times [9], Bianchi type III and Kantowski-Sachs space-times [10], spatially homogenous rotating space-times [11], and cylindrically symmetric static space-times [12]. In [13], the author calculated energy, momentum, angular momentum, and teleparallel Killing vector fields for Brane world black hole in teleparallel theory, and he showed that when the results of energy, momentum, and angular momentum are in agreement with the results of general relativity, teleparallel Killing vector fields are also in agreement with the results of general relativity.…”

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

“…For above mentioned space-time (13) we can write the non trivial tetrad components S μ a and its inverse non trivial tetrad components S a μ [23] S μ a = diag e A(r) , e B(r) , r 2 , r 2 sin 2 θ , S a μ = diag e −A(r) , e −B(r) , r −2 , r −2 sin −2 θ . (14) The corresponding non-zero torsion components can be obtained as [24] T 0 10 =…”

“…As in general relativity, symmetries can be studied in teleparallel theory of gravitation [11][12][13][14][15][16][17][18][19][20][21]. Thus, the study of symmetries in the presence of curvature falls in the area of general relativity while the study of symmetries in the presence of torsion falls in the area of teleparallel theory of gravitation.…”