Analysis of Einstein Field Equations of Static Plane Symmetric Space-Time in General Relativity via Lie Approach

Year 2019, , 14 - 17, 20.04.2019

Adil Jhangeer

https://doi.org/10.33187/jmsm.430767

Abstract

In this study, Einstein field equations of static plane symmetric space-time is discussed by means of Lie approach. Lie point symmetries are computed for the considered equations. Moreover, one dimensional conjugacy classes for symmetry algebra are reported. Additionally, similarity variables are derived which are further utilized to find the solutions for parameters $A$ and $B$ of the considered space-time. Both linearly independent and dependent relationships between the unknown parameters $A$ and $B$ are observed for different cases. At the end, some well-known solutions are presented.

Keywords

Plane symmetric solutions, Optimal system, Lie algebra, Plane symmetric solutions

References

• [1] D. Kramer, H. Stephani, M. MacCullum, E. Herlt, Exact Solutions of Einstein Field Equations, Cambridge University Press, 1980.
• [2] P. J. Olver, Applicationsof Lie Groups to Differential Equations, Springer-Verlag, New York, 1986.
• [3] A. H. Taub, Empty space-time sadmitting a three parameter group of motions, Annals Math., 53( 1951), 472–490.
• [4] H. Bondi, Plane gravitational waves in general relativity, Nature,179 (1957), 1072–1073.
• [5] H. Bondi, E. Pirani, I. Robinson, Gravitational waves in general relativity. III. exact plane waves, Proc. Roy. Soc., 251 (1959), 519–533.
• [6] H. Takeno, On plane wave solution of field equations in general relativity, N. S. Tensor, 7 (1957), 97–102.
• [7] M. Sharif, M. F. Shamir, Plane symmetric solutions in f (R) gravity, Mod. Phys. Lett., A25 (2010), 1281-1288.
• [8] J. Patera, P. Winternitz, Subalgebras of real three-and four-dimensional Lie algebras, J. Math. Phy.,18 (7) (1977), 1449-1455.