Killing Vectors in Spacetime of the De Sitter Invariant Special Relativity

Abstract

In this paper, we use the Killing vector method to formulate the de Sitter/Anti-de Sitter invariant special relativity (dS/AdS-SR). Through solving the Einstein equation with $\Lambda\neq 0$, the basic inertial metric for dS/AdS-SR is determined to be the Beltrami metric $B_{\mu\nu}(x)$. The corresponding Killing equations are system of ten simultaneous partial differential equations of first order. Their most general solutions were obtained, and all the ten independent Killing vectors were found out. These results confirm that the Beltrami metric has maximal spacetime symmetry. The ten Killing-Noether charges are obtained. They are energy, momenta, Lorentz boost and angular momentum in SR-theory with $\Lambda\neq 0$. Consequently, dS/AdS-SR is consistently established for the vacuum with $\Lambda\neq 0$ via Killing vector method rather than the unpopular classical domain theory.