Killing Vectors in Spacetime of the De Sitter Invariant Special Relativity

TL;DR

This paper formulates de Sitter/Anti-de Sitter invariant special relativity using Killing vectors, deriving the maximal symmetry of the Beltrami metric and identifying conserved charges for the vacuum with nonzero cosmological constant.

Contribution

It provides a new formulation of dS/AdS-SR based on Killing vectors, confirming the maximal symmetry of the Beltrami metric and deriving associated conserved quantities.

Findings

Beltrami metric has maximal spacetime symmetry.

All ten Killing vectors were explicitly obtained.

Conserved charges include energy, momentum, and angular momentum.

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.