Some remarks on Einstein-Randers metrics

Abstract

In this essay, we study the sufficient and necessary conditions for a Randers metrc to be of constant Ricci curvature without the restriction of strong convexity (regularity). The classification result for the case $\|\beta\|_{\alpha}>1$ is provided, which is similar to the famous Bao-Robles-Shen's result for strongly convex Randers metrics ($\|\beta\|_{\alpha}<1$). Based on some famous Einstein-Lorentz metrics in General Relativity, such as Minkowski metric, Sitter metric, anti de Sitter metric, Schwarzschild metric, Kerr metric, C-metric, Kasner metric, Levi-Civita metric, Cartor-Novotn\'{y}-Horsk\'{y} metric, etc., many non-regular Einstein-Randers metrics are constructed. Besides, we find that the case $\|\beta\|_{\alpha}\equiv1$ is very distinctive. These metrics will be called singular Randers metrics or parabolic Finsler metrics since their indicatrixs are parabolic hypersurface. A preliminary discussion for such metrics is provided.