Light rays in static spacetimes with critical asymptotic behavior: A variational approach

Abstract

Let $mathcal{M}=mathcal{M}_{0}imes mathbb{R}$ be a Lorentzian manifold equipped with the static metric $langle cdot ,cdot angle _{z}=langle cdot ,cdot angle -eta (x)dt^{2}$. The aim of this paper is investigating the existence of lightlike geodesics joining a point $z_{0}=(x_{0},t_{0})$ to a line $gamma ={ x_{1}} imes mathbb{R}$ when coefficient $eta $ has a quadratic asymptotic behavior by means of a variational approach.

Keywords

• Static spacetime
• light ray
• quadratic asymptotic behavior
• Fermat principle
• Ljusternik-Schnirelman category.