Homogeneous Lorentzian structures of 4-dimensional nilpotent Lie groups

We consider the four-dimensional 2-step nilpotent Lie group $H_{3}\times \mathbb {R}$, equipped with a family of non-flat left-invariant Lorentzian metrics. We give a complete classification of all homogeneous structures for each metric. We then investigate the properties of each structure and prove the existence of a non-canonical homogeneous structure. Finally, we present an example of a naturally reductive, non-flat, left-invariant Lorentzian metric on $H_{3}\times \mathbb {R}$ for which the center is degenerate.