Geodesics in the conformally flat Eisenhart metric

The (4+1) dimensional conformally flat Eisenhart geometry is investigated in this work, stressing the contribution of the stress tensor generating its curvature. The energy-momentum tensor $T^{a}_{~b}$ is traceless and has only one nonzero component. It could be written as an anisotropic fluid with null transversal pressures and nonzero energy fluxes. The null and timelike geodesics are computed in the pure cosmological case when the Eisenhart potential energy is $V(r) = -m\omega^{2}r^{2}/2$, where $\omega$ is related to the cosmological constant $\Lambda$. Although the metric is curved, the radial null geodesics $R(T)$ and $Y(T)$ are straight lines, with finite $R_{max}$ and $Y_{max}$, $Y$ being the 5th coordinate. In contrast, for a radial timelike geodesic, $Y_{max} \rightarrow \infty$ if $T \rightarrow T_{max} = 1/\omega$.