@Ce; @JMN]. The purpose of the present paper is to investigate a variational problem for time-like curves in the conformal compactification $\mathcal{E}^{1,n-1}$ of the Minkowski n-space. In the literature, $\mathcal{E}^{1,n-1}$ is often referred to as the [*compact $n$-dimensional Einstein universe*]{} [@BCDG; @Fr; @GS]. The choice of working within the compact model is motivated by the fact that, in this case, the conformal group is a group of matrices. This greatly simplifies the computational aspects. However, our considerations apply as well in the non-compact case. It ’should be noted that any Lorentzian space-form or any Friedmann–Lemaitre–Robertson–Walker cosmological model can be realized via a conformal embedding, as an open domain of $\mathcal{E}^{1,n-1}$ [@HE].

Lorentzian conformal geometry shares, at least on a formal level, many common features, with Riemannian conformal geometry, a classical topic in differential geometry. The Lorentzian conformal geometry of the Einstein universe can be developed in analogy with the conformal geometry of a sphere (Möbius geometry). For instance, the existence of a canonical conformal invariant arc-element along a generic curve in $S^n$ is a well known fact that goes back to the works [@F; @Ha; @L; @T; @V] of Fialkov, Haantjes, Liebmann, Takasu and Vessiot, all published in the first 