decades of the past century. In [@DMN] the construction of the conformal arc-element has been extended to time-like curves in a $3$-dimensional Einstein universe. In this paper we show how to define a conformal invariant arc-element for a time-like curve in an Einstein universe of arbitrary dimension. The integral of the conformal arc-element defines the simplest conformal invariant variational problem on the space of generic time-like curves of $\mathcal{E}^{1,n-1}$, called the conformal strain functional. This is the Lorentzian counterpart of the conformal arc-length functional in Möbius geometry [@LO2010; @LS; @MMR; @M1; @M2; @MN] and generalizes the homonymous functional for time-like curves in $\mathcal{E}^{1,2}$, previously considered in [@DMN]. Proceeding in analogy with [@MMR] and using the method of moving frames we deduce the variational equations satisfied by the stationary curves, referred to as conformal world-lines. We prove that conformal world-lines in $\mathcal{E}^{1,n-1}$ are trapped into a totally geodesic Einstein universe of dimension $2$, $3$ or $4$. This is the Lorentzian analogue of a similar result in Möbius geometry [@MMR]. The $2$-dimensional case is rather trivial since the trajectories are orbits of $1$-parameter groups of conformal transformations. The $3$-dimensional case is more challenging and it has been partially investigated in [@DMN]. Here, 