The geometry of conformal timelike geodesics in the Einstein universe

This paper studies the geometry of the critical points of the simplest conformally invariant variational problem for timelike curves in the n-dimensional Einstein universe. Such critical curves are referred to as conformal timelike geodesics. The functional defining the variational problem is the Lorentz analogue of the conformal arclength functional in Möbius geometry. We compute the Euler–Lagrange equations and show that the trajectory of a conformal timelike geodesic is constrained into some totally umbilical Einstein universe of dimension 2, 3, or 4. The case of dimension 2 leads to orbits of 1-parameter groups of Lorentz Möbius transformations, while that of dimension 3 has been dealt with in [8]. In this paper, we discuss the case of conformal timelike geodesics in the 4-dimensional Einstein universe whose trajectories are not contained in any lower dimensional totally umbilical Einstein universe. It is shown that such curves can be explicitly integrated by quadratures and explicit expressions in terms of elliptic functions and integrals are provided.