Intrinsic geometry of curves and the Lorentz equation J.H. CALTENCO *), a. LINARES Y M., J.L. LOPEZ-BONILLA**) SecciSn de Estudios de Posgrado e Investigaci6n, Escuela Superior de Ingenierfa Mecdnica y E1dctrica, Instituto Polit~cnico Nacional Edi£ Z, Acc. 3-3er Piso Col. Lindavista, C.P. 07738, M~xico D.F. Received 25 October 2001; final version 4 March 2002 We show that the trajectory of a point charge in a uniform electromagnetic field is a helix if the Lorentz equation governs its motion. Our approach is totally relativistic, and it is based on the use of the Frenet-Serret formulae which describe the intrinsic geometry of world lines in Minkowski spacetime. PACS: 03.50.De, 41.20.-q Key words: Lorentz equation, Frenet-Serret formulae, world lines of classical charged particles. 1 Introduction The Lorentz equation [1-4] in special relativity describes the motion of a charged particle q under the influence of an electromagnetic field represented by the Faraday tensor F, without taking into account the radiation reaction force [3-10]. In this case, when F is constant, we can explicitly obtain [11-14] the exact path of q; we note that only Synge [12] shows that the corresponding world lines are helices in Minkowski space. In Sect. 2 we explain the Frenet-Serret (FS) equations [10,12,14-18] which have all the information about the intrinsic geometry of a trajectory in flat spacetime. Section 3 is dedicated to Faraday tensor [2,3] and its expression in terms of the FS tetrad. In Sect. 4 we employ the Lorentz equation and FS formulae for studying the case of uniform F, resulting thus in the three curvatures of the world lines being constants (helices); therefore, our approach provides an alternative method to Synge's proof [12]. 2 Frenet-Serret equation for a timelike path In special relativity, the 4-space is flat and hence the coordinate system (x r) = (ct, x, y, z), r = 0,..., 3 is such that the squared separation between the events x r and x r + dx r is given by the invariant (c is light velocity in the vacuum) ds 2 = c2dt 2 - dx 2 - dy 2 - dz 2 = ~Tabdxadx b , (1) where we assume the dummy indices convention and ~ = (7?ab)a×a = Diag(1,-1, -1,-1) is the Minkowski metric tensor. ~ allows us to distinguish three types of vectors with respect to its proper inner product [2]: A ~ is null, spacelike or timelike *) E-mail: hcalte~maya, esimez. ±pn.mx **) E-mail: 1opezbj1@hotmail.com Czechoslovak Journal of Physics, Vol. 52 (2002), No. 7 839