V. Dannon showed that spherical curves in E4 can be given by Frenet-like equations, and he then
gave an integral characterization for spherical curves in E4 . In this paper, Lorentzian spherical timelike and spacelike curves in the space time R41 are shown to be given by Frenet-like equations of timelike and spacelike curves in the Euclidean space E3 and the Minkowski 3-space R31. Thus, finding an integral characterization for a Lorentzian spherical R41 -timelike and spacelike curve is identical to finding it for E3 curves and R31 -timelike and spacelike curves. In the case of E3 curves, the integral characterization coincides with Dannon’s.
Let {T, N, B} be the moving Frenet frame along the curve a(s) in the Minkowski space R31. Let a(s) be a unit speed C4 -timelike (or spacelike) curve in R31 so that a'(s)=T. Then, a(s) is a Frenet curve with curvature k
s) and torsion t(s) if and only if there are constant vectors a and b so that (i)T’(s)= k(s){a cos x(s)+b sin x(s)+ òs0 cos[x(s)- x(d)] T(d)k(d)dd}, T is timelike, (ii) T’(s)= k{aex+bex òs0 cosh (x(s)- x(s)- T(d)k(d)dd}, n is timelike, where x(s)+ òs0t(d)dd.