On the differential geometry of time-like curves in Minkowski spacetime

We establish the Serret-Frenet equations in Minkowski spacetime and use them to give a simple proof of the fundamental theorem of curves in Minkowski spacetime. We also derive two theorems that represent Minkowskian versions of a well-known theorem of the differential geometry of curves in tridimensional Euclidean space. We discuss the general solution for torsionless paths in Minkowki spacetime. We then apply the 'four-dimensional Serret-Frenet equations to describe the motion of a charged test particle in a constant and uniform electromagnetic field. and show how the curvature and the torsions of the four-dimensional path of the particle contain information on the electromagnetic field acting on the particle. (c) 2006 American Association of Physics Teachers.