Tensor Analysis - Lecture 10

In this lecture I focus on the relationship between force and acceleration in the Special Theory of Relativity. In the process I discuss the distinction between time-like, space-like and light-like four vectors. We also meet a rather important result - that a non-zero four vector orthogonal to a time-like vector has to be space-like. The force is the rate of change of momentum as in pre-relativity physics, but the more complicated relationship between momentum and velocity here, leads to a more complicated version than force equals mass times acceleration. I show that the force - acceleration relation simplifies in two special cases - the transverse one, where the velocity of the particle is perpendicular to the acceleration; and the longitudinal one - where velocity and acceleration are parallel. I show that the force- acceleration connection is one which involves a "mass tensor", and also show how to invert this relation to obtain acceleration directly from the mass. I end the lecture with an example - that of a relativistic projectile. I examine how this problem differs from the standard non-relativistic projectile motion.