Tensor Analysis - Lecture 09

In this rather longish lecture, I have started by recapitulating the basic properties of Lorentz transformation and 4-vectors that he had discussed earlier in the series. If you are interested in applications of 4-vectors to the Special Theory of Relativity, you may skip ahead and start watching from around 28:45. In the rest of the lecture I describe some physically important 4-vectors, like the 4-velocity and 4-momentum. Using the fact that 4-vectors transform exactly like space-time coordinates I go on to derive transformation laws for velocity and Energy-momentum. I also discuss why relativity demands a re-definition of the Newtonian concept of momentum, and how Lorentz covariance of the conservation of momentum leads to the conservation of energy. I then go on to to define the 4-momentum for the photon, and use its transformation to derive the relativistic formula for the Doppler effect of light. I then re-derive this result using the fact that the inner product of 4-vectors is a scalar, so that it can be evaluated in any inertial frame without changing its value. I then look at a different kind of application of this scalar product of 4-vectors - where we use the fact that the "square" of a 4-momentum is simply related to the particle's mass. Using this, I describe a very simple derivation of the Compton effect.