classical dynamics

Four-vectors and conservation laws in relativity

This lecture was delivered to the final year honors class of 3 year science degree students on 21 November 2017 as part of the Classical Dynamics paper.

In this lecture we will discuss some of the important tools of relativistic mechanics. We will discuss the idea of proper-time, 4-velocity, 4-acceleration, 4-momentum, 4-force and related conservation law of the 4-momentum.

A. Proper-time. 
The proper time is the time interval in the rest-frame of any event. The proper time is related to time-interval in other inertial frame by: tau = (1/gamma)t where gamma  > 1 always.

Gamma is the Lorentz factor or Lorentz boost factor directly related to the speed of an object in speed-of-light units, i.e. beta.

gamma = 1/sqrt{1-v^2/c^2}

Hence proper-time is the smallest possible time interval for an object in motion in among all possible inertial frames of reference and it occurs in the rest frame.

d(tau) < dt

Proper-time is necessary to define other basic quantities in theory of relativity if we are to preserve their basic meaning in terms of the non-relativistic mechanics definitions.

B. Four velocity. 
Four velocity of a particle is the rate of change of 4-displacement …

So, …  is the position vector — or space-time interval in the Minkowski  space — akin to the difference of two 3-dimensional vector in coordinate space, this time with 4 coordinates rather than 3.

The proper-time interval d(tau) is a Lorentz invariant i.e. when we move between arbitrary inertial frames of references given by the Lorentz factor beta or  gamma this interval retains its value — because it retains its form. Any variable which would retain its form under such transformation are said to be Lorentz invariant quantities.

Relativistic Doppler effect

Relativistic Doppler effect. 

There is an apparent shift in the observed frequency of any electromagnetic wave (light) when there is any relative motion between the source of light and the observer. This can be easily determined by using the 4-vector formulation of theory of relativity.

Lets discuss the details of this phenomena under two situations.

A. Source is at rest and observer is in motion. 
Lets us consider two inertial frames S and S’. S’ is moving wrt S, along the x-axis with speed v = (beta) c where the observer is at rest in S’ frame but the source is at rest in the  S frame.