Section 5

Relativity and the Addition Of Velocities In Classical Mechanics

April 22, 2022

Assume a railway carriage travelling uniformly, as a ‘uniform translation’:

  • uniform: constant velocity and direction
  • translation: although the carriage changes its position relative to the embankment yet it does not rotate

A raven is also flying uniformly in a straight line, as observed from the embankment. If we observe the flying raven from the moving railway carriage, the raven’s motion would be of different velocity and direction. But it would still be uniform and in a straight line.

Thus, if a mass m is moving uniformly in a straight line with respect to a non-moving viewpoint K, then it will also be moving uniformly and in a straight line relative to a second moving viewpoint K', provided that the moving K' has a uniform translatory motion with respect to the non-moving K.

It follows that:

  • If the non-moving K is a Galileian co-ordinate system, then every other moving viewpoint K' is a Galileian one, when, in relation to a non-moving K, it is in a condition of uniform motion of translation.
  • Relative to a moving K' the mechanical laws of Galilei-Newton hold good exactly as they do with respect to a non-moving K

Our tenet is thus: If, relative to a non-moving K, a moving K' is uniformly moving without rotation, then natural phenomena run their course with respect to a moving K' according to exactly the same general laws with respect to non-moving K.

This is my principle of Relativity (in the restricted sense). It is obvious in classical mechanics.

However, the recent developments in electrodynamics and optics showed that classical mechanics is insufficient to describe all natural phenomena.

This where relativity comes in.

Classical mechanics cannot explain all physical phenomena. But it still has a considerable measure of “truth” since it supplies us with the actual motions of the heavenly bodies with detail.

The principle of Relativity must therefore apply with great accuracy in mechanics since a broad, general principle that holds with exactness in one domain of phenomena should also be valid for another.

If the principle of Relativity (in the restricted sense) does not hold, then the Galileian coordinate systems K, K', K'', etc., which are moving uniformly relative to each other, will not be equivalent for the description of natural phenomena.

In this case, we should be constrained to believe that natural laws can be formulated in a particularly simple manner, and of course only on condition that, from amongst all possible Galileian co-ordinate systems, we should have chosen one (K0) of a particular state of motion as our body of reference.

We should then be justified (because of its merits for the description of natural phenomena) in calling this system “absolutely at rest,” and all other Galileian systems K “in motion.”

If our embankment were the system K0 , then our railway carriage would be a system K, relative to which less simple laws would hold than with respect to K0.

This diminished simplicity would be due to the fact that the carriage K would be in motion (i.e. “really”) with respect to K0.

In the general laws of nature which have been formulated with reference to K, the magnitude and direction of the velocity of the carriage would necessarily play a part.

We should expect that the note emitted by an organpipe placed with its axis parallel to the direction of travel would be different from that emitted if the axis of the pipe were placed perpendicular to this direction.

Our earth is like a railway carriage travelling at 30 kilometres per second in an orbit round the sun.

If the principle of Relativity were not valid we should therefore expect that the direction of motion of the earth at any moment would enter into the laws of nature, and also that physical systems in their behaviour would be dependent on the orientation in space with respect to the earth.

For owing to the alteration in direction of the velocity of rotation* of the earth in the course of a year, the earth cannot be at rest relative to the hypothetical system K0 throughout the whole year. However, the most careful observations have never revealed such anisotropic properties in terrestrial physical space, i.e. a physical non-equivalence of different directions.

This is a very powerful argument in favour of the principle of relativity.