Introduction To Special Relativity - Einstein's Postulate: The Constancy of The Speed of Light

Einstein's Postulate: The Constancy of The Speed of Light

Einstein's postulate that the speed of light is a constant comes out as a natural consequence of the Minkowski formulation.

Proposition 1:

When an object is travelling at c in a certain reference frame, the spacetime interval is zero.

Proof:

The spacetime interval between the origin-event (0,0,0,0) and an event (x,y,z,t) is

The distance travelled by an object moving at velocity v for t seconds is:

giving

Since the velocity v equals c we have

Hence the spacetime interval between the events of departure and arrival is given by

Proposition 2:

An object travelling at c in one reference frame is travelling at c in all reference frames.

Proof:

Let the object move with velocity v when observed from a different reference frame. A change in reference frame corresponds to a rotation in M. Since the spacetime interval must be conserved under rotation, the spacetime interval must be the same in all reference frames. In proposition 1 we showed it to be zero in one reference frame, hence it must be zero in all other reference frames. We get that

which implies

The paths of light rays have a zero spacetime interval, and hence all observers will obtain the same value for the speed of light. Therefore, when assuming that the universe has four dimensions that are related by Minkowski's formula, the speed of light appears as a constant, and does not need to be assumed (postulated) to be constant as in Einstein's original approach to special relativity.