Postulates of special relativity

Remarks

Add the homogeneity of space and time and the isotropy of space. Then the two postulates force each change of coordinates between inertial frames to be a Lorentz transformation followed by a translation. The idea is that the map must be affine and must send each light cone $\set{x : -c^2 t^2 + \abs{\mathbf{x}}^2 = 0}$ to a light cone. Without the second postulate, the same symmetry arguments still leave the Galilean transformations as a possibility, which is the case of an infinite limiting speed. Time dilation, length contraction and the relativity of simultaneity are consequences of the two postulates.

Going deeper

For a student

The postulates are the physical input; the mathematical output is Minkowski space with its Lorentz group of symmetries. In a course, one first derives the Lorentz transformation from the postulates, then reads off time dilation, length contraction and the velocity-addition formula from it. Maxwell's equations of electromagnetism satisfy the first postulate exactly when frames are related by Lorentz transformations, which was a main historical motivation.