The Paradox of Special Relativity

The postulate that the speed of light is the same in any inertial frame leads immediately to a paradox, described below.

Vermilion emits a flash of light.

Vermilion thinks that the light moves outward at the same speed in all directions. So Vermilion thinks that she is at the centre of the expanding sphere of light.

Again, Vermilion emits a flash of light. But now here also is Cerulean, moving away from Vermilion (at about half the speed of light, apparently).

Vermilion thinks she is at the centre of the expanding sphere of light, as before.

But, says special relativity, Cerulean also thinks that the light moves outward at the same speed in all directions from him. So Cerulean should be at the centre of the expanding light sphere too. But he's not, is he. Paradox!

Can you figure out Einstein's solution to the paradox? Somehow you have to arrange that both Vermilion and Cerulean regard themselves as being in the centre of the expanding sphere of light.

The spacetime diagram below is not a solution to the paradox, but it suggests a way of thinking which leads to the solution.

This is a spacetime diagram of Vermilion emitting a flash of light.

In a spacetime diagram, time goes upward, while space dimensions are horizontal. Really there should be 3 space dimensions, but I've drawn 2 for simplicity.

In a spacetime diagram, the units of space and time are chosen so that light goes one unit of distance in one unit of time, i.e. the units are such that the speed of light is one, \(c = 1\). Thus light moves upward and outward at 45° from vertical in the spacetime diagram.

The lines along which Vermilion and Cerulean move through spacetime are called their worldlines. Each point in 4-dimensional spacetime is called an event. Light signals converging to or expanding from an event follow a 3-dimensional hypersurface called the lightcone. In the diagram, the sphere of light expanding from the emission event is following the future lightcone. There is also a past lightcone, not shown here.

Consider again the challenge problem. The problem is to arrange that both Vermilion and Cerulean are at the centre of the lightcone, from their own points of view.

Here's a clue. Cerulean's concept of space and time may not be the same as Vermilion's.