Simultaneity in Special Relativity

Most of the apparent paradoxes of special relativity arise because observers moving at different velocities relative to each other have different notions of simultaneity.

How can simultaneity, the notion of events ocurring at the same time at different places, be defined operationally?

One way is illustrated in the diagram. Vermilion surrounds herself with a set of mirrors, equidistant from Vermilion. She sends out a flash of light, which reflects off the mirrors back to Vermilion. How does Vermilion know that the mirrors are all the same distance from her? Because the relected flash returns from the mirrors to Vermilion all at the same instant.

Vermilion asserts that the light flash must have hit all the mirrors simultaneously. Vermilion also asserts that the instant when the light hit the mirrors must have been the instant, as registered by her wristwatch, precisely half way between the moment she emitted the flash and the moment she received it back again. If it takes, say, 2 seconds between flash and receipt, then Vermilion concludes that the mirrors are 1 lightsecond away from her.

This is a spacetime diagram of Vermilion's mirror experiment above.

According to Vermilion, the light hits the mirrors everywhere at the same instant, and the spatial hyperplane (shaded greenish in the diagram) passing through these events is a hypersurface of simultaneity.

More generally, from Vermilion's perspective, each horizontal hyperplane in this spacetime diagram is a hypersurface of simultaneity.

Cerulean defines surfaces of simultaneity using the same operational setup: he encompasses himself with mirrors, arranging them so that a flash of light returns from them to him all at the same instant.

But whereas Cerulean concludes that his mirrors are all equidistant from him and that the light bounces off them all at the same instant, Vermilion thinks otherwise. From Vermilion's point of view, the light bounces off Cerulean's mirrors at different times and moreover at different distances from Cerulean. Only so can the speed of light be constant, as Vermilion sees it, and yet the light return to Cerulean all at the same instant.

Of course from Cerulean's point of view all is fine: he thinks his mirrors are equidistant from him, and that the light bounces off them all at the same instant.

The inevitable conclusion is that Cerulean must measure space and time along axes which are skewed relative to Vermilion's. Events which happen at the same time according to Cerulean happen at different times according to Vermilion; and vice versa. Cerulean's hypersurfaces of simultaneity are not the same as Vermilion's.

From Cerulean's point of view, Cerulean remains always at the centre of the lightcone. Thus for Cerulean, as for Vermilion, the speed of light is constant, the same in all directions.