By Hamilton A.J.S.
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Additional info for General relativity, black holes, and cosmology, Edition: web draft
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. 6 How Vermilion defines hypersurfaces of simultaneity. She surrounds herself with (green) mirrors all at the same distance. She sends out a light beam, which reflects off the mirrors, and returns to her all at the same moment. She knows that the mirrors are all at the same distance precisely because the light returns to her all at the same moment.
However, the diagram is accurately drawn, and you should be able to check your answers by measuring. 1. 12, label the worldlines of you and your twin. Draw the worldline of a light signal which travels from you on Earth, hits Alpha just when your twin arrives, and immediately returns to Earth. Draw the twin’s “now” when just arriving at Alpha, and the twin’s “now” just departing from Alpha (in the first case the twin is moving toward Alpha, while in the second case the twin is moving back toward Earth).
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 from the mirrors arrives back 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.