The Emperor's New Mind: Concerning Computers, Minds, and the Laws of Physics (Oxford Landmark Science)

by Roger Penrose

In particular, the reader may be acquainted with what is called ‘the twin paradox’ of relativity: one twin brother remains on the earth, while the other makes a journey to a nearby star, travelling there and back at great speed, approaching that of light. Upon his return, it is found that the twins have aged differently, the traveller finding himself still youthful, while his stay-at-home brother is an old man. This is easily described in terms of Minkowski’s geometry – and one sees why, though a puzzling phenomenon, it is not actually a paradox. The world-line AC, represents the twin who stays at home while the traveller has a world-line composed of two segments AB and BC, these representing the outward and inward stages of the journey (see Fig. 5.19). The stay-at-home twin experiences a time measured by the Minkowski distance AC, while the traveller experiences a time given by the sum16 of the two Minkowski distances AB and BC. These times are not equal, but we find: Fig. 5.19. The so-called ‘twin paradox’ of special relativity is understood in terms of a Minkowskian triangle inequality. (The Euclidean case is also given, for comparison.) AC > AB + BC, showing that indeed the time experienced by the stay-at-home is greater than that of the traveller. The above inequality looks rather similar to the well-known triangle inequality of ordinary Euclidean geometry namely (A, B, and C being now three points in Euclidean space): AC < AB + BC, which asserts that the sum of two s...

This is mind blowing