The Andromeda Paradox

Rietdijk, Putnam, and Penrose independently proposed a thought-experiment to explore the concept of simultaneity in special relativity. The Andromeda galaxy is approximately 2.5 million light years from Earth (≈2.5×m) . Assume, for simplicity, that the galaxy and the Earth remain momentarily at this fixed distance, with no relative motion. Suppose an Earthling is slowly strolling in the direction of Andromeda. Then events on Andromeda that occur, in concept, simultaneously in the Earthling's frame of reference, depend rather sensitively on his or her walking speed. Roughly, an increase in walking speed of one foot per second corresponds to a simultaneous event occurring on Andromeda about an Earth day later! This can be deduced from the Lorentz-transformation equation , where is the time advance on Andromeda, which can be considered simultaneous with an event on Earth occurring at , most conveniently set equal to 0. Note that the observer on Earth can only "infer" what is happening simultaneously on Andromeda. He would not actually "see" what is occurring until 2.5 million years later.
Roger Penrose called this argument the "Andromeda Paradox". To a stationary observer on Earth, the Supreme Galactic Council on Andromeda might be engaged in a debate on whether to attack Earth, whereas to an observer strolling at a leisurely pace of 2 feet per second, the Intergalactic Battle Fleet has already been launched toward Earth. Needless to say, this paradox has stimulated a great deal of controversy, even leading some to doubt the metaphysical underpinnings of the theory of relativity.

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For more information, see the Wikipedia entry for Rietdijk-Putnam argument.
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