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The Michelson-Morley experiment demonstrated that the speed of light is the same in all frames of reference. Special relativity is the correct theory that incorporates this curious fact of nature.
Best beginner Special Relativity Textbooks:
The standard textbook is Spacetime Physics by Taylor, Wheeler
Recommended Further Reading:
In the standard twin “paradox” scenario, with an instantaneous turnaround (which requires an infinite acceleration by the traveling twin that lasts only an infinitesimal time), what makes it SEEM paradoxical is that the traveling twin (“he”) is inertial during what SEEMS like essentially his entire trip … inertial all except the single instant of the turnaround. And SURELY the home twin (“she”) couldn't possibly age any during that single instant in the traveling twin's life, COULD she? While the traveling twin is inertial, he in entitled to use the famous time dilation result, which tells him that the home twin is ageing more slowly than he is. So it SEEMS paradoxical that, when the twins are reunited, he finds that it is the home twin who is the older.
The resolution of the “paradox” is that, according to the traveling twin, the home twin ages by a very large amount during that one instant in his life at his turnaround. The implicit assumption that nothing could happen to the home twin's age during that single instant in the traveling twin's life at the turnaround was wrong.
The above resolution of the twin “paradox” is most often obtained by an analysis usually referred to as the “co-moving inertial frames” analysis. An easier, quicker, and less error-prone method (called the “CADO” method) gets the identical answer that the co-moving inertial frames method gives. The CADO method is different in that it uses novel terminology which is less error-prone, and also in that it uses an especially simple equation called “the CADO equation” (which is derived from the co-moving inertial frames analysis applied to a Minkowski diagram).
The CADO method and the CADO equation are described here:
“Accelerated Observers in Special Relativity”, PHYSICS ESSAYS, December 1999, p629.