Can the Theory of Time Lapse proposed by Einstein proved and under what conditions?

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I assume you mean the idea of time dilation as proposed in his second theory of relativity where time slows down as an object's speed increases. This has been proved in several ways. Here are just three: Certain types of sub-atomic particles called mesons have a particular lifespan that is known and can be measured accurately in particle accelerators. Some mesons are produced in the upper atmosphere by particles there being bombarded by particles and waves found, for example, in the Solar Wind. Physicists know where these mesons are formed and they know the speed at which they travel. These mesons can be detected on the earth's surface but the point is that they shouldn't be able to be detected - they should have decayed long before. However, when the time dilation effect is appplied to them then the figures agree remarkably well - the mesons survive because for them, travelling at high velocity, time slows down and their lifespan increases to a stationary observer measuring them. Secondly, within particle accelerators, as certain particles are accelerated towards the speed of light, not only do their lifespans increase in exactly the amount predicted by Einstein, but also their mass increases in pproportion - again exactly in accordance with Einstein's predictions. Thirdly, during one of the Apollo missions an atomic clock was taken along in the spacecraft which was exactly synchronised with a second clock on earth. Atomic clocks are the most accurate timepieces in the world - accurate to one second in several thousand years. When the ship returned, the clocks were compared, and the clock on the ship was found to have slowed by a tiny fraction of a second. A tiny period of time indeed, but when investigated and put into Einstein's fomulae for time dilation, it was found that the slowing down agreed exactly with Einstein's prediction.
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