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Is there significant time dilation on the other planets orbiting the sun? Would an 'hour' experienced on earth actually be 59 or 61 minutes on other planets, depending on their distance to the sun? What about the moons orbiting Jupiter?

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It depends on what you mean by "significant." None of the orbits of other planets exhibit relativistic effects at the level you specify. You can estimate the overall size of relativistic corrections by squaring the ratio of v/c, where v is the speed of the orbit. For the Earth, that number is .00000001, and it is not more than about a factor of 3 higher for any other orbit. So it's pretty negligible, but that doesn't mean it has no consequences-- over many orbits, for example, Mercury precesses by a larger amount than can be understood without relativity. Also, if one takes the relativistic perspective, the reason the planets follow the orbits they do is because of that tiny time dilation effect-- that small effect is enough to select the orbits we see over other possible ones. So that's "significant," but not like a 59 minute hour.

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