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If we start from the assumption that all frames of reference are valid for describing motion, how can a foucault pendulum either prove or disprove that the earth rotates or is stationary?

Couldn't it just as easily prove that the earth is stationary and that the whole universe (including the earth's atmosphere) rotates? what would be the difference in the observed Foucault pendulum? How can the observation prove one explanation over the other in light of my assumption? Is the rotating Earth model preferred for the sake of simplicity?

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  • $\begingroup$ Would this mean there was no conservation of angular momentum and everything had a preferred spin? $\endgroup$ – DWin Jan 3 '14 at 7:27
  • $\begingroup$ This is basically another way of asking about Mach's principle, and in particular the bucket experiment (search the article I've linked for "bucket"). The question has no answer. $\endgroup$ – John Rennie Jan 3 '14 at 8:19
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Yes, the "rotating Earth" model is preferred for the sake of simplicity. In fact, only in such "inertial systems", the laws of motion have the simple form in which the acceleration is determined by the force calculated via the inverse square law, and so on. All other inertial frames - moving by a uniform velocity and constant direction relatively to the first ones - offer the same simple laws.

The other frames, accelerating or rotating ones relatively to an inertial frame, don't admit a simple description like that. More precisely, their formulae for accelerations contain additional terms, the so-called fictitious forces – the centrifugal and Coriolis' force. These forces may be derived by transforming the simpler laws from the inertial systems to the non-inertial ones.

So the rotating frames – e.g. one in which the Earth is not spinning – are less good. Newton would recommend not to use them at all. Ernst Mach suggested that they are equally good and the fictitious forces result from some relative behavior with respect to distant stars. This view, Mach's principle, motivated Einstein to look for his new theory of gravity.

However, the final product, general relativity, didn't confirm Mach's principle in the original form. One may describe everything in terms of rotating frames – e.g. one in which the Earth is not spinning – but one must also include the corresponding modification of the gravitational fields which guarantee that the fictitious forces will be present.

As a result, one may still show that the spacetime is nearly flat around the Earth and this fact becomes manifest in the inertial (not spinning) frames, e.g. one in which the Earth is spinning by the usual rate. Ernst Mach would deny that the spacetime may carry any information about its being in a spinning state or not. However, according to general relativity, even empty spacetimes do carry this information about the geometry. And the geometry is only (nearly) flat in some (non-rotating) frames. The other frames may be used as well but the laws of motion in these frames will contain additional terms – the fictitious forces – which result from a "not explicitly flat" form of the metric i.e. spacetime geometry.

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  • $\begingroup$ Hi Lubos, thank you for answering my question! Where in the universe would an observer need to be located in order to have an inertial frame of reference, if not on the Earth? $\endgroup$ – user36677 Jan 4 '14 at 20:13
  • $\begingroup$ Hi, any frame that is just shifted by a constant distance away from an inertial frame is still inertial. The in. frames don't care about the absolute location. They don't even care about the absolute constant velocity. They only prohibit acceleration or spinning. - Otherwise there is nothing special about the position of the Earth in the Universe, a fact we have known at least from Copernicus, haven't we? $\endgroup$ – Luboš Motl Jan 5 '14 at 9:19

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