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The notion of "movement" seems to be well understood in physics. In fact, I don't recall any physics text-book defining motion. Special relativity theory says that there is no absolute frame of reference.

Consider a star (such as the Sun). We say that the Sun is moving at so and so speed with respect to the Milky Way, which is moving at so and so speed with respect to the Andromeda Galaxy, and so on. However, this implies that motion only makes sense if there is a background frame of reference.

In a universe, where there is only one star, what would it mean for the star to rotate around its axis? How could we define this axis? And relative to what would it rotate?

If we can say that the star rotates, then there is some absolute frame of reference. In special relativity, the star would stop rotating if it was isolated from the rest of the universe. This seems to indicate that Mach's conjecture is true, and the existence of astronomical objects far away defines our notion of motion.

Is there any formal definition of motion, especially rotation?

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Even without external references, the presence of 'fictitious' forces, such as the coriolis force, would let you know that your planet is rotating. Or you could launch a rocket straight up until its speed slowed to nothing, and then falls back. If it falls back down to the launch pad, you have no rotation. Otherwise, it will fall back somewhere else, or even enter an elliptical orbit.

Mach's conjecture that the distant universe forms a frame of reference for zero rotation is difficult to test since it could be rotating very slowly at a rate we could never detect by experiment. But we know from the Big Bang theory that all those distant parts of the universe were once adjacent and connected to our part of the universe and frictional forces would have damped out any net rotation between our part of the universe and the now distant parts. So I would consider Mach's conjecture a manifestation of the Big Bang.

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However if we can test the planet to be rotating, it would automatically imply an external frame of reference, which can be considered absolute. Say the radius is r and angular velocity w. That means, at any instant, a point on the equator is moving at w*r with respect to something. –  Jus12 Jul 29 '11 at 23:32
    
@ Pete Jackson: could you please answer the question in my above comment? –  Jus12 Aug 26 '11 at 13:27
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