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I thought that I understood the centrifugal force earlier, but I can't seem to grasp how it interacts when considering that everything is relative?

Let's imagine that you are the only one in the entire universe, and that you are spinning with high angular velocity, with the rotation axis pointing in the same direction that your eyes are directed. Surely you would feel the centrifugal force pulling your feet and head apart, wouldn't you?

A problem with this, though, is the following: Since you are the only object in the universe, there's no way to tell if you're rotating. Your angular velocity isn't even defined, since you aren't rotating in relation to anything else.

How, then, can one know how what the centrifugal force is? Is it defined in relation to all the other mass in the universe, in such a way that it's negligible in classical mechanics problems?

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The answer is that not everything is relative. Indeed, you've just shown that it is possible to detect rotation in an absolute sense.

More generally, the principle of relativity says that all inertial frames are equivalent. In other words, it is impossible to detect (or even to define) absolute motion at a uniform velocity; it doesn't make sense to say that you're moving in a straight line at constant speed if you don't tell me with respect to what.

But this doesn't apply when there are accelerations involved. If your velocity is not constant (so you're either changing your speed or rotating, or both), then it's perfectly possible to measure your acceleration, and centrifugal force is an example of this. For example, with the use of a Foucault pendulum it's possible to see that the Earth rotates without looking at the sky. If there were no stars or planets, and the sky was just a black background, we could still use Foucault's pendulum to measure the rotation of the Earth.

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    $\begingroup$ Cool, never thought of it this way. Does this apply equally to small particles and their spin? $\endgroup$
    – user52657
    Commented Jul 2, 2014 at 20:17
  • $\begingroup$ @user52657: Spin is a weird thing. It acts like angular momentum, but (without getting into the complications of QFT) elementary particles don't have a size. They're just points. What this means is that they can't rotate around an axis, and so you need to be very careful when thinking of spin as a regular angular momentum. For the purposes of this question, you should just consider spin as a property like mass or charge, and not think about it as an actual rotation of the particle. $\endgroup$
    – Javier
    Commented Jul 2, 2014 at 20:21
  • $\begingroup$ As a curiosity, in his introduction to special relativity in the Lectures, Feynman discusses exactly this kind of interpretation that “philosophers” have of relativity, that “everything is relative”. It's in one of the three chapters related and you may find interesting to read about it. $\endgroup$
    – pppqqq
    Commented Jul 2, 2014 at 20:32
  • $\begingroup$ I would expect a little bit of discussion around the fact that for certain space-times "the distribution of matter and field energy-momentum (and possibly other information) at a particular moment in the universe determines the inertial frame at each point in the universe"(link again). This means rotation is not always absolute in relativity. $\endgroup$
    – Void
    Commented Jul 3, 2014 at 15:52
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Until general relativity is included, all that Javier says applies.

However, once we apply general relativity and large amounts of moving mass, we find an effect called "frame dragging", which basically means the definition of inertia gets dragged along moving mass. That is, if you are near a massive body, it will seem you are moving inertially if you copy the movement of the body.

This effect was discovered already by Einstein in a very simple example. Einstein considered a massive spherical shell rotating about an axis in a universe with fixed stars on the background. Inside the shell, you would find a state where you don't feel any kind of centrifugal force and a Foucalt pendulum shows no rotation. Then the stars would seem to slowly rotate in comparison to you! However once far outside the shell, the stars would seem again fixed.

The discussion whether this means inertia is really defined by all masses in the universe is lengthy and technical - the statement can be completely proven only for special "universes". However, you can be sure that some situations induced by relativity in a certain sense violate the meaning of absolute rotation and relate it to moving masses.

What about your "empty universe" question? I believe that most theoretical physicists would tell you we actually just don't know what would happen with the meaning of inertia in an empty universe. It might just be that it is in that case defined by you yourself and you never feel the centrifugal force or that there is a meaning of inertia established just by "an empty universe" coming into existence. But for sure, in all everyday situations an absolute meaning of inertia (and rotating) does apply.

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