We know GR turns gravitation induced acceleration into curvature in spacetime. But in GR, what happens to the acceleration due to the centrifugal force? And similarly, the acceleration due Euler or Coriolis force?
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$\begingroup$ You should check how it works in classical mechanics eg. Lagrangian point of view. I suppose even, that studying GR without such background is completely counterproductive $\endgroup$– kakazCommented Apr 19, 2020 at 14:45
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3 Answers
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The law of inertia can be stated by saying that the covariant derivative of the velocity vector is zero for a free particle. The covariant derivative with respect to a certain coordinate differs from the partial derivative by a term involving a Christoffel symbol. In a rotating coordinate system, these terms can be interpreted as describing the centrifugal and Coriolis forces. No curvature is involved, so this is not really GR.
The answer by Constantin is wrong. His example of the bike inside the cylinder presupposes a centripetal force, but we can observe the centrifugal and Coriolis forces regardless of whether there is a centripetal force. The centrifugal and Coriolis forces exist in any rotating coordinate system.
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$\begingroup$ Hi user261328, thank you for your first post, I completely agree! If you want to improve your answer further, it would help to increase readability e.g. by inserting equations instead of just text ("covariant derivative of the velocity vector is zero" etc.). $\endgroup$– zonksoftCommented Apr 19, 2020 at 15:51
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$\begingroup$ @user261328, In a heuristic sense I see your point. I am surprised though with all possible coordinate transformations available in GR, why GR is still sensitive to the difference of the inertia frame vs a particular kind of non-inertial rotation frame. $\endgroup$ Commented Apr 19, 2020 at 16:07
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$\begingroup$ Speaking as a GR-outsider: imho it is a matter of choice to describe the experimental physical phenomena, i.e. the laws were chosen to be constructed that way. $\endgroup$– zonksoftCommented Apr 19, 2020 at 17:01
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If the movement is due to a gravitational phenomenon then the particle will "fall" along a geodesic trajectory and the study of this trajectory will describe the movement and will take care of what we experience as acceleration and therefore forces on the object. You would need to use the geodesics to calculate how we can experience that as a centripetal force or any other (it GR is more complicated and there are other phenomena such as frame dragging problems). For Example, you can check the calculation for a "free" moving object in a Schwarzschild space-time, along the geodesics, and recover the planetary or any other conic trajectory and even profound non-Newtonian corrections. If you want to consider Coriolis you would need to use a more complicated metric. In this free article you find discussions which might help with this:
http://articles.adsabs.harvard.edu//full/1975AcC.....3...97H/0000101.000.html
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In GR, objects follow the geodesic(allowed path induced by energy/matter) only in free fall. In the centrifugal force case(say biker circling in a big cylinder) objects follow the restricted path. And this is not a free fall.
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$\begingroup$ So you are saying the physics of centrifugal phenomenon does not apply in GR? How then can we consider GR being a more useful physics when the Newtonian machines can treat centrifugal phenomenon but GR can't? $\endgroup$ Commented Apr 19, 2020 at 14:53
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$\begingroup$ Also, with all possible coordinate transformations available in GR, how it is possible that GR can still miss centrifugal way of motion? $\endgroup$ Commented Apr 19, 2020 at 15:04