The rotating analogy

Big gap while studying a serious cosmology module, and still (near the end of it) trying to understand how the surface of a solid object, even a simple one like a sphere, can be described in only two dimensions.  It just means the third dimension is implied in the frame of reference used, and it still (one way or another) needs coordinates in three orthogonal senses to define any point on it.  Tensors are all smoke and mirrors, a bit worrying just a couple of weeks from the exam!

But I come back to the rotation analogy.  The hammer-thrower, once he has got his hammer rotating, exerts an original force on the handle.  It does not come and hit him in the face (as it would if not rotating) because the moving mass has to continue moving.  Its inertia in the tangential direction makes it produce an equal and opposite force along the wire, putting the system in equilibrium.  This is an example (are there any others?) of how a force can be exerted on a free body, causing a constant acceleration, yet with no movement in the rotating frame of reference.

For a ball sitting on the ground there is an original force pushing it upwards, but it does not shoot up into the air, why?  Like the hammer, the ball is in equilibrium, and its inertia in some tangential direction makes it produce an equal and opposite force pushing it downwards.

A similar ball falling to Earth is not being 'pulled by the force of gravity', it is just sitting in spacetime with no forces acting on it, though it is accelerating.  The direction of acceleration is towards the Earth's centre, but this is not the tangential inertial direction as it is local to this system, and opposite to a similar experiment in Australia.

Setting the hammer system rotating gives it potential energy, equivalent to gravitational potential energy.  The difference in the analogies is that the forces on the hammer act in only two dimensions around an axis in the third.  It rotates with a certain period, so time is already involved.  For the ball on Earth neither the centre of rotation nor the axis can be readily identified.  But what other mechanism involves a force and an acceleration with no motion in the system frame of reference, and no energy expended?