The energy in rotation

A straightforward mechanics problem: a mass whirling round a central point in a frictionless environment, set in motion and restrained by a string providing centripetal force to counter the tendency (as Newton's 2nd law) for the mass to fly off at a tangent in a straight line.  So the string is imparting a force to the mass, accelerating across the diameter from 0 to V and back to zero, twice each revolution.  The mass is moving, so we have a force and a distance moved, so work is being done and energy is being expended.  Yet we know that once rotating, it will continue to do so indefinitely with no further energy after the start.

Stay with the diameter, and imagine instead the mass not orbiting the centre point, but being pulled to the centre with an impulse, travelling through the centre to the other side and being restrained and re-accelerated inwards again to the other side.  In this case work is being done, as energy lost in the sudden braking and pulling is replenished.  But if the string and pulling effort is replaced by an ideal spring, the system exhibits simple harmonic motion (SHM) and the energy is exchanged periodically with no loss, between the accelerating mass and the expanding spring.

An orbiting mass is just doing SHM in two orthogonal axes, so that the spring is being replaced by the motion in the second axis, there is a constant exchange of energy between the x and y axes, and no new energy is needed to sustain the orbit.  An orbit (this would extend to any ellipse) is SHM where the energy storage is provided not by a physical storage entity, but just by virtue of the mass moving also in the second dimension.

There is an interesting limiting case.  For increasing ellipticity the finite length of the minor axis allows the energy storage to sustain the high-amplitude oscillation of the major axis.  But as the minor axis length tends to zero this storage disappears, and SHM ceases.  The practical limit is the strength of the centripetal force.  In orbital mechanics the natural tendency is for ellipticity to reduce, ie the orbit to become more circular.

So gravity provides the centripetal force to enable bodies to orbit each other.  What provides the centripetal force to enable the rotating engine of gravity?  Whatever it is, it is fundamental.

The properties that rotation shares with gravity seem to be:
- once the system is started, masses are constantly accelerated yet with no energy input required
- unlike SHM (eg with a mass and a spring), for rotation there is a constant force, ie always positive in the direction towards the centre
- rotation involves angular momentum, which is (a) a conserved quantity and (b) has an additional dimension, the axis of rotation, at right angles to the plane of the rotating mass.  It is clear that gravity acts in not two but three dimensions, so the question of the 4th spatial dimension in which its "axis" exists is begged, since it can't be found in the first three spatial dimensions, and time is common to both rotation and gravity.

It is difficult to visualise a 4th spatial dimension and this axis (which is still linear, but in 2-d!)  I think back to earlier posts, postulating a circular axis (to 4d+t toroidal space-time) to avoid it having ends, to avoid needing further dimensions just for the purpose of accommodating it, and to enable the toroidal volume (4d+t equivalent to a 3d+t toroidal surface).  Amongst the properties of a toroid are:
- it has both positve and negative curvature, and can expand in either or both its two radii r and R
- potentially it has net zero angular momentum of rotation "through the hole"
- its shape suggests a time before the hole was created and a time since - the hole creation could be part of the process of inflation, one of the ongoing mysteries.

Yes it seems odd if the universe is just a gigantic smoke ring - yet surprisingly, I have not found anything in study over the last two years to put me off the idea.  And scope is still there to help explain dark energy (the net angular momentum), dark matter (an artifact of GR or 4d+t curvature on the largest scale), and potential shortcut paths for speeding neutrinos.