by wendy » Tue Nov 22, 2005 6:58 am
The thing with "stable orbits" is that there must be a whole class of these, that one can change from one to another. In three dimensions, this is provided by elliptical orbits.
Supposing that gravity is a radiant flux, leads to the proposition that for any sphere, (surface area) * (specific force intensity) = (mass) * (constant) = (flux).
One then finds (surface area) = function(radius), and it is this that dictates the nature of radiant fields, not only in the higher dimensions, but also non-euclidean geometries.
Newton's mechanics are built on Euclidean physics by adding the notions of mass and time. One can add these notions to Hyperbolic geometry, or to 4D geometry, and let the underlying geometry do the talking.
Even the radiant field has its part to play. That the sky is black, is a sign that the heavens is not uniformly peppered with stars.
The only instances where a radiant force leads to set of stable orbits is in N=1, and N=3.
To create a kind of universe where stable orbits exist, then one has to suppose some kind of other element keeps the planets on track. This is not entirely unknown in our world, since the atom would collapse except for quantisation.
It could be that gravity is somehow quantised at a large scale, and that what we're seeing is GM.Gm = Nk, for some very large k, and that the planet is drawn to a stable orbit N. For people and rocks and so forth, we have N=0, so they're stuck on the surface.
I don't know, but even modern physics abounds with things that are not to our nature defined. For example, quarks have a force independent of distance, as like a string-tension. The Bohr atom is quantised, again unexpectedly.
There are of course other things that could conspire to make stable orbits in four dimensions: such as the swirl of clifford-rotation. (note, for example, that the orbit of planets follow the rotation of the sun).
In any case, my premise is to make all of this stuff a SEP (someone else's problem), and deal instead in trying to find what parameters are needed to be solved for. For example, the nature of 4d rotation was made by simply setting up a model and watching the night sky. In this sense, it is more real than supposing a pole etc as we have in 3d.
Even for relativity, Einstein did not have a clean slate to work with. The experimental evidence was already present, just waiting for an explination. Einstein just explained it, with formulae.
What i am doing is trying to find out the most earthlike kind of four-dimensions, and leave it to others, or point in the direction where i think an answer ought lie.
W