Comment Re:If you have a better idea ... (Score 2) 264
First, it's an analogy, not a model, so it doesn't need to be mathematically correct, it only has to be conceptually correct. I don't see the problem there. Conceptually, gravity bends the spacetime around the mass, such that objects moving through the distortion appear to an observer to travel along a path that is not straight (it is straight to the object) in space or in time. The rubber sheet is a perfectly good representation of this concept.
If you want it mathematically perfect, you have problems. First, we don't yet know for certain if you should treat a mass as a point source or one with volume. The ternary star system recently found will help there. Second, in order for something to change state, there must be a force. With springs or rubber, this is a restoring force of known value. It not only removes curvature where there is no mass, it prevents the mass stretching the material to infinity and beyond. The nuclear forces play a role in reproducing some of this. The object cannot collapse further than the point where gravity and the nuclear forces all balance out. But as far as I know, the nuclear forces do NOT prevent spacetime bending infinitely, nor remove the distortion when the mass moves elsewhere. This matters. You cannot produce a mathematically-correct simulation with a deformable surface if you don't know the precise rules governing the deformation and restoration.
Let us imagine, though, that we know Hooke's Constant for spacetime. Ok, you get a material (or invent one) with the same constant. Unfortunately, not quite that simple. Relativistic equations are non-linear. You'd need a material where the forces involved reversibly (important!) altered the material in such a way that at any given instant, Hooke's Constant was correct, but that this constant would be purely an instantaneous value.
Ok, that is doable, we've plenty of adaptable materials. Gives you a geometrically correct solution and therefore the right mathematical results. Messy, though.
Is there an alternative?
Well, yes. This is all about reproducing forces. There is absolutely no rule that says you can use only physical shapes to do this. There are plenty of other forces (eg: electromagnetism) which can substitute for one of the others. Have the "fixed" mass as an electromagnet to encapsulate all the details that exist in spacetime that don't readily transpose to a rubber sheet. Let the sheet model gravity alone. That is what it is supposed to do. The other variables are factored in, so the geometry is still correct, only this time by imposing values rather than letting them naturally be correct.
Problem solved. Nobel prize to the usual address, please.