General Relativity.

Quick Note of Agreement:If you checkout "Gravitation" by John A. Wheeler, et. al., you will find that over 300 different gravitational theories have been proposed as alternatives to General Relativity, and all but one has been eliminated by various experiments or observations. And that was back in the 1980s.There is a great deal of support for General Relativity.

I could go into this in a bit more depth later (going to catch the bus, shortly), but basically what you are dealing with here are two different languages for describing the same phenomena. For example, Newton's gravitational theory could be expressed without reference to "gravitational forces" if it were expressed in the language of curved spacetime. If one were to do this, the geometry of space itself would be Euclidean, but the geometry of spacetime would be curved with the curvature existing between the dimension of time and the dimensions of space. In either case, the phenomena which is described will be the same phenomena, but whether your are expressing it in terms of curved geometry or gravitational forces in a flat geometry may differ -- and this constitutes the form in which the theory is expressed.However, when one expresses Newton's gravitational theory in terms of curved spacetime, the mathematics becomes a great deal more complicated. Expressing the theory in terms of no curvature with gravitational forces keeps the mathematics simple. However, in the case of General Relativity, the language of curved spacetime without gravitational forces is what keep the mathematics simple -- relatively speaking.

Well, simply put, the geometry of space would still be Euclidean in much the same way as a Friedman model with critical mass density. However, there would exist curvature between the temporal dimension and the three spatial dimensions, such that the paths which maximize proper time through spacetime (given the metric tensor) would be the same as the paths of objects in free fall -- the same way as in General Relativity. Although I haven't actually seen the mathematics behind such an approach (I have only heard it briefly described in a textbook on General Relativity), I would assume that you could employ standard cartesian or polar coordinates to space with a metric tensor appropriate to a flat space, but the non-diagonal components where either the left or the right index corresponded to time would be non-zero.As for a "bundle structure" of "hypersurface fibers" on the "time base-space" having a "non-trivial connection," I am not exactly sure what you mean by this. I have heard of "fiber-bundle theory," for example, but I haven't ever heard it in any way applied to a gravitational theory -- unless of course you mean Roger Penrose's "twistor theory" which I suppose could be described as something along these lines.

Unfortunately, I doubt this really brings him any further along. As long as he is thinking of gravity having to be a force, he isn't thinking in the language of curved spacetime. There isn't any force which is curving spacetime -- the curvature of spacetime is a different language in which one expresses one's gravitational theory, as an alternative to a language of gravitational "fields" and "forces." Well, not exactly. If you have a static solution to Einstein's field equations (i.e., the ten equations represented by G=8(pi)T), then there are no gravitational waves. Without gravitational waves, there is no gravitational energy being transmitted. No gravitational energy means no addition to the stress-energy tensor T. Honestly, analogies might be nice, but beyond a certain point, I would recommend picking up the math. General Relativity is a little tough as it is expressed in the language of pseudo-Riemannian geometry. But if one can pick up differentiation, partial differentials, and matrice mathematics, then one can pick up the tensor mathematics of General Relativity (although I am not sure how many people are able to follow through Einstein's derivation of his "field" equations). However, Special Relativity (essentially what corresponds to classical mechanics without gravitation) can be understood without the aid of anything more than algebra and matrices.

Beautiful!Another physics geek like myself. In my case, I taught myself calculus a little early just so that I could follow through the derivation of the Schwartzchild solution. Didn't get much further than that, though, then turned my attention to Quantum Mechanics, but it has been a long time for me as well. In college I ended up into philosophy, and now I am a computer programmer. Currently getting into evolution as much as I can in my offtime, though.Anyway, I will have to check out the geometric approaches. Not sure how far I will get, or when I will be able to get to them, but they do sound interesting.

The stress-energy (or in the case of the Schwartzchild solution, the mass) is the source of the curvature. Just as the electric charge is not the same thing as the electromagnetic field, but is the source of the electromagnetic field, the stress-energy/mass is not the curvature, but is the source of the curvature.As for myself, I kind of like the Friedman solutions. Yet the strong and the weak forces have been integrated, whereas we have yet been unable to cancel out the infinities in the case of gravity. I forget exactly what the difference is.However, gravitational waves are generated only in the case of dynamic solutions. For example, the Kerr solution is the solution for a black hole with angular momentum, but in the Kerr solution, nothing actually changes, therefore no gravitational waves are produced, whereas with two neutron stars revolving around one-another, gravitational waves are produced, and therefore energy is lost from the system in the form of gravitational waves, resulting in the death spiral where they will ultimately collide.See the following:Binary pulsar PSR B1913+16
compiled by Wm. Robert Johnston
last updated 30 August 2004
Binary pulsar PSR B1913+16No gravitons are emitted in the case of the non-dynamic solution, but when the solution is dynamic, gravitational waves are produced, in which case gravitons are produced which result in the loss of energy. The gravitons are the energy, just as photons are energy, and the energy will be equivilent to mass, resulting in additional spacetime curvature.