In the previous pages I showed that Newton was right all along, so there is really no need to 'save' his theory by introducing 'dark matter' in our galaxy, nor is there a need to modify his theory in any way (at least not in the realm of low speed - relativity is another thing). It's hard to assess where, how or why things went wrong, but once you accept that one defining feature of Newtonian gravity is, that the volume of an enclosed mass diminishes in a very predictable way, the conclusion that a ~ 1/r for the mass distribution of our galaxy is inevitable.
We could leave it at that, but I think we'd miss the opportunity te have a fresh look at the basics of his theory.
In my line of thinking 'Volume' is the central concept.
|7)||ΔVol = 2πMGΔt2|
I think the volume-way of thinking is the more fundamental, because it explains (in geometrical terms) why in Newtons original expression we find an 1/r2. In the original theory there is no reason for this square, it could have been a third power, or there could have been an extra (smaller) term. The only reason is, that this appears to be the best description of reality.
Starting of with volumes it is immediately clear why things could not have been any other way - at least not in Euclidean space.
This also severely limits the possibility to tinker with the theory. We do not only want a theory that fits the data, we also prefer to understand why things are the way they are. With a principle of conservation of empty volume I think we do, at least more so than in the original theory.
The irony of Newtonian mechanics is of course, that it is only perfectly valid in complete rest. As soon as things start to move relativity kicks in, length contracts and time dilletates. This is all well described by special relativity. So: does the idea of conserved volume suffer from this? Not nescessarily. If we look at an object with lenght l (and breath b, and heigth h), moving at a speed such that ϒ = 10, we'd see the thing shortened by that factor, and time dilletated by ϒ. So we'd look for 10 seconds at an object with lenght 1/10 * l. The volume of spacetime the object would have to us, would be: 1/10 * l * b * h * 10 = l*b*h. Exactly what someone within the object would measure: he/she would measure 1 second, in an object of l*b*h. So, if we take time into the equasion, for a non-accelerating object (even if it travelles at high speed) the volume of timespace (l*b*h*t) is still preserved for a moving object if we're not taking the mass of the moving object into account.
Since this site is dedicated to the gravity of galaxies, I will not elaborate on this, but as far as I can see, it is possible to express this as a very fundamental principle, that the volume of timespace is always conserved. This also applies to volumes with enclosed mass. Although in that case(as stated above) the xyz-volume decreases, this is being 'compensated' for by a distortion of time near the attracting mass. Loosly espressed as: the closer you get to the mass, the 'slower' the time goes, and this compensates for the contraction of the xyz-volume.
This line of thinking requires a slightly different view on space itself: space is no longer a system of locations where objects reside and events happen, it becomes a 'thing' that can distort and more importantly, it can move. I've done some calculations on these concepts, and it leads me to some kind of theory of general relativity, although - as far as I can see - in some aspects it is a bit different. This is still work in progress. If you are interested, please do not hesitate to contact me.