TrickyDicky said:
OSo I think it is more logical to try to first see (possibility 3) if there is a fundamental problem with linearization of GR in a complex system like a galaxy , to make it approximate the solution from Newtonian theory
We aren't talking about linearization. You don't have to linearize to do perturbation theory. What you have to show is that a small change results in a small response. This is different from linearization. You could have a highly non-linear situation in which perturbation theory works, if the response function is concave or saturates quickly. You can have a linear situation in which the response functions are steep in which perturbation theory doesn't work.
Also you have to be careful about "complexity." Systems with high degrees of freedom can be trivial to mathematically model, whereas systems with low degrees of freedom can be hard to model. Assuming there are no galactic scale magnetic fields, galaxies are quite easy to model. The entire universe is *MUCH* easier to model than smoke from a cigarette or for that matter my wife.
The reason that I think it's unlikely that you are going to find "weird things" with GR is that ultimately GR is a theory that is based on differential geometry and smooth manifolds, and any theory based on smooth manifolds will have smooth response functions if you look at a small enough area. It's possible that a mathematician has formalized this idea.
As it's been mentioned, even in a much , much simpler system like Mercury's orbit we find a small discrepancy, that might be enlarged non-linearly in a galactic system.
What tends to happen when you increase the degrees of freedom is that non-linearities cancel themselves out. If you look at a single atom, it's quite complicated. If you look at a trillion atoms, you have a gas, and any odd behavior within a single atom gets washed out.
There are some pretty standard tests that you can use to see if there is an impact of small changes affect the larger system, and in the case of gravity, small changes get washed out. Now if you are talking about magnetic fields, that's a totally different story. The basic issue is that gravity can be approximated as a scalar potential, and scalar potentials wash out these effects.
If that were so ( the big disparity between GR and Newtonian theory when applied to complex enough systems) then it woud be worth doing the hard math.
Galaxies are fairly simple systems. Big systems are often simpler than small systems. Systems with lots of moving parts are often (and in fact usually) are simpler than systems with few moving parts.
Or do the easy math. You are talking a lot about "if's" and what I'm telling you is that a lot of people have looked at this and found nothing. I think like a physicist and not a mathematician so my logic isn't rigorous, but there are a whole bunch of people that have put some rigor into the arguments that I've made.
Also, it's much easier sometimes, if look at the general situation rather than a specific situation. Mathematicians are useful because they *don't* look at the physical situations. They just tell you how certain rules behave under certain conditions.
If one concludes like you do that this is not the case (but notice that your reasoning is heuristic, you have no formal proof that the quick argument is true beforehand)
I don't have a formal proof, but it's something that mathematicians spend their time doing. If the mathematicians thought that there was something seriously wrong, it would get filtered through the mathematical physicists.
These might look like unlikely hypothesis, but I think they are worth exploring before (or at the very least in parallel with) turning to possibility number one:
Go to
http://adswww.harvard.edu/ and the Los Alamos preprint server and search for MOND and f(r). You will find *hundreds* (and possibly thousands) of papers on modified gravity theories. It's not something that people are ignoring, but there are reasons why dark matter is favored over modified gravity. Right now, modified gravity isn't quite dead with respect to galaxy rotation curves, but it's critically ill, and I'll leave it to you doing some research to figure out why.
a completely new form of matter, never suspected before that would seem more like science-fiction and that so far has not been detected after numerous experiments.
Sure. But right now it's the least bad situation.
But certainly I'm not dismissing this possibility, I only wonder why you and others think the other two possibilities should be discarded so easily.
I hate to be harsh about this, but it's because you aren't aware of the research that has been done, and the effort that has been put into this. Just google for MOND and f(R).
People have looked very, very hard for the possibility that there is some approximation problem or some modified gravity, and haven't found anything convincing. After haven't several hundred people spend about a decade looking for unicorns and finding nothing, you start wondering if they aren't finding things because they don't exist.
This applies to dark matter too. If after another decade or so, we find no sign of dark matter, than people will think of something else. However, the fact that we are starting to see gravitational lensing of something that looks like dark matter does change things. I suspect that within a decade, we'll have very good maps of exactly where the dark matter is.
Some other things...
1) Most of the work on modified gravity has moved away from dark matter to dark energy
2) Even if you were to establish that there is no weird dark matter around galaxies, you'd still have a big problem since cosmological dark matter requires a lot more dark matter than that
