Does Einstein's Theory of Relativity Predict Dark Matter?

[twofish-quant]

You just mentioned some good examples of hunting unicorns.

I'm not sure what your point is. In each of the cases that I mentioned, you calculate the speeds involved, the gravitation fields, and then the observational sensitivity, and from pretty simple algebra, you very quickly figure out that yes, GR matters.

If you do the calculations with galaxy rotation, the numbers just say that it doesn't matter to the limits that we can do detections. If as mental exercise, you just want to do things the hard way, and spend six months trying solve the full Einstein equations for galaxy rotations with the high probability of finding out something that you could have figured out in five minutes, you are free to do so, but I really don't see the point.

Even if you love doing complicated math, you could spend those six months solving the full Einstein equations for something else, like a black hole accretion disk.

 

[TrickyDicky]

If you do the calculations with galaxy rotation, the numbers just say that it doesn't matter to the limits that we can do detections. If as mental exercise, you just want to do things the hard way, and spend six months trying solve the full Einstein equations for galaxy rotations with the high probability of finding out something that you could have figured out in five minutes, you are free to do so, but I really don't see the point.

Ok, here is the point, I am just inverting the order of the possibilities you mentioned:

It also tells you that there are three possibilities 1) dark matter 2) some theory of gravity that is totally unlike Newtonian gravity or 3) some fundamental problem with the quick argument.

So 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, here I do think it is not so straightforward that it is possible and thus my citing Weyl in his 1944 paper criticizing Birkhoff's theorem. 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.
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. 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) then you move on to possibility 2, a problem with Newtonian theory in systems other than our solar system, that makes us think of a new theory, or perhaps about a correction for larger systems, well that wouldn't be such strange thing to happen, we already know Newton theory is incomplete and fails at high speeds and strong gravitational fields, but that doesn't make it wrong just incomplete.

The virial theorem might not work properly for systems like galaxies or clusters. Perhaps at such large dimensions the geometry of space affects the ergodic properties on which the virial theorem heavily relies.

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: 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.
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.

 

[twofish-quant]

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

 

[TrickyDicky]

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.

We are talking about GR and gravity in the weak field limit, I never said that perturbation theory is the same as linearization. When GR is linearized, perturbation methods can be a tool to do it but it's not the same. See for intance this from wikipedia:
http://en.wikipedia.org/wiki/Linearized_gravity

Galaxies are fairly simple systems.

I'm sure in this forum there are people who disagree with this statement,for instance in this thread. https://www.physicsforums.com/showthread.php?p=2776322#post2776322

But I think you are confusing simplicity with the final result of some calculation, galaxies as gravitational systems could be very simple, but a single parameter could change the final result a lot.

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.

Well, let me doubt it, anyway if everybody thought this way, science wouldn't advance much. And I don't mean only formal proofs but different physics approaches to apparently already tried problems.If every scientist (after reading the pertinent literature) think that his specific approach must have already proven false there'd be no theoretical breakthroughs.

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.

Maybe, but I don't think that is the common feeling around here. Hope this statement won't get you in trouble.

Regards

 

[twofish-quant]

I'm sure in this forum there are people who disagree with this statement,for instance in this thread. https://www.physicsforums.com/showthread.php?p=2776322#post2776322

We are talking about the context of gravitational rotation curves. What goes in the core for that is irrelevant. Now if you are talking about other aspects of galaxies, then that's different.

But I think you are confusing simplicity with the final result of some calculation, galaxies as gravitational systems could be very simple, but a single parameter could change the final result a lot.

I care about whether it does or it doesn't. If something maybe can or maybe can't, that's a useless statement. What are the parameters that effect galactic rotation? How does the calculation change in response to the different parameters, and so forth?

Well, let me doubt it, anyway if everybody thought this way, science wouldn't advance much.

The nice things about mathematicians is that when you have a mathematician give you a formal proof, then it's rock solid. What's also cool is when the mathematician gives you a formal proof and then points out about five or six different loopholes in that proof. If you really want to go into the mathematical aspects of general relativity, that's an entire career, but you may find (and I think you probably will find) that what are trying to do just won't work.

And I don't mean only formal proofs but different physics approaches to apparently already tried problems.If every scientist (after reading the pertinent literature) think that his specific approach must have already proven false there'd be no theoretical breakthroughs.

Hardly. Science is a conversation. If you read a paper saying that your approach just won't work, and you disagree, you can spend a few weeks coming up with a rebuttal and things progress. Also, sometimes your approach just won't work because God has determined that your approach just won't work, and you need to try something else. You can often take the work you've done in one area and adapt it in another.

People spend a decade trying to make cosmic deuterium and failed, but it turns out that they could make lithium. The coffin is closing on galactic dark matter but it's still quite open for cosmological ones.

Also if you learn that your approach just won't work, that's usually a cause for celebration, because getting to that point can be grueling. CDM came out of supersymmetry.

Maybe, but I don't think that is the common feeling around here. Hope this statement won't get you in trouble.

Why do you say that? It's not like anyone dogmatically believes in dark matter. Personally, I think most people around here would think it would be totally cool if someone came up with a strong argument that dark matter just won't work. Most physicists I know jump for joy when they figure out that everything they thought they knew was wrong. One of my memories was when the COBE results were coming in, and people were hoping that their *weren't* blackbody anisotropies, because that would mean that the big bang was all wrong.

I knew of a famous physicist that works in both HEP and cosmology, and he says that HEP is a little depressing because everything fits theory unlike cosmology where we really don't understand what is going on.

One other thing, you have to go from "could or should" to "is" or "is not". People have mentioned why they don't think that GR has much of a role in galaxy rotations. This is a challenge to you. If you want to plan the game, you need to respond with a counter-argument. That argument has to be more than "maybe it works" or "it might be important." You need to come with something more solid. In particularly, you need to estimate how *much* of a difference does it make?

Also you can start by explaining the relevance of the papers you cited. I've gotten a list of papers in that discussion, Birkoff was proposing a completely different theory of gravity than Einstein, and Weyl was pointing out that when you linearize Birkoff's theory and when you linearize GR, you get different results. This has nothing to do with the topic under discussion.

The other thing is that if you are new to GR, you are much better off working on problems where GR is *known* to be essential than ones where it's likely that it's not.

 

[TrickyDicky]

I knew of a famous physicist that works in both HEP and cosmology, and he says that HEP is a little depressing because everything fits theory unlike cosmology where we really don't understand what is going on.

Let's hope they get some surprise from the LHC.