Does Einstein's Theory of Relativity Predict Dark Matter?

[RCulling]

I know that Newtonian mechanics predicts dark matter due the the fact that the sun (and other stars) is orbiting around the centre of the galaxy much faster than expected

But i was just wondering if Einstiens general theory of relativity predicted this aswell?
Does it "say" there is as much dark matter as Newtons theory?
or are they about the same?

Just wondering
Thanks : )

 

[Mu naught]

They don't "predict" dark matter... they predict a certain result, which does not match what people observe. So people have come up with the idea that there must be hidden matter so make our observations fit the equations. The other possibility is that there is no dark matter and the equations are not perfect yet.

However there is some other evidence for DM besides just the velocity of stars on the edges of galaxies.

 

[RCulling]

Do the 2 theories calculate the same orbiting velocties?
i.e. as a result of having the same orbital velocities conclude that there is the same amount of DM in the universe (The Galaxy)?

 

[cesiumfrog]

i was just wondering if Einstiens general theory of relativity predicted this aswell?
Does it "say" there is as much dark matter as Newtons theory?

You probably want to look at http://www2.phys.canterbury.ac.nz/~dlw24/" 's work:

The amount of non-baryonic dark matter relative to baryonic matter is decreased, but still significant. Ratios of 3:1 non-baryonic to baryonic matter are typically found as a best fit, for cosmological solutions using boundary conditions consistent with the CMB anisotropies and primordial inflation.

 

[RCulling]

Oh this is a prof from my uni .. Geuss i should go talk to him :O
Thank you guys

 

[Ich]

Do the 2 theories calculate the same orbiting velocties?

Yes. Wiltshire's work has nothing to do with that.

 

[RCulling]

Thanks Ich.. i was looking for ages through his papers and journals at uni and couldn't find anything

Cheers

 

[Ich]

Wow - quick reply. I had to google for cesiumfrog's http://www2.phys.canterbury.ac.nz/~dlw24/universe/summary.html" , and it's about something completely different, and very speculative, too.
Gravity in Galaxies is relatively weak, so it doesn't matter for all practical purposes whether you use GR or the Newtonian approximation. Except that the latter is much easier to apply.

 

[cesiumfrog]

Oh this is a prof from my uni .. Geuss i should go talk to him

That's a great idea! Please do and tell us what you learn.

Wiltshire's work has nothing to do with that.

The OP's question was to what extent does the inferred dark matter ratio depend on whether the observational data (galactic luminosity and redshift curves) are interpreted under the framework of Newtonian physics or GR. Ich, can you site other sources that have addressed this question? The fact is that there has been a difficulty in identifying the theoretically correct averaging procedure to describe grainy matter distributions in GR properly.

it's about something completely different, and very speculative, too.
Gravity in Galaxies is relatively weak, so it doesn't matter for all practical purposes whether you use GR or the Newtonian approximation.

Do you have any evidence to support your claim that the Newtonian approximation does not make any relevant practical difference from GR here?

 

[Ich]

Ich, can you site other sources that have addressed this question?

There has been a paper a few years ago which claimed a difference between GR and Newtonian analysis. It has been shown to be errorneous. I forgot both the paper and the replies, maybe someone else remembers it?

Do you have any evidence to support your claim that the Newtonian approximation does not make any relevant practical difference from GR here?

Sure. The total mass of a galaxy is ~ 5*10^11 Msun, with a Schwarzschild radius of ~0.1 Ly. So we're talking about a potential of order ~10-6 to 10^-5. OTOH, we have velocity measurements with an accuracy of order ~0.1. No way for GR to produce significant corrections.

 

[RCulling]

That's a great idea! Please do and tell us what you learn.

I couldn't get hold of him, he's vbusy (which makes sense :P )
But, i read his journals and papers in the library and couldn't find anything referring to this? Not to say he didn't as all his journals weren't there.. I'm not sure.. Hopefully he will be my PhD supervisor, when i answer this question tehehe

So I will let you know, when i solve it xD

 

[D H]

Do you have any evidence to support your claim that the Newtonian approximation does not make any relevant practical difference from GR here?

To add to what Ich already gave:

The orbital velocities of stars, clusters, and satellite galaxies about a central galaxy should be tiny. Instead, they are just small (compared to c). For example, NGC3198 has orbital velocities that are about 150 km/s from 4 to 30 kpc from the central core. 150 km/s is 0.0005 c, which is too small for any significant relativistic effects to be involved.

 

[RCulling]

150 km/s is 0.0005 c, which is too small for any significant relativistic effects to be involved.

When you put it that way, I understand
Thank you :)

 

[Chronos]

Compelling evidence for dark matter is provided by the bullet cluster papers. That is about as good as observational evidence gets. Zwicky noticed the missing mass problem almost a century ago. Virial theory confirmed his suspicions. MOND is merely a drive by shooting victim. Dark matter is not going away any time soon.

 

[cesiumfrog]

The total mass of a galaxy is ~ 5*10^11 Msun, with a Schwarzschild radius of ~0.1 Ly. So we're talking about a potential of order ~10-6 to 10^-5. OTOH, we have velocity measurements with an accuracy of order ~0.1. No way for GR to produce significant corrections.

150 km/s is 0.0005 c, which is too small for any significant relativistic effects to be involved.

Dark matter is not going away any time soon.

Can any of you experts tell me the correct way to use GR to model a galaxy? Obviously one does not use a Schwazschild metric except in a trivial worst approximation, since that would treat the bulk of the galaxy as a vacuum. Since GR is nonlinear, we know that the field of two stars is not quite simply the sum of the individual fields of either star in isolation. Do any of you know how much difference this makes? Have any of you seen the calculation attempted?

Isn't it obvious to every thinking person that one will arrive at different numbers for the quantity of dark matter depending on whether one interprets that data using one theoretical framework or another subtly nonequivalent framework? So how is it invalid (nay, threatening) to investigate how much those numbers differ? You know, actually bothering to check quantitatively how good the approximation is? I don't think performing such exercises warrants the label "speculative", if anything then "speculative" would be asserting a particular outcome of such exercises without bothering to have anyone perform them carefully. I'm certainly not claiming that Newtonian physics is a terrible approximation: the papers I've seen only reported about a factor of 2 discrepancy, which should be perceived as no threat to the dark matter dogmatists. I'm only advocating placing more weight on nuanced application of scientific method (in this case, reading of published calculations that exist on the topic) than on simple hunch.

Does [GR] "say" there is as much dark matter as Newtons theory? or are they about the same?

 

[twofish-quant]

Since GR is nonlinear, we know that the field of two stars is not quite simply the sum of the individual fields of either star in isolation. Do any of you know how much difference this makes? Have any of you seen the calculation attempted?

What you do is to expand things out into a power series

GR = Newtonian + (something) x + (something) x^2 + (something) x^3 + ...

Then you look at what x is and how big it is, and x is v/c and for the purposes of galaxy rotation it's not big enough to make any difference. The good thing about this sort of argument is that could apply even if GR is wrong.

Also if you show that x is big enough to be important, but x^2 isn't, then you end up with what is called the post-Newtonian formalism and that happens to be linear.

Isn't it obvious to every thinking person that one will arrive at different numbers for the quantity of dark matter depending on whether one interprets that data using one theoretical framework or another subtly nonequivalent framework?

Except that you won't. As long as your theory of gravity approximates Newtonian gravity then it's not going to matter. Now you could be in a situation where your theory of gravity *doesn't* approximate Newtonian gravity in which case you get into the world of MOND models.

What you can show is that as long as things are "subtly" different, things aren't going to matter. Things have to be very different for things to matter.

So how is it invalid (nay, threatening) to investigate how much those numbers differ? You know, actually bothering to check quantitatively how good the approximation is?

People do that. The trouble with that is that it can be done in two paragraphs and it's so easy to do that it's not worth writing a paper about it. Now what *would* be worth a paper is if you could stare at the basic argument and show that it's flawed.

Be my guest :-) :-) :-)

I'm certainly not claiming that Newtonian physics is a terrible approximation:

That's interesting since there is a whole series of papers that argues that Newtonian physics *is* a terrible approximation to galaxy rotations. If you want galaxy rotations to be a gravity effect, you have to argue that Newtonian gravity is wrong at galactic scales. This isn't a crazy thing to do. I mean, we know that Newtonian gravity is very wrong at cosmological scales, and we know it's pretty good at solar system scales.

I'm only advocating placing more weight on nuanced application of scientific method (in this case, reading of published calculations that exist on the topic) than on simple hunch.

You don't want nuance. This is something that you want to be blunt about. You don't want complex math, you want a simple straightforward argument, and that turns out to be that v/c << 1.

Also the calculation to check if GR matters or not is a trivially simple one, and it's so simple that no one is going to publish a paper about it. Now if you can think of a reason why the basic argument is wrong, then *that* would be worth a paper.

 

[twofish-quant]

Also astrophysicists like these sorts of simply arguments because they are quick. If it turns out that GR makes a difference then you are going to be spending the next months/years/decades figuring out the details.

So a five minute argument that says "it's not going to matter" is something that is very useful. 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.

The problem with these arguments is that they never quite get published because it's too easy and not worth publishing.

 

[Ich]

I'm only advocating placing more weight on nuanced application of scientific method (in this case, reading of published calculations that exist on the topic) than on simple hunch.

It strikes me as odd that you rather call us dogmatists than consider the possibility that the problem really is easy enough to be settled with a few numbers. There are weak fields (~10^-6), slow velocities (~10^-3), so perturbation theory will work. That means that the respective corrections are themselves of order 10^-6, 10^-3.
That's not magic, it's science. And if you don't like the result, because there's a dogma or two in your thinking that DM must be non-existent, well, then come up with a better argument. This one won't work.

 

[Geigerclick]

People do that. The trouble with that is that it can be done in two paragraphs and it's so easy to do that it's not worth writing a paper about it. Now what *would* be worth a paper is if you could stare at the basic argument and show that it's flawed.

Be my guest :-) :-) :-)

That was good for a chuckle, thanks. This came immediately to mind:
[PLAIN]http://www.acsu.buffalo.edu/~dpadgett/ackbar.jpg

I find the dark matter result disturbing, but as Ich has said it is because of previous dogmatic thinking on my part. The more I learn about it (often in such forums as this) the more I understand how limited the options are. I would put some money on WIMPs, but in general, if DM didn't exist it would be almost inconceivable at this point. GR is just too good, and observations all seem to agree on the basic issues.

 

[cesiumfrog]

there's a dogma or two in your thinking that DM must be non-existent

I apparently can't repeat this enough: I do not think such a thing. I don't understand how you have misread so extremely.

 

[Ich]

I don't understand how you have misread so extremely.

You tried to explain our refusal of such ideas with psychology. Called us dogmatists. I think it's understandable that I mirrored that word.
But it's ok, so I misunderstood something.

 

[cesiumfrog]

Ich, I'll reiterate my previous post.

If you try to measure a quantity using the same data but two (not exactly equivalent) theoretical frameworks, you will expect to get two not exactly equal results. This much is surely obvious to you, but tell me if you need an explicit example.

Now, my understanding of the OP is that it raises the quantitative question, how much do those numbers differ in the particular case of whether full GR is used to interpret the dark matter evidence. Hence, one would be both technically incorrect and failing to answer the question if one were to react insisting there were literally zero difference.

Now, a very simple argument has been put forward to say that the difference would be (bounded above by) less than one percent, and (twofish-quant might find that arguments which are too little to warrant a separate research article still often tend to be mentioned in passing, and detailed in textbooks or review articles) indeed I understand this to be the consensus view among astronomers. Fine. Even if that argument is flawless, I still think it is worth asking about the actual difference, because I think the process necessary to actually approach this problem fully within GR must be very interesting (indeed, twofish-quant implicitly agrees with this by guessing it could take decades of work for the details to be worked out the first time). But it would seem the expert-reviewed GR literature does already mention attempts to answer this question, so the obvious thing to do is to cite those for the OP.

As it happens, the result I cited describes a difference of more than 1% (and less than 100%; I don't see why you keep referring to "such ideas" like complete nonexistence of dark matter). Since this conflicts with the very simple argument used outside of the hard core general relativity field, we could suggest that experts on GR have not heard the basics that are known of GR. Alternatively, we could suggest that the a person whose qualifications include presiding for a society of professional GR and gravitation researchers might have already encountered (and understood the possible flaws in) such a simple argument. Since this is not the same area of GR as my expertise, I haven't a great deal to add.

 

[Ich]

If you try to measure a quantity using the same data but two (not exactly equivalent) theoretical frameworks, you will expect to get two not exactly equal results.

Exactly. You expect two not exactly equivalent results. The size of "not exactly equivalent" can be estimated by the procedures that twofish-quant and I applied. "not exactly equivalent" is not 10% (measurement uncertainty), and it's definitely not a factor of 5. OP answered.

Hence, one would be both technically incorrect and failing to answer the question if one were to react insisting there were literally zero difference.

Yes. Nobody did so.

I think the process necessary to actually approach this problem fully within GR must be very interesting

I don't think so. It would be a waste of time, nothing more. You'll always do the perturbative approach in such a situation, anything else would be ... insane. Like doing the full quantum mechanical approach in automotive crash test simulations.

But it would seem the expert-reviewed GR literature does already mention attempts to answer this question, so the obvious thing to do is to cite those for the OP.

Well, at least I said that I remembered one attempt to explain DM with GR, and its debunking. I couldn't find the references after a quick search, so I thought it'd be ok if I give an OOM estimation with the same result. Maybe someone with better memory will give you the links.

As it happens, the result I cited describes a difference of more than 1%

No, it doesn't. It says:

The amount of non-baryonic dark matter relative to baryonic matter is decreased, but still significant. Ratios of 3:1 non-baryonic to baryonic matter are typically found as a best fit, for cosmological solutions using boundary conditions consistent with the CMB anisotropies and primordial inflation.

That's nothing to do with the ratio of visible to dark matter in galaxies.

Since this conflicts with the very simple argument used outside of the hard core general relativity field, we could suggest that experts on GR have not heard the basics that are known of GR. Alternatively, we could suggest that the a person whose qualifications include presiding for a society of professional GR and gravitation researchers might have already encountered (and understood the possible flaws in) such a simple argument.

Or, as a third alternative: There's no conflict, because Wiltshire is not talking about galaxy rotation. He's talking about cosmology. (a citation: "Here we take a “dust particle” to be of at least the scale of statistical homogeneity, 100h−1 Mpc or somewhat
larger". Much bigger than a galaxy.)
Maybe you noticed that I gave you numbers related to the potential, even if the deviation due to velocity is bigger. I did so because it is exacly the condition that the potential be << 1 that Wiltshire violates when talking about cosmological averaging effects. The "simple results" do not necessarily apply cosmologically.
I'm convinced that there are no vital averaging effects in cosmology, too, but that's something completely different.

 

[cesiumfrog]

There's no conflict, because Wiltshire is not talking about galaxy rotation. He's talking about cosmology.

Ah, you are correct. It is by analysing other categories of data in the framework of GR that he is dismissing 40% of the dark matter otherwise normally expected to exist in the universe. Looking at PRD 78 084032 (2008), it seems he has not yet tackled rotation curves, but outlines reasons to not presume those to be inconsistent with the same result.

I don't think so. It would be a waste of time, nothing more. You'll always do the perturbative approach in such a situation, anything else would be ... insane. Like doing the full quantum mechanical approach in automotive crash test simulations.

We can agree to disagree on whether it is interesting. For what it's worth, though I wouldn't use QM for every routine crash test, I think it would be interesting to see QM applied once to a giant system (such as a computer - in the context of MWI this might much better test/demonstrate emergence of classicality), especially a mysterious system with proposed quantum explanations (for example, "a person operating a computer" - you can see how that would get the attention of the followers of Penrose). In the case of galaxies, the perturbative approach makes implicit assumptions about the surrounding geometry, the validity of which are questioned in the paper I just mentioned.

 

[Geigerclick]

Didn't I say this was a trap?! No one listens to the Mon Calamari.

 

[Ich]

There has been a paper a few years ago which claimed a difference between GR and Newtonian analysis. It has been shown to be errorneous. I forgot both the paper and the replies, maybe someone else remembers it?

https://www.physicsforums.com/showthread.php?p=2773705#post2773705".
http://arxiv.org/cits/astro-ph/0507619"you find the replies.

 

[twofish-quant]

I think it would be interesting to see QM applied once to a giant system (such as a computer - in the context of MWI this might much better test/demonstrate emergence of classicality)

Been done. One of the reasons that people don't think that dark matter consists of neutrinos is that it turns out that you can't put enough neutrinos around a galaxy because of the Pauli exclusion principle. What you do is to model the galaxy as a hydrogen atom, count the number of atomic states that you can put a neutrino in, and it's not nearly enough.

Also "simple is beautiful" and "complex math is a necessary evil that you need to avoid if possible." When people apply simple scaling or order of magnitude arguments to a physical situation with GR or QM, that's a valid argument and it's a *better* argument that requires pages of obscure math.

Now you can try to invalidate the simple arguments with complex math, but then the ball is in your court. If you think that solving the full GR equations create a situation in which the weak field perturbation fails, you are welcome to publish. The thing about this is that it takes years to develop expertise to the point where you can do this sort of work. A lot of people involved in the field are specifically looking for this sort of thing, and no one has published anything suggesting that as a matter of math, the weak-field arguments are invalid at galactic scales.

especially a mysterious system with proposed quantum explanations (for example, "a person operating a computer" - you can see how that would get the attention of the followers of Penrose).

Penrose is a crank when it comes to things outside his field of expertise. The basic problem with Penrose is that you can come up with easy arguments that quantum mechanics *doesn't* play a major role in neuroscience, so to get around that Penrose invents his own neuroscience.

In the case of galaxies, the perturbative approach makes implicit assumptions about the surrounding geometry, the validity of which are questioned in the paper I just mentioned.

Perturbative approaches to GR assume that the background geometry can be approximated by a flat-space time.

If we are talking about Whitshire's work, no it doesn't. It's a very interesting paper as for as dark energy goes, but when you go into the scale of single galaxies, none of his arguments apply, and going through the math, I didn't get any major change in galactic-scale dark matter. If Whitshire published a paper in which he argued that the perturbative approaches to GR are invalid at the level of single galaxies, that would be very interesting, but he hasn't.

Also, if you go through Whitshire's paper, it's interesting because you can look at the scaling arguments and show that GR will make a difference. Whitshire argues that there may be interesting averaging effects cosmologically, and at those scales v/c starts becoming significant, which says that you do need GR.

 

[Ich]

no one has published anything suggesting that as a matter of math, the weak-field arguments are invalid at galactic scales.

Um, actually, Cooperstock and Tieu did, as I just said.

 

[twofish-quant]

Um, actually, Cooperstock and Tieu did, as I just said.

Cool!

If I had an extra week or so, I'd go through their calculations and figure out why Poisson's theorem stops working for their calculations. One thing that very seriously bothers me is that they are using a stationary metric to model a rotating frame, and that will cause a lot of problems.

Forget general relativity and just think of Newtonian physics. If you try to take a stationary cylindrical coordinate system and apply it to a rotating system, you end up with wrong answers. What I think will happen if you go step by step in the derivation is that you'll find a missing "centrifugal force" term, and the "dark matter" is necessary to provide a force that counterbalances the centrifugal force.

Also I think that the assumption of stars pressure-less dust is also suspect.

This does point out the problem with "non-publication bias." Suppose I do spend a week and manage to convince myself that Cooperstock is wrong. That's not publishable, because it's likely that a hundred other people have gone through the some calculation and figured out the same thing.

 

[TrickyDicky]

In the case of galaxies, the perturbative approach makes implicit assumptions about the surrounding geometry, the validity of which are questioned in the paper I just mentioned.

Cesium, I agree with you about this and also that it's worth the effort to try out GR in the problem about galactic rotation curves. Sure is a hard task, it might even be argued that it'll probably turn out to be practically impossible, but if you don't use GR to try to solve this kind of gravitational problem, what do you use it for? Are we to conform just with schwarschild trivial solutions?. If that were the case, even if conceptually rich, in practice GR would be a bit of a letdown.
The paper by Tieu and Cooperstock may be flawed but that is no reason for giving up this approach.
I think there is a basic misunderstanding when trying to linearize GR (make equivalent the non-linear equations of GR to the linear Newtonian theory in the limit of the weak field) GR, except for the trivial case of Schwarzschild solution with T=0. This was already realized by Weyl many years ago: http://www.jstor.org/pss/2371768

 

[TrickyDicky]

Now if you can think of a reason why the basic argument is wrong, then *that* would be worth a paper.

Certainly, and it's been published, and not just by anybody. Check the link I posted.

 

[Ich]

If I had an extra week or so, I'd go through their calculations and figure out why Poisson's theorem stops working for their calculations.

Just follow the citations, there's a bunch of things wrong with their calculations. Like claiming that the Schwarzschild solution describes an empty spacetime - if it weren't for this nasty spot in the middle where the metric is a little bit discontinuous.
Look especially at Garfinkle's short note, where he brings the same arguments as we did here, supplemented with an implausibility argument regarding "self-sustained" (i.e. source-independent) curvature.

 

[twofish-quant]

if you don't use GR to try to solve this kind of gravitational problem, what do you use it for?

Black holes. Cosmological models. Gravitation waves. High resolution measurements or satellites. GPS. Gravitational lensing.

Sometimes you need GR, but you are just making life difficult for yourself for no good reason if you are using it for things in which it just doesn't make a difference. I'm not sure what the point is.

Are we to conform just with schwarschild trivial solutions?. If that were the case, even if conceptually rich, in practice GR would be a bit of a letdown.

You use the least mathematically complicated tool that you can. If you can get away with the Schwarzschild solution, then you use the Schwarzschild solution. If you don't have to deal with GR at all, you don't. Mathematical complexity is a necessary evil and an annoyance.

The paper by Tieu and Cooperstock may be flawed but that is no reason for giving up this approach.

If you want to go hunting unicorns, then you have to deal with the possibility that there are no unicorns to hunt. The reason for not using GR in galaxy rotation is that you end up with lots of extra math to do, and you don't get anything useful out of doing it. I seriously doubt that Tieu and Cooperstock are the first people to try to do a calculation applying GR to galaxy rotation, except that everyone else that has done this seems to understand GR enough to do the calculation correctly and find that it doesn't make a difference.

Even if you are convinced that that there is some odd GR effect that affects galaxy rotation that itself obvious from the perturbation expansion, you are much, much better off describing this effect as a *general* astrophysical phenonmenon that you can observe with high precision measurements like GPS, and then applying it to galaxy rotation once it's been observed elsewhere.

 

[twofish-quant]

if DM didn't exist it would be almost inconceivable at this point. GR is just too good, and observations all seem to agree on the basic issues.

Curiously this isn't the consensus of the astrophysical community. There are an entire industry within astrophysics proposing various forms of modified gravity. Modified gravity is looking less and less likely for dark matter, but the door is still wide open for dark energy, and it's also settled that what people considered "standard GR" in 1995 (i.e. no cosmological constant or external fields) is just not going to work.

Any currently viable alternative gravity theories can be stated as "GR + fudge factor" The thing about "GR + fudge factor" is that you can't completely rule out the alternative theory. You can only set limits on the size of the fudge factor.

 

[TrickyDicky]

If you want to go hunting unicorns, then you have to deal with the possibility that there are no unicorns to hunt.

Black holes. Cosmological models. Gravitation waves.

You just mentioned some good examples of hunting unicorns.

High resolution measurements or satellites. GPS. Gravitational lensing.

These are derived from the equivalence principle and bending of light concepts that were predicted by Einstein years before the field equations were found in 1915.