New study shows Dark Matter isn't needed? Relativty explains it?

[Lapin Dormant]

ever heard of private messaging?

And if you read the Guidelines you will know that it is NOT private, Not at all.

Funny too, your public profile wouldn't let me read any of your other postings, just a Ruse/psyeudonymforamentor name, are you?

LD
Don't Bother .. .. .. .. .. .. ThinksThanks .. .. .. .., ,.. .. .. I lept .. Figures, NOW it does .. .. .., ,.. .. .. lept again

 

[Garth]

Bringing this thread back on topic...

I'm going to hide behind the skirt here, Garth... Can we say with any confidence what disc model might work? I think the observational evidence is really thin. Another objection: since when did CERN jump into the fray? I don't think that is even relevant to this conversation.

No, we cannot say at the moment what disk model might work with pure GR, though I do think the infinitely thin disk of Cooperstock & Tieu is a workable rough approximation to the real situation.

The point is that until a proper GR analysis is done noboby will know whether any model might work.

However, as I posted in the S&GR forum "Overturning GR contest" thread, given that the mass in a galaxy is in orbit, rather than concentrated at the centre, GR's non-linear effects might well be significant. The orbiting mass's 'kinetic energy' contributes to the density and angular momentum terms of the stress-energy-momentum tensor T_uv which then generates more gravity (curvature) in a way that does not happen in Newtonian theory.

The onus is therefore on those who want to analyse galactic rotation profiles in the Newtonian approximation to prove that the non-linear terms are not significant.

AFAIK this has not been done. As I asked in the "Overturning GR contest" thread: "does anybody know of any previous work" where this has been published?
Garth

 

[EL]

No, we cannot say at the moment what disk model might work with pure GR, though I do think the infinitely thin disk of Cooperstock & Tieu is a workable rough approximation to the real situation.

But, still the thing is that C&T happened to include a singular disk which they didn't mean to, i.e. their model includes an ADDITIONAL source of gravity APART FROM the ordinary matter in the galaxy.
If you would model the galaxy as simply an infinitely thin disk, you would not end up with the same rotation curves as they get in their paper.

Their additional thin disk acts as dark matter, and hence it is not that strange they get the correct rotation curves. They included DM without knowing it...

 

[Garth]

But, still the thing is that C&T happened to include a singular disk which they didn't mean to, i.e. their model includes an ADDITIONAL source of gravity APART FROM the ordinary matter in the galaxy.
If you would model the galaxy as simply an infinitely thin disk, you would not end up with the same rotation curves as they get in their paper.
Their additional thin disk acts as dark matter, and hence it is not that strange they get the correct rotation curves. They included DM without knowing it...

Hi EL! Thank you for your comment.

Yes I do understand what Korzynski is saying. He does not, however, indicate the total mass of this additional thin disk. I cannot believe it is as massive as the DM halo it replaces, 10X the baryonic mass, as it is all within the visible galaxy outer radius and it would affect stellar orbital periods too much. If it is only a small additional component to the total galaxy mass (Cooperstock's value 2.1 x 10¹¹ M_solar) then it might be a reasonable model of the thin galactic disk observed (6 x 10¹⁰ M_solar).

My main point, however, is notwithstanding Cooperstock & Tieu model's validity, the non-linear GR effects may well be significant in galactic rotation profiles and should be investigated thoroughly.

Garth

 

[Garth]

But if you notice the Cooperstock Appendix you'll see their solution for density is
$$\rho=5.64 . 10^{-14}\frac{(N_r^2+N_z^2)}{r^2}$$ kg/m³
so they are back to the old 1/r² Newtonian flat rotation solution!
Garth

 

[SpaceTiger]

But if you notice the Cooperstock Appendix you'll see their solution for density is
$$\rho=5.64 . 10^{-14}\frac{(N_r^2+N_z^2)}{r^2}$$ kg/m³
so they are back to the old 1/r² Newtonian flat rotation solution!

It's not obvious to me that the above equation is even a close approximation to a 1/r^2 dependence, since N depends non-trivially on radius. Did you check this numerically?

 

[Garth]

It's not obvious to me that the above equation is even a close approximation to a 1/r^2 dependence, since N depends non-trivially on radius. Did you check this numerically?

I was using a 'wand waving' OOM approximation:
As $$V(r, z)=\frac{3.10^8}{r}N(r, z)$$ is more or less constant on the flat part of the rotation curve therefore approximately we can take
$$N(r, z) = A.V(r,z).r$$
and at constant z
$$\rho=5.64 . 10^{-14}\frac{(N_r^2+N_z^2)}{r^2}$$ kg/m³
becomes
$$\rho=A_1 + A_2\frac{N_z^2}{r^2}$$
more or less the Newtonian model with appropriate values for the constants A₁ and A₂.
I hope this helps.
Garth

 

[SpaceTiger]

I was using a 'wand waving' OOM approximation:
As $$V(r, z)=\frac{3.10^8}{r}N(r, z)$$ is more or less constant on the flat part of the rotation curve therefore approximately we can take
$$N(r, z) = A.V(r,z).r$$
and at constant z
$$\rho=5.64 . 10^{-14}\frac{(N_r^2+N_z^2)}{r^2}$$ kg/m³
becomes
$$\rho=A_1 + A_2\frac{N_z^2}{r^2}$$
more or less the Newtonian model with appropriate values for the constants A₁ and A₂.
I hope this helps.

Well, firstly, take another look at your final equation. It says that the density approaches a constant value as the radius approaches infinity. This should give you a hint that something is wrong. The basic rotation curve that results from that equation is flat towards the center and then rises linearly as the first term becomes larger than the second term.

What it would seem you did is misinterpret the $N_r^2$ and [itex]N_z^2[/tex]. They're partial derivatives, so to take advantage of the flat rotation curve simplification, you have to differentiate V with respect to r like:

$$\frac{dV}{dr}=AN_rr+AN_zr+AN(r,z)=0$$

which leads to:

$$N_r^2=(-N_z-\frac{V}{Ar^2})^2$$

Plugging this into your density equation will still give you something non-trivial, I'm afraid. Try reading the density profiles from their plots instead.

Edit: Replaced $V_r$ with $\frac{dV}{dr}$ to make it clear that I was taking a total derivative, not a partial.

 

[Garth]

Well, firstly, take another look at your final equation. It says that the density approaches a constant value as the radius approaches infinity. This should give you a hint that something is wrong.

The Newtonian density function that delivers the flat rotation curve is:
$$\rho(r)=\frac{C_0}{(a^2+r^2)}$$
where C₀=4.6x10⁸M_solar and a = 2.8 kpc. of course the density distribution has to be truncated at some radius r = a₀ otherwise the total galactic mass would be infinite. This is basic theory.

Cooperstock &Tieu discuss the matter quite extensively: “It is unknown how far the galactic disks extend. More data points beyond those provided thus far by observational astronomers would enable us to extend the velocity curves further. Presumably a point (let us call it r_f ) is reached where we can set rho to zero. At this point, (2) no longer applies as there are no longer co-rotating fluid elements being tracked. As a result, (9) no longer applies and the w function is no longer constant. Beyond r_f, no further mass is accumulated.“

There is also a change of regime as r tends to zero.

All I did was to give a very ‘rough and ready’ comparison, taking z to be a constant, therefore N becomes N(r), and also taking V to be a constant then N becomes linear in r and N_r a constant..

I was considering only the central ranges to show that $$\rho$$ varies as r^-2. The complete numerical calculation was done by Cooperstock & Tieu.

As I said the main question is whether the non-linear GR effects are significant in galactic rotation, and if so then what of galactic halo DM?

Garth

 

[SpaceTiger]

The Newtonian density function that delivers the flat rotation curve is:
$$\rho(r)=\frac{C_0}{(a^2+r^2)}$$
where C₀=4.6x10⁸M_solar and a = 2.8 kpc. of course the density distribution has to be truncated at some radius r = a₀ otherwise the total galactic mass would be infinite. This is basic theory.

Come on Garth, you're just making this worse. First of all, that's not the relation you got in your above calculation:

$$A_1+\frac{A_2}{r^2} \ne \frac{B_1}{B_2+r^2}$$

for any constants A and B. You can see this by again looking at its limits. Your first expression goes to a constant density at infinity, while this new one goes to zero.

Secondly, that is not the expression to generate a flat rotation curve in Newtonian gravity, it's an expression for which the rotation curve asymptotes to flatness at infinity. A completely flat rotation curve comes from an isothermal sphere:

$$\rho=\frac{A}{r^2}$$

All I did was to give a very ‘rough and ready’ comparison, taking z to be a constant, therefore N becomes N(r), and also taking V to be a constant then N becomes linear in r and N_r a constant..

If this was what you did, it would be inconsistent with your result. That gives:

$$\rho = \frac{const.+N_z^2}{r^2}$$

Now, this is the Newtonian result if $N_z$ is a constant with radius, but there's no reason to assume this should be the case. Taking the radial partial derivative at a constant z does not mean that partial of z is constant with r.

 

[Garth]

$$\rho(r)=\frac{C_0}{(a^2+r^2)}$$

that is not the expression to generate a flat rotation curve in Newtonian gravity, it's an expression for which the rotation curve asymptotes to flatness at infinity.

It's the standard expression that delivers a flat velocity profile at large
r >> a, but modifies the isothermal sphere to give rigid-body rotation at small r << a.

A completely flat rotation curve comes from an isothermal sphere:
$$\rho=\frac{A}{r^2}$$

Which is what I was saying, I know that N_z is not constant in general - however it is zero if in a certain regime in the Cooperstock & Tieu relationship the orbital velocity is taken to be constant.
$$V (r, z) =\frac{3.10^8}{r}N(r, z)=constant$$

so N(r,z) = C.r

therefore N_r = C and N_z = 0,

my A₁ should have been in fact zero (I answered the post hurriedly - thank you for correcting me) so the density expression does approximate to the isothermal sphere,
$$\rho=\frac{A}{r^2}$$

As I have been saying this is only my first approximation to the exact Cooperstock & Tieu equation, 'reverse engineering' it to see how it works and that it is consistent with the standard Newtonian theory. As I said, the equation must be solved properly as in fact they did. And they got that result without a massive external halo - just (with the Korzynski correction) of an extra infinitely thin disk, which might be modelling the observed thin galactic disk.

This diversion has taken attention away from the main question: as I have now raised several times:"Are the non-linear GR effects significant in galactic rotation, and if so, then what of galactic halo DM?"

Garth

 

[EL]

"Are the non-linear GR effects significant in galactic rotation, and if so, then what of galactic halo DM?"

I would love to see this actually worked out in detail. My impression is that almost everyone (including myself) "feels" that the non-linear corrections would be unimportant, or at least not able to replace the dark matter, but that's of course not a reason why they should be so.

 

[Garth]

Thank you EL, what do you think of the Cooperstock and Tieu approach to the problem? I am at a disadvantage in not being able to get a copy of their reference 6 - van Stockum, W.J., 1937. Proc. R. Soc. Edin. 57, 135, which appears to be quite important; does anybody know where it might be downloaded?

Having been rebutted maybe they will present a revised paper for publication during the refereeing process, but it would be unfortunate if such a paper were not accepted and the question of non-linear effects were simply forgotten.

Garth

 

[Turbot]

van Stockum's work is cited all over the 'net in the context of frame dragging, temporal anomalies, etc, but it does not appear that anybody has transcribed or scanned the original work for download.

Here's a paper that might be interesting to people studying DM distribution.

http://www.csun.edu/~vcphy00d/PDFPublications/2004 FDARB.pdf

We give examples of axially symmetric solutions to the field equations in which zero angular momentum test particles, with respect to nonrotating coordinate systems, acquire angular velocities in the opposite direction of rotation from the sources of the metrics. We refer to this phenomenon as “negative frame dragging.”

 

[pervect]

I'm finding the Cooperstock and Tieu paper very hard to follow. The rebuttal paper is much clearer.

Assuming the rebutal paper is correct about the expression for N(r,z), it's very clear that
N_{z,z} is not well behaved at z=0, which leads me to believe the rebuttal paper. (That's the second partial of N(r,z) with respect to z, in case the notation isn't clear).

At this point, though, I am getting different results for the Einstein equation than Cooperstock and Tieu. I've tried a couple of different approaches

1) Ignore C&T's remarks about $\bar{\Phi}$ and just find the Einstein equations for zero pressure in the coordinate basis.

2) Set up an orthonormal basis of one-forms that creates a diagonal metric, and calculate G with respect to this orthonormal basis. I *think* this is most likely what C&T means by a "local" transform. Unfortunately, I still get different results for the Einstein equation.

In the group of equations given in (5) in

http://arxiv.org/PS_cache/astro-ph/pdf/0507/0507619.pdf

the fourth equation follows from the first two, however I get a different result for the third equation.

$$(3 N^2+r^2)(N_r^2+N_z^2) + 2(N^2-r^2)(V_{r,r}+V_{z,z})$$

It could easily be a mistake on my part. Then Van Stockum paper might help clear up what's going on (but I don't have easy access to it).

 

[SpaceTiger]

As I have been saying this is only my first approximation to the exact Cooperstock & Tieu equation, 'reverse engineering' it to see how it works and that it is consistent with the standard Newtonian theory.

We haven't even started on the fact that their galaxy model is not spherically symmetric, which is what is assumed in those various Newtonian limits you (and I) cited.

I don't know why you're pushing this point, though. If their equations reduce to the Newtonian limit, it means their paper is even more wrong than we already thought.

This diversion has taken attention away from the main question: as I have now raised several times:"Are the non-linear GR effects significant in galactic rotation, and if so, then what of galactic halo DM?"

If you're so curious, do the calculation yourself. My intuition tells me that it's not significant enough to solve the dark matter problem in galaxies, so I wouldn't be prone to waste my time on it. If you feel otherwise, then go for it. If you can show it to be significant, you'll be famous.

 

[Mike2]

My intuition tells me that it's not significant enough to solve the dark matter problem in galaxies, so I wouldn't be prone to waste my time on it. If you feel otherwise, then go for it. If you can show it to be significant, you'll be famous.

If the calculation reveals more mass than otherwise thought, than does that mean the process needs to be iterated to now accommodate the added mass of the previous calculation? Would this series of additional iterations converge quickly or would it eventually add up? Thanks.

 

[Garth]

We haven't even started on the fact that their galaxy model is not spherically symmetric, which is what is assumed in those various Newtonian limits you (and I) cited.
I don't know why you're pushing this point, though. If their equations reduce to the Newtonian limit, it means their paper is even more wrong than we already thought.
If you're so curious, do the calculation yourself. My intuition tells me that it's not significant enough to solve the dark matter problem in galaxies, so I wouldn't be prone to waste my time on it. If you feel otherwise, then go for it. If you can show it to be significant, you'll be famous.

As we, perfect as well as myself, have been saying its not so easy to do. But the question stands, and raises an interesting possibility. That is why we have been trying to understand C&T more deeply. Who knows? The solution might even require a scalar field in addition to the matter field to replace the singular disk!

Garth

 

[Garth]

If the calculation reveals more mass than otherwise thought, than does that mean the process needs to be iterated to now accommodate the added mass of the previous calculation? Would this series of additional iterations converge quickly or would it eventually add up? Thanks.

Its not just the extra mass (kinetic energy) that you have to worry about but also time dilation, angular momentum and frame-dragging as well. Even though orbital velocities are only 10^-3c it is not so obvious that the non-linear accumulative effect can be ignored.

Garth

 

[pervect]

The rebuttal paper by Korzynski

http://arxiv.org/PS_cache/astro-ph/pdf/0508/0508377.pdf

does a pretty good job of setting up the problem. They find there is no solution which is both asymptotically flat and static. This puzzled me for a bit, but I think I may see what's going on.

The problem is attempting to find a static solution for a disk of finite thickness with no pressure. This is indeed not possible.

The 4-acceleration of any point above z=0 must have a downward component. The only way to support a static disk of finite thickness is to have pressure in the z-direction.

Probably the best approach would be to try an analysis similar to that by Korzynski, but in 2 dimensions, not three, and keep the zero-pressure assumption.

 

[EL]

Thank you EL, what do you think of the Cooperstock and Tieu approach to the problem?
Having been rebutted maybe they will present a revised paper for publication during the refereeing process, but it would be unfortunate if such a paper were not accepted and the question of non-linear effects were simply forgotten.

I must admit I'm not enough into GR to say in what way one should approach the problem.
Personaly, my intuition tells me the non-linear effects will turn out to be very small, so I don't feel for digging to deep into the problem either. (Probably I will go for SUSY DM instead.) However, I would of course be happy if someone else finally cleared this out! If you decide to give it a try, I wish you all luck, and I'll look forward to the result.

 

[Garth]

The problem is attempting to find a static solution for a disk of finite thickness with no pressure. This is indeed not possible.

The 4-acceleration of any point above z=0 must have a downward component. The only way to support a static disk of finite thickness is to have pressure in the z-direction.

Probably the best approach would be to try an analysis similar to that by Korzynski, but in 2 dimensions, not three, and keep the zero-pressure assumption.

That makes a lot of sense, although it is not altogether immediately clear why the need for pressure in the z-direction is resolved by adding a singular disk at z = 0!

Garth

 

[pervect]

The sign of the contribution from the delta-function density singularity hasn't been explicitly determined in anything I've read. If it turns out to be repulsive, this would explain the finite thickness, but then one wonders why the solution models galactic rotation which requires more (not less) matter.

 

[Chronos]

My admittedly crude intuition insists a rotating, roughly spherical mass will naturally flatten out into a disc-like structure. Deriving the observed features of galaxies appears almost incomprehensively difficult. I see all kinds of complications - classical physics, turbulence, tidal forces, electromagnetism, backreactions and relativistic corrections. Perhaps dark matter represents an approximation of these combined effects.

 

[Garth]

We haven't even started on the fact that their galaxy model is not spherically symmetric, which is what is assumed in those various Newtonian limits you (and I) cited.
I don't know why you're pushing this point, though. If their equations reduce to the Newtonian limit, it means their paper is even more wrong than we already thought.

Yes, that is where the non-linear effects kick in, its not Newtonian.

Newton delivers flat rotation with a spherically symmetric distribution, whereas GR (if C&T are more or less correct) delivers it with a thin axially symmetric distribution both with an r dependence of:

$$\rho(r)=\frac{a}{r^2}$$.

Garth

 

[Garth]

If it turns out to be repulsive, this would explain the finite thickness, but then one wonders why the solution models galactic rotation which requires more (not less) matter.

Hi pervect - is that the rebuttal correction compared to C&T or C&Tcompared to Newtonian?

The C&T solution requires less matter (no DM halo) than the Newtonian.

Garth

 

[Garth]

My admittedly crude intuition insists a rotating, roughly spherical mass will naturally flatten out into a disc-like structure. Deriving the observed features of galaxies appears almost incomprehensively difficult. I see all kinds of complications - classical physics, turbulence, tidal forces, electromagnetism, backreactions and relativistic corrections. Perhaps dark matter represents an approximation of these combined effects.

Thanks Chronos obviously a detailed model able to explain the spiral arms, the central bulge, the warp in the disk and the contribution of the globular clusters and anything else out there in the form of a DM halo is going to be horribly complicated - not a 'back of the envelope' type of calculation!

However it would be good to get the basic flat rotation profile sorted.

Garth

 

[Chronos]

Agreed, Garth. Would you concede that even the 'simple' model is anything but simple? I think if we could get that much right, the details would be mostly easier.

 

[pervect]

Hi pervect - is that the rebuttal correction compared to C&T or C&Tcompared to Newtonian?
The C&T solution requires less matter (no DM halo) than the Newtonian.
Garth

The article in question is http://arxiv.org/abs/astro-ph/0508377

They first point out that the second derivative of N with respect to z is undefined at z=0 because the first derivative changes sign. Their argument that the disk must contains a "shell" of matter is based on the Komar integral - this is equivalent to the Komar mass, which is mass defined in terms of a Killing vector (this concept of mass is valid in any static space-times). They take the limit of the Komar integral (which gives the enclosed mass) for a cylinder which approaches zero volume (by shrinking the height 'a' of the cyliner to zero), and find that the resulting limit as the height a->0 is non-zero

I don't know offhand whether non-zero means positive, or negative. I see a minus sign in (20), but it's easy to lose track of signs.

BTW the concept of Komar mass is the one found on pg 298 of Wald - I've posted about it before, but never under that name - I didn't realize it had a name until just now (it's handy to know it's name).

 

[Garth]

Thank you, yes I have read the rebuttal paper, its good to have it explained so clearly.

Garth

 

[SpaceTiger]

Interesting sidenote. The speaker at my seminar today used this paper as an example of "How not to do GR."

 

[Garth]

Concur - what is needed is an equivalent to the Kerr metric.

Garth

 

[Garth]

And now it seems spiral galaxy rotation curves are not even flat - as C&L were assuming! Comment on "Scalar-tensor gravity coupled to a global monopole and flat rotation curves" by Lee and Lee

Garth