## [astro-ph/0507619] General Relativity Resolves Galactic Rotation Without Exotic Dark Matter

 Authors: F. I. Cooperstock, S. Tieu Abstract: A galaxy is modeled as a stationary axially symmetric pressure-free fluid in general relativity. For the weak gravitational fields under consideration, the field equations and the equations of motion ultimately lead to one linear and one nonlinear equation relating the angular velocity to the fluid density. It is shown that the rotation curves for the Milky Way, NGC 3031, NGC 3198 and NGC 7331 are consistent with the mass density distributions of the visible matter concentrated in flattened disks. Thus the need for a massive halo of exotic dark matter is removed. For these galaxies we determine the mass density for the luminous threshold as 10^{-21.75} kg.m\$^{-3}. [PDF]  [PS]  [BibTex]  [Bookmark]

Discussion related to specific recent arXiv papers
Garth Antony Barber
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### [astro-ph/0507619] General Relativity Resolves Galactic Rota

This would seem to be an important paper in the study of DM galactic halos.
One might be inclined to question how this large departure from the Newtonian picture regarding galactic rotation curves could have arisen since the planetary motion problem is also a gravitationally bound system and the deviations there using general relativity are so small. The reason is that the two problems are very different: in the planetary problem, the source of gravity is the sun and the planets are treated as test particles inthis field (apart from contributing minor perturbations when necessary). They respond to the field of the sun but they do not contribute to the field. By contrast, in the galaxy problem, the source of the field is the combined rotating mass of all of the freely- gravitating elements themselves that compose the galaxy.
We have seen that the non-linearity for the computation of density inherent in the Einstein field equations for a stationary axially-symmetric pressure-free mass distribution, even in the case of weak fields, leads to the correct galactic velocity curves as opposed to the incorrect curves that had been derived on the basis of Newtonian gravitational theory.
(Emphasis mine)

Garth Antony Barber
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### [astro-ph/0507619] General Relativity Resolves Galactic Rota

Well, note also astro-ph/0508377 "Singular disk of matter in the Cooperstock and Tieu galaxy model"
We argue that in this model the gravitational field is generated not only by the galaxy matter, but by a thin, singular disk as well. The model should therefore be considered unphysical.
In fact I do not agree with its conclusion that the Cooperstock & Tieu galaxy model is 'unphysical', the galaxy does have a disk - the galactic disk in which the spiral arms are embedded, it consists of stars, dust and gas including massive molecular clouds. The total galactic mass Cooperstock & Tieu require is 21×10^{10}M_{&#8857;} which is totally realistic.

Chris Vale
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### [astro-ph/0507619] General Relativity Resolves Galactic Rota

The reasoning in this paper looks like circular logic to me. A constant velocity rotation curve is assumed, and the mass distribution is solved for, yielding the familiar \rho ~ 1/r^2 result, which is the same as the Newtonian answer. It does not answer the central issue, which is that the observed Baryon mass distribution doesn't follow this profile, implying the existence of dark matter...

Garth Antony Barber
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### [astro-ph/0507619] General Relativity Resolves Galactic Rota

Thank you Chris, I obtain your result as follows:

'Reverse engineering' the Cooperstock & Tieu relationship in one certain regime, that of flat rotation, the orbital velocity may be approximated as a constant.
$V (r, z) =\frac{3.10^8}{r}N(r, z)=constant$

so $N(r,z) = C.r$

therefore $N_r = C$ and $N_z = 0$, where subscripts are partial differentiation w.r.t. that coordinate.

so the density expression does approximate that of the isothermal sphere,
as C&T derive
$\rho=5.64 . 10^{-14}\frac{(N_r^2+N_z^2)}{r^2} kg/m^3$
then
$\rho=\frac{A}{r^2}$

This is only a first approximation to the exact Cooperstock & Tieu field equation, from which they derived a solution without a massive external halo - just (with the Korzynski correction) of an extra infinitely thin disk, which might be a crude model of the observed thin galactic disk.

Korzynski does not state how massive this singular disk is but I cannot believe it is of the order of the DM halo it replaces as orbital velocities would be affected too much.

The difference between this result and the Newtonian limit is that the non-linear effects achieve flat rotation in a thin axially symmetric solution rather than a spherically symmetric one, and without a massive halo.

Notwithstanding the validity or not of the (corrected) C&T model the general question left to be resolved is:"Are the non-linear GR effects significant in galactic rotation, and if so, then what does that do to Newtonian predictions of the galactic halo DM?"

Garth Antony Barber
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### Re: [astro-ph/0507619] General Relativity Resolves Galactic

Garth Antony Barber wrote:Notwithstanding the validity or not of the (corrected) C&T model the general question left to be resolved is:"Are the non-linear GR effects significant in galactic rotation, and if so, then what does that do to Newtonian predictions of the galactic halo DM?"
The same point is made by Vogt & Letelier in their paper "Presence of exotic matter in the Cooperstock and Tieu galaxy model" astro-ph/0510750
Although the proposed galactic model does not really resolve galactic rotation without the presence of exotic matter, we believe that the idea of treating the non-linear galactic dynamical problem in the context of General Relativity is quite interesting and should be further investigated, specially the rotating models where we have the non-Newtonian effect of dragging of inertial frames; a modest step in this direction is presented in [9].
That reference 9 was to their paper "Relativistic Models of Galaxies" astro-ph/0507406.
We also calculated the first order effects of galactic rotation on the tangential velocity of circular orbits on the galactic plane using an approximate form of the Kerr metric expressed in cylindrical isotropic coordinates. In general, rotation increases the progade tangential velocity and has an opposite effect on the retrogade tangential velocity.

Boud Roukema
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### Re: [astro-ph/0507619] General Relativity Resolves Galactic

Chris Vale wrote:The reasoning in this paper looks like circular logic to me. A constant velocity rotation curve is assumed, and the mass distribution is solved for, yielding the familiar \rho ~ 1/r^2 result, which is the same as the Newtonian answer. It does not answer the central issue, which is that the observed Baryon mass distribution doesn't follow this profile, implying the existence of dark matter...
I agree - equations (12) + (15) show that for a flat rotation curve, the solution presented here is \rho ~ 1/r^2 . I don't see any difference from the newtonian result.

Garth Antony Barber
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### Re: [astro-ph/0507619] General Relativity Resolves Galactic

Boud Roukema wrote: I agree - equations (12) + (15) show that for a flat rotation curve, the solution presented here is \rho ~ 1/r^2 . I don't see any difference from the newtonian result.
But the paper does raise the interesting and unresolved question of whether GR non-linear effects are significant in determining spiral galactic halo DM.

The subject has also been addressed before by others such as Vogt & Letelier in "Relativistic Models of Galaxies" http://arxiv.org/abs/astro-ph/0507406 and earlier authors cited therein.

Garth

Tommy Anderberg
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### [astro-ph/0507619] General Relativity Resolves Galactic Rota

They are back, this time without singular disk:

http://arxiv.org/abs/astro-ph/0512048

This is interesting (and I am envious; at least they "received a large volume of correspondence with interesting questions, comments, suggestions and criticism"...).

Daniel Grumiller
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### [astro-ph/0507619] General Relativity Resolves Galactic Rota

Here is my assessment:

The overall idea by Cooperstock and Tieu (CT), namely to try to take General Relativity (GR) more seriously and thereby get rid of Dark Matter, is laudable. However, it did not work.

The idea to reverse-engineer the mass distribution from the velocity profile is certainly a good idea, at least from a theoretical point of view - this makes the whole approach simpler and IMHO it is equally good to take the velocity profile as input and to predict the mass density distribution or to follow the traditional route and do the opposite. The prediction for the mass density profile then has to be checked with experimental data, of course.

Obviously, flat rotation curves are then no longer a prediction but an input, but to balance this the mass density no longer is an input but a prediction. So no loss on the phenomenological side: one curve is input, another is predicted.

Korzynski did not yet show that the model is unphysical, but he pointed out an important mathematical issue (well-known to any student who recently had exams on electrodynamics), namely that CT do not solve the equation they pretend to solve: their Laplace-like equation has not a zero but some delta-function on the r.h.s.! Things like these happen every day in ordinary electrodynamics - think of the charged disk for instance.

However, Vogt and Letelier demonstrated that the additional disk of matter needed by CT is unphysical. Moreover, one can show easily that this singular disk extends over the whole galaxy, so one cannot "hand-wave" it away by appealing to something like the central galactic black hole.

In short, the original CT model is dead and the "second edition" astro-ph/0512048 is, with all due respect, nonsense.

Actually, Herbert Balasin and I have been toying around since last Summer (after CT) with a similar model.

The good news is that we have found a way that resolves the problems and inconsistencies of the CT model.

The bad news, if you wish to call them bad, is that Dark Matter is still needed, but somewhat surprisingly there is a considerable difference (of about 30%) to the Newtonian calculations!

The results are described in astro-ph/0602519

For the impatient among you here is a brief summary:

Let us suppose, as CT do, axial symmetry and stationarity, together with a corotating perfect fluid which is pressurless. One can find exact solutions to the Einstein equations, because the difference to the CT system of equations is marginal (actually, only one equation differs). To derive this you need hardly more than Wald's book General Relativity.

Now comes the crucial point: in order to solve the Laplace-like equation (Eq. (9) in our paper and Eq. (5) in the first CT paper) one has two possibilities:

A) modes in z-direction oscillate

B) modes in radial direction oscillate

CT chose B) and consequently the modes in z-direction either grow exponentially for large z (and thus are unphysical) or, if they decay, there has to be a singular layer at z=0 (which is what CT arrived at without noticing).

However, it is possible to choose A), which is what we did. In this way one avoids a singular disk at z=0. Now one would like to avoid exponential growth in radial direction, which is possible, but then one has a singular layer at r=0.

But, and this is our physical point here, the axis r=0 is not described very well by the pressurless perfect fluid approximation anyhow. Moreover, around this axis the galactic BH sits and jets are emitted, so the emergence of sources there is perhaps not too surprising. Therefore, we suggested to cut out a cylindrical region around the axis where a different solution of GR has to be pasted, in a sufficiently smooth manner (as to avoid sources in the matching layer). This is possible on general grounds, but it will require sources within the cylinder, read, jets. So in a sense, we predict the necessity of jets.

The main technical result is Eq. (12), which provides the general solution consistent with all boundary- and fall-off conditions (and with the assumed Z_2-symmetry around the disk). It depends on the spectral density C(x) which has to be chosen according to experimental needs. Note that each mode already has a "nice" radial behaviour and falls off for large z and for large radii.

For pedagogical reasons we provide a very simple choice for the spectral density, which fits more or less the observed "flat" velocity profile.

The final point concerns comparison with Newton. To our surprise we have discovered that Newtonian calculations consistently over-estimate the amount of Dark Matter needed to explain the flat rotation curves by about 30% (regardless of the particular choice of the spectral density, as long as it yields rotation curves which are approximately linear in one regime and approximately flat in another regime).

The reason is subtle: although GR may be approximated very well by Newtonian gravity both in the linear and the flat regime independently, the fact remains that in the GLOBAL GR solution there is an integration constant which can only be fixed once. If one fixes it in such a way as to achieve consistency with Newton in the linear regime (which is what people usually seem to do), it turns out that in the flat regime a discrepancy of 33% emerges.

Thus, GR need less Dark Matter than Newton (about 30%), but since Dark Matter is a "500% effect" it is still needed.

So the bottomline is: CT were correct in suspecting that GR might have to say something of relevance on galactic rotation curves, but their mathematical implementation was, unfortunately, erroneous and unphysical. In astro-ph/0602519 Herbert and I have provided some remedy.

Tommy Anderberg
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### [astro-ph/0507619] General Relativity Resolves Galactic Rota

Very interesting, thanks! Reading your paper very quickly (past midnight) two things come to mind:

- You seem to have come up with a cure for the central CDM cusp problem, but to be too modest to emphasize this point.

- Since I'm evil to the core, I would really like to see a discussion of the implications of 30% less dark matter for the concordance model... ;)

Garth Antony Barber
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### Re: [astro-ph/0507619] General Relativity Resolves Galactic

Tommy Anderberg wrote:Very interesting, thanks! Reading your paper very quickly (past midnight) two things come to mind:

- You seem to have come up with a cure for the central CDM cusp problem, but to be too modest to emphasize this point.

- Since I'm evil to the core, I would really like to see a discussion of the implications of 30% less dark matter for the concordance model... ;)
It's only galactic halo DM we are talking about, is it not?, There is still cluster DM and IGM DM - C&L wanted to explain away all DM but I think that was a hope too far.

Garth

Daniel Grumiller
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### [astro-ph/0507619] General Relativity Resolves Galactic Rota

Here are some remarks regarding Tommy Anderberg's interesting points:
You seem to have come up with a cure for the central CDM cusp problem, but to be too modest to emphasize this point.
Perhaps you are right. But we wanted to be cautious and restrict ourselves to the core (no pun intended) results.

Trying to cure the cusp problem with our approach certainly is an attractive possibility. (Actually, there are many further points that would be interesting to address, but we confined ourselves to Letter size) To this end one has to come up with

1.) a better "engineering" of the spectral density C(x)

and

2.) an explicit construction for the matching around the central axis which yields "reasonable" sources
Since I'm evil to the core, I would really like to see a discussion of the implications of 30% less dark matter for the concordance model... ;)
Please correct me if I am mistaken, but I think experiments and observations, although already excellent by comparison to a decade ago, are not yet in the range to see this 30% effect. (Recall that 30% here just means "30% less than Newton would predict")

So our slight modification of the concordance model by no means is a "revolution", but this effect still should be of relevance in the (hopefully not too distant) future.

Tommy Anderberg
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### [astro-ph/0507619] General Relativity Resolves Galactic Rota

Garth Antony Barber wrote:
It's only galactic halo DM we are talking about, is it not?, There is still cluster DM and IGM DM
Good point, but as noted in astro-ph/0512048,
[...] for the dynamics of clusters of galaxies, the virial theorem is used. This is based on Newtonian gravity theory. It would be of interest to introduce a general relativistic virial theorem for comparison. It is only after possible effects of general relativity are explored that we can be confident about the viability or non-viability of exotic dark matter in nature.
In other words, if Newtonian gravity is off by 30% for galaxies, how does it fare for clusters?

Daniel Grumiller wrote:
Please correct me if I am mistaken, but I think experiments and observations, although already excellent by comparison to a decade ago, are not yet in the range to see this 30% effect.
I'd like to know too - observers, please pipe up! Given that the total to luminous mass ratio inferred from galactic rotation curves is 5:1, it would seem to me that a 30% correction should be readily noticeable (taking us to < 4:1), but I may be completely wrong about this.

Garth Antony Barber
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### Re: [astro-ph/0507619] General Relativity Resolves Galactic

Tommy Anderberg wrote:Garth Antony Barber wrote:
It's only galactic halo DM we are talking about, is it not?, There is still cluster DM and IGM DM
Good point, but as noted in astro-ph/0512048,
[...] for the dynamics of clusters of galaxies, the virial theorem is used. This is based on Newtonian gravity theory. It would be of interest to introduce a general relativistic virial theorem for comparison. It is only after possible effects of general relativity are explored that we can be confident about the viability or non-viability of exotic dark matter in nature.
In other words, if Newtonian gravity is off by 30% for galaxies, how does it fare for clusters?
Cluster DM has also been observed by gravitational lensing of distant quasars, do the GR non-linear effects achieve this effect without extra mass?

Garth

Tommy Anderberg
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### [astro-ph/0507619] General Relativity Resolves Galactic Rota

Garth Antony Barber wrote:
Cluster DM has also been observed by gravitational lensing of distant quasars, do the GR non-linear effects achieve this effect without extra mass?
You are the relativist; you tell me. My intuitive answer would be that any GR effect which reduces the need for DM with regard to orbital motion should also mimic the effect of the replaced DM on test particles, zero mass ones included.

But suppose it didn't work out that way. If galaxies are 20% less massive than we thought, but we insist that Newtonian gravity is fine for clusters, the missing mass problem just got 20% worse for clusters. So we throw in - let me guess, 20% more dark matter at the cluster scale. What does that do to structure formation?

Questions, questions...