Authors: Mikolaj Korzynski

Abstract:

Recently a new model of galactic gravitational field, based on ordinary General Relativity, has been proposed by Cooperstock and Tieu in which no exotic dark matter is needed to fit the observed rotation curve to a reasonable ordinary matter distribution. 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.

I think that they are arguing that what you are asserting is true in the usual analysis, but when the co-moving coordinates are used in the analysis this causes w to remain constant at 0 and consequently the field equation becomes non-linear. The justification of this approach comes from the notion of a “gravitational-bound” system, whatever that means. The upshot is that, not withstanding the weak galactic field, the galactic dynamics are non-linear:

…insufficient attention has been paid to the fact that the stars that compose the galaxies are essentially in motion under gravity alone (“gravitationally bound”). It has been known since the time of Eddington that the gravitationally bound problem in general relativity is an intrinsically non-linear problem even when the conditions are such that the field is weak and the motions are non-relativistic, at least in the time-dependent case. Most significantly, we have found that under these conditions, the general relativistic analysis of the problem is also non-linear for the stationary (non-time-dependent) case at hand.

I can’t argue the merits of this assertion. All that I can do is ask that someone who can do so. So far, I don’t think anyone has, have they?

What is the physics (as opposed to what is the math) of what the authors are arguing? It is that the rotation of the galaxy provides a contribution to the metric that is much stronger than the Newtonian contribution. Based on what we know about gravity and various weak field expansions that exist for GR, this is implausible. How do effects that die off at (1/r)^2, and or faster sum up to exceed the (1/r) contribution? Moreover the source of the (1/r)^2 effects is v/c smaller than the source of the (1/r) contribution.

Absolute certitude requires redoing the calculation independently. For something very implausible, it is not worth the effort, at least, for someone like Sean. However, it may be a good exercise to give students.

-Arun

7Normally, the fall-off of w with R = (r2 + z2)^-1/2 is used to derive the total mass of an isolated system. However, w is constant in this system of coordinates by (9) and we cannot do so here. The w constancy does not imply that that the mass is zero. In other (non-co-moving) coordinate systems, w would be seen to be variable. With the field being weak and the system being non-relativistic, the mass is well-approximated simply by the integral of p over coordinate volume.

The constancy of w seems to be the key to their argument for non-linearity, so if non-linearity fails, as Sean asserts, is it because their argument for the constancy of w fails? They write:

It is to be noted that it is the freely gravitating motion of the source material (the stars) in conjunction with the choice of co-moving coordinates (2) that leads to the constancy of w within the source. Had there been pressure, w would have been variable… the non-linearity of the galactic dynamical problem is manifest through the non-linear relation between the functions rho and N. Rotation under freely gravitating motion is the key here. By contrast, for time-independence in the non-rotating problem, there must be pressure present to maintain a static configuration, N vanishes for vanishing w and del^2w is non-zero yielding the familiar Poisson equation of Newtonian gravity.

Is this a valid argument or not? If not, why not? It seems to me that we are trying to dismiss their argument rather than trying to answer it.

Anyway, I see that the paper has been submitted to the Astrophysics Journal. How do referees deal with such?

ds^2 = (1- 2M/r) dt^2 – dr^2/(1- 2M/r) – r^2 ( dθ^2 + sinθ^2 dφ^2)
+ ( 4S sin&theta)/r^2 ( r sinθ dφ) dt

The last term is non-Newtonian and resembles the term with the function N
from the paper. A rough estimation of its value at the periphery of a
galaxy shows it to be much smaller (by v/c, where v is an average stellar
velocity and by geometric factors – most of the mass contributes very
little to the angular momentum) than the Newtonian potential M/r.

I translate the main claim of the paper to be that the periphery of the galaxy is higly non-Newtonian, and motion is dominated by the angular momentum term. The non-linearity claim I translate to the claim that the frame dragging term accumulates as one moves from the center of the galaxy to the periphery until it dominates the Newtonian potential. I do not know enough yet to be able to readily rule it out.

Does the fact that the constituent stars of the galaxy are large rotating systems that contribute to the field, unlike the constituent planets of solar systems, play into the non-linearity of the field equation so significantly, as he claims?

He writes:

Since the field equation for rho is non-linear, the simpler way to proceed in galactic modeling is to first find the required generating potential phi and from this, derive an appropriate function N for the galaxy that is being analyzed. With N found, (12) yields the density distribution. If this is in accord with observations, the efficacy of the approach is established.

Equation 12: (N2r+N2z)/r2 = (8piGrho)/c2

If the non-linearity argument fails, then it doesn’t matter, but if it holds, then is this approach valid, as he claims?

General relativity resolves galactic rotation without exotic dark matter

I too am really anxious to hear from the experts on this. Can you imagine what eliminatiing dark matter from the picture will do? Dark energy now gets bumped up to about 97%, right? Sean, you’ll have to redo all those pie charts! LOL.

It seems to be quantization with two contributions, to a high accuracy.