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.