Re: OT: Dark Matter....does it....matter that is?

Bobby D. Bryant wrote:
> On Tue, 11 Oct 2005, "David Ewan Kahana" <dek@xxxxxxx> wrote:
>
> > Cyde Weys wrote:
> >> Ken Shackleton wrote:
> >> > I have a question about Dark Matter:
> >> >
> >> > Is there any evidence to support its existence? My understanding is
> >> > that it is simply a plug that fills a theoretical gap in gravitaional
> >> > theory with respect to stellar orbits around the galactic centre.
> >> >
> >> > This kinda reminds me of the Ether Theory about a century ago used to
> >> > explain how light propegates through space; or the circles within
> >> > circles used to explain planetary orbits before the elliptical nature
> >> > of planetary orbits was discovered.
> >> >
> >> > Any input would be appreciated.
> >>
> >> Aristocranes says:
> >>
> >> You may be on to something. Here is a reference to a very recently
> >> released paper that seems to be in agreement with what you are saying:
> >> < http://xxx.lanl.gov/abs/astro-ph/0507619 >
> >
> > It looks like an interesting paper to be sure, though I
> > haven't checked all of their maths.
> >

I finally found a little time to check the mathematics.

There's an error. The paper is simply wrong as it stands.
The problem is that the authors don't solve the equations
that they write down.

In particular a function of the form contained in
equation (16) is not a harmonic function in the
half plane (r, z) in cylindrical coordinates.

As a result, the fourth of the constraint equations among
the field equations to this order in the `weak field expansion,'
equation (5d), also eqn (11), is not solved by the form given
in eqn (17).

The proposed solution contains a function whose
derivative is discontinuous at z=0, as a result of which
the rhs of eqn (5d) contains a delta function, singular
at z=0, while the lhs should be equal to zero for all r and z.

Footnote 11 states the following:

`The absolute value of z must be used to provide the
proper reflection of the distribution for negative
z. While this produces a discontinuity in N_z at
z=0, it is important to note that this has no physical
consequence since N_z enters as a square in the density
and N_z does not play a role in the equations. Moreover,
the metric itself is continuous. This is analogous
to the Scwarzchild constant density sphere problem
that leads to a discontinuity in metric derivative
across the matter-vacuum interface in Schwarzchild
coordinates. In principle other coordinates could be
found to render the partial derivatives globally
continuous but this would be counter-productive
as it would unnecessarily complicate the mathematics.
As in FRW, our comoving coordinates simplify the
analysis.'

This footnote serves as a great big red flag.

The problem is that N_zz, the second derivative of N with
respect to z does play a role in the equations. Making a
change of coordinates to eliminate discontinuities is
completely irrelevant, when the metric doesn't satisfy the
field equations in the first place.

We can solve the Laplace equation (11) for N by separation of
variables, that is, by searching for a solution in the
form f(r)*g(z). This immediately yields the result that g(z)
may be either a combination of sines and cosines, or
it may be a combination of exponentials in z. Exponential
solutions in z are physically unacceptable, since they
increase without bound either at positive or negative z,
or at both.

The solutions in f(r) are either ordinary or modified
cylindrical bessel functions depending on whether exponential
or oscillatory behaviour is chosen for g(z). Since oscillatory
behaviour in z must be chosen, it follows that f(r) will
be a combination of modified cylindrical bessel functions
either regular or irregular at r=0.

One could indeed search for a solution to equation (12) using
a superposition of solutions of equation (11), say given
an approximate fit to the distribution of luminous matter in
a galaxy. This requires a bit of numerical work, and it seems
highly unlikely that the velocity field will now come out
constant as a function of r. More likely, something close
to the Newtonian result will come out.

> > But the assumption that ordinary galactic matter could be
> > modelled as a pressure-free fluid seems odd on it's face.
> > I'm not quite sure how that can be reconciled with models
> > of the initial stages of galaxy formation, which, either
> > in the case of Newtonian modelling or in the case of general
> > relativistic modelling always have involved the treatment of
> > self-gravitating fluids which have non-trivial equations of
> > state. Both pressure and viscosity would seem to be important,
> > in fact, unless one has some other way of introducing rotation
> > into the systems.
> >
> > I'ld like to know what the authors have to say about
> > those questions, and whether they've considered them.
>
> Notice that it is "submitted", and thus still may not even pass peer
> review.
>
> Slashdot ran the story Monday and one person replied with a link to
> another unpublished paper, a rebuttal of this one. That author claims
> that this model has the untenable side effect of requiring a singular
> disk at z=0, or something to that effect.
>

The rebuttal sounds correct to me. I'll look for the reference
later.


David

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