[T-A]

> The solitons/defects belong to the poisson equation for newtonian gravity.

No.

You are confusing the scalar potential of gravity in the weak-field approximation with the hypothetical scalar field which acts as its source. Those are two completely different things.

The author does NOT derive or even show a topological defect in the scalar potential. What he does is:

1) Posit the existence of a weird mass distribution (a planar dipole composed of a negative and a positive mass layer, shaped into a sphere).

2) Put this weird mass distribution on the RIGHT hand side of Einstein's equations, where all sources of gravity belong:

https://en.wikipedia.org/wiki/Einstein_field_equations

3) Show that in the weak field approximation, the LEFT hand side of Einstein's equations, which describes the gravitational field, then reduces to the form needed to have flat rotation curves.

At this point, you have the scalar potential of gravity on the LEFT hand side and its weird, unexplained matter source on the RIGHT hand side.

4) Since the weird mass distribution on the RIGHT hand side can not be produced by any known form of matter, the author then proceeds to say that it may be caused by a topological defect of the kind referenced in Section 7.

The references in Section 7 are about topological defects which arise in field-theoretic models of Higgs-like scalars. The only thing those have in common with the scalar potential of weak-field gravity is the word "scalar". The equations of motion which determine their evolution are completely different from those of gravity. Their role in the author's story is to act as SOURCES for the gravitational field, by producing the weird mass distribution he needs.

[barfbagginus]

I'm doing way more than confusing things - all this stuff is mainly new to me!

I'll try to understand your comment over the next few days

BUT what do you think about that weird business with the s scaling factor that supposedly rescales the Delta function so that it takes a finite value at 0? I can't yet prove what kind of animal s is supposed to be. It's like some kind of funky infinitesimal.

[nyssos]

> BUT what do you think about that weird business with the s scaling factor that supposedly rescales the Delta function so that it takes a finite value at 0? I can't yet prove what kind of animal s is supposed to be. It's like some kind of funky infinitesimal.

Strictly speaking, it's nonsense. What they're probably actually doing is taking the limit of the result for shells of nonzero thickness as thickness goes to zero.

[barfbagginus]

That's not what the s factor controls. It's a scaling factor on the intensity of the dirac shell itself, not the radius or positioning.

s acts something like the 1/d, a reciprocal of the dirac delta function. But this requires some careful technical attention. For example it might only invert the Dirac Delta functio on its support at the origin, and leave it zero elsewhere. But even then that notion is problematic. We might require some generalization of the distributions to allow reciprocals of distributions like this.

Another idea is that it maybe be are working in the limit as s goes to zero, like you suggest, but what they are attenuating is the intensity of the shell (without ever literally inverting the delta function)

Read the paper and try to work it out if you know a bit about distributions. It's a fun mystery right now!

[nyssos]

Sorry, my wording was a bit unclear: what I meant was that they were probably taking the limit of

    \int_{0}^{r} s_{i} \* f_{i}(R - r)/r^2 + f_{i}'(R-r)/r

with respect to some suitable sequence of functions `f_{i}` (narrower and narrower gaussians, for instance), for which `s` is actually well-defined.