[barfbagginus]

In my usage, a soliton and a topological defect are synonyms for certain isolated solutions to differential equations. Please replace soliton with topological defect in my statement above and see if it makes more sense.

The idea is there are families of solutions to partial differential equations that are stable, but cannot not deform continuously into any of the "ordinary" solutions without leaving the set of solutions at some intermediate point of deformation. These outlying solutions are topologically isolated Islands in the solution space.

We can think of them as disconnected families of non-standard solutions - solitons.

We can also think about them as topological defects in the set of solutions. We ordinarily expect the solution set of a differential equation to form a path connected set including the trivial solution. If there is some region of solutions that is topologically cut off from the ordinary solutions, we call its solutions topological defects. But these are nothing more than the solutions that can't be smoothly deformed into the zero solution.

[T-A]

The point is not whether you call it a soliton or a topological defect. It's whether the paper's proposal is about a soliton or topological defect "in the gravitational field equations" or in a scalar field model. I'm saying it's the latter: the author is just deriving the gravitational field produced by a topological defect in a scalar field.

[barfbagginus]

Again I consider the terms synonyms.

The solitons/defects belong to the poisson equation for newtonian gravity. I called that a gravitational field equation, since it models gravity as a field.

You called it a scalar field model because it's a scalar field. Your contention is that the gravitational field equations typically refer to Einstein's field equations. I grant that, but also consider poisson's equation as a gravitational field equation. They're both classical field theories for gravity.

Again it's seeming like a bit of a case of potato/potato, rather than advancing the conversation. I'd like you to accommodate me a little better please, and hold off on nitpicking unless it leads to constructive synthesis.

Are you okay with the clarified terminology and my request?

If so, see my other response one layer up, where I asked about problem with the s term in their modified shell solution. We actually have some nits to pick there! I hope to see you there in the other thread!

[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!