[barfbagginus]

Topological defects - also known as soliton solutions - are isolated solutions of PDEs that cannot be continuously deformed into the ordinary solutions.

This is saying that the observed effects of DM could be explained as a bunch of enormous solitons in the gravitational field equations. Each soliton has no mass, but together they explain galaxy rotation curves and DM lensing.

But there is a catch. The authors found a massless soliton in the poisson field equation for newtonian gravity. General relatively transforms into this equation in the limit when space is flat. But we don't know if that solution is still stable in spacetimes - like ours - that are curved by massive objects.

In a curved space time being hit by relativistic particles and gravity waves, these spherical shells could wobble and collapse, fade away, or break apart. Proving this one way or another will be interesting work!

[T-A]

> This is saying that the observed effects of DM could be explained as a bunch of enormous solitons in the gravitational field equations.

From the abstract, first sentence, my emphasis:

"Assuming spherical symmetry and weak field, it is shown that if one solves the Poisson equation or the Einstein field equations sourced by a topological defect, i.e. a singularity of a very specific form, the result is a localized gravitational field capable of driving flat rotation"

In Section 7, the author has this to say about the topological defect:

"although its origin is uncertain (despite being deemed stable in Section 3), there are scalar field solutions which can give rise to it (see Hosotani et al. 2002)"

In short, this is not about "solitons in the gravitational field equations"; it's about topological defects in scalar field models. The author derives their gravitational potential in the weak field limit and shows that it has the the right shape to cause flat rotation curves.

I agree with one thing you say: the author's claim of stability is unconvincing. Finding a time-independent solution, as he does in Section 3, shows that the configuration is static. To prove stability, you need to show that small perturbations of the static solution will shrink.

The author's proposal is a set of "unresolvably closely spaced singular shells" (see paragraph under Eq. (8)). But as he notes under Eq. (4), there is no gravitational force inside a spherically symmetric shell. So what's keeping all these shells concentric? As far as I can see, nothing. If you give one of them an ever so small push, it will glide away from its concentric position until it collides with the containing and/or with the contained shell, whichever comes first. At the point(s) of collision, there is gravitational attraction (see paragraph under Eq. (8) again), so the colliding shells will stick together. Even if the hypothetical topological defects survive that (I wouldn't bet on it), they will then proceed together (because of momentum conservation) until they hit the next set of containing and contained shells, and so on. Each collision makes the whole thing less concentric and increases the gravitational attraction at the points of collision. That sure doesn't look stable to me.

[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.

[barfbagginus]

Please forgive me for double dipping, and read my other response on the soliton/topological defect matter.

Re: stability / realism

My concern comes at a point even before yours - the solution given is in a spherically symmetric and perfectly flat space time without any actual matter to bend the gravitational tensor. If it were moving through space with an undulating and assymmetric mass/gravity field, we might expect the shell itself to deform in random ways as it encounters ripples in space. There's no force that would restore it to spherical symmetry, so it might drift and disperse into some weird blob. The extra degrees of freedom of the curved space may even let the solution wiggle and diffuse into an ordinary solution, so that it no longer represents a topological defect in the solution space.

The shells certainly wouldn't remain concentric in my thinking, and might not need to. In my thinking I don't see that the shells necessarily need to impose forces against each other when they pass through each other. Imagine an onion shell where the onion layers are free to phase through each other and become a bit wibbly wobbly.

Eventually you have a space with many topological singularity walls of this kind of passing through each other. This random "space lasagna" is actually much more realistic than concentric shells. A problem I see though is that we might need concentricity to induce a centrally directed attractive force keeping galaxies together.

Ie, why does random isotropic space lasagna bind together spherical galaxies and clusters?

Actually modeling this more realistic and chaotic version of the topological defect solution could be very exciting! Of course it's no more satisfying than dark matter, because we don't know what would actually cause the singularity lasagna solutions to form in the first place, instead of just ordinary solutions.

But wait there might even be a problem before this!

Re: the dirac delta s term.

In the definition of the modified shell model on the first page, we have a constant term s such that sδ(R-r)=1 if r=R. This might be nonsense.

First let's point out that R-r=0, so this implies sδ(0) = 1. But there is no real number s such that sδ(0) = 1 is finite, since d(0) is not a real number.

But okay, maybe s is a distribution that transforms the dirac delta distribution into the kronecker delta function. If so which distribution is it?

The Fourier transform gives us a clue. Let k be the kronecker delta at 0

k(x) = 1 if x=0, else 0.

s now scales δ into k:

sδ = k

Applying the fourier transform we get

F(sδ) = F(k)

F(s) * 1 = 0

where * is convolution and 1 and 0 are constant functions.

Note that F(k) = 0 because F(0) = 0 and k = 0 almost everywhere.

Now

[F(s)*1](x) = 0 for all x

Which implies

∫F(s) = 0.

So we know that s can't be a constant, and its Fourier transform has a vanishing integral.

This isn't enough for me to reject s, but I'm puzzling as to what it actually is. What do you think about this?

[T-A]

Sorry, saw this only now. The author is assuming that the spherical shell is a topological defect in some Higgs-like field. Generally speaking, if the coupling constant(s) of a field are large compared to gravity, it's reasonable to ignore gravity when looking for solutions to its equations of motion (just like you usually ignore gravity while doing electric engineering). Once you find one, you can switch on gravity and verify that it still holds. In the case at hand, it's all conjecture, since the author just assumes existence of a suitable solution.