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

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 2y ago
> 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 2y ago
  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 2y ago
    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 2y ago
      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 2y ago

> 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 2y ago
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 2y ago
  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 2y ago
    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!