3 ms·

Is this saying that dark matter can be modeled well as defects in the topology of spacetime itself?

Drunk_Engineer 2y ago
IANAA, but this says we don't have to rely on with the notion of dark matter at all. The paper proposes a special type of singularity that apparently has all the properties of galactic rotation and bending light normally associated with dark matter.
T-A 2y ago
No, the (hypothetical) topological defects mentioned in Section 7 of the article are configurations of (hypothetical) scalar fields (like the Higgs field, but with different equations of motion): https://en.wikipedia.org/wiki/Kibble%E2%80%93Zurek_mechanism https://en.wikipedia.org/wiki/Kibble%E2%80%93Zurek_mechanism Modeling dark matter as a scalar field is bog-standard stuff, what's unusual here is the way it's arranged.
barfbagginus 2y ago
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.

IANAA, but this says we don't have to rely on with the notion of dark matter at all. The paper proposes a special type of singularity that apparently has all the properties of galactic rotation and bending light normally associated with dark matter.

T-A 2y ago

No, the (hypothetical) topological defects mentioned in Section 7 of the article are configurations of (hypothetical) scalar fields (like the Higgs field, but with different equations of motion): https://en.wikipedia.org/wiki/Kibble%E2%80%93Zurek_mechanism https://en.wikipedia.org/wiki/Kibble%E2%80%93Zurek_mechanism Modeling dark matter as a scalar field is bog-standard stuff, what's unusual here is the way it's arranged.

barfbagginus 2y ago

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.