The binding of cosmological structures by massless topological defects
academic.oup.com
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I checked, and the publish date is not in fact the first of April.
Its extremely implausible, putting it generously, that such fine-tuned structures could even form. They mention at the end "this paper does not attempt to tackle the problem of structure formation" but that feels like a colossal understatement.
I've studied physics myself, and I understand that sometimes people toy around with implausible theories solely for the sake of it but .... it seems like these peoples brainpower could be much better spent elsewhere.
They are massless spherical shells, i.e. a sphere embedded in 3d space and thus can be continuously deformed to a point by sending the radius to zero.
Spheres are literally the second homotopy group and the homotopy group of flat space is zero. Q.E.D
The distribution used works roughly as if we overlapped two equal positive and negative mass shells. But there are some extra details that ensure there is a net inward force for matter situated on the final shell. We'd have to actually work the math to really understand why that force appears, without hand waving.
Topological defects are solutions to the underlying physical equations that are of a different homotopy class to the vacuum, and these simply aren't in a different homotopy class, as I can smoothly deform them to zero, by sending the radius to zero or sending alpha to zero.
Argued another way: a point charge can be modelled as a dirac delta charge distribution, but nobody would argue that a point charge is a topological defect
Now I also realize that the paper seems to say that the both the ordinary dirac shell solution and their modified shell are TDs, without proving it.
I'd like to work up to proving whether the collapsing modified shell really does homotopy into into a point and then fade away into the vacuum.
But first, I'm struggling with
> nobody would argue that a point charge is a topological defect
It's actually not clear to me that the point mass is not a TD!
Let's try to write the homotopy sending a point mass Mδ₀ to the vacuum solution 0. Let t vary from 0 to 1. Then a possible homotopy is
h(t) = (1-t)Mδ₀
This gives us
h(0) = Mδ₀
h(1) = 0
The problem is that I don't know how to prove continuity of h.
First, I don't know how to compute even the continuity of neighboring Delta functions for t < 1. But that feels intuitively like it should be continuous.
On the other hand, I REALLY don't know how to prove continuity at t = 0, since the function seems to spontaneously collapse from a distribution with a kind of pseudoinfinite value at the origin, into a regular function with the value 0 at the origin.
Using the notation of inner products and test functions, can we prove that it's continuous both for t < 1 and t = 1?
I know that's a bit more technical than we usually get here on HN! I truly appreciate the help!
You can solve the above by remembering that the dirac delta is the limit of a series of functions.
If you take your delta to be lim a -> 0 N(0, a) where N() is the normal distribution, then you can see that in your above equation, you then have two limits. lim t -> 0 and lim a -> 0. By swapping the order of the two (which is a dubious operation), you can send t to zero first then a to zero, and the result is zero.
So in one way, it can be deformed to zero, in another way it can't, because it's 0 times infinity .
However, the thing to focus on is that dirac deltas aren't actually valid points in the solution space of partial differential equations. They aren't functions, and they aren't actually physically real.
Come to think of it, that would probably exclude them from being TDs a-priori. Because a TD must be a solution to an underlying physical equation, and that solution must be deformable to zero. But if it's not a solution to a PDE (because it doesn't live in any valid hilbert space), then it can't be a TD.
Conceptually this is how we would model a field with interacting particles using PDEs, and is as far as I know the reason we solve with distributions and consider them physical in the first place.
The question of whether dirac is homotopic to vacuum would then mean exhibiting a homotopy in D', which is the dual space to the underlying function space, and represents both functions and distributions as functionals on a space of test functions. The topology of D' is the weak-* topology. Given a net N_t of distributions, N_t converges to N if N_t(g) converges to N(g) for all test functions g.
So to make your proof tight, I propose we should lift it into D', where our family u_t of shrinking deltas is already a net. We'd then prove that u_t -> 0 converges for each test function.
I think the result would be much more satisfying and watertight, since your current proof does not really make up its mind about what it's conclusion is.
Now, I am not sure that the distributions need to be a hilbert spaces to solve PDEs in this sense. This is because distributions have derivatives and can be acted on by our lifted differential operators. These solutions are weak in the sense that they are solutions with respect to the test functions. But I also believe that the solutions are faithful, in the sense that any solution corresponding to a function in the base space will also be a solution for the original operator.
Also, if you need our space of distributions to be a Hilbert space, we can pass to distributions over sobolev n space, the space of functions with square integrable nth derivatives. The distributions here end up being hilbert spaces! Not only that but the square integrability condition is pretty mild and still leaves us a lot of very useful functions for physics. I think that some quantum theories use these kinds of Hilbert spaces. Lol, take that with a grain of salt - this is lore to me, not terra firma.
Anyhow, if you want to try to persuade yourself that we can't just throw away delta functions and other distributions, and then try to lift your proof into space of distributions, I think that would be very fun!
I'll try to do that as well!
Ps. Forgive me if I'm Tom Sawyering you or nerd sniping! Please also forgive me for any mistakes I'm making!
Nevertheless, societies still need physicists, and those physicists, by the nature of how incredibly challenging the "industrial" problems are (e.g. designing next generation fusion weapons), need challenging problems. Indeed, they crave them. And so you're sort of stuck solving progressively more esoteric and even contrived problems.
- Celestial Mechanics (support: the descendants of von Braun): dynamical systems, linear algebra, topology, variational calculus, symplectic geometry
- Cryptography (support: CIA, SVR, etc.): number theory, algebraic geometry over finite fields, algebra, combinatorics, and computers.
- Hydrodynamics (support: boomer builders): complex analysis, partial derivative eqn's, Lie groups/algebras, cohomology theory, HPC.
Might there be one small village of pure mathematicians still holding out against the industrial support? (resident druid: Dunforkix)
Lagniappe: https://www.smbc-comics.com/comic/apply
Good Science spreads into Technology where you least expect: https://youtu.be/-OkwGDKoY0o?t=2715
Even the study of Black hole horizons could a have a direct and immediate usefulness for your Fluid Mechanics Engineers: https://youtu.be/-OkwGDKoY0o?t=2851
However, it's not clear how to make progress in these areas.
First, the number of shells is finite. This actually gives us one of the points where we can test this theory:
Namely, if there are discrete shells, their effect on gravitational lensing will introduce artifacts that we may be able to detect. We would observe the edge of a shell as a shearing zone in the gravitational lensing around a massive structure, where the amount of lensing jumps dramatically, significantly magnifying the cosmic background in the shear zone. These would look like rings around gravitational lensing sources.
This discrete lensing artifact - these rings that might appears - could let us reject some dark matter theories, which predict smoother lensing with no rings.
Second the shells themselves do not have positive or negative mass. They have density functions which are a sum of a delta distribution and the derivative of a delta distribution. On the shell singularity itself, the density function breaks down - from being a function into being a distribution. So we can't say it's positive or negative mass - it's a singularity without any mass.
The reason we invoke positive and negative shells is because we can think of distributions as limits of ordinary functions. We can think of the topological defect shell -as if- it was the limit of a pair of positive and negative physical mass shells of positive thickness as they merged together into a shell of zero thickness. In the limit, there is no longer any externally observable mass at any enclosed radius. There's no longer any finite mass density function that properly describes the singular shell's density - the density has become a distribution.
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.
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.
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!
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.
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.
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!
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.
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.
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.
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!
\int_{0}^{r} s_{i} \* f_{i}(R - r)/r^2 + f_{i}'(R-r)/r
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?
Kip Thorne once commented that black holes are “bigger across” than we’d expect from their circumference. Which I think is what those cusp-on-sheet models are showing.
Does that create a topological defect?
I ask because a circle around a cusp in a sheet will get “caught” on the infinite spike; and a circle not around the cusp will end up with weird deformations if it comes close. That suggests there’s something “topological” about such a cusp, akin to there being a hole.
Does that create a topological defect?
No. There are no topological defects in any of the standard black hole spacetimes.
Your intuition is right, this is an example of a topological defect, but it's not directly related to circles having the "wrong" sizes. It's topological in that it's just a property of how your sheet is connected, and can't be changed by continuous deformations. Black holes are not topological defects in the same way, because they're not really "infinite spikes".
But we can formally integrate Delta functions and their derivatives. When we perform the mass integral we end up with the result m = 0.
It's an artifact caused by allowing generalized functions as our density functions
A problem is that we don't know how to physically interpret a distribution as a density function. Mass distributions caused by matter all have positive real valued functions as density functions. So if we have these structures in the universe, their cause is probably not physical matter!
It would be as if god put his thumb on the scale, and his finger print indented a concentric series of shells where the mass density goes completely nonphysical.
https://en.wikipedia.org/wiki/Dirac_delta_function
Maybe an autocorrect issue?
Discussion: https://news.ycombinator.com/item?id=40610133
For example, imagine we live on 2D infinite plane, and once found out, that in some places, moving on for example for 1km, we see same shift as with moving linearly (to be more exact, by some arc or some polygonal line) by other route for 100km (really need much less to prove topological defect, just standard scientific 3-sigma difference from our model is enough), so this mean, we have crease on space-time topology (and possible, some sort of wormhole, or just holes on plane and controllable way to navigate/move from one hole to other and move between hole and our plane).
Sadly I lack the training to take it anywhere formally, but it’s still pleasing to see my notion wasn’t entirely crazy.