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of freedom of the curved space may even let the solution wiggle and diffuse into an ordinary solution, so that it no longer represents a topological defect in the solution space.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?
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.
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?