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The nature of the highly compressed matter that makes up neutron stars has been the subject of much speculation. For example, it’s been suggested that under extreme gravitational compression the neutrons may collapse into quark matter composed of just strange quarks – which suggests that you should start calling a particularly massive neutron star, a strange star.

However, an alternate model suggests that within massive neutron stars – rather than the neutrons collapsing into more fundamental particles, they might just be packed more tightly together by adopting a cubic shape. This might allow such cubic neutrons to be packed into about 75% of the volume that spherical neutrons would normally occupy.

Some rethinking about the internal structure of neutron stars has been driven by the 2010 discovery that the neutron star PSR J1614–2230, has a mass of nearly two solar masses – which is a lot for a neutron star that probably has a diameter of less than 20 kilometres.

PSR J1614–2230, described by some as a ‘superheavy’ neutron star, might seem an ideal candidate for the formation of quark matter – or some other exotic transformation – resulting from the extreme compression of neutron star material. However, calculations suggest that such a significant rearrangement of matter would shrink the star’s volume down to less than the Schwarzschild radius for two solar masses – meaning that PSR J1614–2230 should immediately form a black hole.

But nope, PSR J1614–2230 is there for all to observe, a superheavy neutron star, which is hence almost certainly composed of nothing more exotic that neutrons throughout, as well as a surface layer of more conventional atomic matter.

Nonetheless, stellar-sized black holes can and do form from neutron stars. For example, if a neutron star in a binary system continues drawing mass of its companion star it will eventually reach the Tolman–Oppenheimer–Volkoff limit. This is the ultimate mass limit for neutron stars – similar in concept to the Chandrasekhar limit for white dwarf stars. Once a white dwarf reaches the Chandrasekhar limit of 1.4 solar masses it detonates as a Type 1a supernova. Once, a neutron star reaches the Tolman–Oppenheimer–Volkoff mass limit, it becomes a black hole.

Due to our current limited understanding of neutron star physics, no-one is quite sure what the Tolman–Oppenheimer–Volkoff mass limit is, but it is thought to lie somewhere between 1.5 – 3.0 solar masses.

So, PSR J1614–2230 seems likely to be close to this neutron star mass limit, even though it is still composed of neutrons. But there must be some method whereby a neutron star’s mass can be compressed into a smaller volume, otherwise it could never form a black hole. So, there should be some intermediary state whereby a neutron star’s neutrons become progressively compressed into a smaller volume until the Schwarzschild radius for its mass is reached.

Llanes-Estrada and Navarro propose that this problem could be solved if, under extreme gravitational pressure, the neutrons’ geometry became deformed into smaller cubic shapes to allow tighter packing, although the particles still remain as neutrons.

So if it turns out that the universe does not contain strange stars after all, having cubic neutron stars instead would still be agreeably unusual.

**Further reading:** Llanes-Estrada and Navarro. Cubic neutrons.

This is interesting on several accounts. I go a bit into a description of some math, for this turns out to be important for packing spheres and with the physics of quark-gluon plasmas, which I eventually connect with below.. The first is that this runs into the Kepler conjecture, which is that a stack of plums is most efficiently stacked in a tetrahedral manner. This is intuitively known by grocers but was not proven until the late 1990s by Thomas Hales. There is something called the kissing number, where each sphere touches 12 others, the packing volume which is that 74.048% of the volume is taken up by the fruit. Any other stacking arrangement utilizes the space less efficiently. Musin proved a similar sphere packing theorem for four dimensions, where the fundamental polytope is the tetrahedrachoron, know also as the 24-cell, which has the largest representation as the F_4 exceptional group. F_4 is the centralizer of g_2 exceptional group as the automorphism group of the E_8 group, a group particularly important for heterotic string theory. The minimal sphere packing for E_8 is the Gossett polytope in 8 dimensions, as “partially proven” by Norm Elkies, which means the root-weight space of the group is the group itself. More recently the DeCloux group worked out the entire representation space of E_8.

This connects with quantum error correction codes. Each sphere can represent a letter in a quantum group, and the minimal sphere packing arrangement is the fundamental symmetry for quantum gauge theory in the dimension of the packing. Check out Conway and Sloane’s book “Sphere Packing, Lattices and Groups” for more on this.

I use the term partial proof that Elkies worked out for it involves some computer convergences which are not complete. Interestingly Elkies also wrote the “7th Brandenburg Concerto,” based on some sketches Johan S. Bach wrote down, but never completed. Fredrick the Great of Prussia commissioned 7 concertos, but ended up spending too much money on wars to pay Bach all of his commission, so Bach never completed the last concerto. So Elkies’ concerto based on Bach’s sketches is a “partial concerto.”

This area of mathematics is incredible, and ferocious for those with the temerity to explore it. It get deeply into modular forms and into something called the Leech lattice, or the Mathieu sporadic group M_{24}, which is composed of 3 copies of E_8 or octonions (one vector and 2 spinor versions) in 24 dimensions, which with 2 additional scalars defines the 26 dimensions of the bosonic string, and defines the 16 dimensions of a gauge group + quantum gravity in 10 dimensions. The Leech lattice is also the automorphism group of the Fischer-Griess group, also known as the monster group. It has 256,883 dimensional root space with the order of

2^{46} • 3^{20} • 5^9 • 7^6 • 11^2 • 13^3 • 17 • 19 • 23 • 29 • 31 • 41 • 47 • 59 • 71

= 808017424794512875886459904961710757005754368000000000

? 8 • 10^{53}

The Leech lattice is the automorphism group of the monster group. Forget fearing space aliens and the like, or even asteroid impacts, if you want to know fear then face the monster group. Conway and Sloane outline this in their book.

Now to connect with physics and neutron stars I have been working with the E_8 and the Leech lattice for some time. I have been aware of the fact that underneath lies this monster group. I largely ignored the monster group, until recently. There are some number theory coincidences with this involving rational prime numbers in the Galois quadratic field (more math) and some strange coincidences with number theory. This gets into something called monster-moonshine advanced by Richard Borcherds. It was pointed out by Mark Thosmas there are numerical results reproduce some mass ratios with QCD and the Planck scale. I posted a bit here on UT about hypernova explosions as being due to a chiral breaking symmetry and the prospect for neutron star quark-gluon plasmas to enter a color-flavor locked (CFL) phase. This would cause a 1.42 solar solar mass core of the neutron star to enter into the CFL phase and the magnetic pressure which holds it up against gravity would cease. It would implode into a black hole. The number 1.42 solar mass is one of the “magic numbers” from the monster-moonshine.

This then finally connects to this paper, for if neutrons resist becoming a quark-gluon plasma by this deformation it pushes the neutron star mass limit upwards, where if it is close to 3 solar masses it then permits this sort of physics to hold. We would then maybe expect to see these dwarf black holes. A neutron star that feeds off the material from another star would approach this phase. The result is the core implodes into a black hole and the rest of the neutron star explodes with 2 to 3 orders of magnitude more energy than a SNIa. Evidence for this type of hypernova and the generation of a “dwarf” black hole would be clue that underlying the structure of the universe is the monster group. If so it means that our deepest ideas of the foundations of the universe we may be arriving at are just a scratch on the surface of a far deeper substratum.

LC

Can you elaborate on the monster group and a hyper-nova/dwarf Black Hole discovery would mean for physics as alluded to in your last sentence?

The Long and Winding Road ?? — As Paul McCartney put it.

First it is important to understand a bit of abstract algebra and group theory. A group G is composed of elements e^{ix}, where x is contained in the algebra g that generates the group. So for x and y in g we define the adjoint action ad_x(y) = [x, y], which constructs a representation of the algebra. For G a group it may have an automorphism subgroup aut(G), such that for x in aut(G) and y in G, but not in aut(G) then

ad_x(y) = [x, y] in G/aut(G).

In other words it preserves the group, or equivalently as a map aut(G):G — > G.

I have been working with issues somewhat tangential to the monster group, and more with the modular forms which are related to the lattice or sphere packing of E_8 and the Leech lattice ?_{24}. The Leech lattice or the group it corresponds to is an automorphism of the monster group. What I have worked up is a moduli space of elliptic curves which have an E_8 representation. This is a type of gauge theory, and it has a correspondence between N = 8 supersymmetric BPS and extremal black holes with 3 and 4 qubit entanglements in an 8-qubit space. What has emerged from this is a central role of rational prime numbers on the Galois field Q(i sqrt{d}) of which there are only

{1, 2, 3, 7, 11, 19, 43, 67, 163}

where these form numbers Z + ?_d*Z, Z = set of integers, for d given by the numbers above. For d = 1 it recovers the Gauss prime numbers, which form a nice discrete pattern on the complex plane. This is connected with some pretty fancy mathematics with three forms and Calabi-Yau spaces and the Wiles’ proof of the Taniyama-Shimura conjecture (elliptic curves = modular forms) which is the cornerstone of the proof of the Fermat theorem. Now in the last of rational primes the last one 163 turns out to be really bizarre. It corresponds in these number theory correspondences to the Hardy-Ramanujan number

HR = e^(2? sqrt{163}) 70^2 = 3.377368×10^{38}

This number is a near integer when expanded out in its full form out beyond the decimal point, you can check OEIS. Further, this number has some relationship to the monster group within this so called moonshine. I have to get glibber about things here than I already am, for this leads into some difficult stuff. Physically what happens is that the ratio of the Planck mass to mass of a neutron squared is very close to this number. This also means that if I multiply this H-R number by the Planck mass I get the mass of 1.42 solar mass black hole. There are a lot more elements in this, but this breaks it down to its essentials.

This means that the constants and masses of things are more determined than thought. Rather than there be just a scatter shot of possible values, the masses of particles and coupling constants have far more narrow distributions.

LC

muhahahaha Im back

congratulations

And what happens to quarks? Cubic too? You won’t make a cube from 3 cubic quarks. What about pyramid shapes with triangle base. 😀 What about pancake square shapes? 😀 Now I’m playing Tetris in my head. 😀

A baryon, such as a neutron or proton, has a radius of around 10^{-13}cm. The real fundamental particles such as the electron or a quark has a much smaller size, and probably does not exhibit any structural feature until around 10^{-31}cm. So quarks do not become cubes as a result of this.

LC

Alright, but at least I mentioned tetrahedron which is in your long post. :d Triangle shapes make a construction more durable.

There is one issue along these lines which bothers me. The packing of neutrons in a dense crystal is probably tetrahedral, or equivalently body centered cubic. The model here has the neutrons becoming squashed into cubes, which suggests a primitive cubic lattice. This suggests there is some change in phase with the bulk material structure. I could of course imagine the neutrons becoming squashed into tetrahedra.

This theory may mean that the neutron phase may persist longer than thought. The energy inside the hadron “bag”of quarks and gluons has an energy density ?, which when integrated over the volume of the hadron V ~ 10^{-39}cm^3 defines about 95% of the mass energy of the hadron E = ?V for hadrons composed of low mass quarks u and d, such as the neutron N = (u,d,d). The energy density inside the vacuum is much larger than outside. However, under huge pressure the energy outside the hadron bag increases. That energy E’ = pV/c^2, for p the pressure increases so that eventually E’ = E or the pressure is p = ?c^2. Prior to this case the quark-gluon bag probably is deviated from spherical symmetry, maybe cubic or tetrahedral, but once this condition is met the barrier between the interior of the quark gluon bag and the exterior is removed. This will be the quanrk gluon plasma condition.

LC

The problem with tetrahedra is that they do not tile space. You can get them to pack more efficiently than spheres, but those arrangements have to be set up quite deliberately. I think any arrangement of tetrahedra that arises accidentally would be less efficient than spheres.

The tetrahedron is not a proper tessellation because the regular tetrahedron has a dihedral angle 70.53-deg which is not a multiple of 360. A set of five regular tetrahedral will form a solid with pentagonal symmetry with a gap of 7.35 degrees. So any tiling will involve an irregularity. Yet a tetrahedral lattice has the optimal packing volume, but the “tiling” is not strictly regular. With 20 tetrahedra and by closing these dihedral angle gaps by a slight deformation, we get a regular icosahedron, which corresponds to a 12-coordinated sphere whose vertices correspond to the vertices of the icosahedron. A lattice of tetrahedral in 3-dimension must have some irregular element to it. It is by this definition that tetrahedral do not form a proper tiling of space.

Curiously, if we think of tetrahedral packing in 3-dimensions, and 3 dimensions as a subspace of 4-dinesions, then the duality of chains means that a lattice of irregular tetrahedral is dual to a tessellation of 3-space with truncated octahedral.

LC

Here’s the thing to keep in mind though.

When you talk about tetrahedral packing, what you’re really talking about is two layers, one sitting on top of the other, not a tessalation of tetrahedra (perhaps it’s because of my geology background I’m accustomed to thinking of it this way).

This solves both of the problems you have raised – it explains why the neutrons are cubical, rather than tetrahedral – if you think of a tetrahedral stacking of two layers, then what you’re looking at is an octahedron along one of its threefold axes of rotation, which means, that, in essence, every neutron is part of an octahedron that shares an edge with another octahedron, so every neutron is surrounded on four sides by another layer, and above and below it are similar layers.

The same is true of both Hexagonal Closest Packing and Cubic closest packing (the difference between them is where the neutrons in the third layer go).

More info here: http://www.science.uwaterloo.ca/~cchieh/cact/c123/tetrahed.html

Better illustrated demonstration here: http://upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Close_packing_box.svg/1000px-Close_packing_box.svg.png

I hope that makes sense :Head scratch: (I meant to post this sooner, but this week has not been kind to me).

The tetrahedron is not a proper tessellation because the regular tetrahedron has a dihedral angle 70.53-deg which is not a multiple of 360. A set of five regular tetrahedral will form a solid with pentagonal symmetry with a gap of 7.35 degrees. So any tiling will involve an irregularity. Yet a tetrahedral lattice has the optimal packing volume, but the “tiling” is not strictly regular. With 20 tetrahedra and by closing these dihedral angle gaps by a slight deformation, we get a regular icosahedron, which corresponds to a 12-coordinated sphere whose vertices correspond to the vertices of the icosahedron. A lattice of tetrahedral in 3-dimension must have some irregular element to it. It is by this definition that tetrahedral do not form a proper tiling of space.

Curiously, if we think of tetrahedral packing in 3-dimensions, and 3 dimensions as a subspace of 4-dinesions, then the duality of chains means that a lattice of irregular tetrahedral is dual to a tessellation of 3-space with truncated octahedral.

LC

I’m curious, what would kind of structure would a quark, gluon or electron have? I remember a few articles back you mentioned that the universe is composed of only a few elementary particles and holographic information. I don’t quite understand this. Can you clarify this point, and tie this reasoning back to atomic/subatomic scales LC?

The question about the shape of a particle was put to an experimental test earlier this year. The results are a bit disturbing. String theory suggests that elementary particles are not point like. This saves us from some calculation problems with point-like vertices that do not transform and have divergences. Rather than a one dimensional line tracking the motion of each interacting particle there is a two dimensional sheet, or for a closed string a tube. The “hard” vertex is removed and replaced with something akin to a plumbing joint that connects three pipes.

Most strings are closed, but the Type I string and one sector of the bosonic strings. Open strings have ends, which attach to Dp-brane. The openings are a bit like particles, and they have a dipole structure close to the string length L_s ~ 10^{-31}cm. The closed strings, type IIA type IIB, Heterotic, bosonic, have modes which run clockwise and counterclockwise (left and right) and these have a dipole structure as well. Think of a closed wire with a current. However, if the right and left movers are identical the dipole terms from the two modes cancel. There appears to be no problem, unless there is chiral breaking, which happens with type IIB string. There should then be a dipole structure. The dipole moment p = e*d = e10^{-31}cm, for e a unit of electric charge.

Measurements of the dipole structure of the electron puts it at around 10^{-28}e-cm^{-28} this is within 3 orders of magnitude of the string dipole, and already eliminates theories which for various reasons have larger dipole moments. So this is one of those developments which I think may precipitate a crisis in physics. The absence of a strong signal for the Higgs field is another rising issue. The recent finding that there is no dispersion physics for photons over vast distances crushes a lot of physics that imposes fluctuations on spacetime. Since November 2009 there has been a growing body of evidence and data which suggests that a lot of theoretical physics, in particular phenomenology, is falling apart.

I think there are stringy structures to the universe, but it appears likely that the vast bulk of these developments is going to collapse. The problem we likely face is that we work with a huge amount of excess intellectual baggage of one sort or the other. We appear to be approaching the next great intellectual crisis in our understanding of physical foundations that mirrors what began to happen in the 1880s with the classical world view.

LC

What particularly happened in November 2009?

That was when the FERMI spacecraft found no linear dispersion of light from distant sources. The result was similar to the recent INTEGRAL result. That began to unravel the LQG game.

LC

Got it – thanks.

I should have written that Nov 2009 was when the data was released.

LC

Or it is simply akin to cosmology, where the standard cosmology encapsulated the old big bang expansion with new boundary conditions.

At least the train is moving forward again!

Any collection of spherical objects will retain dead space when compressed so that they touch. To reduce the space you must decrease the radius of the spherical objects, allow them to overlap, change their shape to un-curved rectangular blocks, or a combination of these. A cube would have the least surface area of any un-curved rectangular block.

Supposedly, neutron star particles have already reached the limit on overlap and size reduction. This leaves only un-curved rectangular block shape alteration as the only alternative to reducing volume with the same mass.

Most of this is over my head (way over) however I remember reading once about self-organised criticality and a sand dune collapsing at a critical point as determined by the shape/stacking of the individual sand grains. So, working backwards, does the collapse of a neutron star into a black hole give an indication of the shape/stacking of the neutron particles.

This is very interesting and I love reading about this stuff. The question remains, though, whether there is any way to experimentally or observationally prove that this is what happens to these particles under those extreme pressures. I can’t imagine that we have anything on Earth that could mimic this kind of pressure. Are there any other predictions that come out of the assumption that the neutrons are compressed into a cube that can be tested?

We will have to see, there could always be something: that was what happened with strange stars in the article.