Some speculations on the topological structure of elementary particles – by Sabbir Rahman

Bismillah.  This is from the remarkable Sabbir Rahman, may God increase his knowledge and intelligence even further.


I had something of an epiphany this morning, and wanted to share it with you.

In a recent paper, I explained how classical electrodynamics could perhaps be explained by the motion of a relativistic fluid in a double-sheeted spacetime, with time flowing in opposite directions on each sheet. The 4-dimensional spacetime in which we live is a superposition of these sheets. The fluid particles are identified with neutrinos and antineutrinos, which are rotating (Kerr) black holes formed by the gravitational collapse of gravitational waves. Furthermore, electrons are rotating (Kerr) black holes formed by the gravitational collapse of neutrinos. The motion of the neutrinos is what gives the electron its apparent negative charge (collapse of antineutrinos gives rise to positive charge, such as in the positron).

Now, Kerr black holes have a ring-like singularity which rotates at the speed of light (unlike the pointlike singularity which occurs at the centre of a Schwarzschild black hole – note that the Schwarzschild black hole can be considered as a particular limit of a Kerr black hole where the rotational anglar momentum is zero). In actual fact, the topology of the Kerr black hole is not that of a torus/doughnut, but rather that of a Klein bottle/double torus, which reconnects with itself only after two rotations around the ring. This is what gives the neutrinos and electrons their half-integral spin, and makes them fermions – i.e. they need to be rotated 720 degrees before they return to their original state/orientation in spacetime.

I was also able to explain in the paper how the interactions between electrons and positrons could be exchanged by the exchange of neutrinos. For example, a source electron will ’emit’ a neutrino, which will ‘strike’ the target electron, passing through its ring singularity and being sent backwards in time (i.e. it looks like an antineutrino going forwards in time) to the source electron, where it again passes through the ring singularity, emergin forwards in time to become the originally emitted neutrino!

Thus, the interaction between charged particles comes about through the exchange of a photon, which is nothing but a neutrino forming a closed timelike loop between the source and target charge. Furthermore, the neutrino travelling forwards in time will interact with itself travelling backwards in time, and this will in general result in an oscillatory or twisting motion, which gives rise to the frequency and polarisation of the photon. The existence of these closed timelike loops, i.e. the existence of time-reversing collisions within the context of classical mechanics can explain why the universe is actually quantum mechanical, because it means that at any given time, there are many possible futures which are consistent with the present (and the past), and which future actually occurs can only be determined probabilistically. As I have mentioned before, this can be identified precisely with the transactional interpretation of quantum mechanics. [Note that determinism and causality is therefore lost and a ‘choice’ has to be made at every point in time – which can lead on nicely to a discussion of predestination and free will, but I will not go into such matters now and stick to the underlying physics].

The above nicely can explain the structure of the electrons, neutrinos, photons, electromagnetic waves, etc, and even the foundations of quantum theory – all of which follow essentially from first principles from classical gravity (i.e. Einstein’s general theory of relativity).

This does however require that we keep the entire mathematical solutions of black holes and admit the existence of two spacetime sheets – usually one of these sheets is discarded as ‘unphysical’ as they seem to correspond to ‘white holes’ which are constantly spewing out particles which are not seen in reality. However, this overlooks the fact that the direction of time is reversed in these white hole solutions, so that they actually look like antiparticle black holes. So, whenever something falls into a black hole, it is actually accompanied by its own antiparticle, or looking at it differently, a particle that falls into a black hole is immediately re-emitted travelling backwards in time.

It also requires that we allow for the existence of ‘gravitational charges’. In particular, gravitational waves/antigravitons with negative mass. This however, must follow from the existence of the second (dual) spacetime sheet corresponding to the discarded half of the the black hole solutions. The change in time direction is also accompanied by a change in sign of mass. Thus, gravitational waves travelling forwards in time on the second sheet will have negative mass and collapse to form antineutrinos which spew out antigravitons into the first (base) sheet.

It seems then that we have a relatively complete description of classical (and quantum) electrodynamics, together with a description of neutrinos and photons, and together these pretty much cover the first family of leptonic particles in the standard model. But it is then natural to ask whether, and if so, how, the remaining elementary particles, namely the quarks and the hadrons (such as protons and neutrons and pions etc, which are composed of quarks) fit into this picture.

If the model cannot explain these as well, then, despite all of the successes so far, ultimately fails.

I believe insha’Allah that I now have a possible answer. It is beautiful, appears to be consistent, as one would ideally want from such a new theory, potentially makes a very interesting prediction which could explain the nature of the missing dark matter in the universe.

The clues to the answer are staring us in the face – the lowest mass quarks are the up and down quarks, and they are unusual in having fractional charges of +2/3 for the up quark and -1/3 for the down quark. It is also interesting that the down quark appears to have almost precisely twice the mass of the up quark. Both particles are also spin half fermions.

Now, we know from our model that charge is associated with particles which act as ‘sinks’ of neutrinos or antineutrinos. Also, mass is associated with spacetime curvature. Spin half is associated with the need for the particle to rotate 720 degrees before returning to its original orientation.

Those who have studied complex analysis, will realise that the structure of the electron looks very much like the structure of complex square root function, w=sqrt(z). The solution for w is a double-sheeted covering of the plane with a single branch cut – each complex number has two complex square roots, one on each sheet.

A good way to picture/model this is to take two circular pieces of paper placed on top of each other, and cut them both from the outside to the centre. Now, keeping them on top of each other, paste or tape them together so that a line drawn around the circle of the upper sheet which crosses the cut continues on to the lower circular sheet. The line then makes a rotation around the lower sheet, crosses the cut, and continues on the upper sheet (unfortunately it is impossible to tape the second cuts together, but hopefully you can imagine what this would look like.

There is a nice picture of the complex square root function on Wikipedia:

http://en.wikipedia .org/wiki/ Square_root# Square_roots_ of_negative_ and_complex_ numbers

The third picture, which shows the pasted sheets, is a pretty good representation of the structure of the electron.

The generalisation of this picture to complex cube roots (using three circular sheets of paper) is hopefully obvious, and is described in Wikipedia here:

http://en.wikipedia .org/wiki/ Cube_root

This is actually very similar to my proposed structure for the up and down quarks.

The problem however is that you would need to do three complete rotations (i.e. 1080 degrees) to come back to the starting points in this case, so this would actually decribe a particle of spin 1/3. As far as we know, no such elementary exists.

But there is a simple trick that solves the problem. Instead of starting with three circular sheets, we can cut out one-third of the circle from each sheet (i.e. leaving a 240 degree segment, each of which looks like a ‘Pacman’ shape), and then paste them together. The final picture remains much the same, however cutting out the 120 degree sliver from each circle results in a ‘pinched’ sheet for the particle’s ‘internal’ spacetime – we only have to go round 720 spatial degrees to travel round all three of the quark’s internal sheets. And this is precisely my proposed structure for the up quark. It is a triple-sheeted black hole solution with each sheet ‘pinched’ so that it has a missing or ‘defect’ angle of 120 degrees (technically such topological/ geometrical structures are called ‘quotient spaces’ or ‘orbifolds’) . See for example:

http://en.wikipedia .org/wiki/ Defect_(geometry)
http://en.wikipedia .org/wiki/ Orbifold).
http://en.wikipedia .org/wiki/ Quotient_ space

The ‘pinching’ of course produces curvature and hence mass, and it is necessary in order to be consistent with the double sheeted nature of spacetime – which also implies the spin-half of the quark. But it also explains in a very pleasing way why up quarks have a charge of +2/3 – the 120 degree defect angle means that only 2/3 of the (anti)neutrinos are falling into the singularity compared with the electron, which does not have a defect angle.

The down quark can be described in the same way as the up quark, but this time there is a defect angle of 240 degrees instead of 120 degrees for the up quark.

This means that the internal spacetime is pinched twice as much, which will require twice as much energy/mass/ curvature, and this is pleasingly confirmed by the fact that the mass of the down quark is approximately twice the mass of the up quark. The charge is negative because the down quark is formed from the gravitational collapse of neutrinos rather than antineutrinos.

The quarks can therefore be thought of as being intrinsically triple-sheeted objects which are being forced (by stretching out the angles) to live in a double-sheeted spacetime. This essentially explains why quarks are not found in their ‘naked’ state and are always constituents of composite particles such as nucleons (neutrons and protons) and mesons such as pions. In the standard model, the quarks are also assigned a ‘colour’ charge, of which there are three – namely ‘red’ ‘green’ and ‘blue’ – with their corresponding anti-colours for antiquarks. The observed particles are all colour ‘singlets’, i.e. they are either a combination of a quark and an antiquark of the same but opposite colour (e.g red + anti-red), or of three quarks, one of each colour, i.e. red + green + blue or anti-red + ant-green + anti-blue.

In the proposed model, the three quark colours can be associated with three sheets of which they are constituted. The requirement of observable particles being colour singlets is simply the requirement that the quark composites live naturally on the double-sheeted spacetime. (In actual fact, for mesons, which are made of a quark and an anti-quark, the specific colours assigned are probably irrelevant).

Consider the structure, then, of the lightest hadrons (mesons and baryons). The positively charged pion for example, is a composite of an up quark and an anti-down quark. We can now picture this as being very similar to a positron, but rather than there being a full 360 degree rotation on the two spacetime sheets around the Kerr singularity, there is a 240 degree rotation on the single (e.g. red) pinched internal sheet due to the up-quark, followed by a further 120 degree rotation on the doubly-pinched internal sheet (e.g. anti-red) due to the anti-down quark. There is a total internal rotation of 360 degrees, but an external rotation of 720 degrees, achieved by ‘stretching’ the internal angles as necessary. The same kind of picture can be drawn for the other mesons.

As for baryons such as the proton, which consists of two up quarks and one down quark for a net charge of +1, we have a similar situation, but this time all three ‘coloured sheets’ are used – e.g. 240 degrees of red, 240 degrees of green and 120 degrees of blue (note that the colours are really just labels for the sheets) which are stretched to fill out the complete 720 degree rotation in the double-sheeted spacetime.

We have seen that electrons and neutrinos have no defect angle, while up quarks have a defect angle of 120 degrees (i.e. 1/3 of a rotation) and down quarks have a defect angle of 240 degrees (i.e. 2/3 of a rotation remains). It is natural to ask whether there might also be particles [any name suggestions? – let’s just call it ‘S’ for now] with a defect angle of 180 degrees (1/2 of a rotation).

Such particles would be expected to have bare mass intermediate between that of an up quark and a down quark, i.e. around 3.6 MeV/c^2 (assuming 2.4MeV/c^2 for the up quark and 4.8 MeV/c^2 for the down quark), but there would only be one type with half-integral charge. It would therefore be paired with its own antiparticle in its lightest stable composite state (i.e. S + anti-S), which would therefore be electrically neutral (but could be significantly more massive than its consituents just as the pions are significantly more massive than their quark constituents, and of the order of 140 MeV/c^2). This would therefore be a massive neutral particle which does not interact the other standard model particles through either the elecroweak or strong forces, and could therefore be a potential candidate for the missing dark matter.

On the other hand, charged versions analogous to the proton (S + S and anti-S + anti-S composites of somewhat larger mass) would probably also be predicted. I don’t know if such things have been observed, or whether there is any other reason why they might not be allowed.

Defect angles of 90 degrees (i.e. 1/4 of a rotation) or higher fractions could potentially also exist, but there is presumably an energy/mass cost associated with squeezing in additional spacetime sheets, just as there is a large jump in mass from electron (0.511 MeV/c^2 to pion (140 MeV). There may be other reasons/selection rules which forbid their existence.

While I have discussed the structure a single family of elementary fermions above (electrons, neutrinos, up and down quarks), there are actually three fermion families which exist and need to be explained. They are very likely simply to be excited states of the first family (perhaps corresponding to black holes with more complex singularity structures than the Kerr black hole), and hopefully this will also explain why only three fermion families have been observed.


The most striking prediction of the model is that antimatter has negative gravitational mass (all matter has positive inertial mass), so antimatter should fall upwards in the gravitational field of ordinary matter. Even the Dirac equation predicts that positrons should have negative mass – though this fact tends to be conveniently overlooked by mainstream physicists for reasons not particularly well understood by me.

Despite the widespread belief that antimatter will also fall downwards like ordinary matter, it happens no-one has actually definitively tested whether which way antimatter falls. Fortunately the Aegis experiment (which was due to be carried out this year at CERN, though I do not know its current status), will for the first time insha’Allah be making measurements which will determine whether antimatter does indeed violate the equivalence principle. I have in the past contacted Michael Doser who is the spokesperson for the experiment, and he says that he is expecting either no violation or a very small violation, but will keep an open mind. Obviously, I am hoping for maximal violation! The AEGIS webpage is here,

http://aegis. ch/aegis/ home.html

though it doesn’t seem to have changed much recently.

Another thought that came to mind after posting my earlier message, and which makes me feel rather silly now (with the benefit of hindsight of course) is that I speculated at the end about the possibility of an S + anti-S particle with zero charge, and also S + S and anti-S + anti-S particles with +1 and -1 charge respectively. Well, the only standard model particles which are missing from the first fermion family (I ignore the Higgs for now, which is still hypothetical) , are the Z0, W+ and W-, that is, the three weak interaction intermediate gauge vector bosons, which have charges 0, +1, and -1 respectively, with mases of 91 GeV/c^2 for the Z0 and 80 GeV/c^2 for the W’s. So the model could well be predicting the existence of these three particles – which would be something of a blessing.

The reason why I did not make the connection immediately is because I was expecting the SS particles to be fermions not bosons – but perhaps the 180 degree defect angle turns them into bosons in this context (needs further thought). Also, I was expecting the intermediate gauge vector bosons to be analogous to the photon in some way, which is associated with neutrino/antineutri no exchange. On the other hand, there is no need for photons to carry any kind of topological charge, whereas this is not the case for the weak vector bosons, and this may account for the major difference between the two types of vector boson in the model.

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4 Responses to “Some speculations on the topological structure of elementary particles – by Sabbir Rahman”

  1. Rob Says:

    Prove there is a god and your theory may be correct. There is still not one shread of evidence to suggest that there is a god. Zero. None. Zip. If god created the universe. Who created God. When the universe implodes back into a singularity then re explodes everything that has happend before will happen again. Including me sending you this email. You and i were pre ordained to share this small moment in time that will repeat itself for eternity. So jesus allah budda etc will return. Just not for a few more billion years.
    Regards Rob.

  2. pre owned Proton Says:

    pre owned Proton…

    […]Some speculations on the topological structure of elementary particles – by Sabbir Rahman « UNITY[…]…

  3. James Says:

    Would this model suggest change in Hawking radiation? What would necessitate the initial “imperfections” to actualize in this certain reality.. Anthropic principle? Or perhaps an anthropic macroscopic black hole would create a time loop that would connect our (perhaps near) future with primordial time?

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