Essay Abstract

By analyzing the dimensions of the quantities I found what I called the main sequence. A column with respectively time, length, mass, momentum and energy. Because of symmetry I concluded that in this main sequence the number of quantities must be extended. And therefore that both the Minkowski metric and the relativistic energy-momentum equation had to be extended. They are combined into the 16-dimensional general metric. From this metric I developed an octonion model of gravity. The elements of the two opposite octonions involved are proportional imaginairy quantities. The physical equations have their origin in and are limited by this intrinsic structure of the quantities.

Author Bio

My name is Peter van Gaalen. Born in the Netherlands, may 16, 1967. I studied biology at Leiden university. Now I am working as a software developer. I always have been interested in different disciplines of science. I am worried about the deterioration of nature and the rapidly vanishing of the indigenous people and their cultures. Blaming the 'western' cultural/social sytem that is out of control. Following the program of E.O. Wilson and trying to study the social system from a biological perspective in order to understand it's emergent behaviour.

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7 days later

Joe thanks for your appreciation.

This morning I red: "One of the premises of the CPT theorem is that the background spacetime is flat Minkowski space." (Penrose p818). I realised that in the same line of reasoning about CPT symmetry in my essay, in the octonion model there must be one more symmetry. I call it 'S' from 'substance interchange'. It's the reflexion of marble into wood or visa versa. So the whole symmetry of the gravitomagnetic system is CPTS.

I want to make two corrections in my essay:

page 3: l/c = gmflux must be c l = gmflux. (l = length)

page 8: metric [math]f^2 l_x^2 l_y^2 l_z^2 = t b_x^2 b_y^2 b_z^2[/math]

does NOT turn into [math]b_x^2 f^2 b_z^2 l_y^2 = l_x^2 t^2 l_z^2 b_y^2[/math]

(in the (latex) preview I didn't see the summation signs between the terms in the metrics, but they ought to be there.)

Dear Peter van Gaalen,

I think, mass is causal for quantity that determines the dimensions of quantities as the point mass by point particles are the points expressed in the geometry of the dynamics of elementary particles in quantum mechanics that are zero-dimensional. The geometric expression originated from a point source that is described by complex and hyper complex number systems is the intrinsic structure of the quantities for their dimensions in quantum mechanics and general relativity of Lambda-CDM cosmology.

But in Cluster-matter universe as the mass of matter has two sets of dimensions by the quantities of cluster-mass and elementary-mass, there are two different geometric expressions from a single point source, where both geometric expressions collate. This implies that the Coherent-cyclic model of Cluster-matter universe does not have any shape and thereby the complex and hyper complex number systems applicable for this model differ from that of the Lambda-CDM model of universe.

By this article I perceive that the descriptions you have provided on fundamental physics is much useful for the formulation of hidden variables in the new physics of BSM and Cosmology that is applicable for both models.

With best wishes,

Jayakar

Hello Jayakar,

My system doesn't not say anything about particles wether they are point particles or particles like strings and branes. It only analyses the quantities. So indeed it can be used in different theories.

The concept of 'mass' is a bit confusing in mainstream physics. I often read that "energy is mass" which is a most confusing and wrong statement. Einstein himself was more carefull when he said: "It followed from the special theory of relativity that mass and energy are both but different manifestations of the same thing." Mass is not energy! mass can be converted into energy and energy can be converted into mass. A lot of writers calculate the energy and say that it is equal to the mass. But it's not mass.

In mainstream physics it's said that E^2 - p^2c^2 = m^2c^4. It is confusing what this mass is.

I will show my view:

Energy is a scalar quantity. The total energy of a closed system is conserved over time. It is the sum of potential energy and kinetical energy. Both are scalar quantities. potential energy is the energy stored in the forces between particles. Kinetical energy is the energy because of the motion of particles. 1/2mv^2.

Momentum is a vector quantity. The total momentum of a closed system is also conserved over time. mv.

In the following I will use my own definitions:

Mass is a scalar quantity. I call it the rest-mass. The total rest-mass of a closed system is conserved over time. In analogy with energy, it could be that rest-mass is also the sum of two scalar quantities. A potential mass which is maybe the mass of the underlying particles and an actual mass which is the mass of the particle.

String is a vector quantity. The total string of a closed system is conserved over time. String is connected with the relativistic mass which is also a vector quantity.

string is m/v. Maybe the speed of a reference frame.

Penrose tells that the total mass of a system is not a scalar quantity. It's value depends of the reference frame. m^2c^4 - s^2c^6 I will call the invariant mass.

E^2 - p^2c^2 I call the invariant energy.

Peter

4 days later

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  • Post #6

Hi - what I like in your essay is that at the foundation of the forces of nature, you propose two octonions which, when multiplied, produce an equality between a sum of squares. In contrast to classical relativity that requires some directionless magnitudes to remain invariant (mass, proper time), you are requiring a symmetry relation between your two fundamental octonions, and also between the left-hand and right-hand sides of their product equation (the sum of squares). That looks like a beautiful ansatz to me.

Now, regarding the potential relation to physics your suggesting, I've got my reservations ... Let me just point out one conceptual problem. You propose a superficial relation to the field equations from General Relativity. That is not the proper realm for a comparison, IMHO, since the Einstein equations model the dynamic balance between sources of fields and the fields themselves. In your model, in contrast, field self-interaction is absent (not at last since you don't really have a concept of "field" yet). If you were to compare your ansatz with General Relativity, then you may look at the linearized field equations. There still is a concern (though maybe a smaller one), since the linearized field equations from General Relativity are obtained from a weak field and small velocity approximation, whereas your approach could justify the weak field approximation, however, would need to remain valid for any velocity, I think.

Good luck, Jens

Hello Jens,

"That is not a proper realm for a comparison". You are right with that. I am a total layman with respect to for instance tensor algebra. But I have this vague intuition.

It seems to me that the product of two (different or not) octonions forms a tensor. I do not exactly know what quantities are used in the tensors, but I thought in case of the energy-stress tensor they where energy density and momentum density and stress. those quantities are not in the proportional table with mass and length. But there are more quantity tables with proportional quantities. for instance a table with the proportional quantities mass density, momentum density and energy density. and a table with proportional pressure e.d. All those tables have a general metric.

Maybe the Einstein equation uses quantities from different proportional tables. So therfore I think that there is a link.

And according to Michio Kaku in his book 'hyperspace', the Riemann tensor is about the curvature and torsion of space. I think that the imaginairy units in the tensor have these properties of curvature and torsion.

Greetings,

Peter

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  • Post #8

Ok - I understand you're doing analogy comparisons; highlighting what you feel is similar, and not necessarily equal. Still, you want to compare your formulation with the linearized field equations from GRT, since they do not account for field self-interaction, either. In classical GRT, you can then "bootstrap" up to the covariant form iteratively. If you're curious about the details, I've prepared a summary and reference list about this here ("Spin 2 / GRT: collection of articles"). Yet take that with caution as well: Your approach from the product of two octonions would not result in a spin 2 model (which is good! you don't want that, IMHO spin 2 is a dead end, others may disagree with me), but you're facing something much, much more complicated (which I can't help you with, either, because I'm looking for answers myself).

Thanks, Jens

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  • Post #9

Hi, Jens

I will take a look at the link with your summary.

I am very curious about what you said about that the approach from the product of two octonions would not result in a spin 2 model. Why do you think that? I hope to hear from your investgations.

Thanks for writting and good luck with your searh.

Peter

The last few days I had some thoughts about the light cone in spacetime. According to Yang-mills theory particles are massless and are traveling with the speed of light. So he particles are on the ligthcone. (without symmetry breaking).

My question: If mass and energy are different manifestations of the same thing, why does yang-mills theory describe only particles with energy and with no mass?

I assumme that yang-mills symmetry is written for particles in spactime.

I we turn time into gmfux then we get gmfluxspace. If paricles are described in this gmfluxspace then these particles are also on a cone. This cone has also the 'speed of light'. But we must not think of 'speed'. These particles will have no energy, but they have mass.

If we have gmfluxspace, and we turn space into burst, then we get burstflux. In this burstflux we have the light cone with the speed of light again. And again the particles will have energy, but no mass.

In this burstflux if we turn gmflux into mass, then we get burstmass. Then our cone is not composed of the speed of light anymore. Instead it is composed of the gravitational constant G.

Peter

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  • Post #10

Just quickly re "not spin 2": Your octonion product is a sum of scalars, therefore the sources of your fields can be vectors at best, in those cases making your exchange particle spin 1 ... but I can only underwrite this in the distinct cases where you can project your formulation onto a degenerate form (namely, with four-dimensional Minkowskian or Euclidean signature); for all others, I don't know. When you introduce nonassociativity, you can't simply separate out the propagator of your exchange particle (or at least I can't), so you can't examine it easily for its spin. Best wishes, Jens

9 days later

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  • Post #11

Hi Peter, my name is also Peter van Gaalen. Born in New Zealand after my parents immigrated from Holland.

You can find me on face book.

De Groeten

Peter

14 days later

Reasons for more dimensions:

If we have complex numbers then {1, i} is not the complete set. {1, i , -1, -i} is the complete set. In the description of spacetime Minkowski used the 'hyperbolic quaternion'. The minkowski metric stops with {1, i, j ,k} but where is the other half? Where are {-1, -i, -j, -k} ? The Minkowski metric is not a closed system, but no one seems to care! This is a fundamental question. In case the Minkowsky metric t^2 - l^2 = S^2 the only thing about the invariant of spacetime S^2 is that it's invariant. It doesn't say how many quantities it's composed of. That's why it's more illuminating to write it like: t^2 - l^2 = f^2 - b^2. ( l = lx, ly, lz and b = bx, by, bz).

In case of the relativistic energy-momentum equation m^2 p^2 = E^2. Just look at "mass" on wikipedia. What is the difference between "relativistic mass", "restmass" and "invariant mass"? Why three different kinds of mass? And relativistic mass is not even a scalar quantity! Further if we take m^2 p^2 = E^2 and we translate m^2 p^2 as the product of the quaternion and conjugate then the energy would be the norm. But that makes no sense. Energy is just another quantity that differs from momentum like momentum differs from mass. m^2 p^2 = E^2 s^2 makes more sense. In which m is restmass and s is an vector quantity. Relativistic mass is related to this vector quantity s.

The description of special Relativity was not finished until Minkowski came with his metric about the spacetime-continuum. Relativity is extremely important. A quantum theory without relativity is not complete."What happens inside a black hole?" The einstein equation says that it is a singularity. Because of these infinities general relativity can't be a complete description of the gravito-magnetic system. So why is no one completing classical relativistic mechanics? It's not finished yet! There is no proper understanding of classical relativistic mechanics. Classical mechanics is not finished, but no one seems to care! Most physisists are focused at quantum theories.Without a proper understanding of classical relativistic mechanics, you can't make a quantum theory that also encompasses gravity.

In my model c and G are the same. c and G both displays relativistic effects. So next to light cones we also have cones with G. Physisists even make a difference between special relativity and general relativity, so they didn't understand that the gravitational constant G is the same as the speed of light c. They are not aware of the difference between quantities and proportional quantities. (Minkowski was). They also aren't aware of the concept of periodicity. And they didn't notice that scalar and vector quantities alternate. The "Octonion model of gravity" adresses these issues.

15 days later

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  • Post #13

Dear Peter,

I'm sorry I overlooked your essay. This paper uses different arguments from my own to argue for "extra dimensions" in a consistent manner. My twelve dimensions may be an H4 quaternion of Spacetime and an E8 Octonion of Hyperspace (Superspace). But if we apply Supersymmetry, then it may double to a 24 dimensional (E8xH4)x(E8xH4) model. Lawrence Crowell is working with a similar 26 dimensional model. To first approximation, Lawrence's model has three 8-dimensional E8 Octonions and a G2 2-brane. I propose that one of these E8's separates into a pair of H4 Quaternions. E8 has symmetries of 240=8x(2x3x5) and H4 has similar symmetries of 120=4x(2x3x5).

The symmetries of an Octonion break into 1 scalar, 1 pseudoscalar (imaginary), 5 polar vectors, 5 axial vectors, 10 symmetric tensors, and 10 anti-symmetric tensors. The Einstein Field Equations of General Relativity are 10 anti-symmetric tensor equations. This coincidentally echoes the structure of Octonions. Likewise Maxwell's Equations of Electromagnetism can be written as one Quaternion equation. Thus, combining a Quaternion and an Octonion contains sufficient mathematical formalism to contain Maxwell's Equations and Einstein's Equations.

Have Fun!

Ray Munroe

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  • Post #14

Dear Peter,

The public vote of the contest is over, and I'm not trying to pick on you, but I think you got it exactly backwards with your first sentence "I wondered why differentiating acceleration gives velocity and why dfferentiating velocity gives distance". Actually, we differentiate distance with respect to time to get velocity, and we differentiate velocity with respect to time to get acceleration. I know it is independent of your extra-dimensional discussion. I just thought you might want to fix your introduction.

What if the "marble" quantities arise from the visible four dimensional Spacetime H4 Quaternion (and you omitted electric quantities such as the Coulomb), and "wooden" quantities arise from the hidden eight dimensional Hyperspace E8 Octonion? Our essays might have more in common than appears on the surface.

Have Fun!

Ray Munroe

Hi Ray,

Yes you are right, my mistake. (the third mistake in my essay and there are probably more). Differentiating length gives velocity.

I omitted electric quantites, because I couldn't think of a means in which to include them. In a way electromagnetic flux and electric charge have resemblance with gravitomagnetic flux and mass (That's the reason why I named gmflux 'flux') and just like gmflux = G/c mass, we also have em-flux = K/c electric charge. In my model I did choose gmflux to be the basal proportional quantity, but it is better to take [math]m\sqrt{G}[/math] (I call esu-mass) as the basal proportional quantity because this quantity has the same dimension as [math]e\sqrt{K}[/math] (which I call esu-charge).

We have (from Steven Weinberg article) electric charge Q, electronic hypercharge Y and electronic isospin T. likewise we have respectively the quantities: esu-electric charge e, esu-hypercharge g', and esu-isospin charge g. But I need a metric with electric charge and mass in it. I have some ideas like Dirac: mass can't be negative that's why the root result in (positive mass positive electric charge) and (positive mass negative electric charge). But this is premature.

I tried to read your essay (I like the matrix with rotations and reflexions) but it is difficult for me. Can you explain what is the H4 quaternion and what is the E8 quaternion?

Hi Ray,

It will be great when our model can be unified. But I think that I can't do it because it is difficult for me.

I am trying to develop some algebra's: From the general metric an equation with only scalar quantities:

f2 E2 = (-t)2 (-m)2

This metric inspired me to develop an algebra that unifies complex numbers, split complex numbers and hyperbolic complex numbers. Now I am trying to develop an algebra that unifies quaternions, split quaternions and hyperbolic quaternions, but that is a hundred times more difficult then complex numbers.

Greetz, Peter

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  • Post #17

Dear Peter,

Because electric charge, hypercharge and isospin are all related, you shouldn't need all three quantities. If I understand your ideas, then introducing the Coulomb unit would encompass all three related charges.

Think of H4 as half of an E8. E8 is 8 dimensional. H4 is 4 dimensional and shares the same component symmetries such as Lisi's 3-fold Triality (as well as a 2-fold Duality and a 5-fold Pentality). For more background on H4, check out these references:Coxeter-Dynkin, 120-cell and 600-cell.

In Clifford algebra, the Quaternion has a 4 dimensional (1,4,6,4,1) (scalar, axial vector, tensor, polar vector, pseudoscalar) component symmetry whereas the Octonion is comparable to a complex (real plus imaginary) Quaternion with an 8 dimensional (1,5,10,10,5,1) (scalar, axial vector, anti-symmetric tensor, symmetric tensor, polar vector, pseudoscalar) component symmetry.

I don't know about negative mass. In some regards, anti-matter could almost be treated like a negative mass, but then gravity seems to always be attractive. Consider pair production from gamma gamma -> electron positron. If we treat the positron like a positive mass, then our mass terms don't equate on both sides of the equation. If we treat our positron like a negative mass, then mass could equal (zero) on both sides of the equation.

My model also implies imaginary mass ("scalar fermions" or tachyons mentioned in the essay) and imaginary time (not mentioned in the essay - I'm still trying to understand my 7th dimension - It doesn't fit into M-theory, but everything else does).

Have Fun!

Ray Munroe

Hi Ray,

What I am doing is classical relativistic mechanics, it is not quantum mechanics. But it is interesting that when using 'proportional imaginairy units' we get imaginary units used in quantum mechanics for free. The planck constant in quantum mechanics is written ih (h-bar). My model automatically generates the 'i'. In my model planck constant is -Lh (L = imaginairy unit) it is derived from the product of the proportional imaginary quantities: (-t/phi * LE; ix/phi * iLpx; f/phi * -Lm) phi = phase.

Maybe classical relativistic mechanics and quantum mechanics aren't that different afterall.

What I do not understand is that quantum mechanics is only using one planck constant. According to my model there has to be 8 different planck constants (of which 4 vector quantities so a total of 16). They are combinations of the planck constant, the light constant and the gravitational constant. But they are different quantities!

I don't know if it is possible to incorporate electric charge. Maybe there is an essential difference between the classical relativistic gravitomagnetic quantities and the other quantities. If it is possible, then I don't know how to start. Do I need the sedenion? or do I need complexified octonions? But the most important is to find a classical relationship between electric charge and mass. I tried the Kerr-Newman metric (electric charged rotating black hole). Maybe I should take a look again at that metric.

I think that it is fundamental to ask what is a particle? I think that they are droplets or bubbles of 'generalized spacetime'. A package of quantities.

And what are the waves that come with particles? A wave packet has an imaginary component. what quantitie(s) is this imaginary component composed of?

What is a wave related to an octonion? or what is a wave related to a complex number?

Regarding electric charge, hypercharge and isospin. I don't think that introducing the Coulomb unit would encompass all three related charges. They are different quantities. I think there must be a metric with all three quantities in it.

I don't think the Coulomb unit would encompass the three related charges.

Coxeter-Dynkin is interesting. I have seen those graphs before, but I didn't understand them. I have to take a closer look at them.

You say that your model implies imaginary mass. This is related to another problem of me: on the one hand there is the difference between the (imaginary) signature of the quantities and on the other hand there is the value of the quantities. In my model proportional length is 'ix'. but we can have a positive length and a negative length.

So is the value of mass imaginary? or is the quantity 'imaginary mass'?

Greetz, Peter

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  • Post #19

Dear Peter,

In the Clifford divisor algebras, we progress from real to imaginary to Pauli matrices to Quaternions (Dirac matrices) to Octonions. Imaginary numbers are rooted in all of the more complex algebras. I think it is appropriate that these factors of 'i' keep appearing. It is a phase factor.

Regarding electric charge, hypercharge and isospin, they are related via

Q=T3 plus Y/2.

I don't think you need to include all three charges, but you probably do need an electric charge, an electric flux, a magnetic charge, and a magnetic flux. Do you need to include color charges for the strong nuclear force?

If we supersymmeterize Garrett Lisi's E8 TOE, then we would have 16 dimensions. You discussed 16 dimensions in your essay. If we supersymmeterize my K12' TOE, then we would have at least 24 dimensions. Do the electric quantities make up these extra dimensions?

I am curious about the 8 components of your Planck constant. Theodor Kaluza unified Gravity and Electromagnetism in 5 dimensions. My model has 12 dimensions, but 8 are part of an unseen Hyperspace. Perhaps these 8 unseen dimensions can be effectively represented by one (Kaluza's fifth dimension). Likewise, your 8 components of the Planck constant can be effectively represented by one (the Planck constant that we measure in the laboratory). I need to read your essay more closely to better understand this idea.

Have Fun!

Ray Munroe

Hi Ray,

According to the tables I found there is no table with magnetic charge in it.

Dirac hypothised magnetic charge in the Maxwell equations, but when we take a closer look at what he called magnetic charge, we see that the quantity he had is in fact electromagnetic flux. Below e = 'electric' em = 'electromagnetic'.

In this respect it is interesting that the hall effect shows quantized electromagnetic flux.

[math]K = \frac{\text{em-length}}{\text{e-instant}}, \ \frac{\text{em-flux}}{\text{e-string}}, \ \frac{\text{em-burst}}{\text{e-charge}} [/math]

[math]U = \frac{\text{e-charge}}{\text{em-length}}, \ \frac{\text{e-momentum}} {\text{em-flux}}, \ \frac{\text{e-energy}}{\text{em-burst}} [/math]

[math]\epsilon_{0} = \frac{\text{e-instant}}{\text{em-length}}, \ \frac{\text{e-string}}{\text{em-flux}}, \ \frac{\text{e-charge}}{\text{em-burst}} [/math]

[math]\mu_{0} = \frac{\text{em-length}}{\text{e-charge}}, \ \frac{\text{em-flux}}{\text{e-momentum}}, \ \frac{\text{em-burst}}{\text{e-energy}} [/math]

First I thought that the table with mass could also contain marble quantities and the table with gmflux could also contain wooden quantities. This was because in the electromagnetic tables we had electric flux, electric charge, magnetic flux and, although hypothetical, magnetic charge. Those four have the same dimensionality. Electric charge and electric flux are in the same table and in the same cell. Magnetic charge and magnetic flux are in the same table and in the same cell. Therefore I thought that in the same table all wooden quantities had their marble variant and visa versa.

This view went into trouble because of three reasons: The first reason. Look at the following quantities:

Magnetic induction B (the so called 'magnetic field')

Electric field strength E (the so called 'electric field')

Magnetic field strength H

Dieëlectric displacement D

Now in my mind I had a picture of an electromagnetic wave as electric and magnetic fields undulating. And the quantities that described those two physical objecs would be both marble and would have the same dimensionality, but would be in different tables. The strengths of those fields would be the electric field strength E and the magnetic field strength H. But in the Maxwell equations the fields describing electromagnetic waves are E and B. I was wrong. E and B are in the same table and are both marble, but they don't have the same dimensionality. They differ by velocity which can be seen in for example the Lorenz force: F = q(E + v B). If we want an anology of this in spacetime, then the electric and magnetic fields are just like a 'length field' and a 'time field' alternating.

The second was by analyzing the fluxes. Magnetic flux (em-flux):

[math]\phi_m = \int \int_A B dA.[/math]

Surface area A, the magnetic field B and phim are all marble. There are two different quantities called 'electric flux'. The first electric flux [math]\phi_e = \int E dA .[/math]

is in the same table as em-flux and therefore is marble. phie is one cell below phim. It has not the same dimensionality as em-flux but it rather has the same dimension as em-burst. They differ from each other in the same way as E differs from B. (Later I realised that phie is electric flux in Gaussian units).

The second electric flux is:

[math]\psi_e = \int \epsilon E \cdot dA.[/math]

This electric flux has the same dimensionality as magnetic flux and is in another table. This electric flux remains in the same cell as electric charge. So this had to be the marble counterpart of the wooden electric charge. But this turned out to be wrong. Because of the 'epsilon' in the equation psie is a wooden quantity and not a marble quantity.

Although we had two electric fluxes, neither of them was both marble and provided with the same dimensionality as em-flux. And because magnetic charge was hypothetical I had no reason anymore to suggest that a table could contain both marble and wood.

The third reason. If a table could contain both wooden and marble quantities of the same dimension, then the wooden and marble quantities in the same cell would also have the same planck value. Marble planck units have a lower limit (i.e. planck length, planck time etc.) and wooden planck units have an upper limit (i.e. planck mass, planck energy etc.). But how could a planck unit be both an upper limit and a lower limit?

But if I am right that there are no wooden and marble quantities together in the same cell, then I have to conclude that magnetic charge can't exist next to magnetic flux. In the quantity table there are no cells left for magnetic charge to reside in. Or magnetic charge must have it's own table, but then it has no influence on the magnetic- and electric field anymore, and that was the whole idea behind the conjecture of magnetic charge in the first place.

Let me ask you, what is the planck value of magnetic charge? (I guess you can't find it on the internet)

Greetz, Peter