Hi Joy,

I couldn't make any sense of the other possibilities for S15 either. This is why I said your first suggestion was intriguing, as it makes a sort of sense and doesn't fall over on the first hurdle. For the purely geometric scenario where the S7 fibre of S15 gives topological monopoles in space (S3), the physical scale of the compactified S7 dimensions defines the scale with which all measurements in space are made. But since the S4 basespace of the S7 in the S8 basespace of S15 isn't involved in this, the relativism of measurement makes the S4 genuinely hidden as it cannot be measured using anything that exists in space - like trying to measure a vertical distance with a horizontal ruler.

The underlying reason for your characterisation of my S1*S10 and S15, is that in my case the time dimension is left hanging outside of the unification, whereas in the S15 case all dimensions are unified on an equal footing. The price you have to pay for that is to explain the origin of the time dimension, and since time is the basis for dynamics, it has to be explained first. This would give a sequence of symmetry breakings - i.e. breaking of the equivalence of dimensions in S15:

time dimension: S15 = S7*S8 -> S7*(S1*S7)

Higgs vacuum: (S7=S3*S4)*(S1*(S3*S4)) -> (S3*(S3*S1))*(S1*(S3*S4))

Monopoles would appear at the second breaking - if it is valid to map S7 to the S3 fibre of a space only, which it may not be - so it must come last otherwise the topology of S15 would give additional features that would probably conflict with particle physics. The breaking sequence could give different vacuum options prior to the Higgs vacuum, as only one option would be selected in classical physics and the universe would stay there, i.e. no quantum tunnelling between vacuums as in QT.

Best,

Michael

Tom, Multiwords: "a good way of saying that quantum theory fails to be fundamental". Fair point.

Rick, I find your label of ensemble derivative a bit confusing. I get your point, but it doesn't seem like an ensemble as such. I would note that the space of O is Euclidean R8, and no matter the basis chosen or the operations performed within it, as long as you're still considering the whole space, it's always Euclidean. So I am a bit suspicious of the change from a Euclidean metric signature to a Minkowski type signature within O. In your "Octionion Algebra and its Connection to Physics", your D-Alembertian arises from a current definition - are you sure you haven't lost a complex conjugation between vector and 1-form? If you have, then the Euclidean signature of R8 would be preserved. Physical measurements say there is a spacetime split: speed of light c, energy-mass inter-conversion, electromagnetism ... So if it is going to be physics, a spacetime split of some form is required.

Fred, I have been pondering a similar thing. For a spatial S3 surface expanding through a Euclidean R4, the radial direction is "time-like" in a sense. Also, if you measure spatial position x1 at time t1 for radius R(t1) and spatial position x2 at time t2 for radius R(t2), although you may think that you have measured the spatial distance within S3, because of the expansion of S3 you've actually measured the hypotenuse of a triangle with the spatial separation at time t1 and the increase in radius R(t2)-R(t1) on the other two sides. To transform your Euclidean distance back to being just a spatial distance within S3 at time t1, you have to subtract the expansion. This gives a change in metric signature, ie. ds2=dx2-cdt2, for radial expansion rate c. However, light cones don't make much sense with this view, so it can't be quite right. But it does illustrate how you could get a Minkowski metric signature within a surface travelling along a time dimension in a Euclidean spacetime.

Best,

Michael

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  • Post #452

Michael,

I am confused by your confusion. You do not get to craft differential equations any way you want, they must be generated by successive holistic application of the entire ensemble of partial differentiations as mandated by O algebra, as spelled out in the ensemble definition. This is a different approach than the covariant derivative of tensor calculus and *its* attendant algebra. Perhaps this is where you are confused by being too wrapped up in its orthodoxy. The ensemble definition is good, and the O algebra was properly applied, and no where is there any hint of a split signature, in fact this would be contrary to the definitions of S^n radii so I am quite confused by your wanting to play in O and how it fits in to S15 or whatever yet insisting on a split.

As evidence the ensemble form works, in my essay I presented how the Lorentz hyperbolic rotation comes in by merely assuming the ensemble derivative definition is good, and the two partials in each field component transform in kind, and the "like" field types must have the same basis products to remain within the algebraic invariance requirement. If you need more convincing, I could post later today the derivation of the transformation of rectilinear H to spherical-polar with nothing more than the ensemble derivative definition and H algebra. The ensemble derivative definition is correct.

Rick

Hi John,

There is a time-temperature connection, but it is the other way round. Temperature in thermodynamics is defined by way of the rate of change of entropy with energy, where entropy can be derived in terms of the number of different ways of distributing M objects into N bins, and in Relativity energy is directly associated with the time dimension. So unravelling the definitions, temperature is a measure of the causal distribution of energy within a physical system - thus the definition of temperature is dependent upon the prior specification of what time and causation are.

The underlying cause of the problem we mentioned earlier - present in the paragraph above and in your discussion with Tom - seems to be a reluctance to admit that the maths formulations of science are subject to the same bounds as apply to maths. Maths first defines a language in which a range of true propositions P can be stated, then gives a set of axioms PA and logical rules that establish a formal system in which a range of proposition PD can be derived. When this system includes arithmetic over the counting numbers, then in addition to PD there are extra un-derivable propositions PU that are nonetheless also in P. Science collectively is under the mistaken impression that scientific theories aren't going to be subject to these simple maths constraints.

The axioms of maths PA are so-called because they cannot be derived, and maths science theories aren't going to somehow magically be different: science will have its own axioms that cannot be derived. These are the foundational principles of science - metaphysics - and include issues of time, space, causation, mass etc. - you know, the usual suspects. It is in trying to define or derive these that you encounter self-reference or tautology. For example, try defining space without referring to space, or similarly time. Or try defining causation without using anything dependent upon causation, such as thinking. The so-called Anthropic Principle is an example of a maths tautology (true for all values): IT must be true, otherwise we would not be here to talk about IT. The values of IT for which this is true - e.g. the nature of space and time - are amongst the axioms of science - i.e. metaphysics - that cannot be derived. Given this reluctance to even admit that science has axioms PA - i.e. that science has a metaphysical basis - it is perhaps not surprising that I am having difficulty trying to get the message across that un-derivable propositions PU exist *within* science.

By the way, your comment about the conscious mind being a look-out on a ship is similar to my view of the conscious mind being the captain of the ship. The view seen by the look-out is generally filtered before it is passed onto the captain, hence we suffer from optical illusions because what we see is not the same as what we perceive.

Best

Michael

Rick,

The different formulation is one reason for my confusion, but a space only has one metric for defining a scalar norm:

vector norm |A|^2=g_ij*A_i*A_j

differential norm |ds|^2=g_ij*dx_i*dx_j

D-Alembertian D^2=g_ij*D_i*D_j

Help, these aren't the same metric in your paper, the last one has a sign reversed.

Michael

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  • Post #455

Tom,

I will take this over to your thread, if you don't mind.

Random thought:

Rick frequently compares mathematics to religion (an idea I'm not averse to; as Godel said, if mathematics is a religion, it's the only religion that can prove itself a religion) -- which set me in mind of Einstein's famous statement, "Science without religion is lame; religion without science is blind."

Suppose we were to substitute mathematics for religion and invert the terms:

"Science without mathematics is blind; mathematics without science is lame."

It's true, isn't it? A mathematical statement cannot physically "walk." Without it, though, no scientist can "see."

Tom

"I think radius of SN can be taken to be time in a relativistic sense. At least in the case of S3, S7, S15. Just divide the radius by c and you will have the relative time."

Oooo, nice, Fred!

Tom

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  • Post #459

Michael,

I'm not referring to time to derive time.

Forget time even exists. Just imagine a simple system, such as balls moving around a pool table. The configuration changes. The original configuration no longer exists. Now if we had a camera taking pictures of this system we would have a sequencing of events. The effect of time emerges from that sequencing, otherwise the balls only exist.

The problem for science is that it treats time as a measure of the interval between events. Various examples would be Julian Barbour's winning essay in the nature of time contest(not linking, personal time issue), where he explcitly debunks the idea time exists, then turns around and argues the only measure of time "worthy of the name," would be to use the principle of least action between configuration states of the universe.

Another example would be Edward Anderson's(an FQXI large grant winner) entry in this contest, where we does specifically describes time as "Machian," emerging from action, then explains the intermediate process, between universal and local, that he would use as units of measure.

Then you have Fred Diether's comment above, where he divides the radius of the sphere by c, to derive a unit of duration/time.

All these are actions, creating sequences of events and the way to measure "time" is the duration from one event to the next, BUT duration doesn't transcend this "point" of the present on that vector between one event and the next. It is the state of the present, ie. the action that is occurring between events, like the wave receding and building between the peaks/points of measure.

Thus every clock does run at its own speed, even if they all physically exist in the same space, such as GPS satellites and ground stations. You would think that if time is a vector from past to future, the faster clock would somehow move into the "future" more quickly, but the opposite is true. Because its "burn rate" is faster, it "ages" quicker and thus recedes into the past, ie. loss of energy and thus non-existence, quicker. Such as the twin in the faster frame dying before the other returns.

Time, ie. the sequencing of events, is not causal. Yesterday doesn't cause today anymore than one rung on a ladder causes the next. Tranfer of energy is causal. The sun radiating on a rotating planet causes the sequencing of events called days. So I don't see "the fabric of spacetime," the correlation of duration between particular events and measures of spatial distances, as calibrated using the velocity of light, as causal.

Something exists to the extent it is composed of energy. It begins as the energy coalesces, grows as long as more energy is coming in then is being lost and ceases to exist when all energy is gone. The effect that emerges is a unit of time, be it a day, or a life.

" ...balls moving around a pool table. The configuration changes."

Instantaneously?

Dear All,

Here is a family portrait of Sedenion and Sons. It depicts the three possible Hopf fibrations of S15, giving the normed division algebras C, H, and O:

S1 ---> S15 ---> S14,

S3 ---> S15 ---> S12,

S7 ---> S15 ---> S8.

O still reigns supreme, but it is certainly not the mother of all algebras. Eccentric or not, at best O is the eldest Sedenion brother. Next in line is H, and then his baby brother C.

R is still suckling, so I haven't bothered to include him in the family portrait.

Enjoy,

JoyAttachment #1: sedenion.pdf

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  • Post #463

Michael,

Please re-read my last post first paragraph, for this is the issue. You are trapped in the orthodoxy of tensor calculus and (with emphasis again) *ITS* attendant *ALGEBRA*. Yes, algebra, the algebra of n'th-rank glorified matrices and their assumed form invariance under transformations, in which the concept of differentiation is made good through the addition of Christoffel symbols in the covariant derivative. This algebra *is not* the more robust O algebra, and you should not assume its calculus to be universal. I am not arguing that it is incorrect, for it is not for its intended usage. I am arguing it does not apply to O, and is replaced by my ensemble derivative. You can't divorce the algebra from the analysis, or worse assume there is no algebra. Additionally you are trying to create the D'Alembertian as its own stand-alone form, which I have stated is not appropriate. You must take all the other stuff along with it juxtaposed as O algebra demands. Hope this helps, at least to explain my position.

Would you please address my statement that the radii of S^n are assumed positive definite quadratic forms, and how a split signature is reconciled with this?

Rick

Rick,

"...the algebra of n'th-rank glorified matrices and their assumed form invariance under transformations, in which the concept of differentiation is made good through the addition of Christoffel symbols in the covariant derivative."

That's the most appropriate putdown of (what I find to be extremely ugly) tensor calculus I've ever seen, and it makes me feel even more strongly that geometric algebra really is the better approach. I have been very unhappy with the connections, or Christoffel symbols, (Susskind calls them "Christ awful" symbols) but your description makes even more transparent what an ugly mess tensor calculus is. When one first learns this stuff, it's presented as if it were "the ONLY way", and after a while most of us simply use it, if and when necessary, and do not question it, even after we find out that tensor calculus really isn't enough for general relativity and QM but that we need 'spinor's as well. Thanks for making it crystal clear just what is achieved by these things, that is, rescuing the covariant derivative. Although at some level I already knew this, I could never have summarized it so succintly.

This thread continues to offer a wealth of insight and Rick's comments some of the most insightful.

Edwin Eugene Klingman

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  • Post #465

Tom,

I'm not saying time is an illusion, I'm saying it is an effect.

If those balls were water molecules in a pot, we wouldn't assume some underlaying thermal scale determines how fast they are moving, because there is a thermodynamic effect. Just as with time, there is no real dimension along which these events exist. Those balls exist physically. They are present. If they were the basis of your clock, like a cesium atom and they moved around faster, you would say time has speeded up. Just like the atomic clocks on GPS satellites move faster than ones on the ground. It is not the present moving along a dimension, as with Newton's "absolute flow," from prior to succeeding events, but the physical movement creating change.

It is no more a mystery why the same clock in conditions that affect the dynamic of its process would move at different rates, than a pot of water would be different temperatures under different conditions.

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  • Post #466

I like that comment Ed.

I can't claim to have mastered tensor calculus, but I share your disdain for the fact that workarounds are used to overcome their inherent limitations. Without spinors and the Christoffel symbols; tensors are too lame for the job they are assigned, and it makes you wonder "why am I learning something that doesn't quite work right, anyhow?" Perhaps it's better to ponder "what kind of Math would actually handle this more naturally?"

I guess that's why it's good we have Rick around.

I was going to comment, Joy;

Perhaps it is a good thing that S15 yields only three Hopf fibrations, if those cases link to C, H, and O uniquely - through S1, S3, and S7. But you seem to have gotten that message loud and clear. When you commented "This does not look good," I thought you must know what you are talking about, but then I imagined it could be a blessing instead of a curse. I'm still trying to wrap my head around some of the possibilities implied, but this line of reasoning seems very promising indeed.

Have Fun,

Jonathan

Yes it was me.

My login must have expired. But I claim the words above.

Jonathan

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  • Post #468

Michael,

"By the way, your comment about the conscious mind being a look-out on a ship is similar to my view of the conscious mind being the captain of the ship. The view seen by the look-out is generally filtered before it is passed onto the captain, hence we suffer from optical illusions because what we see is not the same as what we perceive."

Our slightly different views on this are likely due to different life experiences. Personally I've spent my life working with horses, in a extended family situation. Not being the managerial type, I've mostly worked with the horses themselves. In terms of the people, I'm not the captian of this particular, rather organic enterprise and in terms of the horses, it is most effective to zen out. To take a light hold and not worry too much.

The result, in both directions, is that my consciousness has to take a back seat and just plug in, as a bit of circuit breaker, or "lookout." You would be surprised how much information is absorbed when you are not focused on points of concentration.

I have some thoughts about time, which I will share once my words are together.

But briefly; I think the meaning of the term 'arrow of time' changes as the universe evolves. The initial arrow of time may have more to do with the metric signature, but there are also the procedural and entropic aspects of time's passage to consider, the contrast of duration with interval - and so on.

More later,

Jonathan