Quaternionic Rays instead of Complex

[Antizzio]

Not sure what paper you are referring to, but it makes it even more interesting to hear analysis of a cosmologist regarding each of the five contemporary problems mentioned above.

 

[PeterDonis]

As I said, I'm not a cosmologist, but I can see at least one obvious flaw in the model you describe:

In the early universe (near the bounce point) matter density dominates.

This is not true of our actual early universe; it was radiation dominated, not matter dominated. It only became matter dominated roughly around the time it became transparent to radiation and the CMB last scattering surface was formed. (Not exactly the same time--the two events weren't connected, at least cosmologists don't believe they were as far as I know.)

 

[Antizzio]

I used "matter" as opposed to "vacuum", not as opposed to "radiation".

 

[PeterDonis]

I used "matter" as opposed to "vacuum", not as opposed to "radiation".

But matter density and radiation density dilute differently as the universe expands (that's why it's even possible for the early universe to shift from radiation dominated to matter dominated), so they can't both be represented by the same mathematical object in the model (in this case, the Lebesgue measure), since that can only change relative to vacuum in one way.

 

[Antizzio]

can't both be represented by the same mathematical object in the model (in this case, the Lebesgue measure)

Yes they can. Lebesgue measure is just volume of spacetime, so it represents both matter density and radiation density quite successfully.

 

[PeterDonis]

Lebesgue measure is just volume of spacetime, so it represents both matter density and radiation density quite successfully.

This doesn't make sense. Lebesgue mesaure is one function of time. Matter density and radiation density are two different functions of time.

 

[Antizzio]

The concept of "density" means certain amount of stuff in certain volume. That certain volume is defined by Lebesgue measure. In other words, Lebesgue measure defines volume of every piece of spacetime. BTW, Lebesgue measure is not a function of time, it assigns a nonnegative number (called volume) to every subset of spacetime manifold. The point is matter+radiation density rapidly decreases, while vacuum energy density stays the same, so eventually it wins.

 

[PeterDonis]

Lebesgue measure defines volume of every piece of spacetime.

A piece of spacetime doesn't have a 3-volume. Here's what you said earlier:

the Haar volume of 3-spheres remains constant, while Lebesgue volume increases towards Past and Future.

In other words, you're saying "Lebesgue volume" of a 3-sphere increases towards the Future, i.e., with time. (Also towards the past, to the past of the narrowest part of the hourglass, but that's a whole separate thing.)

Ok, so the Lebesgue measure is just a measure of volume (but only with respect to matter or radiation, not vacuum), not density--the density depends on what's in the volume (matter or radiation). Is that a fair statement of the model you're describing?

 

[Antizzio]

A piece of spacetime doesn't have a 3-volume

Of course it does: each 3-sphere slice is a piece of spacetime and does have 3-volume. But yes, that's what I have been trying to describe, essentially.

 

[PeterDonis]

each 3-sphere slice is a piece of spacetime

It's a slice taken out of spacetime, but it has no extent in the fourth spacetime dimension. So it has no spacetime volume.

and does have 3-volume.

In a model where the universe is spatially finite, yes. But it's a 3-volume--a volume of space. It's not a spacetime volume--that would be a 4-volume. See above.

 

[mitchell porter]

That's good to know. Can you cite a reference that covers this? Thanks.

https://www.maths.usyd.edu.au/u/athomas/FunctionalAnalysis/daners-functional-analysis-2017.pdf

Corollary 18.7

btw I tested the ubiquity of this result by googling "functional analysis textbook" and looking for it in the textbooks that had online copies. I found it in the third online textbook listed.

Does all this make sense to a cosmologist?

The problem is that in general relativity, you have a metric field whose value varies throughout space-time, and you have other fields whose values can also vary, and then you have equations (Einstein field equations, geodesic evolution equations) which describe how all the fields vary together - how they interact. Then there is an infinitude of possible space-time geometries which are solutions to these coupled equations, like black hole geometries, expanding universes, black holes orbiting each other in expanding universes, and so on.

What Trifonov is doing is taking the quaternions, which form a four-dimensional continuum, focusing on two ways to define a notion of "volume" for regions of quaternionic space, and then saying maybe one measure behaves like matter density in a common class of cosmological solutions (the FLRW geometries), and the other measure behaves like dark energy density.

Even if this can be made to make sense, it's lacking all the causal details which, in general relativity, not only govern the relationships between matter and geometry in the FLRW universes, but also the other kinds of possible universe, and which also explain everything else about gravity, including phenomena (like planets orbiting or light rays bending or the formation of event horizons) that take place on smaller scales.

Hopefully my point is clear - that even if Trifonov's correspondence makes sense, it is merely mimicking one class of universes, and has nothing to say about falling apples or planetary orbits. This is why it looks like a kind of coincidence.

Furthermore, by performing the fiber bundle decomposition of quaternionic Hilbert space, he's saying each quantum state (a quaternionic ray) has a four-manifold associated with it (the fiber of quaternions by which a quantum state could be multiplied). Assuming that some form of quaternionic quantum mechanics can be made to work, then the quaternionic rays will be associated with particular values of some physical observables, defined by a set of operators indexed by 4d space-time coordinates. But none of that has any evident connection to the fiber of quaternions. As I said, it should equally be possible to work with a different operator algebra describing observables in a space with a different number of dimensions.

 

[Antizzio]

But it's a 3-volume--a volume of space. It's not a spacetime volume--that would be a 4-volume.

3-volume has no absolute meaning in GR (it is frame dependennt), only 4-volume does. 3-sphere slice has no "time extension" in the comoving frame, in all other frames it gets it back again (and does not look like a 3-sphere anymore).

 

[Antizzio]

What Trifonov is doing is taking the quaternions, which form a four-dimensional continuum, focusing on two ways to define a notion of "volume" for regions of quaternionic space, and then saying maybe one measure behaves like matter density in a common class of cosmological solutions (the FLRW geometries), and the other measure behaves like dark energy density.

This makes sense on it's own, it does not need "to be made to make sense". BTW, he does not need to focus on "defining" two notions of volume, they are natural structures already build in. But what about those 5 cosmological problems? Can you take a whack at them, as a cosmologist, since you are responding to my "does all this make sense to a cosmologist?" question?

 

[renormalize]

3-volume has no absolute meaning in GR (it is frame dependennt), only 4-volume does.

The proper 3-volume element $dV=a^{3}\left(t\right)/\sqrt{1-kr^{2}}\,r^{2}\sin\theta\,drd\theta d\phi$ of an observer comoving with the cosmological expansion certainly has physical meaning. What does Trifonov specifically predict for the functional form of the universal scale factor $a\left(t\right)$?

 

[Antizzio]

The proper 3-volume element dV=a3(t)/1−kr2r2sin⁡θdrdθdϕ of an observer comoving with the cosmological expansion certainly has physical meaning.

Because you specified the frame. I believe the scale factor is the only free parameter in his framework.

 

[renormalize]

Because you specified the frame. I believe the scale factor is the only free parameter in his framework.

You do indeed say about Trifonov's theory:

(3) it has a natural FLRW metric (with a single free parameter, scale factor a(t));

But then you go on to state:

(1) In the early universe (near the bounce point) matter density dominates.
(2) As universe expands, matter density (Lebesgue) dilutes rapidly, while vacuum energy density (Haar) stays constant.
Dark Energy: Vacuum energy density catches up to and eventualy overtakes matter density, producing accelerated expansion. That's a purely geometric source of dark energy.
Fine Tuning: Since vacuum energy density is always nonzero in this description, it also seems to produce a geometric solution to the fine tuning problem.
Hubble Tension: The rates of expansion are different before and after the "catch up" point.
Coincidense problem: the position of the "catch up" point.
Mature Structures too early: some of the stuff gets through the bounce point from the "other side".

That description of the universe implies that the scale factor $a(t)$ exhibits a rather definite history in Trifonov's theory. How then can it be a "free parameter"? If it was truly free, it could just as well be chosen to be the $a=\text{constant}$ of Einstein's original static universe.

 

[Antizzio]

That description of the universe implies that the scale factor a(t) exhibits a rather definite history in Trifonov's theory. How then can it be a "free parameter"? If it was truly free, it could just as well be chosen to be the a=constant of Einstein's original static universe.

It free in the sense that his theory does not predict the specific function, it describes a range of theories parameterized by a(t). It only predicts the dimensionality, topology, metric and a few other things. It may not be as breathtaking as assigning "operators phi(x0,x1,x2,x3,x4,x5,.......) to each point in spacetime", but it still deserves attention, don't you agree?

 

[PeterDonis]

3-volume has no absolute meaning in GR (it is frame dependennt)

In a sense, that's true, but not the way you mean. See further comments below.

And in any case, that doesn't change the fact that a spacelike slice cut out of spacetime (however you cut the slice) has a 3-volume, but not a 4-volume.

3-volume has no absolute meaning in GR (it is frame dependennt), only 4-volume does. 3-sphere slice has no "time extension" in the comoving frame, in all other frames it gets it back again

In a coordinate sense, yes, but...

(and does not look like a 3-sphere anymore).

...no, this is not correct. The fact that it is a 3-sphere--a 3-dimensional spacelike hypersurface with the topology $S^3$ and a particular induced metric on it--is not frame-dependent, it's an invariant. And therefore so is the fact that it has a 3-volume, but not a 4-volume. That fact is simply less obvious in a coordinate chart other than the standard FRW coordinates.

 

[PeterDonis]

It free in the sense that his theory does not predict the specific function.

I'm not sure what you mean. You say the theory does predict the metric. That is sufficient to predict the specific function.

 

[PeterDonis]

it still deserves attention, don't you agree?

This question has already been answered: cosmologists so far have paid little or no attention to the theory.

Ultimately it's up to the relevant scientific community to decide what deserves attention. That's something that's going to be resolved here at PF. So if your goal is to get cosmologists to pay attention, you are in the wrong place.

 

[Antizzio]

'm not sure what you mean. You say the theory does predict the metric. That is sufficient to predict the specific function.

It predicts a specific type of metric, it still can have free parameters.

 

[Antizzio]

it has a 3-volume, but not a 4-volume

That's what we need isn't it? Lebesque 3-volume increases, density goes down.

cosmologists so far have paid little or no attention to the theory

Again, no wonder: my yapping about cosmology has not been published or even mentioned anywhere except this particular thread (since you deleted it from Cosmology on this forum). As far as I know Trifonov's papers have none of this stuff either.

 

[renormalize]

As far as I know Trifonov's papers has non of this stuff either.

So you're describing an unpublished personal theory, which is completely off limits here. Didn't you read the rules for Physics Forums before you joined?

 

[Antizzio]

So you're describing a personal theory, which is completely off limits here.

The whole theory is in my "5 contemporary problems in cosmology" post above.

 

[renormalize]

The whole theory is in my "5 contemporary problems in cosmology" post above.

But your post #27 explicitly violates the PF rules:

Speculative or Personal Theories:
Physics Forums is not intended as an alternative to the usual professional venues for discussion and review of new ideas, e.g. personal contacts, conferences, and peer review before publication. If you have a new theory or idea, this is not the place to look for feedback on it or to ask for help developing and publishing it.

That post should be deleted and/or this thread should be closed.

 

[berkeman]

Thread is closed for Moderation.

 

[PeterDonis]

As far as I know Trifonov's papers have none of this stuff either.

Then this thread is personal theory, and will remain closed.