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RECENT POSTS IN THIS TOPIC

Yuri Danoyan: on 7/26/13 at 19:15pm UTC, wrote Jonathan This is additional confirmation of 2D World ...

Anonymous: on 7/25/13 at 23:02pm UTC, wrote Jonathan This is additional confirmation of 2D World ...

Jonathan Dickau: on 7/25/13 at 0:29am UTC, wrote Read it and enjoyed Patrick. Thank you for commenting here. All the Best,...

Lev Goldfarb: on 7/22/13 at 21:32pm UTC, wrote Thanks for the great reference!

Georgina Woodward: on 7/22/13 at 21:09pm UTC, wrote OK Tom, thanks for the suggestion.

Thomas Ray: on 7/22/13 at 14:57pm UTC, wrote Georgina, "If there is resistance, and I think there must be to account...

Patrick Tonin: on 7/22/13 at 8:05am UTC, wrote Hi Jonathan, I have developped a scale-invariant 2D theory. It is very...

Jonathan Dickau: on 7/22/13 at 3:59am UTC, wrote That absolutely appears relevant Yuri. It could all boil down to whether...



FQXi FORUM
November 21, 2018

CATEGORY: Cosmology [back]
TOPIC: Dimensional Reduction in the Sky [refresh]
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FQXi Administrator Zeeya Merali wrote on Jul. 8, 2013 @ 14:53 GMT
Thanks to Jonathan Dickau for spotting this paper http://arxiv.org/abs/arXiv:1305.3153by FQXi member Giovanni Amelino-Camelia and others.

Here's the abstract:

"We explore the cosmological implications of a mechanism found in

several approaches to quantum-gravity, whereby the spectral dimension of

spacetime runs from the standard value of 4 in the infrared (IR) to a

smaller value in the ultraviolet (UV). Specifically, we invoke the

picture where the phenomenon is associated with modified dispersion

relations. With minimal assumptions, we find that UV behaviour leading

to 2 spectral dimensions results in an exactly scale-invariant spectrum

of vacuum scalar and tensor fluctuations, regardless of the equation of

state. The fluctuation production mechanism is analogous to the one

known for varying speed of sound/light models and, unlike in inflation,

the spectrum is already scale-invariant before leaving the horizon,

remaining so after freeze-in. In the light of Planck's recent results we

also discuss scenarios that break exact scale-invariance, such as the

possibility that the spectral dimension runs down to a value slightly

higher than 2, or runs down to 2 but with an extremely slow transient.

We further show that the tensor to scalar ratio is fixed by the UV ratio

between the speed of gravity and the speed of light. Not only does our

model not require inflation, but at its most minimal it seems

incompatible with it. In contrast, we find that running spectral

dimensions can improve the outlook of the cyclic/ekpyrotic scenario,

solving the main problems present in its simplest and most appealing

realisations."

Please discuss here...

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Jonathan J. Dickau wrote on Jul. 8, 2013 @ 15:55 GMT
Thank you Zeeya,

I have long believed that the dimensionality of space and/or spacetime is not fixed, but rather is either evolutive, scale-dependent, or both. The notion that dimensionality is resolution independent (the same at all scales) may be merely a convenient approximation, due to the fact that our local-neighborhood of Euclidean 3-space plus time is so large, compared to our common experience. It is interesting that the authors of this paper chose to focus on the divergence between IR and UV measures of spectral dimension, because this relates it back to our common framework of observation, but the exciting part about it is that they are relating this story to cosmology.

What this work seems to indicate is that dimensionality evolves or emerges from a value near 2-d at the point of the universe's origin, to become 4-d spacetime before now, and in fact while the universe was still young. My essay from last year focused somewhat on this possibility, as I was emphasizing that a constant value for the dimensionality of space is one of those cherished assumptions that might be wrong. The behavior of a universe that's 2-d near the Planck scale and 4-d at the common or cosmic scale is actually attractive for quite a few theories now under consideration. I'll have more to say in another post, a little later.

Have Fun,

Jonathan

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Member Sabine Hossenfelder wrote on Jul. 8, 2013 @ 16:26 GMT
I wrote about this paper here.

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Jonathan J. Dickau replied on Jul. 8, 2013 @ 18:07 GMT
And I invite you to share any comments here, as you see fit.

Please Bee; don't be a stranger to this space.

Your insights are always welcome here.

All the Best,

Jonathan

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Member Sabine Hossenfelder wrote on Jul. 8, 2013 @ 16:27 GMT
Argh. Here's the link http://backreaction.blogspot.de/2013/05/dimensional-reductio
n.html

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Jonathan J. Dickau replied on Jul. 8, 2013 @ 16:40 GMT
Thanks Sabine!

I'm glad I was not the only one who found this paper interesting. I'll take a look at your posting, and fold some reflections into this thread.

All the Best,

Jonathan

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Jonathan J. Dickau replied on Jul. 8, 2013 @ 16:47 GMT
I offer the following link to Sabine's blog;

Dimensional Reduction at Backreaction - Sabine Hossenfelder

Thanks again!

Jonathan

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Jonathan J. Dickau wrote on Jul. 8, 2013 @ 16:36 GMT
To continue a bit..

The property of changing dimensionality is an emergent feature of spacetime for Causal Dynamical Triangulations and Quantum Einstein gravity theory, as is mentioned by the authors. But the idea of a spectral dimension that runs from 2-d near the Planck scale to 4-d as we approach the common scale is attractive to a number of Quantum Gravity theories now under serious consideration, including loops, spin foams, and various theories of causal structure. To constructivist geometers, there is no dimensionality until a means of measurement is constructed, which gives determination a dual face. That is; determination by measurement is construction or creation - at its core.

However at this point; I wonder whether what is most realistic is some sort of bi-metric approach, where there exists an upper and lower bound to what can properly be referred to as dimensionality. I note here that String theory and Supergravity operate in a 10-d or 11-d space respectively, near the Planck scale, and I do not find this to be in conflict with the lower bound of 2-d described by CDT and QEG - so long as a bi-metric cosmos is allowed. But I also wonder whether our spacetime is evolving past the 4-d phase. The fact that the 5-d sphere offers maximal hypervolume would seem to suggest that the end of dimensional evolution would be to a cosmos of five dimensions.

Intriguingly; the authors of this paper note that their result is at odds with most versions of the inflationary universe scenario. However; this may make for a closer reflection of the Planck satellite results, and other recent data, in future cosmological models. It would seem Steinhardt was right to call his own Inflationary model into question, but it remains to be seen which proposed model is the most appropriate replacement.

All the Best,

Jonathan

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Jonathan J. Dickau wrote on Jul. 8, 2013 @ 17:31 GMT
Determining dimensionality has broader implications.

Perhaps the most interesting piece is how dimensional reduction relates to abstraction, forming the basis for cognition and symbolic reasoning. The research of Judy DeLoache found that children below the age of 2 1/2 display 'dimensional confusion' - but once this hurtle is cleared, human children rapidly acquire a capacity for reasoning with symbols.

My insight is that what is learned at that point is a skill with dimensional reduction. The ability to distinguish a 2-d picture from a 3-d scene is a gateway skill to higher learning, because that way the idea of 'writing' information onto a surface actually makes sense. And this notion has made its way into a number of my previous essays, and other published papers before and since their appearance.

But the idea that nature also determines dimensionality dynamically is a kind of validation for the constructivist view that observation and measurement are another face of imagination and construction. This validates also the quantum mechanical view that observation is creation. That is; on some level, learning and creativity are the same - the universe and its inhabitants are at play.

Have Fun,

Jonathan

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Jonathan J. Dickau replied on Jul. 9, 2013 @ 03:46 GMT
I'll talk on this a little further..

The playful exploration of human children in the range of 2 to 2 1/2 years of age helps them calibrate; they are mobile and relate to things but don't yet have their full sense of dimensionality - for objects and their surroundings. Once this sense is acquired, according to the research of DeLoache; there is also an increasing ability to relate to information presented in symbols - and to learn through symbolic reasoning. That is; after their sense of 'dimensional confusion' has gone away, symbolic reasoning is more easily learned.

I equate this insight, in a broad way, with the Holographic principle, and with the specific line of reasoning and insight of Gerard 't Hooft - in his paper on "Dimensional Reduction in Quantum Gravity" that was the most downloaded arXiv Physics paper of all time, and which first inspired the whole range of Holographic Universe papers. I've talked about this connection of the determination of dimension early childhood with holography in several papers, but I thought it was important to mention it here.

I'll talk more about how the holographic principle relates to the process of abstraction, in a later post.

Have Fun!

Jonathan

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Jonathan J. Dickau wrote on Jul. 9, 2013 @ 04:45 GMT
And back to the subject of Cosmology, and the paper under discussion..

Imagine that spacetime and the universe could start out 2-d, and then evolve into a 4-d extension. What if the spectral dimension D could run smoothly from a value of about 2, up to its current observed value of 4, for our local spacetime? Giovanni Amelino-Camelia, Michele Arzano, Giulia Gubitosi, and Joao Magueijo explore this possibility in great detail in the paper 'Dimensional reduction in the sky' which we are discussing here.

But it appears the same authors have just come out with another paper, to further illustrate how gravity behaves under dimensional reduction. And that paper, which is a direct extension of the paper under discussion, is available at this link.

Rainbow gravity and scale-invariant fluctuations

I shall return to this forum after giving a look at this new paper, a visit to Sabine's blog, and reviewing the earlier paper.

All the Best,

Jonathan

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Member Giovanni Amelino-Camelia wrote on Jul. 9, 2013 @ 08:32 GMT
ciao Jonathan

thanks for your interest in our paper.

I should stress that most of the issues we are intrigued by in these papers are presently unmanageable in other scenarios for obtaining UV-running down to 2 spectral dimensions. For one of these scenarios, based on Planck-scale modifications of relativistic properties (such as on-shellness), we managed to do the analysis and we are conjecturing that the results might apply to other scenarios with 2 UV spectral dimensions.

In this respect the second paper "Rainbow grvaity and scale-invariant fluctuations" provides some encouragement: it suggests that the core mechanism is gravity acquiring special properties when the number of spectral dimensions is 2 (and this looks like a property that might indeed be not much dependent on what are the seed features that drive the running).

I do find striking that several approaches to the quantum grvaity problem point toward 2 UV spectral dimensions and exactly 2 UV spectral dimensions is what we found to be required for a scale-invariant spectrum of fluctuations in our setup (to good first approximation observations point toward a scale-invariant spectrum). It might be just a numerological coincidence, but these are the sort of numerological coincidences which I feel are worth pursuing...they might contain important hints...Michele Arzano, Giulia Gubitosi, Joao Magueijo and I are determined to keep trying to find out what is behind all this

best regards

Giovanni

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Jonathan J. Dickau replied on Jul. 9, 2013 @ 19:41 GMT
Thanks greatly Giovanni!

I appreciate your commenting, and I hope you will check in on this thread from time to time, as your research develops. The work of your team is very interesting indeed. That fact that a D of 2 is robust, or appears to be independent of the seed features that drive the running of D, seems especially significant. Perhaps a more literal interpretation of the 'Universe as a Hologram' is in order.

I'll likely say more here, once I read about what Rainbow Gravity has to offer.

All the Best,

Jonathan

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Thomas Howard Ray wrote on Jul. 9, 2013 @ 10:34 GMT
Thanks for the heads up, Jonathan.

"Imagine that spacetime and the universe could start out 2-d, and then evolve into a 4-d extension."

Fundamentally, I think that's the basis for all continuous function models and topologically based extradimensional theories for d > 4. While general relativity cannot avoid the singularity, string theory begins with the negative root, - 2, whose square is positive real, 4. This is one reason that I believe general relativity suffers no loss of generality when viewed as a model finite in space and unbounded in time; the singularity disappears, and the time parameter absent from discrete (Hilbert space) dynamics is restored.

Tom

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Jonathan J. Dickau replied on Jul. 9, 2013 @ 19:48 GMT
Thanks Tom,

Your comment is absolutely relevant and apropos. The continuous running of the parameter D allows topological features to be maintained or transformed as dimensionality evolves or emerges. Topology preserves information.

I'll have to ask you to explain the significance of -2 in String theory, as the meaning of that comment is lost. I do note however, that -2 is the furthest point from the origin in the Mandelbrot Set. I have no idea if that is relevant, though.

More Later,

Jonathan

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Thomas Howard Ray replied on Jul. 10, 2013 @ 11:28 GMT
Hi Jonathan,

I think that the - 2 in the Mandelbrot calculation is a mathematical artifact resulting from the properties of the 2 dimensional complex plane in which the object is constructed.

String theory takes 2 dimensions (the plane or "brane") as the fundamental space in which strings propagate. Witten discards the idea of spacetime and replaces it with a 2-dimensional field theory. That all of the physical world can be described in 2 dimensions eventually comes down to the holographic principle -- the information we can abstract from the surface of a black hole. Because black hole energy increases to the negative inverse (- 1) of temperature, and entropy simultaneously increases to the - 2 power, entropy increases with area (the surface of the black hole) which is r^2, radius squared.

So if we're speaking of a fundamentally geometric theory that applies arithmetic functions, as string theory does, we need all those mathematical artifacts that originate in the complex plane C, as a closed algebra, imaginary roots and all. Otherwise, we would be (mathematically) stuck in the infinitely open plane and could not escape incompleteness. String theory, although based in quantum field theory, is -- like special and general relativity -- mathematically complete.

______

I hope you and the group will forgive me for being self-serving at this point -- I have invested a lot of time and thought into constructing an alternative basis for encoding information in n-finite dimensions of space while nevertheless preserving a time metric. For this discussion, we only refer to 1.3 -- 1.10 in the draft paper linked. The table explains correspondence of the plane to the integer - 2.

All best,

Tom

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Jonathan J. Dickau wrote on Jul. 10, 2013 @ 01:51 GMT
I'll make a general comment here..

One of interesting the things about having a spectral dimension D that runs from 2 to 4, is that it implies spacetime can take on a fractal character in the middle ground (in the regime of changing dimension). As Sabine said in her blog, we need to be fairly specific about how we define dimension, in this context. In CDT; a random walk yields a measure of the local dimensionality, determined by how swiftly a walker returns. And it was found that this measure evolved when the simulation was run from very short to larger scales.

But the authors of the 'Dimensional reduction' paper state that having D as a running parameter can leave variations 'frozen' at the horizon, which makes it much easier to explain cosmological large scale structure fractality and the existence of extreme scale structure such as the Huge-LQG of Clowes and Campusano described in this paper in MNRAS and discussed in this FQXi Forum thread.

Since the notion being researched by Giovanni and his colleagues does not favor the Inflationary Universe scenario, this leaves the door open to other possibilities, where the homogeneity in the FLRW metric is not a requirement. In such a setting; a cluster like the Huge-LQG is not seen as a problem. I'll say more about this later.

All the Best,

Jonathan

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Jonathan J. Dickau wrote on Jul. 10, 2013 @ 02:09 GMT
Another comment here..

In case it escaped notice; I wanted to point out the comment about a scalar to tensor ratio, in the abstract of the paper under discussion. The authors state that this ratio "is fixed by the UV ratio between the speed of gravity and the speed of light."

The implication here, of course is that gravity propagates at a speed faster than light. This was first seriously discussed in a paper in Foundations of Physics by Tom VanFlandern, which appeared not long before his demise. But it appears that, if the Dimensional Reduction paper is on target; VanFlandern may have been right.

Anyhow; this could change the picture for primordial gravity waves, and so on. Yet one more reason to actually build next-generation GW detectors, such as the planned (and scrapped - I think) LISA probe. There is, of course, more to talk about with this subject.

All the Best,

Jonathan

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Jonathan J. Dickau replied on Jul. 10, 2013 @ 02:22 GMT
Hello again folks;

I wanted to post a link, having located the Foundations of Physics paper by VanFlandern and Vigier - that I mentioned above. This paper suggests there may be a more rapid propagation for gravity, than for EM waves.

Experimental Repeal of the Speed Limit for Gravitational, Electrodynamic, and Quantum Field Interactions

I had the pleasure to meet Tom, who was a very colorful and flamboyant fellow, at CCC-2 - the 2nd Crisis in Cosmology Conference. He was one of the lead organizers for that event. Unfortunately; he passed away only months later.

All the Best,

Jonathan

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Thomas Howard Ray replied on Jul. 10, 2013 @ 12:13 GMT
Hi Jonathan,

Van Flandern's abstract reads in part: "If angular momentum conservation is invoked in the geometric interpretation to explain experiments, the causality principle is violated. The field interpretation avoids this problem by allowing faster-than-light propagation of gravity in forward time."

My essay explains why the conservation of angular momentum does not violate causality in a geometric theory of continuous functions.

Another problem with Van Flandern's hypothesis, though, is that of simultaneity. Remember that thought experimenbt where the Sun suddenly disappears, and we don't know it until approximately 8 minutes later? If it were possible for any physical influence -- including gravity -- to propagate faster than the speed of light, it would also be possible to observe action at a distance simultaneously affect two or more spacetime separated cosmic events and we would be able to observe this correlation. As it is, gravity seems to work just as Newton and Einstein described it -- as a function of mass and time reversible dynamics indexed to the speed of light.

Best,

Tom

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Jonathan J. Dickau replied on Jul. 10, 2013 @ 15:13 GMT
Not so fast (or maybe faster) Tom;

The work of Jenkins and Fischbach on varying rates of nuclear decay has resulted in an interesting development. Specifically; it was found there is a seasonal dependence where the rate of decay half-life varies measurably, with a cycle that appears to relate to Sun-Earth distance. But once a greater degree of precision was achieved, it was found that this variation is also observed during a Coronal Mass Ejection or CME.

My guess is that, because these events move a significant amount of mass in a very short time, this ripples or deforms the fabric of spacetime. So if there was a discrepancy between the scalar mode and tensor mode propagation, one would expect that arrival time for such a signal would be less than 8 minutes. This is apparently exactly what is observed. Now Professor Jenkins has a Patent pending for an Early Warning System for CME events - and he even appeared on the Weather Channel to talk about it.

A topological spacetime fabric is the simplest explanation, as light travels around, while gravity acts center to center, and the ripple is felt at the sub-atomic scale.

Regards,

Jonathan

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Edwin Eugene Klingman wrote on Jul. 11, 2013 @ 01:47 GMT
Jonathan,

I discuss the Jenkins and Fischbach data in "The Chromodynamics War" and propose that, based on my model of particle physics, it is due to solar neutrinos. I do not know what the CME mechanisms are, but if they result in an increase of solar neutrinos, this would be consistent with my model. This seems to be much simpler than "a topological space-time fabric".

In fact, I just googled and found: "Aug 30, 2010 - Jenkins and Fischbach suggest that the changes in the decay rates are due to interactions with solar neutrinos, nearly weightless particles ..."

This was what I suggested to them in 2009 when I sent them a complimentary copy of "The Chromodynamics War" upon receipt of permission to publish their data.

I'm also interested in your remarks about possible difference in the speed of light and gravity. Are there any arXiv or viXra papers on this?

Also, if you or Tom would explain the angular momentum problem, I would be interested.

And, FYI, I have gotten another competent physicist interested in Kauffmann's work. (Each one teach one!)

I don't know how you stay so up to date on these topics. Do you sleep?

Best regards,

Edwin Eugene Klingman

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Jonathan J. Dickau replied on Jul. 11, 2013 @ 17:06 GMT
Well actually,

I always seem to be on the go in my dreams, and there are always several things to do or get done when I am awake, but my romance with pure knowledge seems to be the most compelling motivator these days. My recollection is that there were problems with the neutrino hypothesis, but I would have to do some research. However; the subject of a neutrino wind or spin-aligned fields and local connections resulting from the neutrino flux are certainly relevant to this discussion, even if topological factors do come into play.

My point was that it might be fruitful to visualize internal and external modes of connection, corresponding to scalar vs tensor modes of propagation, and a difference between the speed of gravity and the speed of light. That is; there are forces that operate upon a center and forces that operate from within that center. Part of this is driven by my desire to understand the connection of non-commutative and non-associative geometries to Physics. If one examines the quaternion 3-sphere and octonion 7-sphere, their Möbius like surface makes them simply connected; inside-out is outside-in.

Thinking about examples like this makes the ideas in the Dimensional reduction paper easier for me to understand, even if they do not exactly apply. The notion that things can be simultaneously connected by their outsides and their insides is not often discussed, but bears examination.

All the Best,

Jonathan

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Jonathan J. Dickau replied on Jul. 15, 2013 @ 17:00 GMT
I just wanted to point out..

To an extent; one can draw an analogy by equating 'from the inside' with 'probing on a small level of scale' and 'from the outside' with 'probing on a larger level of scale' because the observer or detection platform always has a particular size, and there will be larger spaces 'out there' than whatever space is contained 'in here' - the space from which observations are taking place. This connects up with what Tom Ray said in his essay from last year, that observation always takes place from a center and defines a sense of toward and away.

Have Fun,

Jonathan

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Jonathan J. Dickau wrote on Jul. 15, 2013 @ 17:27 GMT
Hello again,

It appears that the idea for 'Rainbow Gravity' was first set out in a paper by Magueijo and Smolin called Gravity's Rainbow that appeared in 'Classical and Quantum Gravity' in '04. The most important piece of that work to the current effort and the 'Dimensional Reduction..' paper is found in the following excerpt.

"the geometry of spacetime may depend on the energy of a particle moving in it. Thus, spacetime geometry has an effective description; in the language of the renormalization group, geometry "runs." Hence there is no single spacetime dual to momentum space; the dual to momentum space is the energy dependent family of metrics."

That is; the spectral dimension D runs with the energy E we use, or equivalently with the smallness of scale we probe the dimensionality of space in, making spacetime appear 2-d as we approach a UV cutoff (toward the Planck scale) and 4-d toward the infrared, or as we approach the common level of scale. The 'family of metrics' noted in the quote above constitute the 'rainbow' of Rainbow Gravity.

More later,

Jonathan

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Marcel-Marie LeBel wrote on Jul. 21, 2013 @ 14:58 GMT
The team of Jenkins and Fischbach attracted my attention for one reason. Nuclear disintegration are said to be random and impervious to anything. But every other spontaneous phenomenon are subjected to the local rate of passage of time!. Nuclear disintegration may be random but its rate should be subject to local time rate.

This is what I was looking for. I understand the relationship of gravity to the rate of passage of time like Unruh.(1) (Below), they are one and the same.

“A more accurate way of summarizing the lessons of General Relativity is

that gravity does not cause time to run differently in different places (e.g., faster far from the earth than near it). Gravity is the unequable flow of time from place to place. It is not that there are two separate phenomena, namely gravity and time and that the one, gravity,affects the other. Rather the theory states that the phenomena we usually ascribe to gravity are actually caused by time’s flowing unequably from place to place.”

From this, I was expecting time to run at different rates away from the Sun and nuclear disintegration rates to follow.



This was the basis of a gravitational wave detector I was thinking about... Passing gravitational waves (made of variations of time rate) would change in a specific way the variations in the rate of disintegration of radio-nuclide.... using particle emission time of flight to avoid temporary local time distorsions .... and coincidence detectors for direction ... The question is; how good are we at measuring short time interval? Better than “space” intervals?

Marcel,

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Marcel-Marie LeBel wrote on Jul. 21, 2013 @ 15:35 GMT
Forgot the reference (1) to Unruh article. Here it is.

(1) arXiv:gr-qc/9312027v2 17 Dec 1993

Marcel,

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Jonathan J. Dickau replied on Jul. 22, 2013 @ 03:49 GMT
Thank you so much Marcel-Marie..

I appreciate the insights and the time taken to comment. I'll look at the Unruh paper and I think I need to read your essay too, as I saw you are in the contest, and what you say sounds very interesting. I hope you find my essay of interest as well, but I will comment here after looking further. The Gravity Wave detector idea is VERY MUCH of interest, especially to a colleague Dr. Beckwith.

There are some very good clocks now, and that definitely opens up the possibility for some exciting new Physics. I'll comment more on this page, after looking at the above.

Have Fun!

Jonathan

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Lev Goldfarb replied on Jul. 22, 2013 @ 21:32 GMT
Thanks for the great reference!

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Yuri Danoyan wrote on Jul. 21, 2013 @ 16:58 GMT
Johnatan

My be h= 1.054x 10^-27g x sm^2/sec is 2-Dimensional quanta of the Universe wrapped in 3-Dimensional Space.Hint to Holographic universe...?

Yuri

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Jonathan J. Dickau replied on Jul. 22, 2013 @ 03:59 GMT
That absolutely appears relevant Yuri.

It could all boil down to whether light travels as photons or plasmons, but I think there is more to it. From what I recall of your research, there are definitely tie-ins of your work to these new findings,and as Giovanni says above, things are developing quickly in a productive way. Maybe some of the roadblocks to understanding are being removed, and important work that has been held back can get the recognition it deserves.

I will try to get to reading your essay too, and I thank you for leaving your comment here.

All the Best,

Jonathan

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Patrick Tonin wrote on Jul. 22, 2013 @ 08:05 GMT
Hi Jonathan,

I have developped a scale-invariant 2D theory. It is very basic (I am not a theoretical physicist) but it might give you some ideas. I hope it helps.

You can also find my essay here .

Cheers,

Patrick

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Jonathan J. Dickau replied on Jul. 25, 2013 @ 00:29 GMT
Read it and enjoyed Patrick.

Thank you for commenting here.

All the Best,

Jonathan

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Anonymous wrote on Jul. 25, 2013 @ 23:02 GMT
Jonathan

This is additional confirmation of 2D World

http://vixra.org/abs/1212.0030

Yuri

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Yuri Danoyan wrote on Jul. 26, 2013 @ 19:15 GMT
Jonathan

This is additional confirmation of 2D World

http://vixra.org/abs/1212.0030

Yuri

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