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An Extended Essay on whether the Geon model Applies to all Particles
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An Extended Essay on whether the Geon model Applies to all Particles
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Dubbelosix
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An Extended Essay on whether the Geon model Applies to all Particles
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24/08/2017 22:03:35 »
The concept was relatively simple to understand - I had an idea that consisted of an argument along the lines of ''if the curvature tensor must describe spacetime at all scales, maybe a non-trivial spacetime commutation could describe how the curvature can be larger than a specific wave. In some way, I argued, the curvature has to be proportional to a set of terms:
This early attempt was hoping to find some understanding in terms of the curvature of a sphere - I soon dropped this idea and concentrated more on how those Christoffel symbols and the commutation should be interpreted. After some talks with my friend Matti on the subject, it struck me that the same commutation relationship that already exists for Riemann geometry could already explain the commutation.
This did not come so quick though, I know some preliminaries but it took to understand how the commutation can be described by two connections would be the easiest way to approach this. Certain things were clear: For instance,
I knew crucially in its structure, we have a derivative of
known as the covariant derivative or the connection of a gravitational field - it seemed there must have been commutation happening between the derivatives of the field and between the field itself.
The only way to assume a spacetime commutation property, is to insist they follow the usual rules implied:
Its like there are two Gamma matrices with commutative properties! It was by understanding this, we could understand maybe the relationship being described by the following commutation relationship:
It appears, this would be the most natural way to translate the dyamics of the curvature tensor
to look for that wanted spacetime uncertainty relationship. The only natural continuation then, would be to see how to construct the commutation and also expand it for the results of its Christoffel symbols. The commutation relationship is (in a usual convention)
Here we have explicitly wrote out the connections as having commutative properties satisfying our desired inequality. Writing the whole commutation out to find the christoffel symbols, (using differential notation) reveals the following and using general indices:
Pulling it out of its differential notation form, what we really have is
and is just the Riemann tensor, but we hope we have specified, it one that follows the spacetime relatioship
.
What this unveils is that my initial idea of there being some commutator between the connection and the Christoffel symbols was true - but we also have another term, another commutator arising relativistically. The final mathematical model proposed to pave the way towards a unification of two concepts, a non-trivial spacetime uncertainty principle with the concept of geometry:
So... what is the whole point?
Yes, maybe it can describe some way to capture a particle in its own gravitational field, what is the signifiance? This is a question that will require a longer explanation.
Essentially a Geon is a type of black hole - but a very special kind of one. Its not like your usual black hole which consists of a many-system body which radiates thermal energy - we don't know whether a Geon would radiate, but what we do know, is that it may describe all states of matter, according to A. Wheeler, who created the Geon hypothesis, but could not prove it using quantum mechanics.
Is matter really just trapped states of waves?
If so, what happens when particles exhibit wave properties? How does the wave escape, does it escape? As great as these questions are, one at a time... and even then, I cannot answer them all. I am new to this Wheeler model of particles and I am literally, still learning about what he had in mind. We may think it has appealing properties, since it would mean a wave can have a definite radius, avoiding singularity problems. It also stops natural divergence problems of a wave without invoking quantum fluctuations to dampen it.
Was Wheeler a bit of a prophet, or have I subliminally took in where he may have been heading? Here's an interesting quote from Susskind who was remarking on a conversation and things Wheeler had told him:
''The first time I met Wheeler was in 1961, I was an undergraduate... with a somewhat unorthodox academic record. ...The hope was that ...I would be admitted as a graduate student... At the time I was working as a plumber... I was completely enthralled. John was enthusiastically describing his vision of how space and time would become a wild, jittery, foamy world of quantum fluctuations when viewed through a tremendously powerful microscope. He told me that the most profound and exciting problem of physics was to unify Einstein's two great theories—General Relativity and Quantum Mechanics. He explained that only at the Planck distance would elementary particles reveal their true nature, and it would be all about geometry—quantum geometry. ''
The spacetime uncertainty, predicted from both string theory unification attempts and quantum loop gravity attempts, also links it to Planckian dynamics. It's true within our own formalization above, that in the region of the Planck length, commutation is revealing it's true properties related to quantum mechanics, making a quantum geometry. Wheelers vision was much more rich though, he believed quantum fluctuations where part and parcel of the vacuum, arguing for a hypothetical, but reasonable, quantum foam (analogous to how we think about fluctuations arising in the vacuum, making the vacuum not truly empty) and so related to the zero point energy concept, which is nothing more than the continuous residual ground state of the field.
Since gravity is about geometry, this feels like a very natural path to investigate and I would never had considered it, if it was not by linking these two concepts together. The problem with considering though, this is something that can be applied to all particles, should we not be able to detect large curvatures associated to the required curvature at such distances? Interestingly,
propagators can be described entirely in terms of their spacetime curvature which gives rise to speculation whether gravity is a direct alternative to the strong binding force and indeed, that confinement itself arises from spacetime geometry - which means also that the phenomenon of quark duality, that is, that no quark is ever found alone, also has to be described by such a principle. Such a model where curvature plays the role of the strong force was explored in depth by a number of authors, with the idea originating with Abdus Salam, though I am skeptical of the model he chose. But who am I, he's a nobel prize winner right?
For such a model to work, the scale in which the curvature needs to be measured would be the femtometer range - at this scale, gravity can be of magnitude 137 times stronger then electromagnetism matching the strength of the strong force. Even though the femtometer range is below the nanometer range, it is still nothing compared to the Planck scale - though it may seem conjecture to think gravity increases between systems as you reach particle scale, but it not so much rooted in hypothesis as one may suspect!
General relativity already predicts, in a way, for spherical systems, curvature increases as the radius of a system decreases! What does this really tell us about the nature of curvature? It means that the gravitational force produced from a small spherical system could actually be large! So in a way, there maybe hints, from this argument, curvature has two scales, a macroscopic scale and a microscopic one.
If it did increase as a systems radius decreased, it would make the investigation into how a Planck system could capture itself in its own gravitational field, much more easier - after all, if all you need is a femtometer range theory of gravity to describe imagine the kind of curvatures you will be invoking for a Planck like system, which is also spherical?
What about this application to all particles, do I believe in Wheelers vision? I think his vision is nice, but it may suffer that such systems can easily have thermodyamic properties leading to inexorably Hawking radiation. I showed such a possibe relationship. My earliest idea's about cosmology was to study it under a microscope, which led to my idea about how commutation could imply a geometry larger than it's specified wave length. That early study looked at a deBroglie wavelength and how the commutation properties may be seen in light of a time-Temperature relationship, which may be seen with a Boltzmann constant or be seen as related to the entropy of system, entropy seems fitting if we find it follows thermodynamic principles related to black hole dynamics - in fact, it wasn't ad hoc at all, this following interpretation of a wavelength inequality, as it not only related it to the deBroglie wavelength, but rather, a special interpretation of the wavelength in thermal properties allows it to be written in the following way - in this version, we chose to write it as related to entropy ~
The thermal energy is obtained from the kinetic interpretation given by the equipartition theorem as
So yes waves trapped in their own geometry could follow temperature relatioships as well that could imply that generally-speaking that these kinds of objects are not stable: At least, this is the stance I take. You can make your own mind up. Anyway, this small essay, summing up concisely my recent investigations is coming to a close. I am not happy about gauge theories of gravity, I think they are doing the wrong thing, in regards to how relativity treats gravity from first principles; its not a real force (pseudo force) which may be a non-trivial indication that it cannot actually be quantized, like a real force ie (electromagnetism). If gravity really does describe confinement, we may come to understand something strange about further attempts at unification, would this mean gravity would no longer be the seat of importance because it is not a real, fundamental field? No of course not, the very investigation into how to treat gravity, on the quantum scale, is of great importance. No one questions curvature and accelerations could have quantum interpretations, the question is how to do it.
https://en.wikipedia.org/wiki/Thermal_de_Broglie_wavelength
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Last Edit: 24/08/2017 22:15:57 by
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Re: An Extended Essay on whether the Geon model Applies to all Particles
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24/08/2017 22:58:28 »
It resonates with a poetic harmony that Susskind reflects on this unification as strongly as he as in very recent times, only this year and the few past. He has been highlighting that relativity has aspects that is either quantizable or has already been quantized, especially in the context of gauge spin-2 fields and their Lie symmetries. Hopefully, with my continuous display of distain for gauge theories when applied to pseudo fields, will eventually catch on like a rash.
He is right though, but not the way I expect this great professor to follow: That is, only geometry is quantized in respect to commutation laws - but the importance of finding such laws will pave the way to a unified theory, at least, hope springs eternal from my side of things perhaps. There are other, greater people than me who have looked under the microscope of commutation on the field equations and has still not led to a unification of such concepts properly - but I have tried to show guile and craft - I am working from the base of a likely non-trivial relationship from both string theory and loop quantum gravity - finding such relationships, from theories that tend to deviate greatly from each other, surely is an indication there is more to be found when using them.
I hope this ''wiggly, jiggly'' nature to a vacuum, will also explain fluctuations arising from spacetime uncertainty. This is not a wild hope, fluctuations are by definition , strictly related to spacetime distortions in the form of commutation and creation and annihilation operators.
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Re: An Extended Essay on whether the Geon model Applies to all Particles
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24/08/2017 23:02:03 »
My next investigation will be how to view gravity with the last quantum suspect - nonlocality. This threading of space with space, space with matter and matter to matter has to be investigated for a full description of gravity. That will be the most difficult task yet, since we understand even less about quantum non-locality than we do gravity.
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