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On, Revised Work Concerning A Qm Approach To Wheeler's Geons


Dubbelosix

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In a previous investigation, I sought a quantum mechanical explanation of a specified wave captured in a gravitational bubble. Such a system had already been coined a name in physics by Wheeler as a ''Geon.'' Note however, he was unable to find a quantum mechanical description of his otherwise, macroscopic systems but was convinced such a system was capable of being described by quantum theory.

 

In the previous investigation, we created an inequality that could describe in theory a spacetime curvature that was either greater or equal to a deBroglie wavelength. Though I think in principle this approach could be used to describe, all types of waves, it seems more natural in this new following format which directly uses the spacetime uncertainty relationship:

 

[math]\Delta L c \Delta t \geq L^2_P = \frac{G\hbar}{c^3}[/math]

 

This kind of relationship has been uncovered from both string theory (see Tamiaka Yoneya) and quantum loop gravity (though loop gravity has its own different corrections). Spacetime uncertainties, has also had interest over the years, one example of a physicist writing much on the subject, is Prof. L. Crowell. Instead of a deBroglie wavelength, we consider a Planck wavelength which satisfies the same non-commutating properties of the spacetime uncertainty principle.

 

Before we link all these dynamics together, there is a string of important equations in terms of the Christoffel symbols:

 

[math]g^{\theta \theta} = \frac{1}{g^{\theta \theta}} = \frac{1}{r^2}[/math]

 

[math]g^{\phi \phi} = \frac{1}{g^{\phi \phi}} = \frac{1}{r^2\ sin^2\ \phi}[/math]

 

[math]\mathbf{R}_{\theta \theta} = g^{ab}\mathbf{R}_{a\theta b\theta} = g^{\phi \phi}\mathbf{R}_{\phi \theta \phi \theta} = \frac{1}{r^2\ sin^2\ \theta}[/math]

 

[math]\mathbf{R}_{\phi \phi} = g^{\phi \phi} \mathbf{R}_{\theta \phi \theta \phi} = sin^2\theta[/math]

 

[math]\mathbf{R}^{\theta \theta} =  \mathbf{R}_{ij}g^{\theta i}g^{\theta j} = \mathbf{R}_{\phi \phi}g^{\phi \phi} g^{\phi \phi} = \frac{1}{r^4}[/math]

 

[math]\mathbf{R}^{\phi \phi} = g^{\phi \phi} g^{\phi \phi} \mathbf{R}_{\phi \phi} = \frac{1}{r^4\ sin^2\ \theta}[/math]

 

Thankfully, I did not need to work out all these Christoffel symbols, they are standard to a certain topology I am investigating. Since the curvature of a bubble is identical to that of the 2-dimensional curvature of a sphere, then the curvature scalar is provided in literature for such a configuration as:

 

[math]\mathbf{R}g = \mathbf{R}^{ij}g_{ij} = r^2 \frac{1}{r^4} + r^2\ sin^2\ \theta \frac{1}{r^4\ sin^2\ \theta} = \frac{2}{r^2}[/math]

 

It can be deducted from this that the curvature of a sphere descreses as its radius gets larger. This means even a bubble encasing a quantum wave would require very large curvature. 

 

This can be generalized to n-dimensions,

 

[math]\frac{2 - n}{2}\mathbf{R}g = \mathbf{R}^{ij}g_{ij} - \frac{1}{2}n\mathbf{R}g_{ij}g^{ij} = r^2 \frac{1}{r^4} + r^2\ sin^2\ \theta \frac{1}{r^4\ sin^2\ \theta} = \frac{2}{r^2}[/math]

 

since ~

 

[math]\mathbf{G} = \mathbf{R} - \frac{1}{2} n \mathbf{R}[/math]

 

[math]\mathbf{G} = \frac{2 - n}{2} \mathbf{R}[/math]

 

when [math]n=2[/math]. For the special case of [math]n=4[/math] dimensions, [math]\mathbf{G}[/math] becomes negative. The final equation then I present is:

 

 

[math]\mathbf{G} = \frac{2 - n}{2}\mathbf{R}g = \mathbf{R}^{ij}g_{ij} - \frac{1}{2}n\mathbf{R}g_{ij}g^{ij} \approx r^2 \frac{c}{r^4} + r^2\ sin^2\ \theta \frac{c}{r^4\ sin^2\ \theta} = \frac{2c}{r^2} \approx \frac{1}{\Delta L \Delta t} \geq \frac{c^4}{G \hbar}[/math]

 

 

This equation/inequality can be summed up very simply in the following way:

 

[math] \mathbf{G} \geq \frac{c^4}{G\hbar}[/math]

 

where the LHS has an extra factor of speed of light. 

 

Again, the inequality specifically states that the geometry can be larger than a given wavelength - in this case, we have adopted a purely gravitational concept, a Planck wave. The shorter version is a very simple and concise form but I expect the inequality to hold for all kind of waves, unless there is some physical reason larger wavelengths above the Planck scale would not be stable. 

Edited by Dubbelosix
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