On A Proposal Towards A Quantum Mechanical Description Of The Geon

[Dubbelosix]

The main equation of interest today is a special equation that has been not only generalised, but also describes the gravitational features of such a theory in terms of the Christoffel symbols. In the form I have been investigating, we use the contracted form of the field equations to obtain the generalised n-dimensional form. 

 

[math](\frac{2 - n}{2})\mathbf{R}(\mathcal{g}) = \frac{\partial \Gamma}{\partial x} + \Gamma \Gamma \approx \frac{p}{\Delta E \Delta t} \cdot \frac{p}{\Delta E \Delta t} \geq (\frac{p}{\hbar})^2[/math]

 

The inequality may have an interesting interpretation - that being, the curvature in some cases can be larger than a specified wavelength. Such a system can only be realistically thought of from Wheelers model which he described such a class of particles in which a wavelength can be caught up by its own gravitational field. 

 

Wheeler couldn't find a quantum mechanical reason for a Geon, his Geon particles where actually macroscopic systems, specifically, for a full quantum theory, he needed one with not such a large mass - though the large masses have tried to be modelled into microscopic black hole theories, leading to questions of particle stability due to thermal radiation.

 

He did believe though the same principle of catching a wave in its own curvature was capable of being described quantum mechanically and even if such a system does not last for long, it is interesting if we ever find indication that nature could be harbouring a more special class of particle that did not evaporate.

 

I came across the interpretation as a bit of an accident, I considered non-commutation for Friedmann cosmology and then started to think about it in terms of curvature purely. Written in terms of the contracted Christoffel symbols (which describe the gravitational field) the non-commutation in the equation can result in a curvature that is larger than a specified wavelength... in this case, we choose deBroglie's wavelength as an example.

 

(2 - n/2)R(g) = ∂Γ/∂x + ΓΓ ≈ p/ΔEΔt · p/ΔEΔt ≥ (p/ħ)²

 

The Einstein tensor is given by two terms

 

G = ∂Γ/∂x + ΓΓ

 

The covariant derivative

 

∇ = ∂/∂x + Γ(x)

 

The gravitational field is

 

Γ(x) = g^-1 ∂/∂x

 

and the metric is

 

g^-1 = δ^-1 + h

 

Which reads, the metric and a small perturbation denoted by h in normal convention. So this form is valid as well:

 

∇ Γ(x) ≈ p/ΔEΔt · p/ΔEΔt ≥ (p/ħ)²

 

or simply ~

 

∇ Γ(x) ≥ (p/ħ)²

 

which is a neat form, where

 

∇ Γ = ∂Γ/∂x + ΓΓ

 

where the covariant derivative is normally of the Levi-Civita notation.

 

The equation has been generalized to n-dimensions and R is the trace of Ricci tensor. If the curvature is larger than a specified wavelength due to corrections from the uncertainty principle, then this would be a type of Geon, where the gravitational field encases a particle like a bubble in space. Since a deBroglie wave implies a small momentum, the mass should be correspondingly small. 

Edited August 19, 2017 by Dubbelosix