_{1}

^{*}

We take note of the material offered in [1] as to Geometrodynamics as a way to quantify an inter relationship between a quantum style Heisenberg uncertainty principle for a metric tensor and conditions postulated as to a barotropic fluid, i.e. dust for early universe conditions. By looking at the onset of processes at/shorter than a Planck Length, in terms of initial expansion of the universe, we use inputs from the metric tensor as a starting point for the variables used in Geometrodynamics.

We will be using, the inputs from [

We will be examining a Friedmann equation for the evolution of the scale factor, using explicitly two cases, one case being when the acceleration of expansion of the scale factor is kept in, another when it is out, and the intermediate cases of when the acceleration factor, and the scale factor is important but not dominant. In doing so we will be tying it in our discussion with the earlier work done on the HUP but from the context of how the acceleration term will affect the HUP, and making sense of

Namely we will be working with

i.e. the fluctuation

In short, we would require an enormous “inflaton” style

Here, we are using the following approximation as to Kinetic energy in the beginning of the expansion of the universe.

Then, up to first order, we could approximate, with H.O.T. being higher order terms

This Equation (6) will be considerably refined in the subsequent document.

From Equation (48) of [

Here, we can also, from [

This change of energy will then be put into Equation (2) with the result that.

Here the subscript k, as in Equation (8) is by [

Or, if the inequality is strictly adhered to

The smallness of the initial scale factor would be of the order of

i.e. the violation of an uncertainty principle for commences for any situation which implies restraints on

For the problem represented by Equation (10a) to hold it would mean that the following Pre-Planckian Potential energy would be then small when the following Potential energy as given in Equation (11) is much smaller than the Kinetic energy given in Equation (2)

From inspection, for Equation (11) to hold, for our physical system we would want Equation (10) to hold which would mean an extremely small Potential energy, as opposed to the large value of the Kinetic energy given in Equation (4). Hence the role of Geometrodynamics given in Equation (7) and Equation (8), will in the case of a quartic potential imply that Equation (11) as Potential energy is much smaller than the kinetic energy as represented for Pre Planckian space-time physics.

What we are doing is confirming the material given in [

The potential used, the quartic, is the simplest version of the potential systems in [

Since our modeling is not predicated upon the inflationary model of cosmology but which is addressing the issue brought up in [

This has a quasi “quantum mechanical” effective white noise introduced term

This work is supported in part by National Nature Science Foundation of China grant No. 11375279.

Beckwith, A.W. (2016) Addition to the Article with Stepan Moskaliuk on the Inter Relationship of Gene- ral Relativity and (Quantum) Geometrodynamics, via Use of Metric Uncertainty Principle. Journal of High Energy Physics, Gravitation and Cosmology, 2, 467-471. http://dx.doi.org/10.4236/jhepgc.2016.24040