Examination of minimum time step, from modified Heisenberg Uncertainty principle, Inflaton physics and Black hole physics

First, we calculate the minimum length for the creation of a 10^45 Hz relic Gravitational wave. Next, we look Padamababhan’s inflaton physics, and work done by the author for a modified Heisenberg Uncertainty principle for constraints on a minimum time step. Sciama’s work in “Black hole explosions” (1982) gives us a linkage between a decay rate for black holes, in terms of a life time, and the mass, M of the black hole, which when combined with a simple exposition from Susskind and Hrabovsky (2013) for the most basic evolution the time change in energy E(t), which is how we form a first order treatment of the square of a minimum time step Delta t. We then reference what was done by Ng (2008) as far as infinite quantum statistics, for entropy as a particle count, and from first principle get constraints upon entropy production, as a function of boundaries on minimum time step. We assume massive Gravity, and obtain a peak 10^36 Giga Hertz frequency range (10^45 Hertz) for relic Gravitational waves, and Gravitons. Key words, massive gravity, inflaton physics, Infinite quantum statistics, modified Poisson’s equation


Introduction
First of all we calculate a minimum length, r, and this on the basis of a Poissons equation given in [1], and then afterwards we will secondly get to calculation of the time step.This is to obtain a Pre inflationary value of the initial time step, t ∆ , and then from there we will use an order of magnitude estimate as to the initial mass M of a black hole, to lead to our final conditions.

Modified Poissons Equation [1] Applied to Our Problem
We will first of all refer to two necessary and sufficient conditions for the onset of a massive graviton given in [1], and combined with Padmanablan's reference [2].This heavily borrows from [3]. i.e. what we will be doing is to re do the reference calculations given in [1] with ( ) Here, we will be using in the Pre-Planckian potential the inputs from the data usually associated with [2] ( ) In other words, we will be using the inflation given by ( ) If so, then our approximation is to call the Potential in Equation ( 2) to be the same as U in Equation ( 1), and then with re arrangements we come up with the following ( ) Then, after algebra, we have the following The quadratic Equation this engenders is, how to say We will, in the latter part of this document, associate And after defining as such work with Equation ( 7) put into Equation ( 6) we obtain We will from here, insert, in the value of E ∆ from the HUP given in [3] and also compare it to the work given by Susskind and Hrabovsky in [4] and from there get a linkage of a radial distance, inflaton physics, and a time step t ∆ .

HUP, as Used in This Problem
Begin with [3] In [3] we make the assumption, namely ( ) , then Equation ( 9) becomes ( ) This will be further elaborated upon in this document.

Examining the Consequences of Equation (10) and Equation (11)
Using the CRC tables, we obtain Now, then, we can form a force equation, and do it via The expression ( ) is formed from the inflaton, and we get From Padmanablah, using [1] if ( ) 0 a t a t γ = , then by [1] ( ) ( ) ( ) Then, by Equation ( 13) Now, the time derivative of energy, is [4] ( ) ( ) ( ) ( ) ( ) Then, if the derivatives dE/dt become instead incremental Then, using Now, set ( ) t t δ = ∆ , and assume in the Pre-Planckian regime, that ( ) ( ) Then Equation ( 12) becomes We then get a positive r value, and up to a point this is equal to Planck Length ~ 10 −33 centimeters To covert this to time, remove the spatial component on the right hand side of Equation (20) and this is done via We then get a positive r value, and up to a point this is equal to Planck Length ~ 10 −33 centimeters This, above, puts an incredibly tight set of constraints upon the minimum time step.It also assume, that we are looking at a rest mass for the graviton of the order of 10 −62 grams.The above Equation ( 21) is arguing for a peak 10 36 Giga Hertz frequency range (10 45 Hertz) for relic Gravitational waves, and Gravitons.

What Can Be Said Now about the Sciama Black Hole
Explosion Idea? (1982) Sciama [5] Assuming there would be say 10 3 initial primordial black holes, we would then be obtaining 10 55 relic gravitons.
Note that if the size of the presumed t ∆ expanded, this would lead to a commensurate increase in relic gravitons.

Conclusions
We will, as a start, incorporate Ng's infinite quantum statistics idea, of entropy being equivalent to a count of particles, i.e. by [8] ( ) For relic conditions this implies about 10 55 for relic entropy, as opposed to 10 120 today for the existing entropy and this also implies an enormously high initial graviton initial frequency a peak 10 36 Giga Hertz frequency range (10 45 Hertz) for relic Gravitational waves.
Relic graviton frequency, if produced in the beginning of inflation would be by necessity red shifted down, by inflation.i.e. conceivably, if the 10 45 Hz was de facto, it may mean present GW of the value of 10 5 or so, i.e. a 40 magnitude drop in frequency range.
i.e. if the generation of GW were at the very start of inflation, it may mean up to a 60 magnitude drop, and a correspondingly enormously low GW frequency.However, the degree of the drop, i.e. say with a 40 magnitude (low end) to a 62 magnitude drop, would say much about a proof about the degree and magnitude of the rapidity of inflation [9].Which would be of critical import as to cosmology.Having said that there are some serious physical issues to keep in mind while doing this analysis.
in 1982 argued for the lifetime of a black hole, of mass M, that the following