_{1}

^{*}

This paper is with the permission of Stepan Moskaliuk similar to what he will put in the confer-ence proceedings of the summer teaching school and workshop for Ukrainian PhD physics stu-dents as given in Bratislava, as of summer 2015. With his permission, this paper will be in part reproduced here for this journal. First of all, we restate a proof of a highly localized special case of a metric tensor uncertainty principle first written up by Unruh. Unruh did not use the Roberson-Walker geometry which we do, and it so happens that the dominant metric tensor we will be examining, is variation in
δg
_{tt}. The metric tensor variations given by
δg_{rr},
and
are negligible, as compared to the variation
δg
_{tt}. Afterwards, what is referred to by Barbour as emergent duration of time is from the Heisenberg Uncertainty principle (HUP) applied to
δg
_{tt }in such a way as to give, in the Planckian space-time regime a nonzero minimum non zero lower ground to a massive graviton, m
_{graviton}. The lower bound to the massive graviton is influenced by
δg
_{tt }and kinetic energy which is in the Planckian emergent duration of time
δt as (E-V) . We find from
δg
_{tt }version of the Heisenberg Uncertainty Principle (HUP), that the quantum value of the
Δt·ΔE Heisenberg Uncertainty Principle (HUP) is likely not recoverable due to
δg_{tt }≠ Ο(1)~g_{tt} ≡ 1. i.e. δg_{tt}≠ Ο(1) . i.e. is consistent with non-curved space, so
Δt · ΔE ≥ no longer holds. This even if we take the stress energy tensor approximation
T_{ii}= diag (ρ ,-p,-p,-p) where the fluid approximation is used. Our treatment of the inflaton is via Handley et al., where we consider the lower mass limits of the graviton as due to when the inflaton is many times larger than a Potential energy, with a kinetic energy (KE) proportional to
ρ_{w} ∝ a^{-3(1-w)} ~ g*T^{4} , with
g
* initial degrees of freedom, and T initial temperature. Leading to non-zero initial entropy as stated in Appendix A. In addition we also examine a Ricci scalar value at the boundary between Pre Planckian to Planckian regime of space-time, setting the magnitude of k as approaching flat space conditions right after the Planck regime. Furthermore, we have an approximation as to initial entropy production N~S
_{initial(graviton)}~10
^{37}. Finally, this entropy is N, and we get an initial version of the cosmological “constant” as Appendix D which is linked to initial value of a graviton mass. Appendix E is for the Riemannian-Penrose inequality, which is either a nonzero NLED scale factor or quantum bounce as of LQG. Note that, Appendix F gives conditions so that a pre Planckian kinetic energy (inflaton) value greater than Potential energy occurs, which is foundational to the lower bound to Graviton mass. We will in the future add more structure to this calculation so as to confirm via a precise calculation that the lower bound to the graviton mass, is about 10
^{-70} grams. Our lower bound is a dimensional approximation so far. We will make it exact. We conclude in this document with Appendix G, which is comparing our Pre Planckian space-time metric Heisenberg Uncertainty Principle with the generalized uncertainty principle in quantum gravity. Our result is different from the one given by Ali, Khali and Vagenas, in which our energy fluctuation is not proportional to that of processes of energy connected to Black hole physics, and we also allow for the possibility of Pre Planckian time. Whereas their result (and the generalized string theory Heisenberg Uncertainty principle) have a more limited regime of interpolation of final results. We do come up with equivalent bounds to recover
δg_{tt} ~ small-value ≠ *O*(1) and the deviation of fluctuations of energy, but with very specific bounds upon the parameters of Ali, Khali, and Vegenas, but this has to be more fully explored. Finally, we close with a comparison of what this new Metric tensor uncertainty principle presages as far as avoiding the Bicep 2 mistake, and the different theories of gravity, as reviewed in Appendix H.

The first matter of business will be to introduce a framework of the speed of gravitons in “heavy gravity”. Heavy Gravity is the situation where a graviton has a small rest mass and is not a zero mass particle, and this existence of “heavy gravity” is important since eventually, as illustrated by Will [

We reference what was done by Will in his living reviews of relativity article as to the “Confrontation between GR and experiment”. Specifically we make use of his experimentally based formula of [

Furthermore, using [

Here,

Then,

And if one sets the mass of a graviton [

Note that the above frequency, for the graviton is for the present era, but that it starts assuming genesis from an initial inflationary starting point which is not a space-time singularity.

Note this comes from a scale factor, if

We will next discuss the implications of this point in the next section, of a non-zero smallest scale factor. Secondly the fact we are working with a massive graviton, as given will be given some credence as to when we obtain a lower bound, as will come up in our derivation of modification of the values [

The reasons for saying this set of values for the variation of the other metric components will be in the 3^{rd} section and it is due to the smallness of the square of the scale factor in the vicinity of Planck time interval.

Begin with the starting point of [

We will be using the approximation given by Unruh [

If we use the following, from the Roberson-Walker metric [

Following Unruh [

Then, the surviving version of Equation (7) and Equation (8) is, then, if

This Equation (11) is such that we can extract, up to a point the HUP principle for uncertainty in time and energy, with one very large caveat added, namely if we use the fluid approximation of space-time [

Then

Then, Equations (11)-(13) together yield

How likely is

In fact, we have that from Giovannini [

Then, there is no way that Equation (14) is going to come close to

tion as will be discussed toward the end of this article, is not feasible. Finally, we will discuss a lower bound to the mass of the graviton.

To begin this process, we will break it down into the following co ordinates

In the rr,

If as an example, we have negative pressure, with

Having said this, the value of

In order to start this approximation, we will be using Barbour’s value of emergent time [

Initially, as postulated by Babour [

If

Key to Equation (19) will be identification of the kinetic energy which is written as

This is done with the proviso that w < −1, in effect, what we are saying is that during the period of the “Planckian regime” we can seriously consider an initial density proportional to Kinetic energy, and call this K.E. as proportional to [

If we are where we are in a very small Planckian regime of space-time, we could, then say write Equation (21) as proportional to

Here, the initial

This is at the instant of Planck time. We can then ask what would be an initial time contribution before the onset of Planck time. i.e. does Equation (22) represent the initial value of graviton frequency?

This value of the frequency of a graviton, which would be red shifted enormously would be in tandem with an initial time step of as given by [

This value for the initial time step would be probably lead to Pre Planckian time, i.e. smaller than 10^-43 seconds, which then leads us to consider, what would happen if a multi verse contributed to initial space-time conditions as seen in Equation (11) above. If the cosmic fluid approximation as given by Equation (12) were legitimate, and one could also look at Equation (13), then

But, then if one is looking at a multiverse, we first will start at the Penrose hypothesis for a cyclic conformal universe, starting with [

However, in the multiverse contribution to Equation (12) above, we would have, that

So, does something like this hold? In a general sense?

If the fluid approximation as given in Equation (12) and Equation (13) hold, then Equation (27) conceivably could be identifiable as linkable to.

If we could write, say

Then, if each j is the j^{th} contribution of N “multiverse” contributions to a new single universe being nucleated, one could say that there was, indeed, likely an “averaging” and that the causal barrier which Mukhanov spoke of, as to each

If Equation (30) held, then we could then write

Instead, we have, Equation (28), and that it is safe to say that for each collapsing universe which might contribute to a re cycled universe that the following inequality is significant.

Hence, the absence of an averaging procedure, due to a multiverse, would then rule against a causal barrier, as was maintained by Mukhanov, in his discussion with the author, in Marcel Grossman 14, in Italy. Then the possible approximation says of

Would not hold, and that in itself may lead to a breakdown of the Causal barrier hypothesis of Mukhanov, which the author emphatically disagreed with.

A way of solidifying the approach given here, in terms of early universe GR theory is to refer to Einstein spaces, via [

Here, the term in the Left hand side of the metric tensor is a constant, so then if we write, with R also a constant [

The terms, if we use the fluid approximation given by Equation (12) as well as the metric given in Equation (9) will then tend to a constant energy term on the RHS of Equation (35) as well as restricting i, and j, to t and t.

So as to recover, via the Einstein spaces, the seemingly heuristic argument given above. Furthermore when we refer to the Kinetic energy space as an inflaton where we assume that the potential energy is proportional to V, so as to allow us to write

In the case of the general elliptic operator K if we are using the Fulling reference, [

Then, according to [

Then

If the frequency, of say, Gravitons is of the order of Planck frequency as in Equation (22), then this term, would likely dominate Equation (39). More of the details of this will be worked out, and also candidates for the

Why is a refinement of Equation (39) necessary?

The details of the elliptic operator K will be gleaned from [

Finally, as far as Equation (39) is concerned, there is one serious linkage issue to classical and quantum mechanics, which should be the bridge between classical and quantum regimes, as far as space time applicability. Namely, from Wald (19), if we look at first of all arbitrary operators, A and B

As we can anticipate, the Pre Planckian regime may the place to use classical mechanics, and then to bridge that to the Planckian regime, which would be quantum mechanical. Taking [

Then there exists a re formulation of the Poisson brackets, as seen by

So, then the following, for classical observables, f, and g, we could write, by [

Then, we could write, say Equation (40) and Equation (43) as

If so, then we can set, in the interconnection between the Planck regime, and just before the Planck regime, say, by setting classical variables, as given by

Then by utilization of Equation (44) we may be able to effect more precision in our early universe derivation, especially making use of derivational work, in addition as to what is given here, as to understand how to construct a very early universe partition function Z based upon the inter relationship between Equation (44) and Equation (45) so as to write up an entropy based upon, as given in [

If this program were affected, with a first principle construction of a partition function, we may be able to answer if Entropy were zero in the Planck regime, or something else, which would give us more motivation to examine the sort of partition functions as stated in [

We start the process of understanding the consequences of choosing the inflaton

The consequences of the above mentioned appendix entries are, mainly that if we wish to avoid the problems given in Appendix G and Appendix H that we really need to keep in mind the following:

1) Our uncertainty principle is fundamentally different from the Black hole commensurate uncertainty principles cited in Appendix G. They do not take into consideration the possibility that there may be Pre Planckian time, which may immensely impact the fluctuations in the metric tensor.

2) As an exercise, Appendix G shows that a highly restricted parameter space is required if we insist upon making our Pre Planckian uncertainty principle commensurate with the possibility that our metric Heisenberg Uncertainty principle (HUP) is in fact, giving us the flat space result which was brought up by Mukhanov, in Marcel Grossman 14. But it is so restrictive that we doubt it is actually mathematically a useful development

3) Appendix H gives us Equation (H1) which is the Pre Planckian Inflaton, which is of foundational importance in determination of if we have general relativity or some other gravitational theory, i.e. the issue of if there is an additional polarization. But to do that, we have to for reasons given in Appendix G, choose our parameter space, wisely. It is still not clear if there is a connection between Black hole physics, and avoiding the catastrophe of Bicep 2. For that much additional experimental work has to be done.

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

Andrew Walcott Beckwith, (2016) Gedanken Experiment for Refining the Unruh Metric Tensor Uncertainty Principle via Schwarzschild Geometry and Planckian Space-Time with Initial Nonzero Entropy and Applying the Riemannian-Penrose Inequality and Initial Kinetic Energy for a Lower Bound to Graviton Mass (Massive Gravity). Journal of High Energy Physics, Gravitation and Cosmology,02,106-124. doi: 10.4236/jhepgc.2016.21012

We will be looking at inputs from page 290 of [

And using Ng’s infinite quantum statistics, we have to first approximation [

This is due to a very small but non vanishing

We begin with Kolb and Turner [

Then by [

Leading to

If

Next, using [

Then, we can obtain

Right at the start of the Planckian era,

The consequences of this would be that right after the entry into Planckian space time, that there would be the following change of pressure

Then, the change in the k term would be like, say, from Pre Planckian to Planckian space time

This goes almost to zero if the numerator shrinks far more than the denominator, even if the initial scale factor is of the order of 10^{−}^{55} or so.

We are making use of the Padmanabhan publication of [

Then, if

The value of initial entropy,

Here we can view the possibility of considering the following, namely [

Should the N above, be related to entropy, and Equation (8) this supposition has to be balanced against the following identification, namely, as given by T. Padmanabhan [

But should the energy in the numerator in Equation (D2) be given as say by (C2), of Appendix C, we have quintessence. then there would have been quintessence, i.e. variation in the “Einstein constant”, which would have a large impact upon mass of the graviton, with a sharp decrease in

The final value of the frequency would be of a magnitude smaller than one Hertz, so as to have value of the mass of the graviton would be then of the order of 10^{−62} grams [

Leading to the upper bound of the Graviton mass of about 10^{−62} grams [

Equation (D5) has a different value if the entropy/particle count is lower, as has been postulated in this note. But the value of Equation (D5) becomes the Graviton mass of about 10^{−62} grams [

If from Giovannini [

Refining the inputs from Equation (E1) means more study as to the possibility of a non-zero minimum scale factor [

where the following is possibly linkable to minimum frequencies linked to E and M fields [

So, now we investigate the question of applicability of the Riemann Penrose inequality which is [

Riemann Penrose Inequality: Let (M, g) be a complete, asymptotically flat 3-manifold with Non negative-scalar curvature, and total mass m, whose outermost horizon

And the equality holds, if (M, g) is isometric to the spatial isometric spatial Schwartzshield manifold M of mass m outside their respective horizons.

Assume that the frequency, say using the frequency of Equation (E3), and

Assume that we also set the input frequency as to Equation (E3) as according to

Our supposition is that Equation (E6) should give the same frequency as of Equation (D3) above. So if we have in

In doing this, this is a frequency input into Equation (E3) above where we are safely assuming a graviton mass of about [

Does the following make sense? i.e. look at, when

We claim that if this is an initial frequency and that it is connected with relic graviton production, that the minimum frequency would be relevant to Equation (E3), and may play a part as to admissible B fields

Note, if Appendix D is used, this makes a re do of Equation (E8) which is a way of saying that the graviton mass given by [

In either case, Equation (E8) and Equation (E3) in some configuration may argue for implementation of work the author did in reference [

We give this as a plausibility argument which undoubtedly will be considerably refined, but its importance cannot be overstated. i.e. this is for Pre inflationary, Pre Planckian physics, so as to get a lower bound to the Graviton mass. To do this, we look at what [

We will treat, then the Hubble parameter, as

Now from Padmanabhan, [

Then using 463 of [

Next, according to [

Therefore, if

If the scale factor is very small, say of the order of

We will in the future add more structure to this calculation so as to confirm via a precise calculation that the lower bound to the graviton mass, is about 10^{−70} grams. This value of 10^{−70} grams is an approximation, via dimensional analysis and will be improved, by more exact calculations.

We are looking here at what was done in [

Compare that with our given value of

This should be compared with our value of equivalence between these two equations which demands

The collapse to a situation with ourselves recovering the standard Heisenberg Uncertainty relationship for fluctuations of energy is seen in, if Equation (G1) and Equation (G2) are both correct having then that

Here, we want the situation for which we would have any time situation with the fluctuation of time, going to a very small number, and that the inverse fluctuation in time going to infinity would be, trivially due to, if

It’s an equation for

This would be equivalent to, then setting

Then by necessity, we would want to have a situation for which to have a more general situation as given in our document for a

In fact, to reconcile Equation (G1) and Equation (G2) in the case of recovering a

That not only would

The situation as given by L. Crowell in [

Here, R is the radius of a sphere for the origins of an emitted wave, which is in turn requiring R to be extraordinarily small. i.e. we recover the inputs for our analysis of [

The problem with Equation (G9) is that there is no provision given as to Pre Planckian length values, and that it is restricted, dimensionally to Planckian Length and temperature, with no clue given as to what happens before a Planck length.

Quoting from the Authors’ recent publication [

From [

Quote, in [

This fact rules out the possibility of treating gravitation like other quantum theories, and precludes the unification of gravity with other interactions. At the present time, it is not possible to realize a consistent Quantum Gravity Theory which leads to the unification of gravitation with the other forces [

End of quote from [

We then will cite what is in [

Quote, from [

Needless to say we will require careful analysis of the result as given in reference [

This enormous value for the inflaton, initially, needs to be examined further. It further should be linked to Corda’s pioneering work with “gravity’s breath”, i.e. traces of the inflaton as given by [

Next, Avoiding the Bicep 2 mistake. What we can do with Equation (H1)

Following [

Omni-directional gravitational wave background radiation could arise from fundamental processes in the early Universe, or from the superposition of a large number of signals with a point-like origin. Examples of the former include parametric amplification of gravitational vacuum fluctuations during the inflationary era, termination of inflation through axion decay or resonant preheating, Pre-Big Bang models inspired by string theory, and phase transitions in the early Universe; the observation of a primordial background would give access to energy scales of 10 to the 9 power, up to 10 to the 10 power GeV, well beyond the reach of particle accelerators on Earth

Needless to say though, we need above all to avoid getting many multiple stochastic signals, in what we process for primordial gravitational waves, and to use, instead tests to avoid getting dust signals which is what doomed Bicep 2, i.e. as was made very clear in [

i.e. the problem is in avoiding multiple stochastic signals, and this is explained in the conclusion of [

This in turn may allow understanding if our document is commensurate with the considerations given in [