^{*}

This document reviews the Landau-Liftshifts reformulation of General relativity with a representation of the available mass of a graviton. From looking at a conservation law, using the Landau-Liftshifs formulation, we obtain conditions for initial mass, in the Pre-Planckian regime of space-time. In doing so, we also indicate a metric tensor and metric pseudo tensor delineation of causal discontinuity.

We will use the Poisson formulation of the Landau-Liftshifts formulation of General Relativity [

We first begin by a recapitulation of the different models for HFGW as given by Dr. Li et al., 2008 [

This reproduced PRD table [

We are considering using massive gravitons [

1. Our methodology suggests that if we measure relic gravitational waves, that we will have to consider if an initial mass in the evolution of the universe, actually existed [

We also review, by example, if there is a way to use a modified version of the Heisenberg uncertainty principle, as given in [

2. To obtain an inflation, in the onset of expansion of the Universe.

By way of our construction, we will also look at if a causal discontinuity exists as an elaboration of [

Having said that, it is time now to unveil by way of the Poisson reference [

We wish to understand the linkage between dark matter and gravitons. To consider just that, we look at the “size” of the nucleation space, V. V for nucleation is HUGE. Graviton space V for nucleation is tiny, well inside inflation/therefore, the log factor drops OUT of entropy S if V chosen properly for both Equation (1) and Equation (2). Ng’s [^{−35} centimeters). We also specify a “wavelength” parameter

This, according to Ng, [

But

where

Such a linkage would open up the possibility that the density of primordial gravitational waves could be examined, and linked to modeling gravity as an effective theory. The details of linking what is done with Equation (2) and bridging it to Equation (3) await additional theoretical development, and are probably conceptually understandable if the following is used to link the two regimes. i.e. we can use the number of space time operations used to create Equation (2), via Seth Lloyds [

Essentially, what will be done is to use Equation (4) to show linkage between a largely thermally based production of entropy, as implied by Equation (3) and a particle counting algorithm, as given by Equation (2). This due to the problems inherent in making connections between a particle count generation of entropy, and thermal contributions. i.e. two different processes are involved.

To do this go to [

Furthermore, the two equations in Equation (5) have a representation of the GR field equations as

If so then, we can have a simple solution to this above which is of the type

Here, in this situation we have that if we are following the ideas in [

where we have restriction to the zeroth (time component) of the metric tensor, so that

We will be looking at the value of Equation (7) if

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

Then, the surviving version of Equation (9) and Equation (10) 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 by [

Then,

Here, we have a causal discontinuity as given by

We will address the implications of Equation (15) in the conclusion

If we then put in the initial mass, of say from [

With the minimum scale factor a small, but non zero factor by [

Note that having the right hand side of the 2^{nd} line of Equation (17) going to zero is implying that there is an invariance as to the gravitational field pseudo tensor, which may have implications as to the entropy-information transfer from Pre-Planckian to Planckian space-time, however we have to consider the Pre-Planckian to Planckian physics of the more traditional stress energy tensor.

What we will say, is that by [^{nd} line of Equation (17) equals zero, with this reflecting the

Dispersal of the terms alpha and beta, into the non time components of the stress energy tensor would be saying also that

Note that in [

Quote

These gapless Goldstone modes are the quantum carriers of information and entropy. Analyzing their effective theory, we observe the information-processing properties strikingly similar to the ones predicted by the black hole portrait. The energy cost per qubit of information-storage vanishes in the large-N limit and the total information-storage capacity increases with N either exponentially or as a power law.

End of quote

Our idea is that the N limit, of information entropy, is akin to graviton counting, using the information given in Section 2, as given by Ng [

If N being a graviton count is sufficiently large, and the initial inflation can be parameterized, we can understand if or not Equation (19) is a causal structure discontinuity.

Furthermore, understanding Equation (17) to Equation (20) more fully may allow us to choose between the different models given in

Note that usual Randal Sundrum brane theory has a production rate [

As the number of Kaluza Klein gravitons per unit time per unit volume Note that this production rate is for a formula assuming mass for which

Sources | Amplitude | frequency | Characteristics |
---|---|---|---|

HFGW in Quintessence inflationary models | Random background | ||

HFGW in some string theory scenarios | Random background | ||

Solar Plasma | On the Earth | ||

High energy particles, e.g. Fermi Ring | On the center the frequency depends upon the rotational frequency of particles in the Fermi Ring | ||

Stanford Linear Accelerator | On the collision center, the frequency depends upon the self-energy and the Lorentz factor of high energy e^{+}e^{−} beams | ||

LHC-Large Hadron collider | Spectra of high energy gravitons | ||

Nano-piezo electric crystal array, with size of about 100 nanometers | On the wave zone with an effective cross section of or less than 0.01 meters squared, for gravitational radiation |

where R is the assumed higher dimension “size” and, d is the number of dimensions above 4, and typically we obtain

This with additional work may allow us to distinguish between the GW and gravity models as given in reference [

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

Beckwith, A.W. (2017) Initial Mass in Pre-Planckian Space- Timed Defined, and Causal Discontinuity. Journal of High Energy Physics, Gravitation and Cosmology, 3, 9-15. http://dx.doi.org/10.4236/jhepgc.2017.31002