_{1}

^{*}

We initially look at a nonsingular universe representation of entropy, based in part on what is brought up by Muller and Lousto. This is a gateway to bring up information and computational steps (as defined by Seth Lloyd) as to what will be available initially due to a modified Zero Point Energy formalism. The Zero Point Energy formalism is modified as due to Vissers’s setting of an angular plane number in early universe cosmology as k(maximum) ~ 1/(Planck length), with a specific initial density giving rise to initial information content which may permit fixing the initial Planck’s constant, h, which is pivotal to the setting of physical law. This will be in the spirit of Stoica’s removal of initial conditions of non-pathological initial starting points in Cosmology. What we want are necessary and sufficient conditions so h(today) = h(initial). We also in addition make a brief survey into 5th force arguments in gravity which also has a strict entropy interpretation. i.e., how to link gravity, quantum mechanics, and E and M through entropy production.

First of all, we wish to ascertain if there is a way to treat entropy in the universe, initially, by the usual black hole formulas. Our derivation takes advantage of work done by Muller, and Lousto [

the scaling, as given by Camara [

cosmological equation of state of the universe,

One of the questions which have come up in discussion is what is meant by the term scale factor. In Cosmology, a as a scale factor is nearly zero at the start of the universe expansion, and equals 1 in the present era. In this case, a starts with a value just above zero, and obeys the cosmological Friedman equations. Scale factors are used as a convenient measuring convention in part in that the actual radii of the universe, and how it expands are controversial. By way of [

If time in the present era is set as

Notice then that what we are referring to physically is that [

So by definition,

given by [

Time step (1) ~ 1/ square root of

This will be compared to another time step (2) based on [

Time step (2) ~ 1/square root of

Further analysis will be assumed in the case where there is an equality between Equation (3) and Equation (4) so that by [

Doing so will then permit us to make further use of [

While here, will briefly allude to what the Cosmological constant did earlier and its role in present cosmological theory. As given by [

The physical dynamics of how this constant works its way in, is in the following Einstein field equation, as given by page 180 of [

When Equation (6) has no change in its size. Then one is then obtaining, with a nonzero curvature the physics saying that the universe has an invariant spatial domain we could render as, in the spirit of [

Realistically this Equation (7) would have the left hand side scale factor equal to 1, so then we would be having

This is in the case that we have non zero spatial curvature, i.e. if we have zero curvature, i.e. flat space, to have no energy evolution, the above becomes even simpler, i.e. the cosmological constant is negative. For a spatially invariant “repulsive” energy which would be the Left hand side of Equation (9) below

The left hand side of Equation (9) has a density in the case of zero curvature and invariant “repulsive” energy of

For the sake of understanding what G is, we could for the sake of argument, invent a cosmology for which

The supposition that the cosmological constant was put in place was initially a way to have repulsive “anti- gravity” so as to have the static universe, as what was considered the case in 1917. And duplicated above, the dynamic universe also is tied into suppositions that the cosmological constant may be the driving force behind re acceleration of the universe, as given in [

The other situation which will comment upon is a situation for which the “cosmological constant” may in some sense vary over time, i.e., an easy example will be given below by using reference [

Note that by Peebles, N, as a particle number count as in the radiation era, is usually conflated with entropy [^{th} in power magnitude to the number of computational cosmic computer steps taken by a cosmic “computer”. i.e. what we will see later is that the gravitons produced up to the present day will be about 10^90, equal to the 3/4^{th} power of 10^120, where the value of 10^120 is the number of operations necessary to produce the equivalence of initial Planck constant, h(initial) with today’s value of Planck’s constant h(today) [

Note that h(today) was the proportionality constant between the minimal increment of energy of a hypothetical electrically charged oscillator in a cavity that contained black body radiation, and the frequency of its associated electromagnetic wave. In 1905 the value of minimal energy increment of a hypothetical oscillator, was associated by Einstein with a “quantum” of the energy of the electromagnetic wave itself. The light quantum eventually was called the photon. This is what [

In addition we will make the following identification of entropy with the following fifth force argument. The two arguments about entropy will be re enforcing each other, and we will talk about what the two entropies portend to, in our conclusion.

So as this is the introduction, before we go to develop the first part of our introduction, we will briefly access 5^{th} force arguments here. This fifth force argument and what it portends to, will be compared to the main developed argument given above, in terms of its effect upon entropy, in the conclusion.

We start off with a description of both the Fifth force hypothesis of Fishbach [

The generalized charges, Q, as brought up are defined, briefly in Equation (13) in the next few lines, whereas the term ^{th} force, with, as given by [

This second term in the potential, in Equation (13) is going to have, here

We have that Unnishkan shared in Rencontres Du Moriond [

are the masses of Equation (14) with the following relation given to the author by Unnishkan when he gave [

This is where Unnishkan would have to be coherent with the prior formalism identification of charges and of mass, a setting of Equation (16) below to help us make sense of a genuine connection between Electro magnetics and gravity. The Left hand side of Equation (16) is E and M. and up to a point similar to Equation (9) whereas the right hand side of Equation (16) is gravity, similar to the right hand side of Equation (15). Here the term

We argue that the linkage of Equation (16) of magnetism with mechanics, and by default gravity, is similar in part to what Ciufolini and Wheeler wrote up in [

The above relationship as in Equation (15) and Equation (16) with its focus upon interexchange relations between gravity and magnetism is in a word focused upon looking at, if A, the nominal vector potential used to define the magnetic field as in the Maxwell equation, the relationship we will be using at the beginning of the expansion of the universe, is a variation of the quantized Hall effect, i.e., from Barrett [

We will be taking the right hand side of the A field, in the above, and approximate Equation (17) as given by

Then, we have an approximation for writing a modification of [

Equation (19) needs to be interpolated, up to a point. I.e. in this case, we will conflate the n, here as a “graviton” count, initially, i.e. the number of early universe gravitons, then assume that

and we refer to the n of Equation (17) to Equation (19) as being the same as

We will elaborate upon this treatment of entropy in our derivations, and compare this behaviour of entropy in the first part of our introduction, which is for coming up with entropy as far as a way to confirm if or not we can preserve Planck’s constant, ^{th} forces will be in showing the unity of entropy production of classical (gravity) predictions, QM, and electromagnetics in terms of how we could maintain the constancy of physical law, due to evolution equations of physics which would have invariant physical constants, i.e. no change during cosmological evolution.

The term non-singular universe is short hand for an initial starting point as to the expansion of the universe which is not at a singular point of space-time. Reference [

For the record, the usual interpretation of

So, we will assume a linkage between black hole physics event horizons, as defined, and early universe cosmology in the manner brought up by [

We begin first by putting the results of [

The specifics of what were done with

The main import of Equation (20) is that it defacto leads to a “non-dimensional” representation of entropy, but before we do that, it is useful to review what is said about

FWIW, we will provisionally in the regime of z (red shift) > 1100 set for inflation from a Planck time interval up to 10^-20 seconds, when the expansion radii of the universe was about a meter, i.e.

What we will do in later parts of this paper, to get an approximation as to what the actual value of

a. Relevance of Equation (23) to the concept of dimensionless entropy

Cai, in [

We will assume that

While assuming Equation (24) we will through [

The entropy so mentioned, above, is commensurate with the following identification, namely how to link a measure of distance with scale factor

Our starting point for the rest of the article will lie in making sense of the following inputs into the scale factor as the last part of Equation (26) grouping of mathematical relations, namely we will look at time defined via

[

and M field given at the start of creation itself, and of course a cosmological “constant” parameter

The author is for now avoiding a time varying G, as it is creating Partial Differential equations the author has no idea of how to solve, for the time being, so we assume that G is invariant and use the Equation (28) result at this time [

i.e.

Then we use, by [

The linkage to graviton mass, and heavy gravitons will build upon this structure so built up via [

This above formula will de evolve, from a larger value, to having the mass of a graviton approximately as given about 10^−62 grams in the present era [

A specified value of

First, now the treatment of entropy due to early universe Gravitons. In the beginning of this analysis, we start with Ali and Das’s cosmology from Quantum potential article [

Equation (11) should be compared to an expression given by Padmanabhan [

Then the entropy at the end of the electro weak era is, assuming this is commensurate with graviton production, with the value of the Horizon radius at the upper end of Equation (32) above, namely about 1 meter

Given this, we can now consider what would be the magnetic field, initially, and the other parameters as given in the end of the last section. Doing so, if so, we can have frequency as high as

Using inflation, this would be redshifted at a minimum of 11 orders of magnitude, down to about 10^10 Hz today, at the highest end. The nature of the E and B fields, also as fill in would have to be commensurate with what was given in [

Still though, as a rule of thumb, we would have that the MINIMUM value of the magnetic field, in question would have to be [

Why we pursued this datum of an initial nonzero entropy? In a word, to preserve the fidelity of physical law from cosmological cycle to cycle, i.e. the bits we calculated with, came from Seth Lloyd [

Lloyd, sets, in [

The first part of Equation (37) in terms of “bits” is approximately similar to Equation (38), and more tellingly,

The upper part of Equation (38) overlaps, a bit with Equation (37) whereas Equation (36) is only a few orders of magnitude higher than the formal numerical count for the number of operations, # of Equation (39), i.e. the number of bits, given in Equation (39) is similar to the graviton entropy count given in Equation (32), However, most tellingly, the initial non zero graviton count, given when the universe is 1 meter in diameter, or so, is initiated by negative pressure, which we recount, below.

We state, first of all, that with we use Lloyd [

The upshot is that the entropy, at the close of the inflationary era, would be dominated by graviton production as of about the electroweak era, and this would have consequences as far as information, as can be seen by the approximation given by Seth Lloyd [

In the electro-weak era, we would be having Equation (42) as giving a number of ‘computational steps’ many times larger (10 orders of magnitude) than the entropy of the electro-weak,

In addition, making use of the above calculations, if we do so, we obtained that the minimum time step would be of the order of Planck time, i.e. of about 10^−44 seconds, which is very small, but not zero, whereas, again, assuming a 1 meter radii, which we obtain at the end of inflation, with a time step the, at the end of inflation of 10^−20 seconds. This is significant, when the universe had a radii of 1 meter, is about when we would expect r to be about 1 meter to then get us a value of Equation (42) in upper bound, i.e., setting r about 1 meter would allow us to have to have the upper bound value of Equation (41) being that of Equation (36).

This set of number of operations would be about when we would expect Planck’s constant to be set, with the values as given in [

Finally, we assert that the following are equivalent, namely in the pre Planckian era, just before the onset of the big bang

The

tial entropy, i.e., bits, it would be possible to have

We include below the derivation of Equation (43) which is for showing the following equivalences given in Electromagnetism, Quantum Mechanics, Classical Mechanics, and Gravity, through foundational Entropy at the Pre-Planckian level.

Equation (43) is a direct result of the following derivation, namely see the below, with Q, here, a fifth force quantity.

Entropy, Its Spatial Configuration near a Singularity and How We Use This Definition to Work in Effects of Non-Linear ElectrodynamicsThe usual treatment of entropy, if there is the equivalent of an event horizon is, that (Padmanabhan) [

If so, then we have that from first principles, (and here we also will set

Then Equation (7) is re written in terms of [

The following parameters will be identified, i.e. what is

will be set toward the end of the manuscript, with the consequences of the choices made discussed in this document as suggested new areas of inquiry. However, Equation (46) will be linkable to re writing Equation (16) as

If

We should, here, keep in mind that that abbreviation of H.O.T. means higher order terms. Sometimes they are extremely important, and other times that happens not to be the case.

If so, then the E field up to a point will be

To reconstruct

Then

If so, then in Equation (49) becomes

The density, then is read as

The current we will work with, is also then linkable to, by order of magnitude similar to Equation (53) of

Then we get an effective magnetic field, based upon the NLED approximation given by Corda et al. [

Then we can also talk about an effective charge of the form, given by applying Gauss’s law to Equation (53)

This charge, Q, so presented, will be part of the effective 5^{th} force [

cal value of

This will lead to an evaluation of

The value of

following are equivalent and imply each other as given in the grouping called Equation (43). This all is in keeping with [

In [^{th} force arguments will aid greatly in determining the choice of either general relativity or Scalar-Tensor theories of gravity as the origin of early universe primordial gravitational wave, and/or electromagnetic phenomenon. We assert that since [

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

Andrew Beckwith, (2016) Non-Linear Electrodynamics Gedanken Experiment for Modified Zero Point Energy and Planck’s “Constant”, h Bar, in the Beginning of Cosmological Expansion, So h(Today) = h(Initial). Also How to Link Gravity, Quantum Mechanics, and E and M through Initial Entropy Production in the Early Universe. Journal of High Energy Physics, Gravitation and Cosmology,02,168-182. doi: 10.4236/jhepgc.2016.22016