Asking If the Existence of Vacuum Energy to Keep Computational “Bits” Present at Start of Cosmological Evolution, Even If Spatial Radius Goes to Zero, Not Planck Length, Is Possible ()

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^{*}

_{initial}→0 if Stoica actually presents Einstein equations in a formalism which remove the big bang singularity pathology, then the reason for Planck length no longer holds. We present entanglement entropy in the early universe with a shrinking scale factor, due to Muller and Lousto, and show that there are consequences due to initial entangled S

_{Entropy}= 0.3r

^{2}

_{h}/a

^{2 }for a time dependent horizon radius r

_{H}= in cosmology, with (flat space conditions) for conformal time. Even if the 3 dimensional spatial length goes to zero, this construction preserves a minimum non-zero

**L**vacuum energy, and in doing so keep the bits, for computational bits cosmological evolution even if R

_{initial}→0 . We state that the presence of computational bits is necessary for cosmological evolution to commence.

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Cite this paper

*Journal of High Energy Physics, Gravitation and Cosmology*,

**2**, 226-243. doi: 10.4236/jhepgc.2016.22020.

Received 10 December 2015; accepted 15 April 2016; published 18 April 2016

1. Introduction

This article is to investigate what happens physically if there is a non-pathological singularity in terms of Einsteins equations at the start of space-time. This eliminates the necessity of having then put in the Planck length since then they would be no reason to have a minimum non-zero length. The reasons for such a proposal come from [1] by Stoica who may have removed the reason for the development of Planck’s length as a minimum safety net to remove what appears to be unadvoidable pathologies at the start of applying the Einstein equations

at a space-time singularity, and are commented upon in this article. in particular is re-

marked upon. This is a counter part to Fjortoft theorem in Appendix I. The idea is that entanglement entropy will help generate bits, due to the presence of a vacuum energy, as derived at the end of the article, and the presence of a vacuum energy non-zero value, is necessary for comsological evolution. Before we get to that creation of what is a necessary creation of vacuum energy conditions, we refer to constructions leading to extremely pathological problems which [1] leads to minus the presence of initial non-zero vacuum energy. [2] also adds more elaboration on this.

Note a change in entropy formula given by Lee [3] about the inter relationship between energy, entropy and temperature as given by

(1)

As a reviewer has asked about Equation (1) and the inter relationship of a mass m, and acceleration, the key point of this review is to look at if gravitons have a mass, m, in the beginning, and if Equation (1) is used, which the mass of a graviton is proportional to the following

(1a)

The reason why the mass of a graviton is stated as given by Equation (1a) is to presume that the relationship given by Lee [3] , as to any mass, is given by Equation (1) and Equation (1a), so we can relate any presumed

mass linked to gravitons to change in entropy. As to acceleration appearing, the acceleration, is part

of the formula given by Equation (1) and by default Equation (1a) and also by thermodynamic reasoning the generalized temperature

(1b)

If we assume, in the onset of expansion of the universe, that Equation (1b) holds, then we can review the application of Equation (1a) to graviton mass, m, as, and to have acceleration, given by as part of a definition of generalized temperature, given by Equation (1b).

Note that temperature is, in this presentation by Lee [3] presumably a constant initially, i.e. very hot, so then we are really in this presentation, assuming that the acceleration as given by is a constant, so in fact what we are actually reviewing through Equation (1a) is a direct relationship of mass as proportional to entropy, i.e. as

(1c)

i.e. the mass of a graviton is presumed to be proportional to entropy, i.e. in choosing Equation (1c) we are leading up to one of the themes of this document which is that if entropy is proportional to information and note that later, we will be relating entropy, as given, to a numerical count factor, i.e. then in fact, this will lead to a re write of Equation (1c) to read as, if N (count) is a numerical count proportional to the change in Entropy, that

(1d)

This assumes that we are evaluating Equation (1b) as a constant, i.e. that the temperature be fixed, which is leading to the acceleration, which the referee was so concerned about, as a constant, i.e. via the relationship of

looking at as an acceleration factor, and presumably that the delta x factor in acceleration is of the interval of Planck length.

Lee’s formula is crucial for what we will bring up in the latter part of this document. Namely that changes in initial energy could effectively vanish if [1] is right, i.e. Stoica removing the non pathological nature of a big bang singularity. That is, unless entanglement entropy is used.

If the mass m, i.e. for gravitons is set by acceleration (of the net universe) and a change in enthropy between the electroweak regime and the final entropy value of, if for acceleration is used, so then we obtain

(2)

Then we are really forced to look at (1) as a paring between gravitons (today) and gravitinos (electro weak) in the sense of preservation of information.

Having said this note by extention. As changes due to and, t hen a is also altered i.e. goes to zero.

What will determine the answer to this question is if goes to zero if which happens if there is no minimum distance mandated to avoid the pathology of singularity behavior at the heart of the Einstein equations. In doing this, we avoid using the energy situation, i.e. of vanishing initial space-time energy, and instead refer to a nonzero energy, with instead vanishing. In particular, the Entanglement entropy concept as presented by Muller and Lousto [4] is presented toward the end of this manuscript as a partial resolution of some of the pathologies brought up in this article before the entanglement entropy section. No matter how small the length gets, if it is entanglement entropy, will not go to zero. The requirement is that the smallest length of time, t, rescaled, does not go to zero. This preserves a minimum non-zero vacuum energy, and in doing so keep non zero amounts of initial bits, for computational bits cosmological evolution even if.

I think that the common confusion here, is that refers to initial RADII and not to curvature, which was also one of the questions raised by the referee. is a minimum radii and has nothing to do with curvature. This formula, which evidently confused referees, i.e. if refers to a computational bits value which will show up in our manuscript, then our statement is that we have an initial radii of less than Planck Length. As given by

(2a)

Is part of the build up of Equation (3) and should be read by readers so as to understand the significance of what is in this Equation (2a). I.e. does not hold, in general, and we get Equation (2a) only if the value is used which refers to a computational bits value.

Before doing that, we review Ng [5] and his quantum foam hypothesis to give conceptual underpinnings as to why we later even review the implications of entanglement Entropy.

We state unequivocally here, that Equation (2a) has referring to a computational bits value which is Equation (3) and will be part of treating entropy and its evolution

Note that this evaluation is preformed in the Planck time interval, and is the basis of evaluation by our paper.

I.e. the concept of bits and computations is brought up because of applying energy uncertainty, as given by [5] and the Margolis theorem appears to indicate that the universe could not possibly evolve if [1] is applied, in a 4 dimensional closed universe. This bottle neck as indicated by Ng’s [5] formalism is even more striking in the author’s end of article proof of the necessity of using entanglement entropy in lieu of the conclusion involving

entanglement entropy, which can be non zero, even if provided there is a minimum non-zero time length.

2. Review of Ng, [5] with Comments

First of all, Ng refers to the Margolus-Levitin theorem with the rate of operations . Ng wishes to avoid black-hole formation. This last step is not important to our view point, but we refer to it to keep fidelity to what Ng brought up in his presentation. Later on, Ng refers to the with the Hubble radius. Next Ng refers to the. Each bit energy is with

The key point as seen by Ng [4] and the author is in

(3)

Assuming that the initial energy E of the universe is not set equal to zero, which the author views as impossible, the above equation says that the number of available bits goes down dramatically if one sets ? Also Ng writes entropy S as proportional to a particle count via N.

(4)

We rescale to be

(5)

The upshot is that the entropy, in terms of the number of available particles drops dramatically if becomes larger.

So, as grows smaller, as becomes larger.

a) The initial entropy drops;

b) The nunber of bits initially available also drops.

The limiting case of (4) and (5) in a closed universe, with no higher dimensional embedding is that both would almost vanish, i.e. appear to go to zero if becomes very much larger. The quest4ion we have to ask is would the number of bits in computational evolution actually vanish?

3. Does It Make Sense to Talk of Vacuum Energy If Is Changed to ? Only Answerable Straightforwardly If an Embedding Superstructure Is Assigned. Otherwise Difficult, Unless One Is Using Entanglement Entropy Which Is Non Zero Even If

We summarize what may be the high lights of this inquiry leading to the present paper as follows:

a. One could have the situation if of an infinite point mass, if there is an initial nonzero energy

in the case of four dimensions and no higher dimensional embedding even if [1] goes through verbatim. The author sees this as unlikely. The infinite point mass construction is verbatim if one assumes a closed universe, with no embedding superstructure and no entanglement entropy. Note this appears to nullify the parallel Brane world construction used by Durrer [6] . The author, in lieu of the manuscript sees no reason as to what would perturb this infinite point structure, so as to be able to enter in a big bang era. In such a situation, one would not have vacuum energy unless entanglement entropy were used. That is unless one has a non zero entanglement entropy

[4] present even if. See [7] for a smilar argument.

b. The most problematic scenario. and no initial cosmological energy. i.e. this in a 4 dimensional closed universe. Then there would be no vacuum energy at all. initially. A literal completely empty initial state, which is not held to be viable by Volovik [8] .

c. If additional dimensions are involved in beginning cosmology, than just 4 dimensions will lead to physics which may give credence to other senarios. One scenario being the authors speculation as to initial degrees of freedom reaching up to 1000, and the nature of a phase transition from essentially very low degrees of freedom, to over 1000 as speculated by the author in 2010 [9] .

The relevant template for examinging such questions is given in Table 1 as printed.

e. The meaning of octonionic geometry prior to the introduction of quantum physics presupposes a form of embedding geometry and in many ways is similar to Penrose’s cyclic conformal cosmology speculation.

f. It is striking how a semiclassical argument can be used to construct in Table 1. In particular, we look at how Planck’s constant is derived, as in the electroweak regime of space-time, for a total derivative [11] [12]

(6)

Similarly [11] [12]

(7)

The A field so given would be part of the Maxwell's equations given by [10] as, when represents a D’Albertain operator, that in a vacuum, one would have for an A field [11] [12]

(8)

And for a scalar field

(9)

Following this line of thought we then would have an energy density given by, if is the early universe permeability [11]

(10)

We integrate (10) over a specified E and M boundary, so that, then we can write the following condition namely [11] [12] .

(11)

(11) would be integrated over the boundary regime from the transition from the octonionic regime of space time, to the non-octonionic regime, assuming an abrupt transition occurs, and we can write, the volume integral as representing [11] [12]

(12)

Then by applying [11] [12] we get formed by semiclassical reasons. In semiclassical reasoning similar to [10] .

Table 1. Time interval dynamical consequences does QM/WdW apply?

(Constant value) (13)

The question we can ask, is that can we have a prequantum regime commencing for (11) and (12) for if? And a closed 4 dimensional universe? If so, then what is the necessary geometrial regime

of space-time so that the integration performed in (11) can commence properly? Also, what can we say about the formation of (12) above, as a number, gets larger and larger, effectively leading to. Also, with an octonionic geometry regime which is a pre quantum state [13] .

In so many words, the formation period for is our pre-quantum regime. Table 1 could even hold if but that the 4 dimensional space-time exhibiting such behavior is embedded in a higher dimensional template. That due to not removing entanglement entropy as is discussed near the end of this article.

4. If Then If There Is an Isolated, Closed Universe, There Is a Disaster Unless One Uses Entanglement Entropy

One does not have initial entropy, and the number of bits initially disappears. That is if one is not using entanglement entropy, as will be examined at the end of this article.

Abandoning the idea of a completely empty universe, this unperturbed point of space time (of matter-energy) appears to be a configuration for a static point of space time with no perturbation, as may be the end result of applying Fjortoft theorem [14] to the thermodynamic potential as given in [15] , i.e. the non definitive answer for fulfillment of criteria of instability by applying Fjortoft’s theorem [14] to the potential [15] leading to no instability as given by the potential given in [15] may lead to a point of space-time with no change, i.e. a singular point with “infinite” mass which does not change at all. This issue will be reviewed in [16] a different procedure, i.e. a so called nonsingular universe construction. To get there we will first of all review an issue leading up to implementation of [16] .

5. Can an Alternative to a Minimum Length Be Put in? Consider the Example of Planck Time as the Minimal Component, Not Planck Length

From J. Dickau, [17] the following was given to the author, as a counter point to leading to a disaster.

“If we examine the Mandelbrot Set along the Real axis, it informs us about behaviors that also pertain in the quaternion and octonic case-because the real axis is invariant over the number types. If numbers larger than.25 are squared and summed recursively (i.e. ?z = z^2 + c ) the result will blow up, but numbers below this threshold never get to infinity, no matter how many times they are iterated. But once space-like dimensions are added-i.e. an imaginary compoent―the equation blows up exponentially, faster than when iterated.

Dickau concludes:

“Anyhow there may be a minimum (space-time length) involved but it is probably in the time direction”.

This is a counter pose to the idea of minimum length, looking at a beginning situation with a crucial parameter even if the initial time step is “put in by hand”. First of all, look at [4] , if E is M, due to setting c = 1, then

(14)

Everything depends upon the parameter which can go to zero. We have to look at what (14) tells us, even if we have an initial time step for which time is initially indeterminate, as given by a redoing of Mitra’s formula [7] which we put in to establish the indeterminacy of the initial time step if quantum processes hold.

(15)

What Dickau is promoting is, that the Mandelbrot set, if applicable to early universe geometry, that what the author wrote, with potentially going to zero, is less important than a

minimum time length. The instability issue is reviewed in Appendix II for those who are interested in the author’s views as to lack proof of instability. It uses [15] which the author views as THE reference as far as thermodynamic potentials and the early universe.

6. Muller and Lousto Early Universe Entanglement Entropy, and Its Implications: Solving the Spatial Length Issue, Provided a Minimum Time Step Is Preserved in the Cosmos, in Line with Dickau’s Suggestion

We look at [4]

for a time dependent horizon radius in cosmology (16)

Equation (16) above was shown by the author to be fully equivalent to

(17)

i.e.

(18)

So, then one has

(19)

No matter how small the length gets, if it is entanglement entropy, will not go to zero. The requirement is that the smallest length of time, t, re scaled does not go to zero. This preserves a minimum non zero vacuum energy, and in doing so keep the non zero initial bits, for computational bits contributions to evolving space time behavior even if.

7. Reviewing a Suggestion as to How to Quantify the Shrinkage of the Scale Factor and Its Connections with Entanglement Entropy

We are given by [16] if there is a non singular universe, a template as to how to evaluate scale factor a against time scaled over Planck time, with the following results.

(20)

Two time and scale factor values in tandem particularly stand out. Namely,

(21)

Also

(22)

The main thing we can take from this, is to look at the inter-relationship of how to pin down an actual initial Hubble “constant” expansion parameter, where we look at:

(23)

Recall that, which is predicated upon, if the time is close to Planck time the initial maximal density of

(24)

And length given by

(25)

So (24) is implying that the amount of matter in a region of space is initially about

(26)

Using 1 GeV/c^{2} = 1.783×10^{−27} kg means that (26) above is

(26a)

Then if

(27)

It will lead to

(28)

Then, to first order, one is looking at initial entropy to get a non zero but definite vacuum energy as leading to an entanglement entropy of about (just before the electro weak regime) regardless of the situation being in fidelity , or lack of with the physics of [18]

(29)

8. Reviewing the Geometry for Embedding (29) above

In line with Stoica [1] shrinking the minimum length and referring to both (29) and (27), the idea is to use a surface area treatment as to getting the initial entropy values as given in (29). To do so, the author at the situation presented in Figure 1.

The two branes presented in Figure 1 given at and refer to the two Brane world states, especially in line with [19] [20] . The first one, namely is the Brane where our physical universe lives in, and is embedded in. If one uses this construction, with higher dimensions than just 4 dimensions, then it is possible to have a single point in 4 dimensional space as a starting point to a tangential sheet which is part of an embedding in more than 4 dimensions. Along the lines of having a 4 dimensional cusp with its valley (lowest) point in a more than 4 dimensional tangential surface. The second Brane is about centimeters away from the Brane our physical world lives in, and moves closer to our own Brane in the future, leading to a slapping of the two Branes together about a trillion years ahead in our future [19] [20] . The geometry we are referring to with regards to embedding is in the first Brane. [6] uses this geometry to have graviton production which the author has used to model Dark Energy.

Figure 1. As adopted from Reference [6] .

9. Conclusion: Making Computational Bits, via (19)

As stated by Ng. the idea would be to have to give imputs into (3) i.e.

(3)

Here in this case, even if the spatial contribution, due to [1] goes to zero, the idea would be to have the time length non-zero so as to have a space-time version of l non-zero. This would also be in tandem with calling E, in (3) as proportional to, where if the time is Planck time, in minimum value, and in value, one would have before the electro-weak an input into E, which would require an entropy (entanglement).

What remains to be seen is, if there is a geometric sheet in more than 4 dimensions, allowing for non-zero time, as argued for, even if the spatial component goes to zero, according to [1] . We suggest an update as to what was written by Seth Lloyd [21] with

(30)

when [22]

(31)

While doing this, a good thing to do, would be to keep in mind the four dimensional version of vacuum energy as given by Park, [23] namely

(32)

As well as the transition given by a combination of [23] , with [24] , Barvinskey et al.

(33)

Quantifying the above, and giving it experimental proof, via detector technology may allow us to investigate an old suggestion by the author as to four dimension and five dimensional vacuum energy which are given for

small time values, and for temperatures sharply lower than, Beckwith^{ } [22] (2007), where for a positive integer n

(34)

In particular, the author is interested in investigating if the following is true.

Look at an argument provided by Thanu Padmanabhan [25] , leading to the observed cosmological constant value suggested by Park [22] . Assume that, but that when we make this substitution that [26]

(35)

i.e. looking at if

(36)

Now to make it more interesting.

We can replace by. In addition, we may look at inputs from the initial value of the Hubble parameter to get the necessary e folding needed for inflation, according to

(37)

Leading to

(38)

If we set implying a very large initial cosmological constant value, we get in line with what Park suggested for times much less than the Planck interval of time at the instant of nucleation of a vacuum state

(39)

Acknowledgements

This work is supported in part by National Nature Science Foundation of China grant No. 110752.

Appendix I. Fjortoft Theorem

A necessary condition for instability is that if is a point in space time for which for any given potential U, then there must be some value in the range such that

(1)

For the proof, see [12] and also consider that the main discussion is to find instability in a physical system which will be described by a given potential U. Next, we will construct in the boundary of the EW era, a way to come up with an optimal description for U.

Appendix II. Constructing an Appropriate Potential for Using Fjortoft Theorem in Cosmology for the Early Universe Cannot Be Done. We Show Why

To do this, we will look at Padamanabhan [15] and his construction of (in Dice 2010) of thermodynamic potentials he used to have another construction of the Einstein GR equations. To start, Padamanabhan [15] wrote

If is a so called Lovelock entropy tensor, and a stress energy tensor

(1)

We now will look at

(2)

So happens that in terms of looking at the partial derivative of the top (1) equation, we are looking at

(3)

Thus, we then will be looking at if there is a specified for which the following holds.

(4)

What this is saying is that there is no unique point, using this for which (4) holds. Therefore, we say there is no official point of instability of due to (3). The Lagrangian structure of what can be built up by the potentials given in (3) with respect to mean that we cannot expect an inflection point with respect to a

2^{nd} derivative of a potential system. Such an inflection point designating a speed up of acceleration due to DE exists a billion years ago [37] . Also note that the reason for the failure for (4) to be congruent to Fjoroft’s theorem is due to

(5)

Appendix III. Details as to Forming Crowell’s Time Dependent Wheeler De Witt Equation, and Its Links to Worm Holes

This will be to show some things about the worm hole we assert the instanton traverses en route to our present universe. From Crowell [38] ^{ }

(1)

This has when we do it, and frequently, so then we can consider

(2)

In order to do this, we can write out the following with regards to the solutions to Eqn (1) put up above.

(3)

And

(4)

This is where and refer to integrals of the form and. It

so happens that this is for forming the wave functional permitting an instanton forming, while we next should consider if or not the instanton so farmed is stable under evolution of space time leading up to inflation. We argue here that we are forming an instanton whose thermal energy is focused into a wave functional which is in the throat of the worm hole up to a thermal discontinuity barrier at the onset, and beginning of the inflationary era.

Appendix IV. The D’Albembertain Operation in an Equation of Motion for Emergent Scalar Fields

We begin with the D’Albertain operator as part of an equation of motion for an emergent scalar field. We refer to the Penrose potential ( with an initial assumption of Euclidian flat space for computational simplicity) to account for, in a high temperature regime an emergent non zero value for t

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