How a Minimum Time Step and Formation of Initial Causal Structure in Space-Time Is Linked to an Enormous Initial Cosmological Constant

Andrew Walcott Beckwith^*

Physics Department, College of Physics, Chongqing University, Chongqing, China.

DOI: 10.4236/jhepgc.2018.43032                

Abstract

We use a root finder procedure to obtain and an inflaton value due to use of a scale factor if we furthermore use as the variation of the time component of the metric tensor in Pre-Planckian Space-time up to the Planckian space-time initial values. In doing so, we obtain, due to the very restricted values for which are of the order of less than Planck time, results leading to an enormous value for the initial Cosmological constant.

Keywords

Inflaton Physics, Causal Structure, Non-Linear Electrodynamics

1. Framing the Initial Inquiry

Volovik [1] derives in page 24 of his manuscript a description of a total vacuum energy via an integral over three-dimensional space

$E_{Vac}\left(N\right)={\displaystyle \int {\text{d}}^{3}r\cdot \epsilon \left(n\right)}$ (1)

The integrand to be considered is, using a potential defined by $U=\frac{c^{2}m}{n}$ as

given by Volovik for weakly interacting Bose gas particles, as well as

$\epsilon \left(n\right)=\frac{1}{2}U\cdot n^2+\frac{8}{15{\text{π}}^2{\hslash }^3}{m}^{3/2}{U}^{5/2}{n}^{5/2}=\frac{1}{2}\cdot c^2\cdot \left[n\cdot m+\frac{4}{15}\cdot \left(\frac{m^5}{{\hslash }^2\cdot \sqrt{c}}\right)\cdot \frac{1}{n^2}\right]$ (2)

For the sake of argument, m, as given above will be called the mass of a graviton, n a numerical count of gravitons in a small region of space, and afterwards, adaptations as to what this expression means in terms of entropy generation which will be subsequently raised. A simple graph of the 2^nd term of Equation (2) with comparatively large m and with $\hslash =c=1$ has the following qualitative behavior, namely for

$E1=\left[c^2/2\right]\cdot \left[\frac{4}{15}\cdot \left(\frac{m^5}{{\hslash }^2\cdot \sqrt{c}}\right)\cdot \frac{1}{n^2}\right]$ (3)

$E1\ne 0$ when n is very small, and $E1=0$ as $n\to {10}^{10}$ at the onset of inflation. This will tie directly with a linkage between energy and entropy, as seen in the construction, looking at what Kolb [2] put in, i.e.

$\rho ={\rho }_{\text{radiation}}=\left(3/4\right)\cdot {\left[\frac{45}{2{\text{π}}^2{g}_{\ast }}\right]}^{1/3}\cdot S^{4/3}\cdot r^{-4}$ (4)

Here, the idea would be, to make the following equivalence. Namely look at,

${\left[\left[\frac{{\Lambda }_{\text{Max}}{r}^4}{8\text{π}G}\right]\cdot \left(4/3\right)\cdot {\left[\frac{2{\text{π}}^2{g}_{\ast }}{45}\right]}^{1/3}\right]}^{3/4}~S_{\text{initial}}$ (5)

We furthermore make the assumption of a minimum radius of

$R_{\text{initial}}~\frac{1}{\#}{\mathcal{l}}_{Ng}<l_{\text{Planck}}$ (6)

This Equation (6) will be put as the minimum value of r, in Equation (5), where we have in this situation [3] [4]

$\#\text{bits}~{\left[\frac{E}{\hslash }\cdot \frac{l}{c}\right]}^{3/4}\approx {\left[\frac{M{c}^2}{\hslash }\cdot \frac{l}{c}\right]}^{3/4}$ (7)

And if M is the total space-time energy mass, for initial condition and E1 is the main fluctuation in energy we have to consider, if $\Delta E~E1$ , as well as [3] [4]

$S_{\text{initial}}~n_{\text{graviton}}~\text{initial}\text{graviton}\text{count}$ (8)

Then what can be said about the inter relationship of graviton counts, and the onset of Causal structure?

2. Examination of the Minimum Time Step, in Pre-Planckian Space-Time as a Root of a Polynomial Equation

We initiate our work, citing [5] to the effect that we have a polynomial equation for the formation of a root finding procedure for $\Delta t$ , namely if

$\begin{array}{l}\Delta t\cdot \left|\left(\sqrt{\frac{8\text{π}G{V}_0}{\gamma \cdot \left(3\gamma -1\right)}}\cdot \Delta t-1\right)-\frac{{\left(\sqrt{\frac{8\text{π}G{V}_0}{\gamma \cdot \left(3\gamma -1\right)}}\cdot \Delta t-1\right)}^2}{2}+\frac{{\left(\sqrt{\frac{8\text{π}G{V}_0}{\gamma \cdot \left(3\gamma -1\right)}}\cdot \Delta t-1\right)}^3}{3}-\cdots \right|\\ \approx {\left(\sqrt{\frac{\gamma }{\text{π}G}}\right)}^{-1}\frac{48\text{π}\hslash }{a_{\min }^2\cdot \Lambda }\end{array}$ (9)

From here, we then cited, in [5] , using [6] a criteria as to formation of entropy, i.e. if $\Lambda$ is an invariant cosmological “constant” and if Equation (10) holds, we can use the existence of nonzero initial entropy as the formation point of an arrow of time.

${S_{\Lambda }|}_{\text{Arrow-of-time}}=\text{π}\cdot {\left(\frac{{R_c|}_{\text{initial}}~c\cdot \Delta t}{l_{\text{Planck}}}\right)}^2\ne 0$ (10)

This leads to the following, namely in [5] we make our treatment of the existence of causal structure, as given by writing its emergence as contingent upon having

$\left(\frac{{R_c|}_{\text{initial}}~c\cdot \Delta t}{l_{\text{Planck}}}\right)~\vartheta \left(1\right)$ (11)

The rest of this article will be contingent upon making the following assumptions. FTR

$\begin{array}{l}\Delta E~E1\\ S_{\text{initial}}~n_{\text{graviton}}~\text{initial}\text{graviton}\text{count}\\ {\left[\left[\frac{{\Lambda }_{\text{Max}}{r}^4}{8\text{π}G}\right]\cdot \left(4/3\right)\cdot {\left[\frac{2{\text{π}}^2{g}_{\ast }}{45}\right]}^{1/3}\right]}^{3/4}~S_{\text{initial}}\\ \iff {\left[\left[\frac{{\Lambda }_{\text{Max}}{r}^4}{8\text{π}G}\right]\cdot \left(4/3\right)\cdot {\left[\frac{2{\text{π}}^2{g}_{\ast }}{45}\right]}^{1/3}\right]}^{3/4}~n_{\text{graviton}}\\ \Delta E~E1~V_0\\ r~R_{\text{initial}}~\frac{1}{\#}{\mathcal{l}}_{Ng}<l_{\text{Planck}}\end{array}$ (12)

In short, our view is that the formation of a minimum time step, if it satisfies Equation (11) is a necessary and sufficient condition for the formation of an arrow of time, at the start of cosmological evolution we have a necessary and sufficient condition for the initiation of an arrow of time. With causal structure, along the lines of Dowker, as in [7] and given more detail by Equation (12) above as inputs into Equation (10) and Equation (11) i.e. Planck length is set equal to 1 and.

$\begin{array}{l}{\frac{\Delta E\Delta t}{\text{Volume}}~\left[\hslash /\text{Volume}\cdot \left(\delta g_{tt}~a_{\min }^2\cdot {\varphi }_{\text{initial}}\right)\right]|}_{\text{Pre-Planckian}}\\ \underset{\left(\text{Pre-Planckian}\right)\to \left(\text{Planckian}\right)}{\to }{\Delta E\Delta t~\hslash |}_{\text{Planckian}}\end{array}$ (13)

i.e. the regime of where we have the initiation of causal structure, if allowed would be contingent upon the behavior of [5] [8] [9]

$\begin{array}{l}{g}_{tt}~\delta g_{tt}\approx a_{\min }^2{\varphi }_{\text{initial}}\ll 1\\ \underset{\text{Pre-Planck}\to \text{Planck}}{\to }\delta g_{tt}\approx a_{\min }^2{\varphi }_{\text{Planck}}~1\\ \iff {\left(\frac{{R_c|}_{\text{initial}}~c\cdot \Delta t}{l_{\text{Planck}}}\right)~\vartheta \left(1\right)|}_{\text{Planck}}\end{array}$ (14)

i.e. the right hand side of Equation (14) is the square of the scale factor, which we assume is ~10^−110, due to [4] [10] , and an inflaton given by [4] [11]

So, the question well will be leading up to is what does Equation (9), Equation (12), and Equation (13), tell us about graviton production, and the causal foundation condition stated at Equation (14)?

3. Conclusion, So What Is the Root of Our Approximation for a Time Step?

Here our derivation result which satisfies Equation (14) is contingent upon initial ${R_c|}_{\text{initial}}~c\cdot \Delta t$ as an initial event horizon. So, our bubble of space-time is of the order of magnitude of approximately one Planck Length,

$\begin{array}{l}{\frac{\Delta E}{\text{Volume}}|}_{\text{Pre-Planck}}\\ {~\left[\hslash /\left(\Delta t\cdot \left(\text{Volume}\equiv {\left(\frac{l_{Ng}}{\#}\right)}^3\right)\cdot \left(\delta g_{tt}~a_{\min }^2\cdot {\varphi }_{\text{initial}}\right)\right)\right]|}_{\text{Pre-Planckian}}\\ ~\frac{c^2\cdot m_{\text{graviton}}^2}{30{\hslash }^2\sqrt{c}\cdot }\cdot {\left(\text{Volume}\equiv {\left(\frac{l_{Ng}}{\#}\right)}^3\right)}^{-1}\cdot {\left(\frac{1}{n_{\text{boson}}=n_{\text{graviton}}}\right)}^2\\ \underset{\left(\text{Pre-Planckian}\right)\to \left(\text{Planckian}\right)}{\to }\\ {{\Delta E|}_{\text{Planckian}}~\frac{\hslash }{\Delta t}|}_{\text{Planckian}}\cdot {\left(\text{Volume}\equiv {\left(l_{\text{Planck}}\right)}^3\right)}^{-1}{~\frac{\hslash }{\Delta t}|}_{\text{Planckian}}\end{array}$ (15)

A convenient normalization would be to have

$\begin{array}{l}r~R_{\text{initial}}~\frac{1}{\#}{\mathcal{l}}_{Ng}<l_{\text{Planck}}\\ \&{\mathcal{l}}_{Ng}~l_{\text{Planck}}\equiv 1\\ \&r~R_{\text{initial}}~\frac{1}{\#}\end{array}$ (16)

If so then, Equation (14) would read as a causal formation transformation we would give as

$\begin{array}{l}r~R_{\text{initial}}~\frac{1}{\#}{\mathcal{l}}_{Ng}<l_{\text{Planck}}\&{\mathcal{l}}_{Ng}~l_{\text{Planck}}\equiv 1\\ \&r~R_{\text{initial}}~\frac{1}{\#}\&\hslash \equiv c\equiv 1\\ {{\frac{\Delta E}{\text{Volume}}|}_{\text{Pre-Planck}}~\left[1/\left(\Delta t\cdot \left(\text{Volume}\equiv {\left(\frac{1}{\#}\right)}^3\right)\cdot \left(\delta g_{tt}~a_{\min }^2\cdot {\varphi }_{\text{initial}}\right)\right)\right]|}_{\text{Pre-Planckian}}\\ ~\frac{m_{\text{graviton}}^2}{30}\cdot {\left(\text{Volume}\equiv {\left(\frac{1}{\#}\right)}^3\right)}^{-1}\cdot {\left(\frac{1}{n_{\text{boson}}=n_{\text{graviton}}}\right)}^2\\ \underset{\left(\text{Pre-Planckian}\right)\to \left(\text{Planckian}\right)}{\to }{{\Delta E|}_{\text{Planckian}}~\frac{1}{\Delta t}|}_{\text{Planckian}}\end{array}$ (17)

And then we would have the following equation if we make the following further normalization, as to Planck Mass, and Graviton mass, namely Planck Mass ~ 2.17645e−5 grams, whereas M (graviton) ~ 2.1e−62 grams. i.e. If Planck Mass = 1 in normalization, then M (graviton) ~ 10^−57

$\begin{array}{l}{\frac{\Delta E}{\text{Volume}}|}_{\text{Pre-Planck}}\\ {~\left[1/\left(\Delta t\cdot \left(\text{Volume}\equiv {\left(\frac{1}{\#}\right)}^3\right)\cdot \left(\delta g_{tt}~a_{\min }^2\cdot {\varphi }_{\text{initial}}\right)\right)\right]|}_{\text{Pre-Planckian}}\\ ~\frac{{10}^{-114}}{30}\cdot \left(\frac{{\#}^3}{{\left(n_{\text{boson}}=n_{\text{graviton}}\right)}^2}\right)~\frac{{10}^{-114}}{30}\\ \underset{\left(\text{Pre-Planckian}\right)\to \left(\text{Planckian}\right)}{\to }{{\Delta E|}_{\text{Planckian}}~\frac{1}{\Delta t}|}_{\text{Planckian}}~{\rm O}\left(1\right)\end{array}$ (18)

i.e. we would roughly have

$\frac{{10}^{-114}}{30}\underset{\left(\text{Pre-Planckian}\right)\to \left(\text{Planckian}\right)}{\to }{\frac{1}{\Delta t}|}_{\text{Planckian}}~{\rm O}\left(1\right)$ (19)

This outlines the enormity of the change from Pre Planckian to Planckian physics. If this is true, it indicates the enormity of the Pre Planckian to Planckian transformation. If we assume that $a_{\min }^2$ remains invariant, it means that the contribution of the inflaton becomes almost infinitely larger, i.e. $a_{\min }^2$ ~ 10^−110 in size.

So, if we have

${\Delta E|}_{\text{Pre-Planckian}}~\frac{{10}^{-114}}{30}\cdot \left(\frac{{\#}^3}{{\left(n_{\text{boson}}=n_{\text{graviton}}\right)}^2}\right)~\frac{{10}^{-114}}{30}~V_0$ (20)

and if ${\Delta E|}_{\text{Pre-Planckian}}~\frac{{10}^{-114}}{30},$ so that we have

$\begin{array}{l}\Delta t\cdot |\left(\sqrt{\frac{8\text{π}\cdot \left({\Delta E|}_{\text{Pre-Planckian}}\right)}{\gamma \cdot \left(3\gamma -1\right)}}\cdot \Delta t-1\right)-\frac{{\left(\sqrt{\frac{8\text{π}{\left({\Delta E|}_{\text{Pre-Planckian}}\right)}_0}{\gamma \cdot \left(3\gamma -1\right)}}\cdot \Delta t-1\right)}^2}{2}\\ +\frac{{\left(\sqrt{\frac{8\text{π}\cdot \left({\Delta E|}_{\mathrm{Pr}e-\mathrm{P}lanckian}\right)}{\gamma \cdot \left(3\gamma -1\right)}}\cdot \Delta t-1\right)}^3}{3}-\cdots |\approx {\left(\sqrt{\frac{\gamma }{\text{π}}}\right)}^{-1}\frac{48\text{π}}{a_{\min }^2\cdot \Lambda }\end{array}$ (21)

As

$\begin{array}{l}\Delta t\cdot |\left(\sqrt{\frac{8\text{π}\cdot \left(\frac{{10}^{-114}}{30}\right)}{\gamma \cdot \left(3\gamma -1\right)}}\cdot \Delta t-1\right)-\frac{{\left(\sqrt{\frac{8\text{π}\left(\frac{{10}^{-114}}{30}\right)}{\gamma \cdot \left(3\gamma -1\right)}}\cdot \Delta t-1\right)}^2}{2}\\ +\frac{{\left(\sqrt{\frac{8\text{π}\cdot \left(\frac{{10}^{-114}}{30}\right)}{\gamma \cdot \left(3\gamma -1\right)}}\cdot \Delta t-1\right)}^3}{3}-\cdots |\approx {\left(\sqrt{\frac{\gamma }{\text{π}}}\right)}^{-1}\frac{48\text{π}}{a_{\min }^2\cdot \Lambda }\end{array}$ (22)

Or more approximately as

$\begin{array}{l}\Delta t\cdot \left|\left(\sqrt{\frac{{10}^{-114}}{\gamma \cdot \left(3\gamma -1\right)}}\cdot \Delta t-1\right)-\frac{{\left(\sqrt{\frac{{10}^{-114}}{\gamma \cdot \left(3\gamma -1\right)}}\cdot \Delta t-1\right)}^2}{2}+\frac{{\left(\sqrt{\frac{{10}^{-114}}{\gamma \cdot \left(3\gamma -1\right)}}\cdot \Delta t-1\right)}^3}{3}-\cdots \right|\\ \approx {\left(\sqrt{\frac{\gamma }{\text{π}}}\right)}^{-1}\frac{48\text{π}}{a_{\min }^2\cdot \Lambda }\end{array}$ (23)

Now, set ${\Lambda }_{\text{initial}}=\Lambda$

$\begin{array}{l}\Delta t\cdot \left|\left(\sqrt{\frac{{10}^{-114}}{\gamma \cdot \left(3\gamma -1\right)}}\cdot \Delta t-1\right)-\frac{{\left(\sqrt{\frac{{10}^{-114}}{\gamma \cdot \left(3\gamma -1\right)}}\cdot \Delta t-1\right)}^2}{2}+\frac{{\left(\sqrt{\frac{{10}^{-114}}{\gamma \cdot \left(3\gamma -1\right)}}\cdot \Delta t-1\right)}^3}{3}-\cdots \right|\\ \approx {\left(\sqrt{\frac{\gamma }{\text{π}}}\right)}^{-1}\frac{48\text{π}}{\left(a_{\min }^2~{10}^{-110}\right)\cdot {\Lambda }_{\text{initial}}}\\ \approx \Delta t\le t_{\text{Planck}}~1\\ \iff \frac{48\text{π}}{\left(a_{\min }^2~{10}^{-110}\right)\cdot {\Lambda }_{\text{initial}}}~\Delta t\le t_{\text{Planck}}~1\\ \iff {\Lambda }_{\text{initial}}\ge {10}^{112}\end{array}$ (24)

This is on the order of the Cosmological constant, as computed by [12] and Peskins, in [13] so that the Pre Planckian Cosmological constant would have an enormous value on par with the Quantum field theory estimate of the Plancks constant, in Pre Planckian space-time [12] [13] .

This so happens to be consistent with Equation (5) of our document. It also has some similarities with the ideas given in [14] .

Finally this should be seen in the light of [15] [16] [17] which establish a non-linear electrodynamic treatment of initial singularities, which the author views credible, as an alternative to [18] and the Penrose Singularity theorem.

Acknowledgements

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

Conflicts of Interest

The authors declare no conflicts of interest.

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