Note here that when this happens, we have two equally admissible solutions for the scale factor, minimum, and the consequences; if # is a real number, then we have a contradiction with what is called Theorem 3, Hawking (1967) as cited on page 271, of [8] we have that

Theorem 3: If ${R}_{ab}{K}^{a}{K}^{b}\ge 0$ for every non space-like Vector K

1) The strong casuality condition holds on $\left(\stackrel{\u02dc}{M},g\right)$ ,

2) There is some past-directed unit timelike vector W at a point p, and a positive constant b, such that if V is the Unit tangent factor to the past directed timelike geodesic through p, then on each geodesic the expansion $\theta \equiv {V}_{;a}^{a}$ of these geodesics becomes less than −3c/p, within a distance b/c from p, where $c=-{W}^{a}{V}_{a}$ , i.e. then there is a past incomplete non space-like geodesic through p.

One does not have a curve violating the causality conditions as given as an assertion by Hawkings and Ellis, 1973. i.e. there is, if this occurs at the causal boundary, instead, a bifurcation point at the surface of the causal set, with real and imaginary components, but the incompleteness of the non space geodesic through a point p, if it is on the surface of the causal surface, as defined by Equation (13) is not due to a point p-. It is well known that certain Kerr black hole models, as in page 465 of Ohanian and Ruffini [14] involve the use of ${g}_{tt}~0$ for their horizon surfaces and the definition of a plate disc singularity surface but we are instead employing, $\delta {g}_{tt}~{a}_{\mathrm{min}}^{2}\cdot {\varphi}_{\text{initial}}.$

i.e. precisely because we have avoided using ${g}_{tt}~0$ as was done in the Kerr black holes, as given in [14] but instead have the $\delta {g}_{tt}~{a}_{\mathrm{min}}^{2}\cdot {\varphi}_{\text{initial}}$ plus the situation we wish to avoid, that of instead looking at

${a}_{\mathrm{min}}^{4}<0\Rightarrow {a}_{\mathrm{min}}~\frac{\pm \sqrt{\#}\cdot \left(1+i\right)}{\sqrt{2}}$ , that a causal surface, would be formed on a

sphere of space time which would in itself violate the 3^{rd} Penrose theorem.

4.2. So What Happens If ${\Lambda}_{initial}\gg {\Lambda}_{Today}$ ?

The second case to consider would be if we have, instead of today’s version of the cosmological constant, a large valued initial cosmological constant, in which then

$\begin{array}{l}\frac{\gamma}{4\text{\pi}G}{\left(\sqrt{\frac{8\text{\pi}G{V}_{0}}{\gamma \cdot \left(3\gamma -1\right)}}\cdot \Delta t-1\right)}^{2}~\frac{{\hslash}^{2}}{{a}_{\mathrm{min}}^{4}\cdot {\left(\Delta t\right)}^{2}\cdot \left(\frac{{\Lambda}_{\text{initial}}{c}^{2}}{3}\right)}\\ \&\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\Lambda}_{\text{initial}}\gg {\Lambda}_{\text{Today}}\end{array}$ (17)

We argue that then, there is no reason for assigning a singularity, but it would in line with [4] , i.e. assigning an almost infinite value for the initial cosmological constant.

Different variants of the above can be imagined, and of course one should be considering [16] in the reformulation of the Causal structure boundary idea. In addition the points brought up as to [17] to [21] of the nonlinear electrodynamics cosmology should be utilized as a refinement as to the Hubble parameter as outlined in Equation (5) above.

4.3. Otonion Geometry and Non-Commutativity as a Future Project to Be Combined with Our Present Inquiry?

We should close with one reference as to the Octonionic geometry program as follows. We may be seeing instead of just our roof finder iterations, as outlined above, an exploration into non commutative geometry. This is what I am referring to, and it is from [22] .

From [22]

Quote:

i.e.

The change in geometry is occurring when we have first a pre quantum space time state, in which, in commutation relations [23] (Crowell, 2005) in the pre Octonion space time regime no approach to QM commutations is possible as seen by.

$\left[{x}_{j},{p}_{j}\right]\ne -\beta \cdot \left({l}_{\text{Planck}}/l\right)\cdot \hslash {T}_{ijk}{x}_{k}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{does}\text{\hspace{0.17em}}\text{not}\to i\hslash {\delta}_{i,j}$ (18)

Equation (18) is such that even if one is in flat Euclidian space, and i = j, then

$\left[{x}_{j},{p}_{j}\right]\ne i\cdot \hslash $ (19)

In the situation when we approach quantum “octonion gravity applicable” geometry, Equation (18) becomes

$\left[{x}_{j},{p}_{j}\right]=-\beta \cdot \left({l}_{\text{Planck}}/l\right)\cdot \hslash {T}_{ijk}{x}_{k}\underset{\text{Approaching-flat-space}}{\to}i\hslash {\delta}_{i,j}$ (20)

End of quote

We assert that the issues as of Equation (18) to Equation (20) if done in higher dimensional analogues, taking into account non commutative initial geometry as outlined in [23] in time, if twinned directly with an analysis of Equation (15) to Equation (17) may in time help us delineate the future of space time research in the early universe.

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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