_{1}

^{*}

Using a root finder procedure to obtain
we use an inflaton value due to use of a scale factor
if we furthermore use
. From use of the inflaton, we initiate a procedure for a minimum scale factor, which would entail the
, for a sufficiently well placed frequency ω. If the Non Linear Electrodynamics procedure of Camara et al. of General relativity was used, plus the modified Heisenberg Uncertainty principle, of Beckwith, and others,
* i.e .* we come due to a sufficiently high frequency a case for which
implies a violation of the Penrose singularity theorem,
*i.e* . this is in lieu of
. If this is not true,
* i.e.* that the initial
, then we will likely avoid
for reasons brought up in this manuscript.

Here, the idea would be, to make the following equivalence, namely look at, [

[ [ Λ Max r 4 8 π G ] ⋅ ( 4 / 3 ) ⋅ [ 2 π 2 g ∗ 45 ] 1 / 3 ] 3 / 4 ~ S initial (1)

We furthermore, make the assumption of a minimum radius of [

R initial ~ 1 # l N g < l Planck (2)

We will initially be assuming that the cosmological constant remains at a constant value, as it is today, and does not change over time (which is the situation given in [

This Equation (1) will be put as the minimum value of r, where we have in this situation [

# bits ~ [ E ℏ ⋅ l c ] 3 / 4 ≈ [ M c 2 ℏ ⋅ l c ] 3 / 4 (3)

And if M is the total space-time energy mass, for initial condition [

S initial ~ n graviton ~ initial graviton count (4)

M then will be defined by the mass of a massive graviton [

Δ E ~ [ ℏ / Δ t ⋅ ( δ g t t ~ a min 2 ⋅ ϕ initial ) ] | Pre-Planckian ~ H ( Hubble ) ~ 4 π G B 2 3 c 2 μ 0 ⋅ ( 1 − 8 μ 0 ω B 2 ) + Λ c 2 3 ~ 4 π G B 2 3 c 2 μ 0 ⋅ ( 1 − 8 μ 0 ω B 2 ) + Λ Today c 2 3 (5)

This will lead to

a min ~ ± # ⋅ ( 1 + i ) 2 (6)

whereas if Λ ≫ Λ Today , Equation (6) likely will not hold, and we also state that Equation (6) is a violation of the Penrose singularity theorem as written up in [

a ~ a initial t γ ϕ = γ 4 π G ln [ 8 π V 0 G γ ( 3 γ − 1 ) ⋅ t ] (7)

While adhering to a potential in line with

V = V 0 exp [ { − 16 π G γ } ⋅ ϕ ( t ) ] (8)

We next then go to the results given in [

We initiate our work, citing [

Δ t ⋅ | ( 8 π G V 0 γ ⋅ ( 3 γ − 1 ) ⋅ Δ t − 1 ) − ( 8 π G V 0 γ ⋅ ( 3 γ − 1 ) ⋅ Δ t − 1 ) 2 2 + ( 8 π G V 0 γ ⋅ ( 3 γ − 1 ) ⋅ Δ t − 1 ) 3 3 − ⋯ | ≈ ( γ π G ) − 1 48 π ℏ a min 2 ⋅ Λ (9)

From here, we then cited, in [

S Λ | Arrow-of-time = π ⋅ ( R c | initial ~ c ⋅ Δ t l Planck ) 2 ≠ 0 (10)

This leads to the following, namely in [

( R c | initial ~ c ⋅ Δ t l Planck ) ~ ϑ ( 1 ) (11)

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

That we will drop most of the terms in the expansion of Equation (9) and instead of a huge infinite expansion of terms, pick instead using

Δ t ⋅ ( 8 π G V 0 γ ⋅ ( 3 γ − 1 ) ⋅ Δ t − 1 ) ≈ ( γ π G ) − 1 48 π ℏ a min 2 ⋅ Λ (12)

This is assuming here that the terms in ( 8 π G V 0 γ ⋅ ( 3 γ − 1 ) ⋅ Δ t − 1 ) are extremely

small, which permits us to come up with a quadratic expression of the term Δ t which is of course useful as to what we do next, i.e.

If we make use of the Peebles relationship [

We then use the Peebles result [

Δ E ~ [ ℏ / Δ t ⋅ ( δ g t t ~ a min 2 ⋅ ϕ initial ) ] | Pre-Planckian ~ H ( Hubble ) ~ 4 π G B 2 3 c 2 μ 0 ⋅ ( 1 − 8 μ 0 ω B 2 ) + Λ c 2 3 ~ 4 π G B 2 3 c 2 μ 0 ⋅ ( 1 − 8 μ 0 ω B 2 ) + Λ Today c 2 3 (14)

The key result is that we have a quadratic expression for the Δ t term, as indicated by (12) with the result that there is a solvable expression in terms of Δ t , so that then, we can take the square of the terms of Equation (14) with using the expression of Equation (7) above, in order to obtain after using an expansion of Ln x, (if 0 < x < 2) from [

γ 4 π G ( 8 π G V 0 γ ⋅ ( 3 γ − 1 ) ⋅ Δ t − 1 ) 2 ~ ℏ 2 a min 4 ⋅ ( Δ t ) 2 ⋅ ( 4 π G B 2 3 c 2 μ 0 ⋅ ( 1 − 8 μ 0 ω B 2 ) + Λ Today c 2 3 ) (15)

This is also reflecting the ideas given in reference [

If that happens, due to a very high frequency value for gravitational waves, and a small cosmological constant, we then have

a min 4 < 0 ⇒ a min ~ ± # ⋅ ( 1 + i ) 2 (16)

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 [

Theorem 3: If R a b K a K b ≥ 0 for every non space-like Vector K

1) The strong casuality condition holds on ( M ˜ , g ) ,

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 θ ≡ 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 [

i.e. precisely because we have avoided using g t t ~ 0 as was done in the Kerr black holes, as given in [

a min 4 < 0 ⇒ a min ~ ± # ⋅ ( 1 + i ) 2 , that a causal surface, would be formed on a

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

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

γ 4 π G ( 8 π G V 0 γ ⋅ ( 3 γ − 1 ) ⋅ Δ t − 1 ) 2 ~ ℏ 2 a min 4 ⋅ ( Δ t ) 2 ⋅ ( Λ initial c 2 3 ) & Λ initial ≫ Λ Today (17)

We argue that then, there is no reason for assigning a singularity, but it would in line with [

Different variants of the above can be imagined, and of course one should be considering [

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 [

From [

Quote:

i.e.

The change in geometry is occurring when we have first a pre quantum space time state, in which, in commutation relations [

[ x j , p j ] ≠ − β ⋅ ( l Planck / l ) ⋅ ℏ T i j k x k and does not → i ℏ δ i , j (18)

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

[ x j , p j ] ≠ i ⋅ ℏ (19)

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

[ x j , p j ] = − β ⋅ ( l Planck / l ) ⋅ ℏ T i j k x k → Approaching-flat-space i ℏ δ 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 [

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

Beckwith, A.W. (2018) How a Minimum Time Step and Formation of Initial Causal Structure in Space-Time May Void the Penrose Singularity Theorem, as in Hawkings and Ellis’s 1973 Write-Ups. Journal of High Energy Physics, Gravitation and Cosmology, 4, 485-491. https://doi.org/10.4236/jhepgc.2018.43026