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This paper uses the “Fjortoft theorem” for defining necessary conditions for instability. The point is that it does not apply in the vicinity of the big bang. We apply this theorem to what is called by T. Padmanabhan a thermodynamic potential which would become unstable if conditions for the applications of “Fjortoft’s theorem” hold. In our case, there is no instability, so a different mechanism has to be appealed to. In the case of vacuum nucleation, we argue that conditions exist for the nucleation of particles as of the electroweak regime, due to injecting material from a node point, in spacetime. This regime of early universe creation coexists with the failure of applications of “Fjortoft” theorem in such a way as to give necessary and sufficient conditions for matter creation, in a way similar to the Higgs Boson.

We first start off with a review of the classical Fjortoft theorem [

From [

Fjortoft theorem:

A necessary condition for instability is that if z ∗ is a point in spacetime for

which d 2 U d z 2 = 0 for any given potential U , then there must be some value z 0

in the range z 1 < z 0 < z 2 such that

d 2 U d z 2 | z 0 ⋅ [ U ( z 0 ) − U ( z ∗ ) ] < 0 (1)

For the proof, see [

To do this, we will look at Padamanabhan [

If P c d a b is a so called Lovelock entropy tensor, and T a b a stress energy tensor

U ( η a ) = − 4 ⋅ P a b c d ∇ c η a ∇ d η b + T a b η a η b + λ ( x ) g a b η a η b = U gravity ( η a ) + U matter ( η a ) + λ ( x ) g a b η a η b ⇔ U matter ( η a ) = T a b η a η b ; U gravity ( η a ) = − 4 ⋅ P a b c d ∇ c η a ∇ d η b (2)

We now will look at

U matter ( η a ) = T a b η a η b ; U gravity ( η a ) = − 4 ⋅ P a b c d ∇ c η a ∇ d η b (3)

So happens that in terms of looking at the partial derivative of the top (2) Equation, we are looking at

∂ 2 U ∂ ( η a ) 2 = T a a + λ ( x ) g a a (4)

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

[ ∂ 2 U ∂ ( η a ) 2 = T a a + λ ( x ) g a a ] η 0 a ∗ [ − 4 ⋅ P a b c d ( ∇ c η 0 a ∇ d η 0 b − ∇ c η ∗ a ∇ d η ∗ b ) + T a b ⋅ [ η 0 a η 0 b − η ∗ a η ∗ b ] + λ ( x ) g a b ⋅ [ η 0 a η 0 b − η ∗ a η ∗ b ] ] < 0 (5)

What this is saying is that there is no unique point, using this η ∗ a for which (5) holds. Therefore, we say there is no official point of instability of η ∗ a due to (4). The Lagrangian structure of what can be built up by the potentials given in (4) with respect to η ∗ a 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 [

[ ∂ 2 U ∂ ( η a ) 2 = T a a + λ ( x ) g a a ] ≠ 0 , for ∀ η ∗ a choices (6)

To use Equation (6) properly, we use the material, in our reasoning from [

We state that (6) tells us is that there is an embedding structure for early universe geometry, some of which may take the form of the following diagram.

Padmanabhan [

E = 1 2 k B ∫ d n T l o c (7)

And the n value as in (7) is given by

d n = 32 π ⋅ P c d a b ⋅ ε a b ⋅ ε c d ⋅ d A (8)

where P c d a b is a so called Lovelock entropy tensor, and ε a b a bi normal on the codimension −2 cross section, and then entropy is stated to be

S ∝ ∫ ∂ ν d n ∝ ∫ ∂ ν 32 π ⋅ P c d a b ⋅ ε a b ⋅ ε c d σ d D − 2 x (9)

The end result, is that energy is induced via the temperature T l o c , while [

T l o c = N a μ n μ 2 π = localaccelerationtemperature (10)

Also, the change in n can be given by, if l P is the Planck’s length value [

Δ n = σ d 2 x / l P (11)

Looking at (9) and (11) we state that the change in number count given in (11) is really a holographic surface phenomena, with N defined [

N = E / [ ( 1 / 2 ) k B T ] (12)

The upshot is that we can, as implied by Ng [

S ~ n (13)

While having a change in n as due to a change in the spatial surface of spacetime as given in (11), we have to realistically infer that the local acceleration temperature (10) is from another pre universe construction and that local instability is ruled out by (5) and (6). This leads us to ask as to what would be an acceptable way to form the formation of mass, i.e. say the mass of a graviton, via external factors introduced into our universe prior to the Electroweak era, in cosmology. To do that, look at if there are two branes on the A d S 5 space-time so that with one moving and one stationary, we can look at

Using [

[ ∂ t 2 + k 2 − ∂ y 2 + 3 y ⋅ ∂ y ] h • = 0 (14)

Using [

h • = H i j = e i j ⋅ exp [ i ⋅ ω ⋅ t ] ⋅ ( m ⋅ y ) 2 ⋅ A ⋅ J 2 ( m ⋅ y ) (15)

e i j is a polarization tensor, and the function J 2 ( m y ) is a 2^{nd} order Bessel function [

h = { exp [ i ⋅ ω ⋅ t ] ⋅ ( m ⋅ y ) 2 ⋅ A ⋅ J 2 ( m ⋅ y ) } ⋅ ( 1 + π 4 ⋅ ( m ⋅ l ) 2 ) (16)

With the factor of ( 1 + π 4 ⋅ ( m ⋅ l ) 2 ) coming in due to a boundary condition

upon the wall of a brane put in, i.e. looking at [

− 2 ⋅ ∂ y H i j = κ 5 ⋅ π i j ( T ) → 0 (17)

This will be in our example set as not equal to zero, in the right hand side, but equal to an extremely small parameter, namely

∂ y H i j | y = y b = κ 5 ⋅ π i j ( T ) ~ ξ + (18)

With this turned into

∂ y h | y = y b ~ δ + (19)

The right hand side of (19) represents very small brane tension, which is understandable. Then using [

∂ y h | y = y b = ∂ y { exp [ i ω t ] ⋅ ( m y ) 2 ⋅ A ⋅ J 2 ( m y ) } ⋅ ( 1 + π 4 ⋅ ( m ⋅ l ) 2 ) | y = y b ~ δ + (20)

and

J 2 ( m y ) = ( m y ) 2 2 2 × 2 ! ⋅ ( 1 − ( m y ) 2 2 2 × 3 + ( m y ) 4 2 4 × 2 ! × 3 × 4 − ( m y ) 6 2 6 × 4 ! × 3 × 4 × 5 + ⋯ ) (21)

The upshot is, that afterwards,

( m y ) 4 2 2 × 2 ! ⋅ 1 y [ ( 1 − ( m y ) 2 2 2 × 3 + ( m y ) 4 2 4 × 2 ! × 3 × 4 − ( m y ) 6 2 6 × 4 ! × 3 × 4 × 5 + ⋯ ) − ( 2 ⋅ ( m y ) 2 2 2 × 3 + 4 ⋅ ( m y ) 4 2 4 × 2 ! × 3 × 4 − 6 ⋅ ( m y ) 6 2 6 × 4 ! × 3 × 4 × 5 + ⋯ ) ] = δ + ⋅ exp [ ∓ i ω t ] A ⋅ [ 1 − π 4 ⋅ ( m ⋅ l ) 2 ] (22)

Should the term

δ + ⋅ exp [ ∓ i ω t ] A ⋅ [ 1 − π 4 ⋅ ( m ⋅ l ) 2 ] → δ + → 0 0 (23)

Then, (22) is acting much as in [

( m y ) = f ( t ) ⇔ m ≡ f ( t ) y (24)

In doing all of this, keep in mind the mathematical information given in [

Needless to say, (22) can only be solved for, numerically, i.e. fourth order polynomial solutions for quartic equations still give over simplified dynamics, especially if (24) holds, and makes things more complicated. This is all being done to keep fidelity with respect to [

It is now then time to do a recap and to organize how such speculation can be vetted using experimental procedures. To do this were cap what can be said about traces massive gravitons can be detected, prior to our conclusion which may lead to an explanation of the following entropy formula [

S g w = V ⋅ ∫ ν 0 v 1 r ( ν ) ⋅ v 2 d v ≅ ( 10 29 ) 3 ⋅ ( H 1 / M P ) 3 / 2 ≈ 10 87 - 10 88 (25)

This is a bridge to future projects which should be kept in mind, i.e., could our formulation of graviton physics lead to identification of gravitons, in the early universe as the main driver of graviton physics being the primary entropy generator as suggested by [

First of all, we review the details of a massive graviton imprint upon h i j , and then we will review the linkage between that and certain limits upon h • as read from Hinterbichler [

h 00 ( x ) = 2 M 3 M Planck ⋅ exp ( − m ⋅ r ) 4 π ⋅ r (26)

h 0 i ( x ) = 0 (27)

h i j ( x ) = [ M 3 M Planck ⋅ exp ( − m ⋅ r ) 4 π ⋅ r ] ⋅ ( 1 + m ⋅ r + m 2 ⋅ r 2 m 2 ⋅ r 2 ⋅ δ i j − [ 3 + 3 m ⋅ r + m 2 ⋅ r 2 m 2 ⋅ r 4 ] ⋅ x i ⋅ x j ) (28)

Here, we have that these h i j values are solutions to the following equation, as given by [

( ∂ 2 − m 2 ) h μ ν = − κ ⋅ [ T u v − 1 D − 1 ⋅ ( η u v − ∂ μ ∂ v m 2 ) ⋅ T ] (29)

To understand the import of the above equations, set

M = 10 50 × 10 − 27 g ≡ 10 23 g ∝ 10 61 - 10 62 eV M Plank = 1.22 × 10 28 eV (30)

We should use the m massive-graviton ~ 10 − 26 eV value in (29) above.

In reviewing what was said about (27), (28) we should keep in mind the overall Fourier decomposition linkage between h • , h i j which is written up as

h i j ( t , x ; k ) = 1 ( 2 π ) 3 / 2 ∫ d 3 k ∑ • = + , ⊗ e i k ⋅ x e i j • h • ( t , y ; k ) (31)

The bottom line is that a simple decomposition involving a basis in two polarization states, of + , ⊗ will have to be amended and adjusted, if one is looking at massive graviton states.

In addition further developments as to (31) could influence giving a semi classical interpretation as to entropic origins of gravity, along the lines brought up by both ’t Hooft, indirectly [

The experimental gravity considerations are covered in [

Reference [

As far as [

Finally, in lieu of [

The author urges readers to look at [

This work is supported in part by National Nature Science Foundation of China grant No110752.

Beckwith, A. (2018) Showing Fjortof’s Theorem Does Not Apply for Defining Instability for Early Universe Thermodynamic Potentials. Asking If Nucleated Particles Result at/before Electro-Weak Era Due to Injection of Matter-Energy at the Big Bang? Journal of High Energy Physics, Gravitation and Cosmology, 4, 48-59. https://doi.org/10.4236/jhepgc.2018.41006

From [

From [

XIX. A generalized problem to making quantization of the Einstein field equations elucidated by first principles. (This reflects our evaluation of [

H E = H T + v a ′ ⋅ ϕ a ′ g ˙ ≈ { g , H E } (A1)

Equation (A1), in a Poisson bracket formulation, was used by Dirac to transform to a set of quantization conditions, in pages 25 to 43 of. The problem is, that it is difficult to come up with constraint equations, as given in the top level of Equation (A1).

The following is easy to do, if you ignore constraints

d 〈 P 〉 d t = − i ℏ 〈 Ψ | [ P , H ] | Ψ 〉 = − 1 i ℏ 〈 Ψ | [ P , H ] | Ψ 〉 → 3 dim → 1 dim − ∫ Ψ ∗ ⋅ d V ( x ) d x ⋅ Ψ d x → 3 dim → Any dim 〈 − d V ( x ) d x 〉 ~ 〈 − ∇ → V 〉 ≡ F ( force ) (A2)

Try doing this, to have equivalence with Equation (A1) and match that with Equation (A2) the equations below given as (A3) to Equation (A4), i.e. what is so difficult is to put in a Hamiltonian system, for gravity, which is complicated. Worse than that, we do not have a quantum mechanical equivalent, and this due to the difficulties in terms of finding a quantum mechanical equivalent to the Poisson brackets { p N , H ( Hamiltonian ) } ≈ 0 which is readily transferrable to the Friedman equation, i.e. so far a quantum bridge between quantized versions of Equation (A3) and Equation (A4) does not exist, right now. As we wrote in [

So as given in [

p N & { p N , H ( Hamiltonian ) } ≈ 0 (A3)

This is, according to Kieffer, the Poisson brackets, equivalent to the following.

What we are looking at is, if we set the Lapse function, N, as = 1.

a ˙ 2 = − 1 + a 2 ⋅ ( ϕ ˙ 2 + Λ 3 + m 2 ⋅ ϕ 2 ) ⇔ ϕ ¨ 2 + 3 a ˙ a ⋅ ϕ ˙ + m 2 ϕ = 0 (A4)

Here, the ϕ is a scalar field (here, called a “homogeneous field”), m is a mass term, and a the scale factor, and Λ the cosmological constant. If m is set equal to zero, this has a simple m = 0 solution with

p ϕ = a 3 ⋅ ϕ ˙ = κ = const & ϕ = ± 1 2 ⋅ arcosh κ a 2 (A5)

It cannot be solved analytically, if m is not equal to zero. Now as to a general problem between the Solvay 1927 conference methods and the application to GR will be alluded to, next.

Dirac claims the bridge from Poisson brackets to the situation represented by Equation (A5) always involves a carefully set extended Hamiltonian situation, i.e. see his discussion in 33 to page 35 of [

Having said, this, we will next go to the problem of Quantum Geometrodynamics. Before going to it, a notice as to the problems of bridging to general relativity using conventional Quantum mechanics, will be raised as a bridge to the use of H A D M Ψ = 0 which makes a plausible bridge to the Fluid equation of general relativity, [

( a ¨ a ) = − 4 π G 3 c 2 ⋅ ( ε + 3 P ) (A6)

This requires two equations, namely,

( a ¨ a ) = − 4 π G 3 c 2 ⋅ ( ε + 3 P ) due to ε ˙ + 3 ( a ˙ a ) ⋅ ( ε + 3 P ) = 0 and a ˙ 2 = 8 π G 3 c 2 ⋅ ε a 2 − κ c 2 R 0 2 ( GR-Friedman ) as opposed to , if U = const . a ˙ 2 = 8 π G 3 c 2 ⋅ ρ a 2 + 2 U r s 2 ( Newtonian-Friedman ) (A7)

The derivation of the acceleration equation for GR, using the two equations cited is in [

In addition we will derive the Fluid equation also used, which is the same form used in Equation (A5) making a linkage to relativity and quantum mechanics, possible, if one uses the following steps, as given on page 59 of [

We then will get a clean derivation of the so called fluid equation, used in Cosmology. This fluid equation, which has the same form used in both GR and Newtonian physics may be in principle linkable to the quantization program outlined in Equation (A5). So with that, we go to the interactions given in Equation (A8) below.

V ( t ) = Volume ( universe ) = 4 π r s 3 a 3 3 & V ˙ = V ⋅ ( 3 a ˙ a ) E = V ( t ) ⋅ ε ( t ) E ˙ = V ⋅ ( ε ˙ + 3 a ˙ a ε )

& First law thermo ( universe ) E ˙ + P V ˙ = 0 ⇒ V ⋅ ( ε ˙ + 3 a ˙ a ε + a ˙ a P ) = 0 ⇒ ε ˙ + 3 a ˙ a ε + a ˙ a P = 0 (A8)

The GR and classical physics forms of the fluid equation, so derived, in Equation (A8) and the results at the bottom of Equation (A7) would allow us to make connection, with a lot of work to the sort of reasoning used in Equation (A5) above, but due to the difference in the Friedman equation, in classical and GR form, as noted in Equation (A6), it would be using the Solvay methods, extremely difficult to make connection between an acceleration equation, using scale factors, as given in Equation (A6) and Equation (A7) with the Equations (A5) (58) connection between classical and quantum mechanics with respect to an acceleration of the universe acceptable in both GR and quantum form.

We can state though that a bridge to the Fluid equation, as given in [

End of our quote from [

As a close to this, all this, in terms of quantum gravity should also keep in mind issues brought up in [