_{1}

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We utilize how Weber in 1961 initiated the process of quantization of early universe fields to the problem of what may be emitted at the mouth of a wormhole. While the wormhole models are well developed, there is as of yet no consensus as to how, say GW or other signals from a wormhole mouth could be quantized or made to be in adherence to a procedure Weber cribbed from Feynman, in 1961. In addition, we utilize an approximation for the Hubble parameter parameterized from Temperature using Sarkar’s H ~ Temperature relations, as given in the text. Finally, after doing this, we go to the Energy as E also ~ Temperature, and from there use E (energy) as ~ signal frequency. This gives us an idea of how to estimate frequency generated at the mouth of a wormhole.

We bring up this study first a result given by Weber, in 1961 [

The behavior of frequency, versus certain conditions at the mouth of a wormhole may give us clues to be investigated later as to polarization states relevant to the wormhole [

In doing all of this, the idea is that we are evolving from the Einstein-Rosen bridge to a more complete picture of GR which may entail a new representation of the Visser “Chronology protection” paper as in [

Using [

Ψ Later = ∫ ∑ H e ( i I H / ℏ ) ( t , t 0 ) Ψ Earlier ( t 0 ) d t 0 (1)

The approximation we are making is to pick one index, so as to have:

Ψ Later = ∫ ∑ H e ( i I H / ℏ ) ( t , t 0 ) Ψ Earlier ( t 0 ) d t 0 → H → 1 ∫ e ( i I H FIXED / ℏ ) ( t , t 0 ) Ψ Earlier ( t 0 ) d t 0 (2)

This corresponds to say being primarily concerned as to GW generation, which is what we will be examining in our ideas, via using:

e ( i I H FIXED / ℏ ) ( t , t 0 ) = exp [ i ℏ ⋅ c 4 16 π G ⋅ ∫ Μ d t ⋅ d 3 r − g ⋅ ( ℜ − 2 Λ ) ] (3)

We will use the following, namely, if Λ is a constant, do the following for the Ricci scalar [

ℜ = 2 r 2 (4)

If so then we can write the following, namely: Equation (3) becomes, if we have an invariant Cosmological constant, so we write Λ → alltime Λ 0 everywhere, then,

e ( i I H FIXED / ℏ ) ( t , t 0 ) = exp [ i ℏ ⋅ c 4 ⋅ π ⋅ t 0 16 G ⋅ ( r − r 3 Λ 0 ) ] (5)

Then, we have that Equation (1) is rewritten to be:

Ψ Later = ∫ ∑ H e ( i I H / ℏ ) ( t , t 0 ) Ψ Earlier ( t 0 ) d t 0 → at wormhole ∫ exp [ i ℏ ⋅ c 4 ⋅ π ⋅ t 0 16 G ⋅ ( r − r 3 Λ 0 ) ] Ψ Earlier ( t 0 ) d t 0 (6)

[

Ψ Earlier ( t 0 ) ≈ Ψ H H ∝ exp ( − π 2 G H 2 ⋅ ( 1 − sinh ( H t ) ) 3 / 2 ) (7)

Here, making use of Sarkar [

H = 1.66 g ∗ T temp 2 / M Planck (8)

We assume initially a relatively uniformly given temperature, that H is constant.

So then we will be attempting to write out an expansion as to what Equation (6) gives us while we use Equation (7) and Equation (8), with H approximately constant.

We will be considering how to express Equation (6) and in doing this we will be looking at having a constant value for Equation (8). If so then,

Ψ Later = ∫ exp [ i ℏ ⋅ c 4 ⋅ π ⋅ t 0 16 G ⋅ ( r − r 3 Λ 0 ) ] exp ( − π 2 G H 2 ⋅ ( 1 − sinh ( H t ) ) 3 / 2 ) d t 0 (9)

Then using numerical integration, [

Ψ Later → t M → ε + ∫ 0 t M e i ⋅ ( α ˜ 1 ) ⋅ t − ( α ˜ 2 ) ⋅ ( 1 − sinh ( H t ) ) 3 / 2 d t ≈ t M 2 ⋅ ( e i ⋅ ( α ˜ 1 ) ⋅ t M − ( α ˜ 2 ) ⋅ ( 1 − sinh ( H ⋅ t M ) ) 3 / 2 − 1 ) α ˜ 1 = [ c 4 ⋅ π 16 G ℏ ⋅ ( r − r 3 Λ 0 ) ] , α ˜ 2 = π 2 G H 2 (10)

Notice the terms for the H factor, and from here we will be making our prediction.

If the energy, E, has the following breakdown,

H = 1.66 g ∗ T temp 2 / M Planck ⇒ E ≈ k B T Temp ≈ ℏ ⋅ ω signal ⇒ ω signal ≈ k B ⋅ M Planck H ℏ ⋅ 1.66 g ∗ (11)

The upshot is that we have, in this, a way to obtain a signal frequency by looking at the real part of Equation (11) above, if we have a small t, initially (small time step).

Equation (11) would imply an initial frequency dependence. What we are doing next is to strategize as to understand the contribution of the cosmological constant in this sort of problem. I.e., the way to do it would be to analyze a Kieffer “dust solution” as a signal from the Wormhole. I.e., look at [

Δ ω signal Δ t ≈ 1 (12)

If so then we can assume, that the time would be small enough so that:

Δ t ≈ ℏ 1.66 g ∗ k B ⋅ M Planck H (13)

If Equation (13) is of a value somewhat close to t, in terms of general initial time, we can write

ψ n ˜ , λ ( t , r ) ≡ 1 2 π ⋅ n ˜ ! ⋅ ( 2 λ ) n ˜ + 1 / 2 ( 2 n ˜ ) ! ⋅ [ 1 ( λ + i ⋅ t + i ⋅ r ) n ˜ + 1 − 1 ( λ + i ⋅ t − i ⋅ r ) n ˜ + 1 ] (14)

Here the time t would be proportional to Planck time, and r would be proportional to Planck length, whereas we set

λ ≈ 8 π G V volume ℏ 2 t 2 → G = ℏ = l Planck = k B = 1 8 π t 2 ≡ 8 π t (15)

Then a preliminary emergent space-time wavefunction would take the form of:

ψ n ˜ , λ ( Δ t , r ) ≡ 1 2 π ⋅ n ˜ ! ⋅ ( 2 ⋅ 8 π ⋅ ( Δ t ) − 1 ) n ˜ + 1 / 2 ( 2 n ˜ ) ! ⋅ [ 1 ( 8 π ⋅ ( Δ t ) − 1 + i ⋅ Δ t + i ⋅ r ) n ˜ + 1 − 1 ( 8 π ⋅ ( Δ t ) − 1 + i ⋅ Δ t − i ⋅ r ) n ˜ + 1 ] (16)

Just at the surface of the bubble of space-time, with t Planck ∝ Δ t , and r ∝ l Planck .

This is from a section, page 239 of the 3^{rd} edition of Kieffer’s book, as to a quantum theory of collapsing dust shells. And so then we have the following procedure as to isolate out the contribution of the Cosmological constant. Namely, take the real part of Equation (16) and compare it with the Real part of Equation (10).

Another way to visualize this situation, and this is a different way to interpret Equation (15). To do so we examine looking at page 239 of Kieffer, namely [

〈 E 〉 κ = n , λ = ( κ = n ) + 1 / 2 λ → λ ≈ 1 / ℏ ω ℏ ω ⋅ ( ( κ = n ) + 1 / 2 ) (17)

What we can do, is to ascertain the last step would be to make Equation (16) in a sense partly related to the simple harmonic oscillator. But we should take into consideration the normalization using that if ℏ = l P = G = t P = k B = 1 is done via Plank unit normalization [

Ψ 1 , κ = n = 0 ≈ ω π ⋅ [ 1 ω + i ⋅ ( t + r ) − 1 ω + i ⋅ ( t − r ) ] (18)

Or,

Ψ 2 , κ = n = 0 ≈ 1 π 8 π t ⋅ [ 1 8 π t + i ⋅ ( t + r ) − 1 8 π t + i ⋅ ( t − r ) ] (19)

With, say

ω ≈ 8 π t (20)

And this in a setting where we have the dimensional reset of Planck Units [

ℏ = l P = G = t P = k B = 1 (21)

To do this we, first of all, consider the real part of Equation (18) when n is set equal to zero, and the state should be the same as the real part of Equation (10). Then do the same with the real part of Equation (19) and compare it with Equation (10) in order to isolate out a bound to the cosmological constant.

Doing so, leads to the following situation, and keep in mind that we are using Equation (21) for regularization as well as setting

r ≡ B ⌢ ⋅ r P → r P → 1 B ⌢ (22)

If so, then we have the following bounding as far as the value of the cosmological “constant”, namely,

Ψ Later → t M → ε + ∫ 0 t M e i ⋅ ( α ˜ 1 ) ⋅ t − ( α ˜ 2 ) ⋅ ( 1 − sinh ( H t ) ) 3 / 2 d t ≈ t M 2 ⋅ ( e i ⋅ ( α ˜ 1 ) ⋅ t M − ( α ˜ 2 ) ⋅ ( 1 − sinh ( H ⋅ t M ) ) 3 / 2 − 1 ) Ψ 1 , κ = n = 0 ≈ ω π ⋅ [ 1 ω + i ⋅ ( t + r ) − 1 ω + i ⋅ ( t − r ) ] α ˜ 1 = [ c 4 ⋅ π 16 G ℏ ⋅ ( r − r 3 Λ 0 ) ] , α ˜ 2 = π 2 G H 2 (23)

We will be looking at comparing the real values of Equation (23) in order to obtain a bound on the cosmological constant, and in doing so we have employing the following Equation (11), Equation (20), Equation (21), Equation (22) and Equation (23) in order to get a bound on the Cosmological constant as given by:

Λ 0 ≈ B ˜ − 2 − 16 π ⋅ B ˜ − 2 ⋅ ( α ˜ 2 ⋅ ( 1 − sinh ( H ⋅ B ˜ ) ) 3 / 2 ) − 16 π ⋅ B ˜ − 2 ⋅ cos − 1 [ 2 ⋅ 8 3 / 4 ⋅ π 1 / 4 8 π + ( 1 + B ˜ ) 2 − 2 ⋅ 8 3 / 4 ⋅ π 1 / 4 8 π + ( 1 − B ˜ ) 2 ] (24)

In doing this, taking into account the Planck units and their normalization, we also need to keep in consideration the frequency, which we will denote here as:

ω signal ≈ k B ⋅ M Planck H ℏ 1.66 g ∗ → ℏ = l P = G = t P = k B = 1 H 1.66 g ∗ ≈ T temperature 2 (25)

Whereas what we will be doing, after we obtain a frequency of a signal near the mouth of a wormhole is to use the following scaling of frequency, near Earth Orbit from this wormhole. First if the wormhole is right at the start of the Universe [

( 1 + z initial era ) ≡ a today a initial era ≈ ( ω Earth orbit ω initial era ) − 1 ⇒ ( 1 + z initial era ) ω Earth orbit ≈ 10 25 ω Earth orbit ≈ ω initial era (26)

If we are say far closer to the Earth, or the Solar system, then we would likely see [

10 ⋅ ω Earth orbit signal ≈ ω wormhole mouth signal (27)

Our derivation so far is to obtain the initial signal frequency for Equation (26) and Equation (27). Our next task is to obtain some considerations as to the Polarization, of say GW to observe and look for, in conclusion of this document.

We will be referencing [

Γ ≈ exp ( ω signal / T temperature ) (28)

Whereas we have from [

Γ ≈ exp ( − E / T temperature ) (29)

Whereas if we assume that there is a “negative temperature in Equation (28) and say rewrite Equation (29) as obeying having:

( ω signal / T temperature ) ≈ ( − E / T temperature ) (30)

This is specifying a rate of particle production from the wormhole. And so then, whereas what we are discussing in Equation (28) and Equation (29) is having a rate of, from a wormhole mouth, presumably from graviton production. If as an example, we are examining the mouth of a wormhole as being equivalent of a linkage between two black holes, or a black hole—white hole pair, we are presuming a release from the mouth of the wormhole commensurate with looking at [

Γ ∝ exp ( − 8 π M ⋅ ω ⋅ [ 1 + β 4 ⋅ ( m 2 + 4 ω 2 ) ] ) (31)

Whereas we define the parameter β via a modified energy expression, as in [

E ˜ = E ⋅ ( 1 − β ⋅ ( p 2 + m 2 ) ) (32)

Our Equations (28) and (29), which are for wormholes, should encompass the same information of Equation (31) which would be consistent with a white hole [

Our next step is to ask if this permits speaking of say GW polarization in the mouth of a wormhole.

To do this, first of all, note that in [

In the case of extending b to become the “shape” of the mouth of a wormhole, we would likely be using [

b ( r ) = [ r 0 γ − 1 γ + γ ⋅ ( 8 π G ) ω ˜ ˜ 1 / γ γ − 1 γ ⋅ ( r 3 − r 0 3 ) ] γ γ − 1 → r → r 0 r 0 (33)

Whereas we need to keep in mind the equation of state for pressure and density of [

p = ω ˜ ˜ ( r ) ⋅ ρ (34)

The long and short of it is as follows. Following [

ρ α ˜ = M ( 4 π α ˜ ) 3 / 2 ⋅ exp ( − r 2 / 4 α ˜ ) (35)

Whereas the b coefficient in the case of Noncommutative geometry is chosen [

b ( r ) = 2 r s π ⋅ γ ⌢ ( 3 2 , r 2 4 α ˜ ) ≡ 2 r s π ⋅ ( r 2 4 α ˜ ) 3 / 2 ⋅ Γ ˜ ( 3 / 2 ) ⋅ e − 3 / 2 ⋅ ∑ k = 0 ∞ ( ( r 2 4 α ˜ ) k Γ ˜ ( ( 3 / 2 ) + k + 1 ) ) (36)

This is called the incomplete lower gamma function, with Γ ˜ being a gamma function [

From here, using that Equation (36) is to be included in the following metric, as given by the coefficient [ α ˜ ] = [ r 2 ] in terms of dimensional analysis is chosen so that the dimensions of [ α ˜ ] = [ r 2 ] are chosen to contain M as mass in a wormhole. i.e. the denominator of Equation (35) ( 4 π α ˜ ) 3 / 2 is chosen so that M is within the volume of space so subscribed. And this is for line element [

d S 2 = − exp ( − 2 Φ ( r ) ) d t 2 + d r 2 1 − b ( r ) / r + r 2 ⋅ ( d θ 2 + ( sin 2 θ ) d φ 2 ) (37)

If we refer to black holes, with extra dimension, n, of Planck sized mass, we have a lifetime of the value of about:

τ ~ 1 M * ( M BH M * ) n + 3 n + 1 → M BH ≈ M Planck 10 − 26 seconds M * ≈ isthelowenergyscale , whichcouldbeaslowasafewTeV , (38)

The idea would be that there would be n additional dimensions, as given in Equation (38) which would then lay the door open to investigating [

In order to do this, we will be estimating that the temperature would be of the order of Planck temperature, i.e. using ideas from [

ω p T p ≡ G k B 2 ℏ → ℏ = G = k B = 1 1 (39)

If so, then there would be to first order the following rate of production,

Γ rateofproduction ≈ e ≈ 2 - 3 (40)

Some of the considerations given in this could be related to [^{−5} grams per black hole, producing say 10^{57} gravitons, produced per black hole of mass about 10^{−62} grams per black hole [

Having said, that what about frequencies? Here, if we have a wormhole throat of about 2 - 3 Planck lengths in diameter, with a frequency of emitted gravitons of about 10^{19} GHz initially, it is realistic, using the following, to expect in many cases a redshift downscaling of frequencies of about 10^{−18}, if the wormholes are close to the initial near singularity, so then that we could be looking at approximately 10 to 12 GHz, on Earth, for frequencies, of initially about 10^{19} GHZ. So then note at inflation we have:

( 1 + z initial era ) ≡ a t o d a y a initial era ≈ ( ω Earth orbit ω initial era ) − 1 ⇒ ( 1 + z initial era ) ω Earth orbit ≈ 10 25 ω Earth orbit ≈ ω initial era (41)

In our situation, the figure would likely be instead of 10^{25} times Earth orbit detected frequency, something closer to 10^{18} to 10^{19} times Earth orbit GW frequencies detected as given by [^{19} GHz GW signals would be about h ~10^{−26} and this could change an order of magnitude given instrument sensitivity. In any case it would be well worth our while to look closely at [

The author declares no conflicts of interest regarding the publication of this paper.

Beckwith, A. (2021) Looking at Quantization of a Wave Function, from Weber (1961), to Signals from Wavefunctions at the Mouth of a Wormhole. Journal of High Energy Physics, Gravitation and Cosmology, 7, 1037-1048. https://doi.org/10.4236/jhepgc.2021.73062