stephaneww

The end of the quantum vacuum catastrophe ?

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Hello

I do not succed to use the latex on this version of the forum, I opted for a text image. Excuse my English too, please;)

 

 

vaccum cata.jpg

Edited by stephaneww
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 coincidentally Strange posted a rather interesting paper on this topic a while back with Unruh as one of thee authors.

here

http://www.scienceforums.net/topic/105873-interesting-idea-to-explain-dark-energy/

" in essence the paper boils down to the following. By this same kind of mechanism, the violent gravitational effect produced by the vacuum energy density is confined to Planck scales, and its effect at macroscopic scales the accelerating expansion of the Universe, due to the weak parametric resonance is so small that, it is only observable after accumulations on the largest scale the cosmological scale".

https://arxiv.org/pdf/1703.00543.pdf

he shows this argument via the parametric resonance behaviors which he describes extensively throughout the paper.

 

On a strictly personal view of the above paper, I think its one of the better solutions to the cosmological constant problem I have read in years (which is a huge quantity of papers lol)

I reread the above so editted this post. It was extremely tricky to read on my phone lol very blurry.

I'm going to think about your relativistic treatment when I can focus on it ie the portion on the pure relativistic treatment

 

 

 

 

 

Edited by Mordred

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Forgot to add this will help with the latex for this site

http://www.scienceforums.net/topic/108127-typesetting-equations-with-latex-updated/

 

Actually I don't see anything wrong the above, It looks like your in essence setting [latex]8\pi G=1[/latex] under wicks rotation which if I recall is the Einstein Hilbert treatment but have to double check that.

By the way its nice to finally see a proper speculation post. Thank you for that +1

Under wicks rotation you apply an effective finite cutoff which once again reminds me of the Einstein Hilbert canonical treatment.

I have to do some digging but I recall reading about issues with using Wicks rotation on Euclidean space if I recall  its the conformal factor problem involved with Wicks and Euclidean space

Edited by Mordred

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9 hours ago, Mordred said:

 coincidentally Strange posted a rather interesting paper on this topic a while back with Unruh as one of thee authors.

here

http://www.scienceforums.net/topic/105873-interesting-idea-to-explain-dark-energy/

" in essence the paper boils down to the following. By this same kind of mechanism, the violent gravitational effect produced by the vacuum energy density is confined to Planck scales, and its effect at macroscopic scales the accelerating expansion of the Universe, due to the weak parametric resonance is so small that, it is only observable after accumulations on the largest scale the cosmological scale".

https://arxiv.org/pdf/1703.00543.pdf

he shows this argument via the parametric resonance behaviors which he describes extensively throughout the paper....

 

Hello Modred

I know this paper and even have notice that the pulse of cosmological constant was interesting for the vacuum catastrophe on a french forum (here) the 17 December 2016  :P

... but I'm not enought qualified to understand the QFT, so the Unruh paper

Thank tou for your +1

 

edit : can you please indicate me the correct latext example for \omega please, i don't succed again...

Edited by stephaneww
latext, date

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I usually prefer to latex in the older forum method( old habits being hard to break) lol

Enclose your latex with [latex]\Omega[/l$tex]

Replace the $ sign in the above. I used it to prevent it from activating. Once you replace the dollar sign above with [/latex]  you will get

[latex]\Omega[/latex]

The new forum software is tricky. Once you type in the [latex] enclosure you must refresh the page to see if you did it correct.

The latex doesn't activate until you renew the page. So it can fool you into thinking it didn't work. So refresh the page after your attempts

Edited by Mordred
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2 hours ago, stephaneww said:

...I know this paper and even have notice that the pulse of cosmological constant was interesting for the vacuum catastrophe on a french forum (here) the 17 December 2016  :P...

read :

...I know this paper and even have notice that the pulsation  [latex]\omega_{\Lambda}[/latex] of cosmological constant was interesting for the vacuum catastrophe on a french forum (here) the 17 December 2016  :P...

i try latex (and succed :cool:), thanks for your explaination

 

Edited by stephaneww

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Your welcome, glad to see you got latex working.

I agree the paper is extremely interesting on the pulsation and informative in the confinement of the different scales. Planck, macro and cosmological.

The beauty of the paper is that it shows that the old cosmological problem isn't a problem if you examine the limits of the effective action due to the HUP.  

I am still studying several of the details in the paper myself. As you have noted, its fairly intensive in detail. Not something one can easily understand by simply reading it. 

The problem of using Wicks rotations on Euclidean space however is still an issue. The issue comes down to renormalization on path divergences when your under significant curvature. The Wicks rotation works well under Euclid space, not so great under curvature.

Most of the papers detailing this is intensive in group algebra, so if your weak on QFT will be pointless to include. I will see if I can dig up a more appropriate level.

I'm a little busy this weekend but would like to assist you on your modelling approach, so once I get a chance I will probably provide some further details to assist you in moving forward into your approach.

In particular we will need to test if your approach gives the equation of state w=-1 for the Lambda term under scalar modelling. Though I see nothing above that shouldn't, one cannot assume that lol.

 

Edited by Mordred

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59 minutes ago, Mordred said:

...

I'm a little busy this weekend but would like to assist you on your modelling approach, so once I get a chance I will probably provide some further details to assist you in moving forward into your approach.

In particular we will need to test if your approach gives the equation of state w=-1 for the Lambda term under scalar modelling. Though I see nothing above that shouldn't, one cannot assume that lol.

...

Your help, expertise and assist is more than welcome. For my part I can not go beyond my first post: I do not have sufficient skills. But I would gladly read all your remarks and your final opinion

Edited by stephaneww

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On 09/09/2017 at 6:45 AM, Mordred said:

...

Under wicks rotation you apply an effective finite cutoff which once again reminds me of the Einstein Hilbert canonical treatment.

...

 

On 09/09/2017 at 6:07 PM, Mordred said:

...

The problem of using Wicks rotations on Euclidean space however is still an issue. The issue comes down to renormalization on path divergences when your under significant curvature. The Wicks rotation works well under Euclid space, not so great under curvature.

...

I'm a little busy this weekend but would like to assist you on your modelling approach, so once I get a chance I will probably provide some further details to assist you in moving forward into your approach.

In particular we will need to test if your approach gives the equation of state w=-1 for the Lambda term under scalar modelling. Though I see nothing above that shouldn't, one cannot assume that lol.

 

Hello Mordred

I searched a little for the "canonical treatment Einstein Hilbert" on Google and found this document :

http://www.tapir.caltech.edu/~chirata/ph236/2011-12/lec33.pdf

I have found a "[latex]8 \pi [/latex] ", and even a" [latex] -8 \pi [/latex] "on page 4 and page 6, which gives me hope for a favorable result for your w = - 1 test, and for bypass the problem of rotation Wick...

Enjoy your weekend, although I am looking forward to reading you. :rolleyes:

Edited by stephaneww
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The last paper is your best approach, start with the more standard QFT approaches under group.

Then look at degree freedom reductions, after  that with higher dimensional symmetries exc.

I hadn't had much time, to look much further than what I already posted, though you have found a non wick rotation method, that avoids the conformal divergence issues.

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I can not go beyond what I already posted as I said: ... I do not have enough skills


Although I am impatient, I have been working on the issue long enough to wait over a month if you still want to assist me ^_^


Help from other people is also welcome

 

Edited by stephaneww

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On 09/09/2017 at 1:04 AM, stephaneww said:

Hello

with forum latex this time :rolleyes:

I would like to know if what I am going to propose below has a meaning, and if the famous [latex]8 * \pi[/latex] find their place in this context:

in relativistic quantum mechanics, a pulse [latex] \omega [/latex] is given by the conservation relation of the energy:

[latex] \hbar *\omega =m_0c^ 2 [/latex]

[latex] \hbar [/latex]: reduced Planck constant


the relativistic part of the average energy density by volume  (in Planck units) is:

[latex]\frac{m_p*c^2}{l_p^3}=4,633*10^{113} \text{ Joules / }m^3[/latex]


[latex] m_p [/latex]: mass of Planck
[latex] c [/latex]: speed of light in a vacuum
[latex] l_p [/latex] length of Planck


- for the quantum part of this relation it is possible, it seems (see below), to calculate a quantum average energy density by volume  of the cosmological constant with its pulsation (in [latex] s ^{- 1} [/latex]) in a Planck context.

(I use a value of cosmological constant [latex] : \Lambda= 9,992 *10^{-36} s^{-2}[/latex]. So [latex]\omega_{\Lambda}=3,131*10^{-18}s^{-1}[/latex]  )

[latex]\frac{\hbar*\omega_{\Lambda}}{l_p^3}=7,900 *10^{52}\text{ Joules / }m^3[/latex]

I do not know if it is permissible: I propose an elevation of this squared calculation in order to restore the cosmological constant to its relativistic unit (we want to put the cosmological constant back to its unit the [latex] s ^{- 2} [/latex] of the blow we end up with [latex] \text {Joules}^2 / m^6 [/latex])

[latex]\left( \frac{\hbar*\omega_{\Lambda}}{l_p^3}\right)^2=6,234*10^{105} \text{ Joules}^2 / m^6[/latex]


- and to find the value of the average energy density by volume  of the cosmological constant in a pure relativistic context, we simply do:

[latex]\left( \frac{\hbar*\omega_{\Lambda}}{l_p^3}\right)^2  / \left( \frac{m_p*c^2}{l_p^3}\right) /(8*\pi)= [/latex]


[latex] \frac{6,234*10^{105} \text{ Joules}^2 / m^6}{4,633*10^{113} \text{ Joules /m^3}}/(8*\pi)=5,354*10^{-10}\text{ Joules / }m^3[/latex]


[latex] 1 / (8 * \pi) [/latex] may be because an imaginary time has been rotated [latex] s ^{- 2} [/latex] ( I known that there is something near  this exist, for exemple with Wick's rotation, but I do not know if it's allow here)

and a perfect equality with the average density of energy  by volume of the cosmological constant, within the framework of the standard cosmological model =

[latex] \rho_\Lambda.c^2 = \frac{c^4 \Lambda}{8\pi G} = \frac{F_P \Lambda}{8\pi}[/latex]

, for the latter formula, [latex]\Lambda[/latex] is in [latex]m^{-2}[/latex].

The conversion to [latex] m ^ {- 2}[/latex] is obtained by [latex]\Lambda\text{ in } m^{-2}=\Lambda\text{ in }s^{-2}/c^2[/latex]

thank you in advance to those who can tell me if something is wrong.

Stephane

 

 


Hello

A university professor objected that there might be a problem of equality being perfect.
So I pushed a little more my work to see if he was right.

I use this value of cosmological constant [latex] : \Lambda= 9,992 *10^{-36} s^{-2}[/latex] for the following calculations.

[latex]\left( \frac{\hbar*\omega_{\Lambda}}{l_p^3}\right)^2  / \left( \frac{m_p*c^2}{l_p^3}\right) /(8*\pi)= [/latex]

[latex]\left( \frac{\hbar*\omega_{\Lambda}}{l_p^3}\right)^2 * \left( \frac{l_p^3}{m_p*c^2}\right) /(8*\pi)= [/latex]

[latex] \frac{\hbar^2*\Lambda}{l_p^3*(m_p*c^2)} /(8*\pi)= [/latex]

[latex] \frac{\hbar*t_p*\Lambda}{l_p^3} /(8*\pi)= [/latex]

[latex]5,354*10^{-10}\text{ Joules / }m^3=[/latex]

[latex] \rho_\Lambda.c^2 = \frac{c^4 \Lambda }{8\pi G} = \frac{F_P \Lambda}{8\pi}[/latex]

, for the latter formula, [latex]\Lambda[/latex] is in [latex]m^{-2}[/latex].

[latex]F_P[/latex] : Planck force.

The conversion to [latex] m ^ {- 2}[/latex] is obtained by [latex]\Lambda\text{ in } m^{-2}=\Lambda\text{ in }s^{-2}/c^2[/latex]

These simplifications show that[latex] \Lambda[/latex] is on both sides of the equality. It seems normal that equality is perfect. Am I right? wrong ? It remains to verify that we do not turn in circles and this [latex]8\pi[/latex] (it comes from this ? :
https://en.wikipedia.org/wiki/Planck_force#Planck_force_as_a_tension_constant_of_the_space_time_fabric
 

Edited by stephaneww
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Ok the professor is probably referring to matter/radiation density equality. Which is something all the above isn't addressing.

If you take the FRW metric for the critical density formula, you will get a value in the same order of magnitude of the 10^-10 joules/cubic metre.

However that value is a combined value of various contributors. The contributors being matter/radiation and the cosmological constant.

 What you have above is an assumption of strictly the scalar field without isolating the individual equations of state for matter/ radiation and the cosmological constant.

Now radiation and matter will evolve while the volume increases. They will evolve at two distinct rates.

The cosmological constant however will not evolve under volume changes.

This will give you the gist of matter/radiation equality see the bennchmark values at the bottom which is derived with H_0 being 70 km/s/Mpc.

Different H_0 values will affect these values...

judging from the format I am going to take a stab that these are on regards to chapters 5 and 6 of "Introductory to Cosmology" by Barbera Ryden. In this chapter link, 

https://www.google.ca/url?sa=t&source=web&rct=j&url=http://www.astronomy.ohio-state.edu/~dhw/A5682/notes5.pdf&ved=0ahUKEwiqj82OkcTWAhVJ1WMKHYuOAMcQFggqMAM&usg=AFQjCNFK8DuR-vm-kyK94ROpORXBwKNVyg

 

However that is a guess, as its one of the few textbooks that utilizes a benchmark model in its teaching methodology

( I don't want to go too far yet as equations of state and the evolution of matter/radiation with the thermodynamic laws) gets complex real fast.

But here is a start point

[latex]\Omega_{total}=\Omega_{matter}+\Omega_{radiation}+\Omega_{\Lambda}[/latex]

The first two densities evolve differently as the universe expands while the last remains constant.

As far as we have been able to determine.

Here I posted this Heuristic approach to the equations of state and how they arise in another thread and also provided some details on redshift corrections involving matter/radiation equality and evolution of both at redshifts beyond Hubble Horizon.

[latex]DU=pdV[/latex].

 

First take the first law of thermodynamics.

[latex]dU=dW=dQ[/latex]

U is internal energy W =work.

As we dont need heat transfer Q we write this as [latex]DW=Fdr=pdV[/latex]

Which leads to [latex]dU=-pdV.[/latex]. Which is the first law of thermodynamics for an ideal gas.

[latex]U=\rho V[/latex]

[latex]\dot{U}=\dot{\rho}V+{\rho}\dot{V}=-p\dot{V}[/latex]

[latex]V\propto r^3[/latex]

[latex]\frac{\dot{V}}{V}=3\frac{\dot{r}}{r}[/latex]

Which leads to

[latex]\dot{\rho}=-3(\rho+p)\frac{\dot{r}}{r}[/latex]

We will use the last formula for both radiation and matter.

Assuming density of matter

[latex]\rho=\frac{M}{\frac{4}{3}\pi r^3}[/latex]

[latex]\rho=\frac{dp}{dr}\dot{r}=-3\rho \frac{\dot{r}}{r}[/latex]

Using the above equation the pressure due to matter gives an Eos of Pressure=0. Which makes sense as matter doesn't exert a lot of kinetic energy/momentum.

For radiation we will need some further formulas. Visualize a wavelength as a vibration on a string.

[latex]L=\frac{N\lambda}{2}[/latex]

As we're dealing with relativistic particles

[latex]c=f\lambda=f\frac{2L}{N}[/latex]

substitute [latex]f=\frac{n}{2L}c[/latex] into Plancks formula

[latex]U=\hbar w=hf[/latex]

[latex]U=\frac{Nhc}{2}\frac{1}{L}\propto V^{-\frac{1}{3}}[/latex]

Using

[latex]dU=-pdV[/latex]

using

[latex]p=-\frac{dU}{dV}=\frac{1}{3}\frac{U}{V}[/latex]

As well as

[latex]\rho=\frac{U}{V}[/latex]

leads to

[latex]p=1/3\rho[/latex] for ultra relativistic radiation.

Those are examples of how the first law of thermodynamics fit within the equations of state. There is more intensive formulas involved. In particular the Bose-Einstein statistics and Fermi-Dirac statistics

Edited by Mordred
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1 hour ago, Mordred said:

Ok the professor is probably referring to matter/radiation density equality. Which is something all the above isn't addressing.

If you take the FRW metric for the critical density formula, you will get a value in the same order of magnitude of the 10^-10 joules/cubic metre.

 

In fact, the professor was referring to a calculation that runs in circles.

I think I take the FWR metric ( sorry for not mentioning it before ), Hubble constant = 67,74 km*s^-1/Mpc, for the critical density, and a rate of the cosmological constant =  69,11%. There is no problem, you can change the values a little, the formulas should remain correct.

 

1 hour ago, Mordred said:

The cosmological constant however will not evolve under volume changes.

 

I know that : [latex] \rho_\Lambda * c^2 [/latex]  are two constant in this model

What we have to validate is whether we can extract a pulsation from the cosmological constant and use this pulsation in quantum mechanics. And also find the origin of this [latex] 8 * \pi [/latex] .

Is the wikipedia link, about Planck force, interesting for
[latex] 8 * \pi [/latex] ?

 

Edited by stephaneww
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I think we may have a translation error. 

"in circles"

as opposed to " under rotations". ie symmetry relations

Please confirm. Ie linear and and angular momentum being primary examples.

Edited by Mordred

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no I confim Professor said "in circles". He thought that the perfect egality was suspect.

Or my english is not enought good for the translation I use Google :unsure:

 

17 minutes ago, Mordred said:

as opposed to " under rotations". ie symmetry relations

Please confirm.

A person on a French forum spoke to me about the rotation of Wick. I understood the principle and its usefulness in QFT but I was not able to go much further in my understanding of these rotations.

Edited by stephaneww

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Ok I think I know what he may be describing but will have to think about it before I go further. Particularly in regards to 8pi.

Edited by Mordred

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Oooh, what a surprise ... I thought it was very simple. lol ^_^

Begin by the link Wikipedia, it's seems to be easier :

Planck force turn out to be actually "a tension constant of the space time fabric"

Edited by stephaneww
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On 09/09/2017 at 5:07 PM, Mordred said:

Your welcome, glad to see you got latex working.

I agree the paper is extremely interesting on the pulsation and informative in the confinement of the different scales. Planck, macro and cosmological.

The beauty of the paper is that it shows that the old cosmological problem isn't a problem if you examine the limits of the effective action due to the HUP.  

I am still studying several of the details in the paper myself. As you have noted, its fairly intensive in detail. Not something one can easily understand by simply reading it. 

The problem of using Wicks rotations on Euclidean space however is still an issue. The issue comes down to renormalization on path divergences when your under significant curvature. The Wicks rotation works well under Euclid space, not so great under curvature.

Most of the papers detailing this is intensive in group algebra, so if your weak on QFT will be pointless to include. I will see if I can dig up a more appropriate level.

I'm a little busy this weekend but would like to assist you on your modelling approach, so once I get a chance I will probably provide some further details to assist you in moving forward into your approach.

In particular we will need to test if your approach gives the equation of state w=-1 for the Lambda term under scalar modelling. Though I see nothing above that shouldn't, one cannot assume that lol.

 

I found an analogue of a wick rotation in geometric algebra terms... I'll see if I can find it again. 

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Hello

I use this value of cosmological constant [latex] : \Lambda= 9,992 *10^{-36} s^{-2}[/latex] for the following calculations.

[latex]\left( \frac{\hbar*\omega_{\Lambda}}{l_p^3}\right)^2  / \left( \frac{m_p*c^2}{l_p^3}\right) /(8*\pi)= [/latex]

[latex]\left( \frac{\hbar*\omega_{\Lambda}}{l_p^3}\right)^2 * \left( \frac{l_p^3}{m_p*c^2}\right) /(8*\pi)= [/latex]

[latex] \frac{\hbar^2*\Lambda}{l_p^3*(m_p*c^2)} /(8.\pi)= [/latex]

[latex] \frac{\hbar*t_p*\Lambda}{l_p^3} /(8.\pi)= [/latex]

[latex]5,354*10^{-10}\text{ Joules / }m^3=[/latex]

[latex] \rho_\Lambda.c^2 = \frac{c^4 \Lambda }{8\pi G} = \frac{F_P \Lambda}{8\pi}[/latex]

, for the latter formula, [latex]\Lambda[/latex] is in [latex]m^{-2}[/latex].

[latex]F_P[/latex] : Planck force.

The conversion to [latex]m ^ {- 2}[/latex] is obtained by [latex]\Lambda\text{ in m^{-2}}=\Lambda\text{ in s^{-2}/c^2}[/latex]

These simplifications show that[latex] \Lambda[/latex] is on both sides of the equality. It seems normal that equality is perfect. Am I right? wrong ? It remains to verify that we do not turn in circles and this [latex]8\pi[/latex] (come from this ? :
https://en.wikipedia.org/wiki/Planck_force#Planck_force_as_a_tension_constant_of_the_space_time_fabric

Hello I found something new and  pleasing I think :....

[latex] \frac{\hbar.t_p}{l_p^3.8.\pi}* \Lambda (\text{ en }s^{-2})= [/latex]

[latex] \frac{\hbar}{c.l_p^2.8.\pi}* \Lambda (\text{ en }s^{-2})=  \frac{\hbar.c^2}{l_p^2.8.\pi}* \frac{\Lambda (\text{ en }m^{-2})}{c} =[/latex]

[latex]\frac{\hbar.c}{l_p^2}*\frac{ \Lambda (\text{ en }m^{-2})}{8.\pi}=[/latex]     [latex](I)[/latex]

we have :

[latex]\frac{c^4 \Lambda (\text{ en }m^{-2}) }{8\pi G}[/latex]     [latex](II)[/latex]

[latex](I)=(II)[/latex]

so


[latex]\frac{\hbar.c}{l_p^2}=\frac{c^3}{G}  \text{ in kg/s }[/latex] which seems to be an interesting unit for quantum mechanics in astrophysics and relativistic quantum mechanics.

so we have :

[latex]G=\frac{c^3.l_p^2}{\hbar}[/latex]

Is it an important data for quantum gravity ???

You can note, if as I think it is a correct approach to quantum gravity, this one is entirely deterministic ...

... And that approach of the catastrophe of the vacuum by decomposing the cosmological constant with a pulsation seems correct.

Thanks in advance for your specialist looks.

48 minutes ago, stephaneww said:

 

 

Hello I found something new and  pleasing I think :....

[latex] \frac{\hbar.t_p}{l_p^3.8.\pi}* \Lambda (\text{ en }s^{-2})= [/latex]

[latex] \frac{\hbar}{c.l_p^2.8.\pi}* \Lambda (\text{ en }s^{-2})=  \frac{\hbar.c^2}{l_p^2.8.\pi}* \frac{\Lambda (\text{ en }m^{-2})}{c} =[/latex]

[latex]\frac{\hbar.c}{l_p^2}*\frac{ \Lambda (\text{ en }m^{-2})}{8.\pi}=[/latex]     [latex](I)[/latex]

we have :

[latex]\frac{c^4 \Lambda (\text{ en }m^{-2}) }{8\pi G}[/latex]     [latex](II)[/latex]

[latex](I)=(II)[/latex]

so


[latex]\frac{\hbar}{l_p^2}=\frac{c^3}{G}  \text{ in kg/s }[/latex] which seems to be an interesting unit for quantum mechanics in astrophysics and relativistic quantum mechanics.

so we have :

[latex]G=\frac{c^3.l_p^2}{\hbar}[/latex]

Is it an important data for quantum gravity ???

You can note, if as I think it is a correct approach to quantum gravity, this one is entirely deterministic ...

... And that approach of the catastrophe of the vacuum by decomposing the cosmological constant with a pulsation seems correct.

Thanks in advance for your specialist looks.

Edited by stephaneww
latex, correction in formula

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2 hours ago, stephaneww said:

so we have :

[latex] G=\frac{l_p^2.c^3}{\hbar}[/latex]

Is it an important data for quantum gravity ???

You can note, if as I think it is a correct approach to quantum gravity, this one is entirely deterministic ...

... And that approach of the catastrophe of the vacuum by decomposing the cosmological constant with a pulsation seems correct.

Thanks in advance for your specialist looks.

oops, it arises from the Planck length squared.  but  it seems possible to not use the QFT

 
Remains to be confirmed if the approach (demonstration?) of the problem of the vacuum catastrophe approached by the pulsation of the cosmological constant is acceptable. I would not make any hypothesis considering the incompleteness of my basic knowledge
Edited by stephaneww
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On 27/09/2017 at 2:35 AM, Mordred said:

...for ultra relativistic radiation.

Those are examples of how the first law of thermodynamics fit within the equations of state. There is more intensive formulas involved. In particular the Bose-Einstein statistics and Fermi-Dirac statistics

 

On 09/09/2017 at 6:07 PM, Mordred said:

...I'm a little busy this weekend but would like to assist you on your modelling approach, so once I get a chance I will probably provide some further details to assist you in moving forward into your approach.

In particular we will need to test if your approach gives the equation of state w=-1 for the Lambda term under scalar modelling. Though I see nothing above that shouldn't, one cannot assume that lol.

 

Hello Modred. I'm not sure I have understand :

Is it valid equation of state (that I do not master) w = -1 or demonstration is to be done ?

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The equation of state w=-1 for the cosmological constant is valid and agrees with observational evidence thus far.

 What is not mentioned is that it is the same equation of state for an incomparable fluid which can be mathematically defined under the Euler hydrostatic equations. ( Though few textbooks will mention this either).

An equation of state in essence gives us a dimensionless value for the energy density to in essence, pressure influence. Via the ideal gas laws.

Later tonight I will try to find the mathematical proof for the W=-1 with regards to Lambda. They can be tricky to find.

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