A Curious Interpretation Of Weak Equivalence Relatioship With Fluctuations

[Dubbelosix]

F​rom a previous post on cosmic seeds, (link below) I showed a theoretical approach to unifying the primordial ground state fluctuation in an expansion picture to dynamically spread over spacetime causing a significant early gravitational clumping giving rise to the large scale structure:

 

 

 

[math]\frac{\dot{R}}{R}\frac{\ddot{R}}{R} + \frac{\hbar c}{mR} \int k\ \dot{k} = \frac{8 \pi G}{6}(\rho + \frac{P}{c^2})\frac{\dot{T}}{T}[/math]

 

 

Today I propose a new theoretical interpretation  of the fluctuations in terms of the equivalence principle. That argument can be composed in form of a temperature equation:

 

 

[math]T = \frac{\hbar a}{ k_Bc} = \frac{\hbar c}{k_B}  \sqrt{\Lambda} = \frac{\hbar c}{k_B}  \sqrt{k \cdot k}[/math]

 

​

​Here, [math]\Lambda[/math] is the cosmological constant and [math]k^2[/math] is the wave number squared. When the cosmological constant is identified in the Friedmann equation its not hard to see they are identical in their dimensions, being (1/length^2). An argument for doing this may also lye in how cosmologists have viewed the cosmological constant, as being something related to the energy of the vacuum. Similarly, Sakharov has proposed fluctuations may even form the basis of gravity itself. Whether or not this is true, one thing is for certain, when we think of vacuum energy, physicists do tend to think about spacetime in terms of the fluctuations inside of it.

 

 

Dividing the equation by [math]mR[/math] and distributing the Boltzmann constant we get

 

 

[math]\frac{k_BT}{mR} = \frac{\hbar a}{ mcR} = \frac{\hbar c}{mR}  \sqrt{k \cdot k}[/math]

 

 

We already have a term which is getting similar to our Sakharov fluctuation term. Simplifying [math]\sqrt{k \cdot k} = k[/math] and integrating one such term through and the cosmological constant for the first two terms, we get

 

 

[math]\frac{k_BT}{mR} \int \Lambda = \frac{\hbar a}{ mcR} \int \Lambda = \frac{\hbar c}{mR} \int  k\ dk[/math]

 

​(the reason we cannot integrate the dk all the way through each term is because the wave number only makes sense for electromagnetic radiation in most cases. In this case, we can pick and chose which coefficient works in this case, because we identify the wave numbers of the electromagnetic radiation with the cosmological constant, a hypothesis, but an attractive one based simply on how we view both subjects [math]\sqrt{\Lambda} = k[/math] so a possible relation may be established).

 

and

 

[math]\frac{k_B\dot{T}}{mR} \int \sqrt{\Lambda} = \frac{\hbar \dot{a}}{ mcR} \int \sqrt{\Lambda} = \frac{\hbar c}{mR} \int  k\ \dot{k}[/math]

 

 

I propose, only, that this is a curious set of relationships that can be formed. If it satisfies the Sakharov term correctly (as it appears it does in the form found in the modified Friedmann equation) then it must link the fluctuations of the electromagnetic field in some intrinsic way to the temperature of the system, which can easily be identified as the background temperatures, and how those fluctuations can equivalently be related to the thermal background of spacetime for some accelerating observer.

 

In terms of these set of equivalences that were formed in the last equation, Arun (et al.) have noted that large curvature (denoted in [math]\sqrt{\Lambda}[/math] ) and large temperatures and strong gravity are all related, especially in the context of an early cosmology. Fluctuations could very well play a cosmological role in the curvature of spacetime. They certainly may have ties with the cosmological constant as we covered not long ago.

 

 

 

 

 

 

references:

 

https://arxiv.org/ftp/arxiv/papers/1205/1205.4624.pdf

 

http://www.scienceforums.com/topic/30282-cosmic-seeds/

Edited July 19, 2017 by Dubbelosix