[BilboGrabbins (OP)]

In the past I wrote a theory on a mechanical model for dark energy. In short, if space was a medium that was accurately described by the fluid equation of state, could the flow of spacetime then varies from place to place due to the fluids local pressure inside some quantum entangled bubble. Such ideas of space being entangled with other places in space (and maybe time) is a key idea in  ER=EPR and many of the leading pioneers who original devised it think it holds one last quantum importance in unifying gravity would be those pesky black holes. As GM Jackson admits here:

http://gmjacksonphysics.blogspot.com/2017/02/using-fluid-mechanics-to-explain.html?m=1

Using the Bernolli equation as it is, is largely useless for the dynamics of expanding space. I'll argue he's right in some ways. And while his math looks sound, but it is as clear as mud that he doesn't explain the significance of his results all too well, a bit like me sometime.

If he took a longer thought over it, he should have soon came to the conclusion I reached. Just make a relativistic version. I've seen various derivations of Bernoulli's constraints, believing at least three don't apply to our universe.

When researching sonoluminescense, for two months, I compared many of these equations with the Friedman equation. I noticed, because both theories involve bubble cavity mathematics, which allows bubbles to spin, expand and contract.
I suppose the popping of these analogue bubbles could be associated to a vacuum decay, another hot topic in astrophysics.

Not drifting off to far now, coming back to my investigations of seeing the symmetries of both theories, I derived a specific form of the Reighlegh-Plesset equation as



* we had two R's remaining in the third expression from the left so it has been turned into a gauge. Also, a surface tension appeared, which if we assume no vacuum decay, can be neglected for the cosmological model.

You can read on its derivation in these four articles.

<Mod edit: Plug to personal site violates terms>

Continuing now, Bernoulli's principle can be stated in a few different forms, but at least one of them did first catch my eye. It's a well-known equation, and I reinvisioned, might be a better word, since a I didn't derive it. Like all his fluid equations, they involved the gravitational acceleration.
In order to reinterpret it under a framework for general relativity, I get -- first, let's refresh our mind on Bernoulli's original principle:



At the face of it, it looks all too familiar as an equation satisfying adabatic laws. Adiabatic models preserve the quantum model of conservation, wheteas the Diabatic models entertain violations of the conservation laws, which is a completely unfounded assumption on cosmological models, the late Lloyd Motz once pointed out, and author of the first mathematical model for a non conservstion by use of a third derivative on the Friedmann equation.

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So a relatively simple operation replacing the acceleration notation for the gravitational field itself (ie. The connections of the field in which are described by the Christoffel symbols).

The Bernoulli equation, from its derivation section, is



The funny thing is, if I want a Bernoulli (modified equation) capable of being sufficient to satisfy those that define cosmology, thrn I need redefine the constant as actually the cosmological constant. In itself, as Jackson pointed out, the last equation isn't useful for cosmology and the flows of streams.

So let's make it useful!

We bring back to attention the following equations and suspect them





The first equation is much more superior. Also, by making the constant the cosmological kind, means it should have energy terms. This requires a simple distribution of a mass term on both equations. But since the first is all we care about:



What's interesting about this equation, is because unlike that has been noticed before, viscocity [v] of a fluid which is an interesting factor as it would decrease any uniform flow. Do wr asdume this term is surely approaching zero in the evudence of universal acceleration?
Except, I don't believe the universe is accelerating at all, and I think scientists have misinterpreted the data very badly! This would be out of context here, so if you want me to elaborate, say so and I will in another post.

But are there any other parameters which might strike the mind important if we want to have an acceptable form of a cosmic safe Bernoulli principle? When I invrstigated mechanical reasons, I used some of the equations we have seen in this post way back in the past. So its ironic to me I'd be dealing with it again.

It's called the Ricci flow. Indid do some massive investigations into in the space. If space is like a fluid, then such behaviour can be found in the Ricci flow, which (is the heat equation) for a Reimannian manifold.

This doesn't actually mean that the flow of heat is attributed simultanously with the glow of space, but it can happen.
Instead, it means that curvature flows, just like heat.

I came to my first attempt at independently deriving the four dimensional case, which seemed easy enough. This was when I realized the temperature was missing. But I couldn't use the math in mind to make ot happen unless I get the radius terms back like R'/R. So that it could be reinterpreted as T'/T due to their symmetries and dimensionlessness. Reareanging my equation we now argue



And so while making a new kind of Friedmann equation, so I made use of the temperature gauge as a coefficient on the viscocity from R/R(0).  This way, the thickness associated to some region of flowing space can slso vary in temperature.
To really finish off neatly, the equation makes time part of the left hand side with the last term on the right have a dependency of time on the pressure, we might ask, what does temperature depend on?
Well, temperature can depend on time, since the scale factor can be invited - We cuttently think it had a very hot and dense past, but we cannot say for sure what happened before it - nevertheless, the viscosity of space may be seen intrinsically related to the density of an element volume of space? All we will be missing is Einsteins curvature constant [k]. The constant in his theory was additive to the RHS



And so in our equation as



And I make a final surprise, not to make temperature time dependent, but as a function of the scale factor.