Is it possible to explain by energy conservation the apparent expansion of the Universe with galaxies
seemingly repelling each other, as a compensation for the potential energy lost in the formation of galaxies by gravitational implosion? In short: Local implosion balanced by global explosion. Let us explore this natural and simple idea.

Let us thus as a basic cosmological model consider the Euler equations including gravitational forces expressing conservation of mass, momentum and total energy of gas in three-dimensional space over time.

We thus seek the density $\rho$ , momentum $m=\rho u$ with $u=(u_1,u_2,u_3)$ the velocity, heat (internal) energy $e$ and gravitational potential $\varphi$ as functions of space-time coordinates $(x,t)$ , where $x=(x_1,x_2,x_3)$ are space coordinates and $u_i$ is the velocity in the $x_i$ -direction. The Euler equations for $(\rho, u, e,\varphi )$ read:

$\dot \rho +\nabla\cdot (\rho u)= 0$
$\dot m+\nabla\cdot(m u)+ \nabla p-G\rho\nabla\varphi = 0$
$\dot e +\nabla\cdot (e u)+p\nabla\cdot u = 0$
$\Delta\varphi =\rho$ ,

where $v_{,i}=\frac{\partial v}{\partial x_i}$ is the partial derivative with respect to $x_i$ , $\dot v =\frac{\partial v}{\partial t}$ is the partial derivative with respect to time $t$ , $p$ is the pressure and $G$ the gravitational constant. Further, $\nabla\cdot v=\sum_iv_{i,i}$ denotes the divergence of $v=(v_1,v_2,v_3)$ , $\nabla w=(w_{,1},w_{,2},w_{,3})$ is the gradient and $\Delta$ the Laplacian. We here leave the gas law defining the pressure $p$ in terms of the conservation variables $(\rho ,m,e,\varphi )$ to be determined.

We imagine that the gas in an initial rest state with uniform density and temperature is subject to a velocity perturbation which triggers a redistribution of matter into a set of galaxies with high density formed by gravitational attraction/implosion separated by voids of low density. This is an unstable process in the sense that perturbations of the gravitational potential $\varphi$ are amplified into concentration-thinning of the density $\rho =\Delta\varphi$ by the differentiation, with feed-back to the gravitational potential.

In the formation of a galaxy, the gravitational potential energy is decreased as matter is falling towards the center of the galaxy, and conservation of total energy requires compensating energy as increased internal energy of the galaxy and/or kinetic energy in the form of

increased velocity inside galaxies
increased velocities of whole galaxies.

Here 2. could be experienced as repulsion between galaxies. Loss of gravitational energy by implosion would thus be compensated by increase of kinetic energy of galaxies by repulsion expressed by the pressure $p$ , which would be determined by a gas law keeping the mean value of the net force $\nabla (p-G\varphi )$ small to allow the kinetic energy to reach a steady state. The repulsive force could be the force of dark energy. This comects to the view of George Chapline, who considers general relativity to be bogus along with black holes and instead supports an idea of dark-energy star.

The gas law would thus act to conserve the total energy, and may be compared with the gas law of incompressible gas acting to maintain incompressibility.

Notice that after the formation of galaxies the repulsive force would be shut off followed by inertial expansion as explored in Accelerating Expansion without Dark Energy?

Notice further that this model does not require any Big Bang, which is difficult to motivate, since the initial state is an innocent uniform rest state with uniform distribution of density and temperature. The Universe is then set in motion and formed through the instability inherent in the second order differentiation of the Laplacian.

8 Comments

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Michele
December 21, 2011

It seems to me (I’m quoting from memory) that the pressure power is div(pu). Is there any reason needing the pressure gradient is neglected?
I think that the “velocity perturbation” must be very strong. In this case would be due to a simple pressure wave?
Michele

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claesjohnson
December 21, 2011

This is because the energy equation is expressed in terms of the internal energy and not as is more usual in terms of the total energy as the sum of internal and kinetic energies.

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Michele
December 22, 2011

OK. Thank’s for your enlightenment.
Michele

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Richard T. Fowler
December 27, 2011

Claes,

I am working on a response to your puzzle from the 23rd. But there is something I do not understand about the present article, “Dark Energy: Repulsion from Implosion?” which is referenced by your puzzle. I have two specific questions regarding my present confusion.

QUESTION #1.
You have, I believe, argued in the past that the idea of dark energy is an ad-hoc assumption without physical evidence or theoretical support, and is something which, if it did exist, would violate various other laws and principles/suppositions made by the standard model. Therefore, you apparently argue that no reasonable person can believe that it exists.

Thus far, I have no problem with any of this reasoning, and find your arguments along those lines quite persuasive.

But in the present article, you write:

——-
Here 2. [increased velocities of whole galaxies] could be experienced as repulsion between galaxies. [. . .] The repulsive force could be the force of dark energy. This comects to the view of George Chapline, who considers general relativity to be bogus along with black holes and instead supports an idea of dark-energy star.

[. . .]

Notice that after the formation of galaxies the repulsive force would be shut off followed by inertial expansion as explored in Accelerating Expansion without Dark Energy?
——-

Taken as a whole, your words seem to imply that, in the cosmological model to which you subscribe, “dark energy” simultaneously does and does not exist.

QUESTION #1 IS: How do you explain this apparent contradiction?

QUESTION #2.
I am also working on a response to the radiation part of your puzzle. However, because of the tying-in through the Euler equations of your work on motion and classical wave dynamics with the cosmological ideas expressed with this article on dark energy, I feel somewhat lost at the moment.

Reading the present article on “dark energy”, I get the feeling that I still do not understand your physical models sufficiently to comment in great detail about them.

I still believe that most of your radiative and thermodynamic conclusions, taken together (I’ll call them your “energy model”), offer a more sensible and more accurate approach to the evidence than either the standard model or the various string hypotheses. (Or, for that matter, than the ideas of certain “Slayers”, from whom you were wise to distance yourself.)

But I get the feelng that you believe that your energy model is somehow inseparable from this cosmology — a cosmology which makes very little sense to me. Actually far less sense than the prevailing one. If indeed these two models — energy and cosmology — are inseparable, then that would seem to imply that I cannot possibly be in agreement with your energy model, even though I think I am. Thus, there would have to be something with the energy model that I am not understanding, which if I did understand, would force me to reject all of your conclusions as being internally inconsistent.

I’d be more inclined to suspect that your energy model is simply not dependent on the cosmology you are advocating.

QUESTION #2 IS: Am I right in this suspicion?

Thank you for your time in considering these two questions.

Richard T. Fowler
RTF

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claesjohnson
December 27, 2011

Good to hear that you are working on the puzzle. Yes, you are right, I now think that maybe there is something like dark energy, as my post of today on God’s equation illustrates. Maybe this post also gives some answer to your 2nd question.

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- Richard T. Fowler
  December 28, 2011
  
  Your clarifications are appreciated; more grist for the thought mill!
  
  FYI, I used to be able to download your pdf’s from your .edu website with my slow telephone connection, but something was changed and now the server times out, even for a small pdf of ~2.5 mb. I have to get myself to another computer sometime in the coming day, so that I can get downloads of these publications. I think I have a copy of “Climate Thermodynamics” somewhere, but not your other pdf’s. Also the google books are very cumbersome to read, because I have to load them page-by-page; so I have not had a close look at those. I once skimmed the 37-mb publication on finite computation, and found it extremely interesting; but that was at the other computer, so I am operating at a distinct disadvantage. Hopefully I will rectify it soon by downloading relevant items to a memory stick. Until then, I am afraid that vague generalities will have to suffice from me; so until I have something more substantial to contribute, I will hold my peace. Perhaps someone else will venture a solution in the meantime.
  
  RTF
  
  Reply
  - Richard T. Fowler
    December 28, 2011
    
    Sorry, I should have said your KTH website, not “.edu”.
    
    RTF
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January 1, 2012

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