fredag 7 februari 2014

The Universe as Weakly Compressible Gas subject to Pressure and Gravitational Forces

Our Milky Way Galaxy as a weakly compressible gas modeled by the Euler equations expressing conservation of mass, momentum and energy, combined with constitutive laws connecting pressure and gravitational forces to mass density.

After the excursion on Newton vs Einstein in recent posts, we are back to a cosmological model in the form of Euler's equations for a compressible gas subject to Newtonian gravitation: Find $(\rho ,m, e ,\phi ,p)$ depending on a Euclidean space coordinate $x$ and time $t$ , such that for all $(x,t)$:
  • $\dot\rho + \nabla\cdot (\rho u ) =0$       (or $\frac{D\rho}{Dt} = -\rho\nabla\cdot u$)
  • $\dot m +\nabla\cdot (mu) +\nabla p + \rho\nabla\phi =0$
  • $\dot e +\nabla\cdot (eu) +p\nabla\cdot u +\rho\nabla\cdot m=0$,
where $\rho$ is mass density, $u=\frac{m}{\rho}$ is matter velocity, $p$ is pressure, $\phi$ is gravitational potential, and $e$ is internal energy as the sum of heat energy $\rho T$ with $T$ temperature and gravitational energy $\rho\phi$and the dot indicates time differentiation and
  • $\frac{D\rho}{Dt}=\dot\rho +u\cdot\nabla\rho$
is the convective time derivative of $\rho$, see Many-Minds Relativity 20.3 and Computational Thermodynamics Chap 32.

These equations express conservation of mass $\rho$, conservation of momentum $m$ with $\nabla p$ pressure force and $-\nabla\phi$ gravitational force, and conservation of internal energy $e$. These laws of conservation are complemented with constitutive laws connection $p$ and $\phi$ to density of the following form:

A1: Weakly compressible gas ($\delta$ small positive constant):
  • $\Delta p =\frac{\nabla\cdot u}{\delta}= - \frac{1}{\delta\rho}\frac{D\rho}{Dt}$

A2: Compressible perfect gas ($0 < \gamma < 1 $): 
  • $p=\gamma \rho T$.
B: Newton's law of gravitation:
  • $\Delta\phi =\rho$ with $\phi =0$ at infinity.            
We observe
  1. Similarity of $\nabla p$ and $\nabla\phi$ in momentum equation. 
  2. Similarity between A1 and B connecting $\Delta p$ to $-\frac{D\rho}{Dt}$ (or $-\rho$) and $\Delta\phi$ to $\rho$.
  3. $p \ge 0$ and $\phi \le 0$.
Here 1 can be seen as the Equivalence Principle (equality of heavy and inertial mass) expressing that there is no difference between gravitational and other forces (pressure) in Newton's 2nd law expressing conservation of momentum.

Further, 2 expresses that the constitutive laws A1 and B both can be viewed as action at distance if $\rho$ is viewed as the cause, but represent local action of differentiation if $\rho$ is viewed as the effect. 

For a weakly compressible gas described by A1, there is no need per se to identify a cause-effect relation between $p$ and $\rho$; it is enough to say that $p$ and $\rho$ are connected in a certain way expressing a form of "perfect harmony". 

In the same way, there is no need per se to identify a cause-effect relation between $\phi$ and $\rho$; it is enough to say that $\phi$ and $\rho$ are connected in certain way expressing a form of  "perfect harmony" in the spirit of Leibniz.

The relation $\Delta\phi =\rho$ is explored in Newtonian Matter and Antimatter with $\Delta\phi > 0$ identifying matter and $\Delta\phi < 0$ antimatter. Mind-boggling!

It may be that the above Newtonian model is just as good (or even better) than Einstein's equations as cosmological model, because it is understandable and may contain relevant large scale physics without the mysteries of "curved space-time fabric" of general relativity.

Second Law of Thermodynamics

Viscosity solutions of the Euler equations satisfy the following 2nd law:
  • $\dot K = W_p + W_\phi - D$
  • $\dot E = - (W_p + W_\phi ) + D$,  
where at a given time instant $t$
  • $K=\int\frac{\vert m\vert^2}{2\rho}\, dx$ is total kinetic energy
  • $E=\int e\, dx$ is total internal energy 
  • $W_p=\int p\nabla\cdot u\, dx$ is total work performed by pressure 
  • $W_\phi =\int \phi\nabla\cdot m\, dx$ is total work performed by gravitation
  • $D>0$ is turbulent dissipation.
The sign of the turbulent dissipation $D$ gives an irreversible transfer of kinetic energy $K$ into internal energy $E$, which defines the direction of time. Transfer from internal energy requires $W_g+W_\phi >0$, that is expansion with $\nabla\cdot u > 0$ if $W_\phi=0$ and contraction with $\nabla\cdot u < 0$ if $W_p=0$. Note also that
  • $W_\phi =- \int \phi\dot\rho\, dx =-\int\phi\Delta\dot \phi\, dx =\frac{d}{2dt}\int\vert\nabla\phi\vert^2\, dx$.
Notice that the above 2nd law does not involve the notion of entropy, and thus avoids the trap of common formulations of the 2nd in terms of an ill-defined concept without physical realization. The above 2nd law only uses the physical concepts of kinetic and heat energy, work and dissipation.

Choice of coordinate system

The Euclidean coordinate system could be fixed to distant fixed stars with the origin at the projected center of the universe. A rotation of this system would show up as extra centrifugal forces, which could explain the apparent accelerated expansion of the universe.

Uni-directional transfer of energy - Alternative pressure laws

Note that A1 implies that $W_p= - \int\delta\int\vert\nabla p\vert^2\, dx < 0$ showing uni-directional transfer of energy from kinetic energy to internal energy. Transfer in the other direction can appear with the stronger effect of compressibility in A2. Of course, combinations of A1 and A2 are thinkable, as well as a pressure analog of B in the form A3: $-\Delta p \sim e$ with $e=\rho T$ and $T$ temperature.

Recall that $W_\phi = \frac{d}{2dt}\int\vert\nabla\phi\vert^2\, dx$, thus with a positive contribution to kinetic energy $K(t)$ over time.

Basic dynamics

The basic dynamics of the above Euler model can be described as local gravitational collapse with gravitational energy being transformed first into kinetic energy and then into heat energy, followed by pressure build up setting a limit to concentration of matter, leading to cosmic web of matter separated by voids:

Visible matter would correspond to $\Delta\phi$ being singular (points, lines, surfaces), while dark matter would correspond to  $\Delta\phi$ being smoothly distributed (and positive).

PS Note that the momentum equation can be formulated:
  • $\rho\dot u +\rho u\cdot \nabla u + \nabla p + \rho\nabla\phi =0$, that is
  • $\dot u +\rho u\cdot \nabla u + \nabla\phi = F$ or    $\frac{Du}{Dt} + \nabla\phi = F$,
where $F$ (with here $F = - \frac{1}{\rho}\nabla p$) can be viewed as a definition of mass density normalized force $F$ in terms of acceleration $\frac{Du}{Dt}$ and gravitational potential gradient $\nabla\phi$. This connects to the common notion of gravitation(al force) as acceleration, with acceleration acting locally in space and time. Again, the basic question is the (lack of) cause-effect in the relation $\Delta\phi =\rho$, where we are not compelled to view $\rho$ as the cause and $\phi$ the effect from instant action at distance, and thus may instead view $\rho$ as the effect of the Laplace operator acting locally in space and time on $\phi$, or simply bypass the question by viewing $\rho$ and $\phi$ to be connected in "perfect harmony by the relation $\Delta\phi =\rho$ without cause-effect dynamics.  


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