New Newtonian Cosmology: Energy Balance

In regions of both positive and negative mass density gravitational collapse concentrates mass and ignites stars.

This is a continuation of the previous post on New Newtonian Cosmology: Positive and Negative Mass. We consider the following simplified fluid model with zero pressure:

• $\rho =\Delta\phi$                                                                                      (1)
• $\frac{\partial\vert\rho\vert}{\partial t} + \nabla\cdot m=0$                          (2)
• $\frac{\partial m}{\partial t}+\nabla\cdot (mu)= -\rho\nabla\phi$          (3)             

where $\phi (x,t)$ is gravitational potential, $\rho (x,t)$ is mass density, $m$ is momentum, $u(x,t)=\frac{m}{\vert\rho\vert}$ is matter velocity, $x$ is space coordinate in a Euclidean coordinate system and $t$ is a time variable. 

Here (1) expresses that mass density $\rho$ of variable sign represents presence of matter with positive and negative mass density assigned by a primordial gravitational potential $\phi$,  (2) expresses positive and negative mass conservation and (3) is Newton's 2nd Law connecting change of momentum to gravitational force $-\rho\nabla\phi$, which amounts to attraction between mass densities of the same sign and repulsion between mass densities of different sign. 

The model satisfies the following energy balance obtained by multiplying (3) by $u$ and integrating over $x$ :

• $\int\frac{\partial}{\partial t}(\vert\rho\vert\frac{\vert u\vert^2}{2})dx=\int\frac{\partial}{\partial t}(\frac{\vert\nabla\phi\vert^2}{2})dx$

and so by integrating in time

• $\int\vert\rho\vert\vert u\vert^2\, dx=\int\vert\nabla\phi\vert^2dx + C=-\int\rho\phi dx + C$,

with $C$ a constant, showing that over time kinetic energy is balanced by gravitational potential energy. Note that the gravitational potential generating positive mass is negative and vice versa. 

We see that from an initial zero state with an initial highly oscillating perturbation of $\phi$ generating, by differentiation substantial mass density of variable sign $\rho =\Delta\phi$, regions of positive and negative mass can form supplied by kinetic energy from gravitational collapse (into galaxies and stars), regions which repel each other. A region of positive mass may then represent the Universe we know or can see, while its negative mass density companion has disappeared out of sight. 

This is a scenario without massive initial input like that of a Big Bang, which appears to include forms of both dark matter and dark energy. The energy settting the system in motion comes from gravitational concentration/collapse not a Big Bang.

You can play with this model in Leibniz World of Mathematics Model Workshop under 21. Cosmology.