New Newtonian Cosmology: Fluid Model with Positive and Negative Mass

Cosmic web from Thesan Simulation.

Here is a clarification of the model of New Newtonian Cosmology discussed in recent posts based on primordial gravitational potential generating mass density of possibly variable sign with attraction between masses of the same sign and repulsion between masses of different sign. Compare with Don't Dismiss Negative Mass.

From an initial perturbation of an initial state with zero potential and mass density, a system with non-zero mass density may emerge with regions of positive and negative mass density concentrating by gravitational attraction while moving away from each other by repulsion. This model does not require a massive initial Big Bang and connects to both dark matter and dark energy.  The model is similar to the N-body models used in the Millennium Simulation and Thesan Simulation. 

We consider a New Newtonian cosmological model in the form of Euler's equations for a compressible gas subject to Newtonian gravitation: Find $(\phi ,m,e,p)$ depending on a Euclidean space coordinate $x$ and time $t$, such that for all $(x,t)$:

• $\vert\dot\rho\vert + \nabla\cdot m =0$                                                           (1)
• $\dot m +\nabla\cdot (mu) +\nabla p + \rho\nabla\phi =0$                              (2)
• $\dot e +\nabla\cdot (eu) +p\nabla\cdot u =0$,                       (3)

where $\phi$ is gravitational potential, $\rho =\Delta\phi$ is mass density, $m$ is momentum, $u=\frac{m}{\vert\rho\vert}$ is matter velocity, $p$ is pressure, $e=\rho T$ is internal heat energy with $T$ temperature, and the dot indicates time differentiation, cf. Computational Thermodynamics Chap 32. Here $x$ is space coordinate in a fixed Euclidean coordinate system, and $t$ is a local time coordinate which  does not have to be globally synchronized.

The primary variables in this model are the gravitational potential $\phi$ and the momentum $m$ connected through (2) expressing conservation of momentum or Newton's 2nd law. We view matter density $\rho =\Delta\phi$ as being derived by local action of the differential operator $\Delta$ and that 

$\rho (x)$ can have variable sign representing positive and negative mass. The connection $\rho =\Delta\phi$ is analogous to the corresponding connection between charge density of variable sign and electric potential in electromagnetics. The model is complemented by a constitutive equation for the pressure.

The essential components of this model are:

1. Newton's law of gravitation $\rho =\Delta\phi$ connecting mass of variable sign to gravitational potential.
2. Gravitational force is given by $\rho\nabla\phi$. 
3. Conservation of momentum (2), mass (1) and total energy.

The essential features are:

• No instant gravitational action at distance required with $\phi$ primordial and $\rho =\Delta\phi$ derived quantity.
• Global clock synchronisation not needed because all action is local.
• Equivalence of inertial and gravitational mass by (2).
• $\Delta\phi$ of variable sign opens to positive and negative mass.
• No limit on matter speed.
• No electro-magnetics or nuclear physics so far included in the model.

The gravitational force $\rho\nabla\phi$ expresses:  

• Positive masses attract each other, negative masses attract each other.
• Positive and negative masses repel each other.  

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