Neo-Newtonian Cosmology 1

                                            Cosmic web of strings of positive mass.

The post Creating the Universe by Plucking a String 1  presented a new interpretation of Newton's Law of Gravitation as Neo-Newtonian Cosmology, which offers answers to the following questions viewed to motivate giving up classical Newtonian cosmology for that of Einstein: 

• What is the physics of instant action at distance?                                      (Q1)              
• Why is an infinite Universe with non-zero uniform mass distribution impossible?  (Q2)

Concerning Q2 already Newton noted that in an infinite Universe with uniform mass distribution the gravitational force at each point by symmetry will be zero, which means that the gravitational potential $\Phi$ is constant, which means that mass density $\Delta\Phi$ with $\Delta$ the Laplacian differential operator, is zero (see The Dynamics of Newtonian Cosmology by Alan Guth and Was Newtonian Cosmology Really Inconsistent? by Peter Vickers): 

• The only Universe with uniform mass distribution is an empty universe!

This very disturbing realisation together with lack of answer to Q1 has served as key motivation to replace Newton by Einstein in modern physics,  see p 96 in Relativity: The Special and General Theory by Einstein.

Let us now recall answers to Q1 and Q2 within Neo-Newtonian Cosmology: 

The basic idea is to view gravitational potential $\phi (x,t)$ with $x$ a Euclidean coordinate and $t$ a time coordinate, to have a primordial role from which mass density $\rho (x,t)$ is "created" by the Laplacian differential operator $\Delta$:

• $\rho (x,t) = \Delta \phi (x,t)$ for all $x$,         (G1)

assumed to act without time delay for all $t$. Mass is thus created locally for each $x$ by differentiation as an instant local operation acting at each time instant $t$.  

This is to be compared with the standard view that gravitational potential $\phi$  is created from primordial mass density as solution of the differential equation:

• $\Delta\phi (x,t)=\rho (x,t) $ for all $x$,           (G2)

represented by the integral formula

• $\phi (x,t) =-\frac{1}{4\pi}\int\frac{\rho (y,t)}{\vert x-y\vert}dy$

which requires instant action at distance and so represents a main mystery of modern physics.

We recall that the Laplacian is invariant under an orthogonal change of coordinates and so the creation process (G1) is independent of the choice of Euclidean coordinate system, which can be seen as a sign of possible physicality. 

(G1) allows "creation of something big from something small" in the following way: Let us then connecting to Q2 start from a gravitational potential $\Phi =0$ satisfying $\Delta\Phi =0$ thus with corresponding zero mass density. Let $\Phi$ be subject to a small scale small amplitude oscillatory perturbation $\phi$ creating small scale large amplitude oscillatory mass density $\rho =\Delta\phi$ by the action of second order differentiation with $\rho =\rho_++\rho_-$ decomposed into positive mass density $\rho_+$ and negative mass density $\rho_-$.  

The action of gravitational force $-\nabla\phi$ will cause attraction between mass densities of the same sign and repulsion between mass densities of opposite sign, and so will segregate the small scale variation of $\rho$ into Universa of positive mass and Universa of negative mass which repel each other and so recede. 

By the action of the Laplacian on a small scale small amplitude oscillating perturbation of a zero state gravitational potential $\Phi =0$, large scale large amplitude finite Universa $U_+$ with positive mass $\rho_+$ can been created, balanced by a corresponding negative mass Universa $U_-$ at eventually large distance. 

It is thus possible to envision a scenario where a small perturbation of a zero gravitational potential in an infinite universe by the action of differentiation + gravitational attraction/repulsion in particular generates  a finite large scale Universa with positive mass as the one we happen to experience. 

Does this mean that the scenario really starts from zero? Not quite, because starting from $\Phi =0$ satisfying $\Delta\Phi =0$ requires some spatial structure to express the Laplacian. We can see this spatial structure as a string under tension and the small scale small amplitude perturbation as a small amplitude high frequency excitation of the string. This is a form of cosmological string theory.

The creation process thus starts with an infinite spatial structure under tension but without excitation, from which the Universe we are living in is created with "a little pluck of a string".

The scenario opens to the existence of dark matter identified by $\Delta\phi$ of small magnitude but large extension, and dark energy as influence on $U_+$ from $U_-$. 

Answer to Q1: Mass is created from gravitational potential by instant local action. No need of instant action at distance. 

Answer to Q2: Start from an empty infinite Universe with gravitational potential $\Phi =0$ satisfying $\Delta\Phi =0$ and introduce a small perturbation with mass density $\rho =\Delta\phi$ of variable sign from which finite Universa of positive and negative mass emerge and diverge.

Summary: It is natural to think of the infinite zero potential/mass universe as eternal without beginning and the plucking of a string initial perturbation as the beginning of universa with non-zero mass density as a very very gentle little big bang,