New Newtonian Cosmology vs Cosmic Inflation

Cosmic Inflation is a theory put forward by in particular Alan Guth explaining the origin of Big Bang as a false vacuum releasing negative pressure in a decay from a metastable to a more stable state driving a quick exponential expansion named inflation.  The time scale is small: from $10^{−36}$ seconds to between $10^{−33}$ and $10^{−32}$ seconds after the Big Bang. Mind boggling! 

The New Newtonian Cosmology discussed in recent posts connects to Cosmic Inflation with a different view on negative pressure as repulsion between mass density $\rho (x)=\Delta\phi (x)$ of different sign created from a fluctuation of gravitational potential $\phi (x)$ with $\Delta$ the Laplacian differential operator with respect to a spatial coordinate $x$, with following energy balance

• $\int\vert\nabla\phi\vert^2dx = - \int\rho\phi dx$.          (E) 

The energy driving Cosmic Inflation is supposed to come from a false vacuum decaying into lower energy,  leaving a question about the formation of the false vacuum?  

In New Newtonian Cosmology expansion comes from repulsion between mass densities of different sign appearing from a gravitational potential fluctuation acting on a tensed drum skin, leaving a question about the origin the tension of the drum skin?

Which question has a better answer? Both answers connect to observations of a uniform very sparse flat (zero curvature) Universe as the result of an expansion still going on.  

In New Newtonian Cosmology the compression comes from a micro-scale small fluctuation of a gravitational potential $\phi (x)$ combined with the local action of the Laplacian creating substantial mass density $\rho (x)$ of variable sign subject to gravitational repulsion $-\rho\nabla\phi$ and with substantial contribution to gravitational potential energy as the right hand side of the energy balance (E) by the action of the gradient $\nabla$. The Laplacian and the gradient thus act against a spatial background which can be seen as a drum skin before hit. 

The prerequisite in New Newtonian Cosmology is thus a three-dimensional Euclidean space with gradient and Laplacian differential operators in which substantial gravitational potential energy is created from a gravitational potential fluctuation into receding Universa of positive and negative mass. The accumulated gravitational potential energy is then used to create the World in gravitational collapse. The Creator thus appears to have been an applied mathematician. 

What can be said about the false vacuum of Cosmic Inflation? Here is an answer by Sabine Hossenfelder.

Unified Field Model of Gravitation and Electromagnetism

This is a continuation of the post New Newtonian Cosmology - Universe from Nothing. We know from experience that creation of material goods from nothing is not possible. To make a soup we need ingredients. So it is natural to ask from where the ingredients came to the Big Bang as the postulated incredibly hot primordial soup as the origin of the Universe we are part of. But we get no answers from physicists proclaiming Big Bang. 

Another possibility is that the Universe came out of virtually Nothing from the equation

• $\Delta (\phi + \psi ) = \rho + \epsilon$ 

where $\psi =-\phi$ and $\epsilon =-\rho$ as formally 

• $\Delta 0 = 0$ as Nothing out of Nothing,

where $\Delta$ is the Laplacian differential operator with respect to a Euclidean coordinate $x$, $\phi (x)$ is a gravitational potential with corresponding mass density $\rho (x)$, and $\psi (x)$ is an electric potential with corresponding charge density $\epsilon$. 

The potentials generate a gravitational force $F_g$ expressing Newton's Law of Gravitation and an electric force $F_e$ expressing Coulomb's Law:  

• $F_g=-\rho\nabla\phi$                            (1)
• $F_e =\epsilon\nabla\psi = \rho\nabla\phi$      (2)

with the force balance: 

• $F_g+F_e=0$. 

We consider an initial zero state as cancelling small-amplitude micro-scale fluctuations $\phi$ and $\psi =-\phi$ generating by the action of the Laplacian large-amplitude micro-scale cancelling fluctuations $\rho$ and $\epsilon =-\rho$ both of variable sign, while the corresponding forces have the following characteristics:

• $F_g$: attraction/repulsion between mass density of same/opposite sign.
• $F_e$: repulsion/attraction between charge density of same/opposite sign.

By the action of the gravitational force, macro-scale collections of large-amplitude positive and negative mass density can form by attraction, which by repulsion separates into widely separated supermacro-scale Universa with either positive or negative mass density, where our Universe has positive mass. 

The action of the electric force is opposite with the effect that large-amplitude micro-scale fluctuation into positive and negative charge will be maintained in the form of atoms, while macro-scale electric effects cancel to zero. It is natural to assume that the initial fluctuation has atomic scale. 

More precisely, atoms can form on micro-scale by attractive electric forces while microscope repulsive gravitational forces can cause an expansion in space separating electric and gravitational effects into short and long range.

We thus find that from a unified zero state split into balancing gravitational and electric potentials an evolution may follow by balancing gravitational and electric potentials/forces into a Universe with positive mass density subject to gravitation as a large-amplitude macro-scale force, combined with a variable-sign charge density subject large-amplitude micro-scale electric force.  

The basic ingredient to this soup is the Laplacian differential operator capable of generating balancing potentials with corresponding initially balancing mass/charge densities subject to initially balancing gravitational/electric forces. Gravitational force then comes out to act on macro-scale while Coulomb electric force acts on micro-scale. 

This can be seen as a unified field model including both gravitation and electrics generated from a common source balancing to zero. Magnetics is then to be added into electromagnetics with light/radiation as new phenomenon.

The micro-scale of Coulomb electric force connects to Schrödinger's equation for quantum mechanics and it remains to understand how the split into micro-scale of electrics and macro-scale of gravitation is formed.

We see that that Newtonian gravitational forces and electric Coulomb forces may have a common origin so that in particular there is no incompatibility between Newtonian gravity and electromagnetism/quantum mechanics.  On the other hand, Einstein's theory of gravitation is viewed to be incompatible with quantum mechanics, an idea which has traumatised modern physics. It appears to be possible to get out of the the trauma by returning to Newtonian gravity, but this not allowed since progress requires that Einstein has dethroned Newton.  

One-Stage of scenario

Mass density from gravitational potential with Newtonian gravitational force:

• Initial small-scale mass density $\rho (x)$ of variable sign expands by repulsion into macro-scale.
• Regions of positive and negative mass density form which reced by repulsion. 
• A Universe with positive mass density forms separated from negative counterpart.
• Gravitational collapse within Universe form galaxies of stars with planets. 

Charge density from electric potential with electric Coulomb force:

• Initial small-scale charge density $\epsilon (x)$ of variable sign representing protons and electrons contracts by attraction into micro-scale of Hydrogen atom as a proton kernel surrounded by electron cloud. 
• Gravitational collapse gives energy to fusion of electron and proton into neutron as electron kernel surrounded by proton cloud, and atomic kernels of protons and neutrons surrounded by electron cloud.
• Atomic kernels formed by attractive gravity force (strong force) stronger than repulsive Coulomb force on micro-micro scale.   

The main idea is thus

• variable sign mass and charge energy densities initially emerge from balancing gravitational and electric potential fluctuations with balancing gravitational and electric forces
• gravitational forces drive expansion of mass density into macro-scale
• electric forces drive contraction of charge densities into micro-scale
• altogether forming a Universe of positive mass density with gravitational attraction on macro-scale combined with electric forces acting on variable charge densities on micro-scale
• combined with attractive gravitational force on micro-micro-scale (strong force).     

Two-Stage Scenario

Instead of a initial one-stage scenario, it is possible to speculate about a two-stage scenario where first mass density of variable sign is created by a gravitational potential micro-scale fluctuation, which expands by repulsion between mass densities of different sign into a macro-scale Universe of positive mass density with its negative counterpart disappearing out of sight while keeping some gravitational connection.  

In a second stage charge density of variable sign in the form of protons with charge +1 and electrons with charge -1 which do not overlap, may be created by an electric potential micro-scale fluctuation, which is maintained by attraction between protons and electrons. Protons and electrons do not annihilate or disintegrate and so keep their unit charge expressed as charge conservation. The second stage then connects to the first stage by assigning positive mass density to both protons and electrons with ratio of 1836 to 1. The scene is then set for formation of atoms followed by gravitational collapse into stars and galaxies and heavier atoms. 

The spatial extension of both charge and mass density may vary and be different for both protons and electrons depending on circumstances, with a proton having less extension than an electron in an atom. Different charge and mass density extensions may explain why electric forces dominate on atomic and gravitational forces dominate on macro-scale. 

In a Hydrogen atom a point-like proton is surrounded by an electron density cloud. It is possible to think of a neutron as a point-like electron surrounded by a proton cloud. 

The strong nuclear force may possibly be seen as a micro-micro scale gravitational force resulting from a very small mass density extension of a proton and neutron. 

The micro-scale is in precise terms expressed in the form of Real Quantum Mechanics with atoms as non-overlapping clouds of positive and negative charge densities held together by attractive electric forces.  

Summary

1. Gravitational potential fluctuation $\phi (x)$ creates mass density $\rho (x)=\Delta\phi (x)$ of variable sign. Repulsion between mass densities of different sign drives expansion into large scale separation of a Universe with positive mass density or matter from a counterpart with negative mass density. 
2. Electric potential fluctuation $\psi (x)$ creates charge density $\epsilon (x)=\Delta\psi (x)$ of variable sign. Attraction between charge densities of different sign drives contraction into micro-scale of protons and electrons of unit charge emerging as gravitational matter and combining into Hydrogen atoms. 
3. Gravitational collapse of a minor part of the matter in Universe creates by fusion from Hydrogen visible galaxies of stars, while the remaining major part remains as invisible gravitational dark matter.  

      

 

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.

Credibility Gap of Exact Prediction from Theory

Precession of Mercury 0.001 degrees per century.

Suppose you have a theory about a complex physical system, like our Solar system or atom physics or global climate,  which you want to sell to the World by showing that it gives predictions in exact agreement with observations. Not in fair agreement with observation, but exact agreement. 

Then you have a problem because observation is subject to error and you cannot expect that your theory about a complex system, which is not known in detail, can be in exact agreement with observations subject to error. This problem can be handled if the observation is known to you before you make your "prediction" from theory, in which case you can fix your model parameters and computation so that your theoretical model appears to agree exactly with observation. Not only in fair agreement, but in exact agreement. But this would not be real theoretical science, just data fitting.

So you should be cautious when you meet predictions from theory which exactly agree with observations (which cannot be exact), and ask yourself if the exact agreement is the result of a posteriori data fitting, or if the prediction was made without prior knowledge of the observation.

Here is a key example: As seen from Earth the precession of Mercury's orbit is measured to be 5600 arc seconds per century (one arc second=1/3600 degrees). Newtonian gravity, taking into account all the effects from the other planets and the Sun predicts a precession of 5557 arc seconds per century. There is a (very small discrepancy) of 43 seconds of arc per century. Newtonian gravity thus showed to be in very good agreement with observation for the complex System system not known in exact detail.  

When Einstein came up with his General Theory of Relativity GR as an improvement of Newtonian gravity, he made a simple computation which exactly delivered the missing 43 arc seconds. Exactly. Einstein then started from the Newtonian prediction of 5557 and added a contribution from GR by a back-of-the-envelope computation and arrived at exactly the missing 43 arc seconds. Einstein did not compute 5600 by applying GR to the complex Solar system, which is impossible because GR is so complicated, but used GR only to make a simple adjustment of Newtonian gravity doing the main job of predicting 5557. 

In any case Einstein managed to sell that GR predicts/computes the precession of Mercury in exact agreement with observation, which Newtonian gravity does not, and so GR has to replace Newtonian gravity. This has become the fundamental pillar of modern physics. Convincing?

The story is the same with Einstein's GR prediction of the aberration of light in exact agreement with  Eddington's 1919 Solar Eclipse observation. 

Another key example is the prediction of the fine structure constant of Quantum ElectroDynamics QED in exact agreement with measurement (up to 0.000000001), used to claim that QED is the most accurate physical theory even more precise than Quantum Mechanics, which is always in exact agreement with observation, although computations in both cases are exceedingly difficult even for simple systems.   

But the gravitational constant is not known better than 0.0003 and there is no theoretical prediction, unless it boils down to data-fitting.   

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. 

New Newtonian Cosmology: Universe from Nothing

New Newtonian Cosmology Magics

In recent posts I have tested the idea that mass of density $\rho (x)$ is "assigned" to matter through a gravitational potential $\phi (x)$ by $\rho (x)=\Delta\phi (x)$, where $\Delta$ is the Laplacian differential operator with respect to a Euclidean space coordinate $x$. The gravitational potential $\phi (x)$ also generates a gravitational acceleration $-\nabla\phi (x)$ acting on matter, where $\nabla$ is the gradient with respect to $x$, with $-\rho (x)\nabla\phi (x)$ the corresponding gravitational force density. I refer to this as New Newtonian Cosmology.

The idea is that mass density can arise from a small oscillation $\phi (x)$  of a zero ground state of the gravitational potential "creating" by differentiation an amplified oscillating mass density of variable sign with repulsion between matter of positive and negative mass densities into separated domains or Universa with positive and negative mass density. 

Focussing on a Universe with positive mass density $\rho (x)=\Delta\phi (x)$, we can see it arising form amplification of a small oscillating perturbation $\phi (x)$ of a zero gravitational potential. If we take into account the extremely small average mass density of the visible Universe (around 8 protons per $m^3$), we see that an even smaller original perturbation $\phi (x)$ would have been enough. 

It is thus possible to see a primordial Universe with very small evenly distributed matter/mass being created from a very small perturbation of a gravitational potential. In a next step of formation gravitation would then concentrate distributed matter into galaxies and stars, supplied with kinetic and heat/nuclear energy balanced by a decrease of gravitational potential energy. 

New Newtonian Cosmology thus offers a creation process without the massive input of energy required in a Big Bang. The key steps of "creation from nothing" are

1. Extremely small distributed positive mass density from an even smaller gravitational potential perturbation combined with gravitational separation from negative mass density.
2. Extremely strong concentration of positive mass density by gravitational collapse. 

This process can be complemented with a partially analogous process of generation of positive and negative charge from a perturbation of an electric potential supported by energy from gravitational collapse. 

In the previous post we observed that the "nothing" is not really nothing since we start with a zero state $\Phi =0$ of the equation $\Delta\Phi =0$ assuming the existence of Euclidean space.  

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

A Stage for Big Bang as Separation: $0 =\Delta\phi$

The previous post presented a stage for the creation of gravitational mass and electric charge as the fundamental components of the World, in the form of the Laplacian differential operator $\Delta$ with respect to a Euclidean space coordinate $x$ acting on a potential $\Phi (x)$ satisfying the equation 

• $\Delta\Phi (x) = 0$ for all $x$.     (1)

With $\Phi (x)$ vanishing for large $x$ the potential $\Phi (x) = 0$ for all $x$ and so represents a null state like a guitar string at rest before plucking or a stage prepared for a play.   

Suppose now the null state $\Phi$ is subject to a highly oscillating perturbation $\phi (x)$ and consider the  assignment:

• $\rho (x) =\Delta\phi (x)$ for all $x$,    (2)

where $\rho (x)$ can be large even if $\phi (x)$ is small because differentiation makes small oscillations big, and so a substantial $\rho (x)$ can emerge from a small perturbation. 

It is possible to view this as a form of Big Bang with creation seemingly out of nothing.  Here (1) sets a stage of possibilities and (2) represents realisation of possibilities as creation of $\rho (x)$ representing  mass or charge density. 

Note that (2) amounts to a "separation" into positive and negative mass/charge out of a zero state, and so the "creation" of $\rho (x)$ is the result of a process of separation, maybe easier to understand than direct creation out of nothing: God separated the light from the darkness.

Also recall that Hesiod's Chaos has been interpreted as the gaping void above the Earth created when Earth and Sky are separated from their primordial unity. The prerequisite for creation is a Euclidean space with a Laplacian differential operator prepared for separation.  

Even if this model "explains" creation of mass and charge, the question remains of how the geometry of Euclidean space with Laplacian as the stage (1) is prepared. Maybe Euclide knew but didn't tell.  

Note if the "creation" happened once it may happen again. Separation does not require massive input of energy and the required energy to form create matter/radiation can come from gravitational collapse in a net zero game, once positive and negative mass have been separated into new Universa.  

For a more detailed specification see the previous post New Newtonian Cosmology and chap 32 in Computational Thermodynamics. In particular, notice conservation of total energy as kinetic energy plus heat energy minus gravitational energy, where gravitational energy thus can feed kinetic and heat energy. This gives an answer to the question from where the energy comes in a conventional Big Bang scenario. 

A notable aspect in the case of separation into positive and negative charges connecting to Real Quantum Mechanics, is that of charge conservation as a result of the conservation law of electromagnetics: 

• $\frac{\partial\rho}{\partial t}+\nabla\cdot J =0$

where $J=\rho v$ current with $v$ charge velocity, from which follows that 

• $\int_{\Omega}\rho dx = constant$, 

where $\Omega$ is the domain where $\rho (x)>0$ or where $\rho (x)<0$ separated by a boundary where $\rho (x)=0$. In particular this motivates why in Real Quantum Mechanics a proton meets an electron with vanishing charge densities, answering a question raised in this post. The number of protons which is equal to the number of electrons does not change over time.  

 

New Newtonian Gravity vs Electromagnetism

Assignments of different capacities by the Creator 

Newtonian Gravity and Electromagnetism include the same mathematical model in the form $\rho (x)=\Delta\phi (x)$ with $\Delta$ the Laplacian with respect to a space coordinate $x$ with the following interpretations:

• Gravitation: $\rho (x)$ mass density and $\phi (x)$ gravitational potential.
• Electromagnetism: $\rho (x)$ charge density and $\phi (x)$ electric potential.

In New Newtonian Gravity we view $\phi$ to be primordial from which $\rho$ is assigned/created by differentiation as an instant local operation. It is natural to take the same view in electromagnetism with charge created from electric potential by differentiation. The corresponding force laws have opposite signs resulting in opposite attractive/repulsive forces between mass/charge (Newton's Law of Gravitation and Coulombs Law): 

• Gravitational force $F(x)=-\nabla\phi (x)$: attraction/repulsion between mass of same/opposite sign.
• Coulomb electrical force $F(x)=\nabla\phi (x)$: repulsion/attraction between charges of same/opposite sign.

Gravitational mass densities of different signs repel each other leaving a Universe with only positive mass density. Charge densities of different sign attract each other leaving a Universe with both positive and negative charges as protons and electrons. 

An important feature of an assignment process like $\rho (x)=\Delta\phi (x)$ is lack of self-gravitation in the sense that that there is no net gravitational force acting on a body from the gravitational mass assigned to the body by the gravitational potential as being primordial. A body cannot itself assign a net gravitational force acting on itself.  You cannot lift yourself in the hair. The Creator/Natural Selection gives life which cannot create itself all by itself. 

Another important feature of the New Newtonian Gravity translated to electromagnetism is that the attraction between positive and negative charges bringing protons and electrons into contact must be balanced by some mechanism maintaining separation and preventing annihilation. This is a core element of Real Quantum Mechanics including contact conditions between protons and electrons. 

Fundamental Theorem of Calculus as Fundamental Physics

Recent posts have presented New Newtonian Gravity as a new way of viewing the connection between gravitational potential $\phi (x)$ and gravitational mass density $\rho (x)$ through the assignment $\rho (x)=\Delta\phi (x)$, where $\Delta$ is the Laplacian differential operator acting with respect to $x$ as an Euclidean space coordinate. Gravitational force at $x$ is given as $\nabla\phi (x)$ with $\nabla$ the gradient with respect to $x$. 

The gravitational potential $\phi (x)$ is here viewed to be primordial somehow generating mass density $\rho (x)$ by differentiation as a process of local instant action, to be compared with the classical view with mass density primordial generating gravitational potential by instant action at distance lacking physics.

The connection/assignment $\rho (x)=\Delta\phi (x)$ is the result of:

1. Observing that gravitational force $F$ is conservative (work independent of path) shows that $F=\nabla\phi$ for some $\phi$.
2. Conservation of gravitational flux/force $F$ of the form $\nabla\cdot F = S$ with $S$ source of $F$.
3. Assignment $\rho = S$. 

Here 2 can be seen to express the Fundamental Theorem of Calculus in the form of Gauss Theorem:

• $\int_\Gamma F\cdot n ds =\int_\Omega \nabla\cdot F dx$,

with $\Omega$ a domain with boundary $\Gamma$ with outward unit normal $n$, which expresses

• total flux out of a domain = total source inside domain 

as a law of conservation. 

In the case of one space dimension Gauss Theorem reduces to the Fundamental Theorem in the form 

• $F(b) - F(a) = \int_a^b\frac{dF}{dt}dt = \sum dF$  

which expresses that the total change of $F(x)$ as $F(b)-F(a)$ is equal to the integral/sum of little changes $dF$ or that "the whole is the sum of its parts".

The connection $\rho (x)=\Delta\phi (x)$ thus is the result of: 

1. Observation that gravitational force is conservative.
2. Conservation of gravitational flux.
3. Assignment of mass density. 

Is it possible that 1 or 2 as conservation laws can be violated? This connects to the question of existence of dark matter arising from observing gravitational flux/force for which the corresponding source appears to be missing because it cannot be seen. Insisting on conservation means that the source must be there as an invisible source = dark matter. Giving up conservation means giving up science with a resort to mysticism.

A conservation law reflects a scientific principle as a form of book-keeping where everything is recorded and balanced so that In = Out. With this view 2 is true a priori without need of experimental verification:

If something is missing, then that something must exist upon more careful inspection. There must be some dark energy explaining to observation of gravitational force/flux. Or it is just a mystery.

There is also very good reason to believe that Nature requires conservation in order to exist. Constant input without corresponding output leads to blow-up, and constant output without input leads to extinction. Total energy must be conserved. 

Recall that the 1st Law of Thermodynamics states that total energy is conserved. In particular, kinetic energy is transformed into heat energy by friction and turbulent dissipation accounted for as losses as parts  of an energy budget. Whatever disappears in one column of the budget as loss appears as gain in another column.    

Altogether, there seems to be very good reasons to believe that New Newtonian Gravity captures real physics and so little/no reason to believe that Einsteinian Gravity can offer something better.   

Einstein questioned Newtonian Gravity and wave nature of light, but he never questioned Conservation of Energy. Why?   

New Newtonian Gravity Including Dark Matter

This is a continuation of previous posts on the gravitational potential $\phi (x)$ as primordial giving mass density $\rho (x)$ to matter by differentiation as instant local action 

• $\rho (x) =\Delta\phi (x)$      (1)

where $\Delta$ is the Laplacian differential operator with respect to a 3d space coordinate $x$. It is assumed that $\rho (x)$ is non-negative expressing absence of negative mass density. 

This is a non-standard view to be compared with the standard view with mass density primordial from which the gravitational potential is created as solution to the differential equation $\Delta\phi =\rho$ which is a global summation process appearing as instant action at distance. 

Both views, referred to as classical Newtonian gravity and new Newtoninan gravity formally express Newtonian gravity but they represent different physics. The big seemingly insurmountable problem of oldNewton since the days of Newton, is the physics of instant action at distance, which does not appear in newNewton. 

Further, newNewton opens to a distinction between visible matter and invisible matter or dark matter, through the smoothness or $\phi(x)$ that is the size of $\Delta\phi (x)\ge 0$:

1. $\phi (x)$ produces visible matter where $\Delta\phi (x)$ is large.
2. $\phi (x)$ produces invisible dark matter where $\Delta\phi (x)$ is small-moderate.     

You find here an very instructive discussion of the experimental evidence of dark matter including the following graph of mass distribution over a flat galaxy cluster with spotted high peaks of visible matter rising over a background of invisible dark matter (with clear separation): 

Notice that the total mass of the dark matter as present everywhere can be much larger than visible matter with pointed presence. 

I have argued that Newtonian gravity may represent a priori knowledge about the World which leaves Einsteinian gravity without role, with new Newtonian gravity in a natural way including dark energy.    

Explaining Spiral Galaxy Rotation Speed with or without Dark Matter

A spiral galaxy is flat and so may not need dark matter to explain constant rotational speed.

This is a continuation of previous posts on the gravitational potential  $\phi (x)$ as primordial giving mass density $\rho (x)$ to matter by the specification $\rho =\Delta\phi$ where $\Delta$ is the Laplacian differential operator with respect to a space coordinate $x$ of dimension $d$. 

If $d=3$ then $\phi (x)=\frac{1}{4\pi\vert x\vert}$ generates a unit point mass at $x$. The corresponding gravitational force is given by $\nabla\phi (x)\sim -\frac{1}{\vert x\vert^2}$. Balancing inertial and gravitational force for a body moving around the unit point mass in a circle with radius $R$ with speed $V$ gives the following relation:

• $V \sim \frac{1}{\sqrt{R}}$    (1)

If $d=2$ then $\phi (x)=-\frac{1}{2\pi}\log{\vert x\vert}$ generates a unit point mass at $x$. The corresponding gravitational force is given by $\nabla\phi (x)\sim -\frac{1}{\vert x\vert}$. Balancing inertial and gravitational force for a body moving around the point mass in a circle with radius $R$ with speed $V$ gives the following relation:

• $V \sim 1$      (2)

We see a speed decay with increasing $R$ of $V\sim\frac{1}{\sqrt{R}}$ if $d=3$, while $V=constant$ independent of $R$ if $d=2$. 

Observations of the velocities of spiral galaxies agree with (2), which has a logic because spiral galaxies are flat disc-like connecting to $d=2$, see figure above. 

The reason planets and stars are spherical is that gravitation is balanced by pressure acting in all directions. The reason galaxies are flat is that gravitation is balanced by rotational inertial forces.  

The motivation to introduce dark matter is that observations do not agree with (3), asking for a spherical halo of dark matter to give extra gravitational force to motivate non-decay of speed. But no dark matter has not been found. 

The explanation of agreement with observations with $d=2$ is elementary, and is also made here. But how is it possible that this has escaped the minds of modern physicists? Or has it been discovered as being too simplistic? Is a spiral galaxy really a 2d object? Does the specification of mass density according to $\rho =\Delta\phi$ really take place in 2d? The creation process of mass density from gravitational potential may carry an answer.

Note that $d=2$ gives a form of MOND as Modified Newtonian Dynamics, which with the gravitational potential being primordial perhaps can be motivated as a geometrical effect (to be returned to). Or it is simply that if $\Delta\phi (x)$ is not intense enough, then $\rho (x)$ is not visible as dark matter.  More in the next post.

Also recall that with the gravitational potential being primordial, there is neither instant action at distance to explain, only local action.

Newtonian gravity may be seen as a form a priori knowledge in the sense of Kant as knowledge which can be deduced without experimental input. It lives up to Einstein's ideal of a mathematical model without parameter to determine by experiment. 

Compare with Euler CFD as parameter-free model of fluid flow.

Observation Obsession of Modern Physics

Do you believe that what you cannot see or hear does not exist?

Modern physics in the form of quantum mechanics is troubled for several reasons by the fundamental role given to observation including measurement as the expression of an instrumental antirealist view focussing on only what you can see or directly observe/measure, as compared to a wider realist view of what is.  

First, the measurement process is viewed to interfere with the system state being observed/measured, which is the problem of "collapse of the wave function" into a specific result from many possibilities as an effect of the measurement process. This is the observation of a dead or alive Schrödinger cat decided by the process of opening the Schrödinger box. Totally mysterious! 

Second, the Heisenberg Uncertainty Principle states that "complementary observables" such as position and velocity/momentum cannot both be measured with precision. Mysterious!

Third, the interpretation of the wave function as a probability is filled with mystery. What is the meaning of making one measurement of a an observable which is a probability? What do you learn by tossing a dice once?

Fourth, focussing on direct observation/measurement leads to the extreme view that the Sun evaporates to nothing when it settles in the evening, only to reappear every morning. Totally mysterious!

This aspect connects to the previous post on the connection between gravitational potential $\phi (x)$ and mass density $\rho (x)$ through the relation $\rho =\Delta\phi$ where $\Delta$ is the Laplacian differential operator with respect to $x$ as Euclidean space coordinate. 

The standard view is to consider mass density to be primordial because we can directly see the Sun, while we cannot directly see the gravitational potential of the Sun. This view comes with the unresolvable problem of instant action at distance. Very troubling!

But if we give up the antirealist instrumental view, then we may consider the gravitational potential $\phi$ to be primordial from which mass density $\rho$ is obtained by differentiation as an instant local operation, which admittedly has mysterious aspects but does not suffer from the unresolvable troubles of the standard view.

Summing up, we have thus found arguments against a philosophy in the spirit of Niels Bohr focussing on what can be measured rather than a wider ontology of what is.   

What is Wrong with Newtonian Gravity?

This is a continuation the previous post on the Universality of Newtonian Gravity. If Newtonian Gravity is Universal, how come that this is not what modern physics is saying? In what sense was Newton wrong?

Here are the main answers to the question What is Wrong with Newtonian Gravity?:

1. Newtonian Gravity is an “action-at-a-distance” theory. Newton himself was deeply concerned about this.
2. Newtonian Gravity is not compatible with the world of special relativity. 
3. Newtonian Gravity is contradicted by observation of the perihelion advance of Mercury and gravitational waves.
4. Newtonian Gravity does not interact with propagation of light. 

Let us now analyse the validity of these arguments. In our non-standard view with gravitational potential primordial, there is no action at distance and so 1. is not an issue. 

Special relativity is a theory empty of real physics as shown in Many-Minds Relativity and so 2. is neither an issue. 

The claimed disagreement of Newtonian Gravitation with observations of the perihelion advance of Mercury may very well depend on neglect of proper influence from other planets and from the Sun. Until all such aspects have been taken properly into account, it is impossible to say that Newtonian Gravitation is wrong and so 3. is not necessarily an issue.

Newtonian Gravitation concerns dynamics of particles/bodies with mass and so say nothing about interaction with massless light, and so 4. is neither necessarily an issue. 

Summing up, we find that the reasons behind viewing Newtonian Gravity to be wrong are weak. The price to pay by dismissing it is very high, since Newtonian Gravity appears as the only Theory of Everything in  physics.  

Universality of Newton's Law of Gravitation

Newtonian Gravitation and Quantum Mechanics is a Theory of Everything ToE.

This is a continuation of previous post on the connection between gravitational potential $\phi (x)$ and mass density $\rho (x)$ through the relation $\rho =\Delta\phi$ where $\Delta$ is the Laplacian differential operator with respect to $x$ as Euclidean space coordinate. 

We take here a non-standard view and consider the gravitational potential $\phi$ to be primordial, from which mass density $\rho =\Delta\phi$ is given to matter by differentiation as an instant local action. 

This is to be compared with the standard view with $\rho$ primordial and $\phi$ obtained by solving the equation $\Delta\phi =\rho$ as a global integration/summation process requiring instant global action or instant action at distance.

Here instant local action, like establishment of contact forces, does not pose the problem of instant action at distance, which may be unsolvable. 

Let us now see how the connection $\rho =\Delta\phi$ can be established viewing $\phi$ as primordial. 

The first step is to see that $F=\nabla\phi$ represents gravitational force field and as such is a conservative force, which means the the work required to move a body from one point to another is independent of the path. As an example the work required to lift a body in the gravitational force field of the Earth only depends on change of level and not path with motion without change of level not requiring work. The gravitational force field is thus given as the gradient of a potential.

The next step is to show that $\nabla\cdot F=\rho$, where $\nabla\cdot F$ is the divergence of $F$. We recall from Calculus  that 

• $\int_\Omega \nabla\cdot F\, dx =\int_\Gamma F\cdot n\, ds$

for a volume $\Omega$ with boundary $\Gamma$ with outward unit normal $n$ with $dx$ volume element and $ds$ boundary surface element.  Viewing $F$ as a flux, $F\cdot n$ is the flux out of $\Omega$ and $\nabla\cdot F$ is the corresponding source inside $\Omega$. 

We can now (simply) define gravitational mass density $\rho =\nabla\cdot F=\nabla\cdot\nabla\phi =\Delta\phi$ to be the source of the gravitational field given by the potential $\phi$, and we can from this relation derive Newton's Law of gravitation in its familiar form as did Laplace in his Celestial Mechanics. 

We see that Newton's Law of gravitation reflects (i) observation that a gravitational force field is conservative, together with (ii) definition of gravitational mass density as the source of this field. 

Newton's Law of Gravitation thus comes out from assuming that a gravitational force field is conservative, which is supported by observation, together with definition of gravitational mass. 

Connecting to Newton's 2nd Law of motion with inertial mass the same as gravitational mass, brings universality to cosmic interaction.

We conclude that Newton's Law of Gravitation can be seen much like an a priori statement about physics in the sense of Kant. We compare with the a priori statement that the length of the circumference of a circle with radius $1$ is equal to $2\pi$, which has universal validity. 

This gives Newton's Law of Gravitation universal validity leaving Einstein's General Theory of Relativity without mission. In particular, it resolves the main dilemma of modern physics of the perceived incompatibility between Quantum Mechanics and General Relativity, since Newtonian gravitation is fully compatible with Schrödinger's quantum mechanics as a Theory of Everything ToE.  

We recall Niels Bohr:

• How wonderful that we have met with a paradox. Now we have some hope of making progress.          

How wonderful that we have found incompatibility of Quantum Mechanics and General Realativity. Now we can make some progress...

Newton's Cosmic Model: Irony of Modern Physics

• From the same principles, I now demonstrate the frame of the System of the World. (Newton)

This is a continuation of the previous post on the primordial role of a gravitational potential $\phi (x)$ as giving gravitational mass $\rho (x)=\Delta\phi$ to matter through the Laplacian operator $\Delta$ as instant local action by differentiation with $x$ a 3d Euclidean space coordinate. This is Newton's Law of Gravitation describing gravitational force.  

We obtain a model of cosmic interaction as a set of point masses represented by $\rho (x,t)$ moving according to Newton's 2nd Law of Motion $F=am$ connecting inertial force $F$ to acceleration $a$ and mass $m$, subject to gravitational force $\nabla\phi (x,t)$ with $t$ a time coordinate. You can follow the dynamics by starting  Cosmic Interaction on Leibniz World of Math and clicking in point masses. 

Combining Newton's Law of Gravitation and 2nd Law of Motion we thus have a mathematical model capable of describing cosmic interaction on any scale as point masses all "falling freely" with gravitational and inertial forces balancing to net zero force. This is the weightless zero-stress state so happily experienced by Stephen Hawking when visiting Virgin Galactic: 

We understand that it is the gravitational potential/force which creates the dynamics of cosmos through the key creative steps of 

1. Giving gravitational mass to matter through $\rho =\Delta\phi$ according to Newton's Law of Gravitation. 
2. Stipulating that inertial mass = gravitational mass to invoke Newton's Law of Motion.    

This is Newton's Cosmic Model describing cosmic interaction on all scales. It is a computable model of extreme simplicity with only one parameter $G$ connecting gravitational force to mass and distance, yet with a universal scope. This is a triumph of classical physics which has not been matched by modern physics. 

We can compare with the Standard Model of particle physics as an uncomputable model of extreme complexity including 19 parameters with a very limited scope:

Newton's Cosmic Model is not a part of modern physics based on the Standard Model without gravitation and Einstein's General Theory of Relativity with gravitation viewed to be incompatible. The unmatched triumph of classical physics is thus not a part of modern physics. This is deeply ironic. It does not help that Einstein ask for mercy: Newton forgive me! It is a travesty.