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.