Mass: Inertial = Passive Gravitational = Active Gravitational

Maybe, it is now time for Newton to return? Maybe I was wrong? In any case: Newton forgive me!

The basic model of Neo-Newtonian Cosmology takes the following form in terms of gravitational potential $\phi$ as primordial, mass density $\rho$, momentum $m$ and $u=\frac{m}{\rho}$ material velocity all depending on a Euclidean space coordinate $x$ and time coordinate $t$ with the dot on top representing differentiation with respect to time:

• $\rho = \Delta\phi$                                    (inverse square law in differential form)
• $\dot\rho +\nabla\cdot m=0$                       (conservation of mass)
• $\dot m +\nabla\cdot (um)+\rho\nabla\phi=0$    (conservation of momentum: Newton's 2nd Law)  

A basic property of this model is conservation of energy in the sense that total energy $KE + GE$  with $KE$ total kinetic energy and $GE$ total gravitational energy, does not change over time: 

• $\frac{d}{dt}(KE+GE) = 0$        (conservation of total energy)

with 

• $KE = \int \rho\frac{u^2}{2} dx$
• $GE = -\frac{1}{2}\int\vert\nabla\phi\vert^2dx$. 

In this model we have identified 

• inertial mass = passive gravitational mass                        (Galileo's Equivalence Principle EP)
• passive gravitational mass = active gravitational mass     (Newton's EP)

with 

• inertial mass = mass $\rho$ in $\rho\frac{u^2}{2}$
• passive gravitational mass = mass $\rho$ in $\rho\nabla\phi$
• active gravitational mass = mass $\rho$ in $\Delta\phi =\rho$. 

In other words, the model has only one form of mass appearing in three different contexts with equality expressed as Galileo's EP and Newton's EP as fundamental features of Newtonian mechanics in the form of Neo-Newtonian Cosmology.

We recall that Einstein used Galileo's EP as a Postulate for his General Relativity (probably also with less exposure Newton's EP). 

Leibniz' Principle of Sufficient Reason shows that Galileo's EP (universality of free fall) cannot be false since sufficient reason is lacking, while violation of Newton's EP would violate conservation of total energy. 

Conclusion1: The basic model of Neo-Newtonian is simple, universal and computable, and it is unthinkable that it is not fundamentally correct. There is no sufficient reason to replace it by General Relativity, which is the opposite of both simple and computable. Recall that a main dilemma of modern physics is the apparent incompatibility/contradiction between its fundamental pillars of Einstein's relativity and Schrödinger's quantum mechanics, while Newton and Schrödinger are fully compatible. More precisely, the incompatibility of the pillars means that modern physics lacks foundation and that is a root cause of the present crisis witnessed by everybody including Sabine Hossenfelder in next post. Contradictory physics cannot be real physics. You cannot turn both left and right at the same time without breaking apart.

Conclusion2: Kinetic energy KE can increase from decreasing GE from increasing $\vert\nabla\phi\vert^2$ from concentration of mass as the engine of gravitational collapse, which gives an endless source of kinetic energy. 

Question: Neo-Newtonian Cosmology and Einstein's Cosmology/General Relativity both start from Galileo's EP + Newton's EP. Does it mean that Einstein agrees with Newton? Probably not, since the basic idea of modern physics is that Einstein does not. The difference appears to be that Einstein demands Lorentz invariance as his incarnation of a Principle of Relativity, while Newton is happy with Galilean invariance. So we have Einstein-Lorentz against Newton-Galileo. What is your bet? Newton-Galileo reigned from 1600 to 1955, while Einstein-Lorentz took over in the 1970s along with the decline of modern physics into our days. Time for Renaissance of Newton-Galileo?

PS1 Recall that the Lorentz transformation lacks physical meaning and so must Lorentz invariance, while Galilean invariance expresses real Newtonian physics. 

PS2 Recall that the above model naturally suggests low intensity distributed $\rho =\Delta\phi\ge 0$ as dark matter and naturally extends to negative $\rho =\Delta\phi$ of as origin of dark energy. Here ordinary matter (protons and electrons) are assigned Inertial = Passive Gravitational Mass, while it is possible (but not necessary) to make a distinction for dark matter with only a role as Active Gravitational Mass thus without Inertial = Passive Gravitational Mass (recall this post). This would mean making a distinction between matter and mass with ordinary matter being assigned mass as Inertial = Passive Gravitational Mass, while dark matter would not be assigned any and so represent mass but not matter..  

Visible Matter vs Invisible/Dark Matter

The apparent gravitational presence of invisible/dark matter is a key open problem of modern cosmology. 

Recent posts seek an answer as Neo-Newtonian Cosmology NNC starting with a primordial gravitational potential $\phi$ as a rapidly oscillating small amplitude perturbation of a zero potential from which mass density 

• $\rho\equiv\Delta\phi\ge 0$ 

is created through the action of the Laplacian differential operator $\Delta$ acting in a Euclidean infinite space. 

We assume a decomposition $\rho =\rho_v +\rho_{iv}$ with $\rho_v$ as substantial and localised representing visible matter, and $\rho_{iv}$ as small and distributed representing invisible/dark matter with clear separation. With $u=u_v+u_{iv}$ as total matter velocity decomposed into velocity $u_v$ of visible matter and $u_{iv}$ that of invisible dark matter, NNC takes the following concise form:

1. $\rho_v+\rho_{iv} = \Delta\phi$                           (inverse square gravitational law in differential form)
2. $\dot\rho_v =-\nabla\cdot (\rho_v u_v)$                   (conservation of visible matter)
3. $\dot\rho_{iv} =-\nabla\cdot (\rho_{iv} u_{iv})$                 (conservation of invisible/dark matter)
4. $\dot u_v=-\nabla\phi$                                    (Newton's 2nd Law for visible matter) 
5. $\dot u_{iv}=-\nabla\phi$                                  (Newton's 2nd Law for invisible/dark matter) 

We see that the total gravitational force $-\nabla\phi$ acts on both visible matter and invisible/dark matter in qualitatively different ways, with precise motion of visible matter as high density + localised and imprecise motion of invisible/dark matter as low density + distributed. 

The observed velocity distribution of visible matter of spiral galaxies can now be understood as driven by gravitational force from $\nabla\phi$ resulting from presence of high density localised visible matter together with presence of  low density distributed invisible/dark matter.

The key assumption is that visible matter as high density localised in the form of protons and electrons carries electromagnetics, while invisible/dark matter as low density distributed of unknown form, does not. 

Here high density localised represents stars and galaxies separated by large apparent voids, while low density distributed may represent a large halo around a galaxy, see below. 

It is natural to extend to dark energy as $\Delta\phi <0$ as small + distributed with an effect of anti-gravity or repulsion between matter of different sign (and attraction for same sign).

NNC naturally explains an expanding Universe from repulsion between matter of different sign.

Questions: 

• Why does the dark matter halo not gravitationally contract? Is the density too small? Is the rate of contraction very small? Is it possible that the invisible/dark matter does not react to gravitational force, only contributes to gravitational potential? 
• Is the visible galaxy formed from a primordial halo by gravitational contraction of in its center into visible matter? 

Neo-Newtonian Cosmology 3

Recent posts 1+2 in this series with further details in Time to Welcome Newton Back outlines a possible Neo-Newtonian Cosmology based on an expanded form of Newton's Theory of Gravitation with mass of both positive and negative sign with a corresponding gravitational force which is attractive/repulsive for mass of equal/opposite sign, where we happen to live in a visible region of positive mass with only gravitational attraction. 

Invisible regions of negative mass would then supply the mysterious dark energy estimated to contribute 70% of the total energy of the visible Universe filling the very large observed voids apparently without positive mass possibly driving the observed expansion of the visible Universe. 

The idea of an Anti-Universe to our Universe with all signs switched, with negative mass and anti-protons/electrons et cet, was suggested by Hannes Alfvén, Swedish Nobel Laureate in Physics 1970, in his book Worlds-Anti Worlds: Antimatter in Cosmology 1966. 

The possibility of dark energy in the form of negative mass has been proposed by Massimo Melli:

• Most of the substance of the Universe appears to consist of dark energy and dark matter, but until now scientists don't have the faintest idea about their origin and ignore what they are. 
• I believe that dark energy and dark matter could consist of tachyons formed during the inflationary phase of the early Big Bang, when time was slow or zero due to the extremely high density of the Universe. 

Melli refers to Big Bang, but one can equally well speculate about creation of mass of variable sign from a perturbation of gravitational potential as suggested in NNC 1+2 as a physically more possible little big-bang, with electro-magnetics similarly resulting from a perturbation of an electric potential creating charges of variable sign.  

It remains to explain why we can see presence of positive mass through interaction with electro-magnetics but not the negative mass supposedly filling the voids, then apparently without interacting with the electro-magnetics of our universe.  

Key Question: Why does negative mass not interact with visible light? 

We recall that atomic electrons (of positive mass) oscillating between two energy levels in an atom radiate electro-magnetic waves in the form of light, which can be absorbed into oscillating electrons in our eyes and so make a star visible to us. This gives a connection between positive mass and visible light.

In any case large voids are present, which does not fit with a universe of only positive mass. 

It is conceivable that large voids of negative mass can explain the observed expansion of the visible Universe. 

We recall the basic model of Neo-Newtonian Cosmology of immense simplicity and generality expressing the interplay between gravitational potential $\phi$ as primordial, mass density $\rho$ and $u$ material velocity all depending on a Euclidean space coordinate $x$ and time coordinate $t$ with the dot on top representing differentiation with respect to time:

• $\rho = \Delta\phi$                    (inverse square law in differential form)
• $\Delta\dot\phi =-\nabla\cdot (\rho u)$    (conservation of mass)
• $\vert\rho\vert\dot u=-\rho\nabla\phi$            (Newton's 2nd Law)  

Here $\rho$ is created from $\Delta\phi$ by fast-local-action, changes of $\phi$ in time are created  from changes of $\rho u$ in space by slow-global-action (see previous post) and changes of $u$ from gravitational force $-\nabla\phi$ by fast-local-action. We see three concise simple equations in three variables $\phi$, $\rho$ and $u$ describing all of celestial mechanics. Amazing! How can you ever imagine to throw away this mathematical model? Compare this post: Unthinkable that the inverse square law is incorrect.

Answer to Key Question: Dark matter as mass $\rho =\Delta\phi$ with $\vert\rho\vert$ too small to carry electro-magnetics, can by presence as negative mass in large voids drive observed expansion and as positive mass in spiral galaxies drive observed rotation. 

 

Dark Matter as Axions as 85% of All Matter?

Sabine Hossenfelder in Exploding stars made of dark matter could heat up universe informs us about some new speculations about the physics of dark matter, believed to make up 85% of all matter in the Universe, in the form of    

• axions or axion particles 

able to form 

• axion stars

able to explode and so able to  

• heat surrounding gas 

which could be a detectable phenomenon. Sabine ends asking how it is possible that physicists can be paid for this kind of speculation. 

Compare with the idea I have suggested that matter with density $\rho (x,t)=\Delta \phi (x,t)$ is formed from a gravitational potential $\phi (x,t)$ locally in space-time with coordinates $(x,t)$ from differentiation expressed by the Laplacian differential operator $\Delta$, and that dark matter corresponds to large regions where the potential is smooth in the sense that $\Delta \phi (x,t)$ is not large enough to create matter which is visible. 

It is conceivable that such large regions could concentrate gravitationally and even form stars which could explode as in the above scenario. Is anyone willing to pay for this idea? Does it make sense? 

Yes, anti-matter does anti-gravitate!

Sabine Hossenfelder asks in a recent post at Backreaction:

• Why doesn’t anti-matter anti-gravitate?

According to the gravitational model presented in my app NewMath: Darkenergy, anti-matter does anti-gravitate as a result of the gravitational law

• $\Delta\phi = \rho$

where $\phi (x,t)$ is gravitational potential and $\rho (x,t)$ mass density depending on a Euclidean space coordinate $x$ and $ t$ is a time coordinate, where $\rho (x,t)$ can be both positive and negative, and positive $\rho (x,t)$ signifies presence of normal matter and negative $\rho (x,t)$ signifies presence of anti-matter, at $(x,t)$.

This model is explored under the following categories on this blog

• dark energy
• dark matter.

Here is an output from as simulation with anti-matter (green) repelling normal matter (red) into expansion, as an explanation of the observed expansion of the universe.

Hopefully, you are now encouraged to download the app and test the model!

And God Said Gravitational Potential!

The question if we live in a simulation is discussed by Ethan Siegel with reference to the 2016 Isaac Asimov Memorial Debate: Is the Universe a Simulation?

The debate connects to my app on Dark Energy at App Store allowing you to play with a cosmological model where gravitational potential $\phi$ is primordial from which mass $\rho$ is connected through the equation $\rho =\Delta\phi$ of local differentiation with the Laplacian differential operator $\Delta$.

In this model a fluctuation of $\phi$ around a zero initial state can create massiv positive and negative mass $\rho$ through the action of differentiation with the power of amplifying fluctuations, thus generating (seemingly out of nothing) universa of positive and negative mass which repel each other and drift apart.

What we can see may thus be one universe of positive mass with its negative counterpart since long gone beyond sight, but with the repellation still felt as dark energy.

Further, visible matter can be connected to $\Delta\phi$ being singular (like a delta-function), while dark matter may correspond to $\Delta\phi$ being smooth.

In the debate the idea of gravitation as primordial "operating system" is touched upon, but is not penetrated since the common view is that it is mass which is primordial from which gravitation magically is created by magic instant action at distance. Turning this common view around making gravitation/gravitational potential primordial, may open to understanding both dark matter and energy. What do you think?

Many Big Bangs: Universe Bigger Than You Think

Astronomer Royal Lord Rees has made a statement:

• There may have been more than one Big Bang, the Astronomer Royal has said and claims the world could be on the brink of a revolution as profound as Copernicus discovering the Earth revolved around the Sun.
• Many people suspect that our Big Bang was not the only one, but there’s a whole ensemble of Big Bangs, a whole archipelago of Big Bangs.
• The theory is still highly controversial, but Lord Rees said he would ‘bet his dog’ on the theory being true.

This fits with the view I have presented in posts on a new view on gravitation, dark matter and dark energy, with gravitational potential $\phi$ viewed as primordial from which matter density $\rho$, which may be both positive and negative, is generated by 

• $\rho = \Delta\phi$, 

through local action in space of the Laplacian $\Delta$. In this model a Big Bang corresponds to a small local fluctuation of $\phi$ around zero, which generates an much bigger fluctuation of matter density by the action of the Laplacian. 

In this model substantial matter may be generated locally from small fluctuations of gravitational potential opening the possibility of an endless number of Big Bangs seemingly created out of nothing. 

You can test the model in the app Dark Energy on App Store.

Spiral Galaxy Formation in Extended Newtonian Gravitation

1. Cosmological Model 

This is a continuation of previous posts on dark matter and The Universe as Weakly Compressible Gas subject to Pressure and Gravitational Forces, which post we recall:

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

• $\dot\rho + \nabla\cdot (\rho u ) =0$       (or $\frac{D\rho}{Dt} = -\rho\nabla\cdot u$)
• $\dot m +\nabla\cdot (mu) +\nabla p + \rho\nabla\phi =0$
• $\dot e +\nabla\cdot (eu) +p\nabla\cdot u +\rho\nabla\cdot m=0$,

where $\rho$ is mass density, $u=\frac{m}{\rho}$ is matter velocity, $p$ is pressure, $\phi$ is gravitational potential, and $e$ is internal energy as the sum of heat energy $\rho T$ with $T$ temperature and gravitational energy $\rho\phi$, and the dot indicates time differentiation and

• $\frac{D\rho}{Dt}=\dot\rho +u\cdot\nabla\rho$

is the convective time derivative of $\rho$, see Many-Minds Relativity 20.3 and Computational Thermodynamics Chap 32.

These equations express conservation of mass $\rho$, conservation of momentum $m$ with $\nabla p$ pressure force and $-\nabla\phi$ gravitational force, and conservation of internal energy $e$. These laws of conservation are complemented with constitutive laws connection $p$ and $\phi$ to density, of the following form:

A1: Weakly compressible gas ($\delta$ small positive constant):

• $\Delta p =\frac{\nabla\cdot u}{\delta}= - \frac{1}{\delta\rho}\frac{D\rho}{Dt}$

or

A2: Compressible perfect gas ($0 < \gamma < 1 $):

• $p=\gamma \rho T$.

B: Newton's law of gravitation:

• $\Delta\phi =\rho$ with $\phi =0$ at infinity.            

We observe

1. Similarity of $\nabla p$ and $\nabla\phi$ in momentum equation. 
2. Similarity between A1 and B connecting $\Delta p$ to $-\frac{D\rho}{Dt}$ (or $-\rho$) and $\Delta\phi$ to $\rho$.
3. $p \ge 0$ and $\phi \le 0$.

Here 1. can be seen as the Equivalence Principle (equality of heavy and inertial mass) expressing that there is no difference between gravitational and other forces (pressure) in Newton's 2nd law expressing conservation of momentum.

Further, 2. expresses that the constitutive laws A1 and B both can be viewed as action at distance if $\rho$ is viewed as the cause, but represent local action of differentiation if $\rho$ is viewed as the effect. 

For a weakly compressible gas described by A1, there is no need per se to identify a cause-effect relation between $p$ and $\rho$; it is enough to say that $p$ and $\rho$ are connected in a certain way expressing a form of "perfect harmony". 

In the same way, there is no need per se to identify a cause-effect relation between $\phi$ and $\rho$; it is enough to say that $\phi$ and $\rho$ are connected in certain way expressing a form of  "perfect harmony" in the spirit of Leibniz.

The relation $\Delta\phi =\rho$ is explored in Newtonian Matter and Antimatter with $\Delta\phi > 0$ identifying matter and $\Delta\phi < 0$ antimatter, with dark matter where $\Delta\phi$ is smooth and visible matter where $\Delta\phi$ is singular, typically as a sum of multiples of delta functions representing matter in point form.  We refer to such a model as Extended Newtonian Gravitation. 

2. Galaxy Formation

We start from a spherical distribution of matter of low density of dark matter (a halo) with $\Delta\phi$ a smooth function, which we assume to be in static equilibrium with the the gravitational force balanced by a weak pressure force with $\nabla p = - \rho\nabla\phi$. 

Starting from this halo of low density dark matter, we assume that some visible matter (stars) is formed by concentration of dark matter by gravitational attraction into point masses with $\rho$ becoming large locally with the result that the gravitational force $\rho\nabla\phi$ can no longer be balanced by a weak pressure force $-\nabla p$. This is an effect of the different action of pressure and gravitational force, with pressure scaling with surface and gravitational force with volume.

The combined effect of the presence of a halo of dark matter and gravitational collapse of visible matter as a system of point masses, may then create a spiral galaxy of visible matter surrounded by a halo of dark matter, which is the standard view of the nature of a spiral galaxy, with in particular a characteristic distribution of velocity of visible matter as roughly independent of the distance to the galaxy center as an effect of the dark matter halo. 

It thus appears that an extended Newtonian model with $\Delta\phi$ of variable sign and concentration may be sufficient to explain essential aspects of galaxy formation, for which Einstein's equation equation is useless.   

Mystery of Dark Energy, Dark Matter and Inflation Uncovered?

In a series of posts (listed under Newtonian mechanics) I have explored the idea of viewing the gravitational potential $\phi (x,t)$ as primordial physical entity, with $x$ a Euclidean space coordinate and $t$ a time coordinate $t$, from which the distribution of matter with density $\rho (x,t)$ is given by (with scaling to unity):

• $\Delta\phi (x,t) =\rho (x,t)$          (1)     

by the local operation of differentiation of the Laplacian in space $\Delta$. This view possibly opens  to an understanding of the mystery of both dark energy, dark matter and cosmic inflation. Recall that (1) as the essence of Newton's theory of gravitation, also is Einstein's equation in flat Minkowski space-time.

To lift the curtain, imagine an initial state with $\phi =0$ and thus $\rho =0$ and suppose $\phi$ at time $t=0$ is  subject to a local perturbation resulting in a corresponding fluctuating $\Delta\phi$ with as much positive matter where $\Delta\phi (x,t)$ is positive, as negative matter where  $\Delta\phi (x,t)$ is negative. Suppose the initial configuration consists of negative matter surrounded by positive matter in spherical symmetry, thus a shell/ball of negative matter (red) surrounded by positive matter (blue), as in the above picture of a youtube-clip.  Note that positive matter attracts positive matter, but repels negative matter, and vice versa.

Watching the clip you see starting from rest negative (red) matter contracting under attraction and positive matter (blue) expanding under repulsion from negative matter overpowering attraction from positive matter.

If we now think of our Universe as consisting of positive matter, we thus find an expanding Universe as if driven by a mysterious force named dark energy and we thus are led to connect the origin of dark energy to negative matter.  Note that the core of negative matter oscillates between contraction and expansion, while the region of positive matter continues to expand on a much longer time scale.

We may further connect regions of positive matter where $\Delta\phi (x,t)$ is smooth as regions filled with dark matter, and then singular concentrations of $\Delta\phi (x,t)$ as visible matter.  

In this model we start from zero, and Big Bang is represented by a small perturbation of the gravitational potential $\phi$ with large output $\Delta\phi$ as balancing positive and negative matter. It is then the nature of the Laplacian as local differentiation, as an illposed or unstable mathematical operation, which can turn a small perturbation into something big like a Universe. Big Bang thus does not come out as a big local explosion, for which a reason is lacking, but rather as the effect of a small perturbation of a zero-state which is inflated by the Laplacian.

In particular, the inflation of the Universe connects to the relation $\Delta\phi =\rho$ and not as in cosmic inflation theory to yet another mystery.

Finally, with matter derived from a gravitational potential by local differentiation, the unsolved problem of the nature of action at distance does show up at all: All action is local and there is no action at distance and nothing of that sort to explain.

Is then our expanding Universe of positive matter powered by an oscillating Universe of negative matter, with plus and minus adding to zero, resulting from an initial perturbation of a gravitational potential through the action of a Laplacian? Yes, why not? Compare with Genesis 1:1-4 with the initial perturbation just a little twist of God's finger:

The Universe as Weakly Compressible Gas subject to Pressure and Gravitational Forces

Our Milky Way Galaxy as a weakly compressible gas modeled by the Euler equations expressing conservation of mass, momentum and energy, combined with constitutive laws connecting pressure and gravitational forces to mass density.

After the excursion on Newton vs Einstein in recent posts, we are back to a cosmological model in the form of Euler's equations for a compressible gas subject to Newtonian gravitation: Find $(\rho ,m, e ,\phi ,p)$ depending on a Euclidean space coordinate $x$ and time $t$ , such that for all $(x,t)$:

• $\dot\rho + \nabla\cdot (\rho u ) =0$       (or $\frac{D\rho}{Dt} = -\rho\nabla\cdot u$)
• $\dot m +\nabla\cdot (mu) +\nabla p + \rho\nabla\phi =0$
• $\dot e +\nabla\cdot (eu) +p\nabla\cdot u +\rho\nabla\cdot m=0$,

where $\rho$ is mass density, $u=\frac{m}{\rho}$ is matter velocity, $p$ is pressure, $\phi$ is gravitational potential, and $e$ is internal energy as the sum of heat energy $\rho T$ with $T$ temperature and gravitational energy $\rho\phi$, and the dot indicates time differentiation and

• $\frac{D\rho}{Dt}=\dot\rho +u\cdot\nabla\rho$

is the convective time derivative of $\rho$, see Many-Minds Relativity 20.3 and Computational Thermodynamics Chap 32.

These equations express conservation of mass $\rho$, conservation of momentum $m$ with $\nabla p$ pressure force and $-\nabla\phi$ gravitational force, and conservation of internal energy $e$. These laws of conservation are complemented with constitutive laws connection $p$ and $\phi$ to density of the following form:

A1: Weakly compressible gas ($\delta$ small positive constant):

• $\Delta p =\frac{\nabla\cdot u}{\delta}= - \frac{1}{\delta\rho}\frac{D\rho}{Dt}$

or

A2: Compressible perfect gas ($0 < \gamma < 1 $): 

• $p=\gamma \rho T$.

B: Newton's law of gravitation:

• $\Delta\phi =\rho$ with $\phi =0$ at infinity.            

We observe

1. Similarity of $\nabla p$ and $\nabla\phi$ in momentum equation. 
2. Similarity between A1 and B connecting $\Delta p$ to $-\frac{D\rho}{Dt}$ (or $-\rho$) and $\Delta\phi$ to $\rho$.
3. $p \ge 0$ and $\phi \le 0$.

Here 1 can be seen as the Equivalence Principle (equality of heavy and inertial mass) expressing that there is no difference between gravitational and other forces (pressure) in Newton's 2nd law expressing conservation of momentum.

Further, 2 expresses that the constitutive laws A1 and B both can be viewed as action at distance if $\rho$ is viewed as the cause, but represent local action of differentiation if $\rho$ is viewed as the effect. 

For a weakly compressible gas described by A1, there is no need per se to identify a cause-effect relation between $p$ and $\rho$; it is enough to say that $p$ and $\rho$ are connected in a certain way expressing a form of "perfect harmony". 

In the same way, there is no need per se to identify a cause-effect relation between $\phi$ and $\rho$; it is enough to say that $\phi$ and $\rho$ are connected in certain way expressing a form of  "perfect harmony" in the spirit of Leibniz.

The relation $\Delta\phi =\rho$ is explored in Newtonian Matter and Antimatter with $\Delta\phi > 0$ identifying matter and $\Delta\phi < 0$ antimatter. Mind-boggling!

It may be that the above Newtonian model is just as good (or even better) than Einstein's equations as cosmological model, because it is understandable and may contain relevant large scale physics without the mysteries of "curved space-time fabric" of general relativity.

Second Law of Thermodynamics

Viscosity solutions of the Euler equations satisfy the following 2nd law:

• $\dot K = W_p + W_\phi - D$
• $\dot E = - (W_p + W_\phi ) + D$,  

where at a given time instant $t$

• $K=\int\frac{\vert m\vert^2}{2\rho}\, dx$ is total kinetic energy
• $E=\int e\, dx$ is total internal energy 
• $W_p=\int p\nabla\cdot u\, dx$ is total work performed by pressure 
• $W_\phi =\int \phi\nabla\cdot m\, dx$ is total work performed by gravitation
• $D>0$ is turbulent dissipation.

The sign of the turbulent dissipation $D$ gives an irreversible transfer of kinetic energy $K$ into internal energy $E$, which defines the direction of time. Transfer from internal energy requires $W_g+W_\phi >0$, that is expansion with $\nabla\cdot u > 0$ if $W_\phi=0$ and contraction with $\nabla\cdot u < 0$ if $W_p=0$. Note also that

• $W_\phi =- \int \phi\dot\rho\, dx =-\int\phi\Delta\dot \phi\, dx =\frac{d}{2dt}\int\vert\nabla\phi\vert^2\, dx$.

Notice that the above 2nd law does not involve the notion of entropy, and thus avoids the trap of common formulations of the 2nd in terms of an ill-defined concept without physical realization. The above 2nd law only uses the physical concepts of kinetic and heat energy, work and dissipation.

Choice of coordinate system

The Euclidean coordinate system could be fixed to distant fixed stars with the origin at the projected center of the universe. A rotation of this system would show up as extra centrifugal forces, which could explain the apparent accelerated expansion of the universe.

Uni-directional transfer of energy - Alternative pressure laws

Note that A1 implies that $W_p= - \int\delta\int\vert\nabla p\vert^2\, dx < 0$ showing uni-directional transfer of energy from kinetic energy to internal energy. Transfer in the other direction can appear with the stronger effect of compressibility in A2. Of course, combinations of A1 and A2 are thinkable, as well as a pressure analog of B in the form A3: $-\Delta p \sim e$ with $e=\rho T$ and $T$ temperature.

Recall that $W_\phi = \frac{d}{2dt}\int\vert\nabla\phi\vert^2\, dx$, thus with a positive contribution to kinetic energy $K(t)$ over time.

Basic dynamics

The basic dynamics of the above Euler model can be described as local gravitational collapse with gravitational energy being transformed first into kinetic energy and then into heat energy, followed by pressure build up setting a limit to concentration of matter, leading to cosmic web of matter separated by voids:

Visible matter would correspond to $\Delta\phi$ being singular (points, lines, surfaces), while dark matter would correspond to  $\Delta\phi$ being smoothly distributed (and positive).

PS Note that the momentum equation can be formulated:

• $\rho\dot u +\rho u\cdot \nabla u + \nabla p + \rho\nabla\phi =0$, that is
• $\dot u +\rho u\cdot \nabla u + \nabla\phi = F$ or    $\frac{Du}{Dt} + \nabla\phi = F$,

where $F$ (with here $F = - \frac{1}{\rho}\nabla p$) can be viewed as a definition of mass density normalized force $F$ in terms of acceleration $\frac{Du}{Dt}$ and gravitational potential gradient $\nabla\phi$. This connects to the common notion of gravitation(al force) as acceleration, with acceleration acting locally in space and time. Again, the basic question is the (lack of) cause-effect in the relation $\Delta\phi =\rho$, where we are not compelled to view $\rho$ as the cause and $\phi$ the effect from instant action at distance, and thus may instead view $\rho$ as the effect of the Laplace operator acting locally in space and time on $\phi$, or simply bypass the question by viewing $\rho$ and $\phi$ to be connected in "perfect harmony by the relation $\Delta\phi =\rho$ without cause-effect dynamics.  

  

The Higgs: Searching for an Elephant by Microscope

Modern physics is based on two supposedly incompatible theories for the four forces of physics acting on different scales

1. gravitational force: macroscopic: relativity theory: cosmology 
2. electromagnetic, weak and strong forces: microscopic: quantum mechanics: atoms  

The incompatibility has made a unified theory impossible and has driven modern physics into absurdities such as searching for the origin of macroscopic gravitation on microscopic atomistic scales.

The 2013 Nobel Prize for the Higgs particle falls into this tradition: The idea is that mass or matter as the subject of macroscopic gravitation is generated from subatomic interactions through the Higgs particle with the Higgs field as an endless ocean in which the Universe is floating. 

Since the manifestation of mass is gravitation, it means to search for macroscopics in microscopics, that is searching for an Elephant using a microscope (quantum loop gravity and string theory) . It does not seem to me to be a constructive approach.   

A different approach is sketched in my pet theory described in Newtonian Gravitation of Matter and Antimatter exploring the possibility that 

• the basic element of the Universe is a gravitational field $\phi (x,t)$ depending on a space coordinate $x$ and time coordinate $t$
• matter and antimatter of density $\vert\rho (x,t)\vert $ is created by differentiation with respect to $x$ of the field $\phi (x,t)$ through the Laplace operator $\Delta$: $\rho =\Delta\phi$ with matter where $\Delta\phi >0$, antimatter where $\Delta\phi < 0$ and vaccum where $\Delta\phi =0$
• a gravitational force $F$ arises as the space gradient $\nabla$ of the field $\phi$: $F = \nabla\phi$.    

It is thus the gravitational field $\phi$ which 

• creates visible matter where $\Delta\phi$ is positive and singular
• creates visible antimatter where $\Delta\phi$ is negative and singular
• creates dark matter and antimatter where $\Delta\phi$ is smooth  
• separates matter and antimatter by gravitational repulsion
• concentrates matter by gravitational attraction.

What about that? Hint: The Hen and the Egg of (Dark) Matter. This is something completely different from the Higgs as an "explanation" of the origin of mass.

  

No Solutions to New Years Puzzle

The New Year Puzzle has not received any convincing solutions and therefore additional clues are offered in new posts on The World As Computation:

• Repulsion between Matter and Antimatter
• Laplace's Equation as God's Equation
• Newtonian Gravitation of Matter and Antimatter
• Newtonian Unified Field Theory?

The Hen and the Egg of Dark Matter

The 2011 Nobel Prize in Physics brings up the question of dark energy, which may not exist, while the other chamber of the Heart of Darkness, that is dark matter, may exist. This connects to the question of The Hen and Egg of Gravitation in Many-Minds Relativity presented as follows:

The gravitational potential P is related to distribution of matter M by the differential equation

• Laplace P = M

where Laplace is the Laplace operator. This equation can be read in two ways, with P the hen and M the matter:

1. Given M, the potential P is obtained by solving the differential equation Laplace P = M.
2. Given P, the matter distribution M is obtained by applying Laplace to P.

Here 1 is a non-local operation (solving a differential equation) reflecting action at distance, while 2 is a local operation (differentiation) without action at distance.

The usual approach is 1 with the complication of requiring spoky action at distance.

In the non-standard approach 2 there is no action at distance, and it also opens to an interpretation of matter as ordinary matter at points of singularity of Laplace P, where the potential P has "spikes", and dark matter where P is smooth (as Laplace P).

In this case the hen of the gravitational potential is the primary entity from which the egg of matter is generated. It may be easier to explain how a hen can lay an egg, than to describe how an egg may become a hen. In science the most efficient explanation is sought.

As concerns dark energy, it may be that inertia dominates on intergalatical scales and if so the present observed seemingly accelerating expansion may simply be a reflection of a Big Bang initial state with a certain expansion distribution without any need of any dark energy driving the expansion. Gravitation would then be important only on galactical scales. This fits with the observed lumpiness of the universe with large voids between galaxies.