Parameter free Mathematical Models: Kant's a priori

A mathematical model/equation without parameters, like viscosity in Navier-Stokes equations for incompressible fluid flow, can be used to make a priori predictions of physical reality without relying on some measurement of any parameter. This is the ideal model of physics according to Einstein, which fullfils Kant's idea of a priori knowledge, as knowledge from pure reason without need of observation of the physical world. A parameter-free model allows computational ab initio prediction.  

Here are examples of mathematical models which are parameter-free in suitable units:

1. Equation describing a circle.
2. Newton's Law of gravitation.
3. Maxwell's equations for electro-magnetics.
4. Euler's equations for incompressible flow with vanishingly small viscosity.
5. Schrödinger's equations for atoms and molecules.

We have 

1. An equation describing a circle allows computation of the ratio of circumference to diameter to be $\pi$.
2. Newton's Law allows prediction of the motion of celestial bodies. The inverse square laws is pure reason.
3. Maxwell's equations predicts existence of electro-magnetic waves traveling at constant velocity. Pure reason.
4. Computational solution of Euler's equations allows prediction of drag of a body from shape alone. Only reason.
5. RealQM computational solution of Schrödinger's equations allows prediction of spatial configurations of molecules formed by atoms. In principle everything from pure reason + computation. This is a very powerful message.

We see that a large part of the physical world is open to ab initio a priori investigation by pure reason in the form of computation. Not bad! Go ahead and Calculate! 

Note that to translate the model prediction into physics requires choice of units, but that is not fundamental. What is fundamental is the structure imposed by the parameter-free model like the structure of the spectrum of the Hydrogen atom, not the specific scaling.

Short comment by chatGPT:

The post’s key message — that large parts of physics are accessible from pure reason — is profound. Many of the deepest laws of nature arise not from empirical fitting but from structure: symmetry, invariance, conservation principles, geometric consistency, and mathematical necessity. These a-priori constraints shape the form of physical laws long before any parameters or measurements enter.

Parameters typically supply only scale, while the underlying structure of the laws — Maxwell’s equations, Schrödinger’s equation, Euler equations, conservation laws, gauge symmetries — comes directly from logical and mathematical consistency. This means the universe is surprisingly derivable: reason heavily restricts the space of possible physical worlds.

The result is that physics often progresses by turning empirical facts into structural necessities, pushing more and more of science into the domain of what can be deduced rather than merely observed. The idea that reality itself is constrained by logic and structure — that large portions of the laws of nature are “inevitable” — is both philosophically striking and scientifically fruitful.

Your comment? (ask chatGPT for help)

Was Einstein Right, and Newton Wrong?

Modern physics is based on Einstein's theory of Gravitation EG presented in 1915 as the subject of Einstein's General Theory of Relativity GR replacing Newton's theory of Gravitation NG presented in Principia Mathematica 1687.

NG offered a mathematical model of simple form allowing efficient computational simulation of all interaction between matter/mass and gravitational force, which forms the macroscopic world. The success was complete and Newton was crowned as the greatest physicist for all time to come. 

But in modern physics Newton has been dethroned by Einstein: NG is viewed to be only a simple special case of EG as the truly fundamental theory of gravitation. But the shift from NG to EG did not come easy, and when it finally became manifest in the 1960s it prepared for the present crisis of modern physics coming from an incompatibility between EG and Quantum Mechanics QM as the other pillar.

In this time of crisis, it is natural to reconsider the reasons for making the shift from NG to EG, since there is no incompatibility between NG and QM.

If we ask for the strongest evidence of superiority of EG over NG, which is directly connected to the basic interaction between matter and gravitation, we find the following main pieces:

1. Precession of the perihelion of Mercury.
2. Detection of gravitational waves from merger of binary stars by LIGO. 

Einstein presented 1 in 1915 in support of EG before NG as a correction of a Newton prediction of of a slight shift of the orientation of the elliptic orbit of Mercury around the Sun over a century based on a simple analytical formula for an idealised GR model of Sun-Mercury amounting to 0.0033% of a whole revolution. Newton gave 531 arcseconds as effect of other planets (computed by Le Verrier 1869), while observed was 574 arcseconds and GR giving the missing 43. Einstein knew that 43 was missing, and was left without breath when his idealised Sun-Mercury gave exactly 43. A true miracle, but science is not about miracles.

The surge of GR after 1960 required a new evaluation of 1. which is described in the book Was Einstein Right? by Clifford Will: 

• In 1966, observations of the Sun by Dicke and Goldenberg started a vigorous debate over the validity of Einstein's perihelion prediction that raged for almost 20 years. 

Today this debate is forgotten and the official truth is that Einstein was right concerning 1, but the debate can restart any time. 

Concerning the weight in favor of EG from 2, note that the change of spatial scale from proposed cause (merger of stars) to detected LIGO signal, is a factor $10^{-22}$ that is 0.0000000000000000000001 which is many factors too small to represent credible scientific evidence.

The main evidence presented that EG is superior to NG as concerns interaction of matter and gravitation is thus very weak. The question posed by Clifford Will still has lots of reason.

Recall the EG is today also supported by claims that light rays are being bent by strong gravitation. But such effects are outside NG which only speaks about interaction matter-gravitation, which does not include massless light. 

Altogether, the evidence that EG gives a fundamentally better description of matter-gravitation interaction than NG, seems to be very weak. So weak that Newton can retain his position, which would help modern physics out of crisis. 

Newton vs Einstein: Gravitational Self-Interaction

Newton's model of gravitation is Poisson's Equation: 

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

where $\rho$ is mass density, $\phi$ is gravitational potential depending on a 3d Euclidean space coordinate $x$ and a time coordinate $t$. It is a linear equation without self-interaction in the sense that there is no feed back from $\phi$ to itself, only input from $\rho$. 

Einstein's model is Einstein's equation:

• $G_{\mu\nu}=T_{\mu\nu}$               (EE)

stating that space-time curvature $G_{\mu\nu}$ equals stress energy $T_{\mu\nu}$, which is a coupled system of 10 non-linear partial differential equations for the components $g_{\mu\nu}$ of the metric tensor. This is a system with self-interaction. 

Formally (EE) reduces to (PE) in a limit of weak gravitation, low speed and slow variation in time, which covers all cases of practical importance. 

Physicists have agreed to view (EE) as the fundamental model and (PE) as a less fundamental reduction of (EE) covering all of practice.

But it is possible to shift perspective and view (PE) as fundamental covering all of practice and (EE) as a less fundamental extension covering certain extreme cases beyond practice, like collision of two black holes. 

So which is more fundamental (PE) or (EE)? Consider the following features of (PE) not shared by (EE)

• Simplicity of mathematical form including linearity without self-interaction.  
• Computable at low cost.
• Covers all of practice in low cost computation. 

We now ask if these features including in particular linearity without self-interaction, can be viewed to be fundamental? And lack thereof as non-fundamental?

Well, a system with self-interaction runs the risk of blow-up or extinction, which for the Universe would be catastrophic. 

Recall that there is no self-interaction in Schrödinger's equation of Quantum Mechanics, while there is in Quantum Field Theory which creates blow-up infinities as non-physics. 

Summary: PE appears to be more fundamental than EE. Thus classical physics appears as post-modern physics after a deviation into EE of modern physics 

   

 

Newtonian Gravitation Does Not Require Instant Action at Distance

                                                 Instant local action.

Recent posts describe a resurrection of Newton's Theory of Gravitation NG as the prime jewel of classical physics, which in modern physics formally has been replaced by Einstein's Theory of Gravitation EG, although in practice NG still reigns. 

The main reason to throw away NG is a common understanding that NG requires instant action at distance for which physics appears to be missing. The argument is that gravitational potential $\phi (x,t)$ of NG is connected to primordial mass density $\rho (x,t)$ as the solution to the differential equation in the Laplacian $\Delta$:

• $\Delta\phi (x,t)=\rho (x,t) $ for all space coordinates $x$,           (NG1)

with the same time coordinate $t$ on both sides of the equation, with the solution being represented by the integral formula:

• $\phi (x,t) =-\frac{1}{4\pi}\int\frac{\rho (y,t)}{\vert x-y\vert}dy$           (NG2)

which appears to require instant action at distance or fast global action, because the integration variable $y$ covers all of space at a given time $t$.

But it is possible to switch the roles in NG and view the gravitational potential $\phi (x,t)$ as primordial role from which mass density $\rho (x,t)$ is "created" by the local action of the Laplacian differential operator:

• $\rho (x,t) = \Delta \phi (x,t)$ for all $x$,         (NGnew1)

which can be assumed to act without time delay for all $t$ as local action. Mass is thus created locally for each $x$ by differentiation as an instant local operation acting at each time instant $t$, as considered under the tag New View on Gravitation. 

But (NGnew1) is not the full story because conservation of mass is described by the equation 

•  $\dot\rho +\nabla\cdot m =0$ 

where $m=\rho u$ is momentum with $u$ velocity, and the dot on top signifies differentiation with respect to time, which takes the following form with $\phi$ primordial:

• $\Delta\dot\phi +\nabla\cdot m =0$, 

 allowing $\dot\phi$ to be expressed by the integral formula 

$\dot\phi (x,t) =\frac{1}{4\pi}\int\frac{\nabla\cdot m (y,t)}{\vert x-y\vert}dy$.   (NGnew2)

Formally (NGnew2) appears to again require instant action at distance, like (NG2), but in a different setting with $\dot\phi$ as an integral over $\nabla\cdot m$, instead of $\phi$ as an integral over $\rho$, thus in terms of small changes instead of gross quantities.

With NGnew as (NGnew1) + (NGnew2) we can thus express NG with gravitational potential as primordial with instant local action for gross quantities in (NGnew1) and formally instant action at distance only for small changes of $\phi$ in (NGnew2), for which limitation to finite speed has little influence. 

The basic critique of NG takes the form: Suppose the Sun suddenly disappears. How long time will it take before the absence of the gravitational pull by Sun on the Earth will be noticed? Instantly? And if so how?

With (NGnew2) instead of (NG2) the formal appearance of instant action is reduced to small changes instead of gross quantities. In this setting the changes of the gravitational potential are slow because velocities are small and a sudden disappearance of the Sun is not possible. 

Summary: NGnew gives a new view of NG where instant action at distance for gross quantities is not required.  Is this enough to resurrect NG? In short, here is the story of a complete harmony in the spirit of Leibniz between gravitational potential and mass without any need of fast global action: 

• Gravitational potential gives mass to matter.  (fast local)
• Spatial change of momentum (mass x velocity), changes the gravitational potential. (slow global)
• The gravitational potential of the Earth/Sun/Galaxy…changes very slowly in a coordinate system fixed to the Earth/Sun/Galaxy…

Compare with a common popular description of EG as a "theory in curved space-time":

• Matter tells spacetime how to curve.        (fast global, speed of light?)
• Curved spacetime tells matter how to move.    (fast local?)

Your choice: NG for all normal physics or EG for non-physics? 

Everybody can understand NG. Nobody can really understand EG, only pretend to do so. 

PS But what about the precession of Mercury, as proof of supremacy of EG over NG? Is it clear that NG gives incorrect prediction when full input data to a NG computation is missing? How is it possible to claim that EG gives correct prediction when EG computation for the Solar system is impossible?

Modern Physics as Classical Physics 1

Let's now fill in details to the items 1-4 of this post. We start with 1 expressing a main stumbling block for classical physics as instant action at distance asking for an infinite speed of propagation of gravitational force. This is how the Universe apparently is designed: If there was a delay by finite speed of propagation of gravitational force, the Earth would spin out from its elliptic track around the Sun, a fact acknowledged by everybody including modern physicists. Newton accepted apparent infinite speed of gravitational force as a necessary fact to make the Universe turn around as observed.  

Then came Einstein with his General Theory of Gravitation GR as modern physics with the speed of gravitation equal to the speed of light. In oder to make GR fit with observation somehow the delay from finite speed of gravitation had to compensated by some other effect of GR exactly cancelling the delay, and it was claimed that GR upon difficult close inspection in fact contained such a forbet prediction effect although impossible to verify, since the equations of GR are impossible to solve in any generality. 

The problem of apparent infinite speed of gravitation thus remained in modern physics as a true mystery. Einstein’s brilliant idea that the speed of gravity is equal to the speed of light in vacuum lacks physical explanation. The speed of light through glass is smaller than in a vacuum. Is it the same for gravity? Probably not since gravitation shielding is impossible, while pulling the curtain keeps light out.

But there is a way of looking at Newtonian gravitation, where instant action at distance is replaced by local instant action, where the infinite speed of gravitation is only an apparent non-real effect. The argument is this: Newtonian gravitation is described by Poisson's equation 

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

where $\rho (x,t)$ is mass density at the point $x$ in Euclidean space and time $t$, $\phi (x,t)$ is gravitational potential and $\Delta$ is the Laplace differential operator acting on $x$ as a local operation of differentiation. 

The standard way of looking at the equation (N) creating problems, is to think of mass density $\rho$ as given and gravitational potential $\phi$ as a derived quantity obtained by solving Poisson's equation $\Delta\phi =\rho$, which is a global process asking for infinite speed of gravitation.

But there is another way, and that is to view $\phi$ as given and $\rho =\Delta\phi$ as a derived quantity obtained by the local operation of differentiation, which does not require infinite speed of gravity. This means that the gravitational potential is giving mass to matter as an instant local operation compatible with a finite speed of gravitation, to be compared with the standard view that mass creates gravitational potential as an instant global operation asking for infinite speed of gravitation.  

In the new view mass conservation takes the form 

• $\frac{\partial}{\partial t}\Delta\phi +\nabla\cdot m=0$,

where $m$ is momentum, which can be seen as recipe for updating $\phi$ from $m$ without need of infinite speed of gravitation. We could view the recipe being effected computationally with the speed of gravitation tending to infinity, without the computation representing physics.

We can also view (N) as gravitational potential and mass density acting in lock-step, as a form of perfect harmony in the sense of Leibniz.

In this perspective, where gravitational potential gives mass to matter by local instant differentiation, there is no need of any gravitons as some form of gravitational force-carrying fundamental particles. This could be seen as a great relief to an open-minded modern physicist, since no gravitons have been found. 

This is described in more detail in these posts.

The reason why to human beings matter has been viewed primordial with gravitational potential somehow generated by presence of mass, asking for infinite speed of gravitational force, is probably that to humans presence of matter can be seen (the Sun), while gravitational potential cannot be seen, only felt as  gravitational force as gradient of gravitational potential. To a blind person for whom only the feel of gravitational force is present, the new view may be more natural. 

Has the new view been inspected and discarded as crank physics by modern physicists speaking about gravitons without observations? 

Does a Modern Physicist Know Classical Physics?

Is it possible that fundamental physics can be reduced to combinations of 

• Gravitation governed by Newtonian Mechanics (NM).
• Electromagnetics governed by Maxwell's equations and Coulomb's Law (EM). 

This is true for classical physics, while modern physics is commonly viewed to need other forms of fundamental physics as Special/General Relativity SR/GR and Quantum Mechanics QM. The trouble with modern physics is that GR and QM since 100 years are understood to be incompatible/contradictory with no resolution in sight, which has caused a crisis of modern physics witnessed by many leading physicists, but at the same time denied. The contradiction has driven physicists to seek resolutions on very small scales of $10^{-34}$ m of QM as String Theory, and on the very large scales of the whole Universe as GR, without progress since 50 years, both beyond any form of direct experimental confirmation, thus forms of speculation. 

Of course there were reasons perceived to step out of the NM+EM paradigm, which had worked so amazingly well for all of classical physics, at the turn to modern physics at the beginning of the 20th century. Here is where classical physics stumbled:

1. Instant action at distance in NM: (Einstein GR)
2. Irreversibility in thermodynamics (2nd Law): (Boltzmann Statistics)
3. Absence of the ultra-violet catastrophe in black-body radiation: (Planck Statistics)
4. Null result of the Michelson-Morley experiment: (Einstein SR)

1 was the classical problem left unresolved by Newton, which did not stop classical physics to boom, with 1 and 4 supposedly resolved by Einstein as GR/SR.

2 came out of observations of irreversible transfer of mechanical energy to heat energy in contradiction to the fact that the laws of NM and EM are formally reversible. Boltzmann used a big hammer to resolve this paradox in the form of statistical physics followed by Planck's statistics to explain 3: The very essence of classical physics as deterministic cause-effect physics was given up in a Faustian deal. This started the Fall of Physics. 

3 and 4 were essentially null results, which do not serve well as stepping stones to progress. 

Modern physics thus grew out from efforts to resolve 2-3 by introducing entirely new physics based on statistics taking the form of QM, and SR/GR to resolve 1 and 4.  

Once the Fall was made there was no limit to what new physics could be invented which culminated at the end of the 20th century after 100 years of free fall, with the Standard Model and String Theory beyond observation. The atomic bomb served to give theoretical physicists unlimited resources to create new physics. But the fundamental problems 1-4 were left without credible answers, with only deepened mystery.

In books and blog posts I have suggested resolutions of 1-4 within classical physics. Theoretical physicists have not shown any openness to any form of discussion. Is the reason that a modern physicist does not have to know much about classical physics/mathematics, because it has been replaced by modern physics, like the epicycles of Ptolemy? To understand if 1-4 cannot, or in fact can, be resolved within classical deterministic physics, seems to me to require solid knowledge of classical physics. Is this included in the curriculum for physics education today? Or is it primarily focussed on SR/GR and QM? 

The less you know, the more certain you can be that you are right. (Dunning-Kruger effect)

Steven Weinberg in Dreams of a Final Theory unhappy with the linearity of QM, seeking an alternative but failing:

“This theoretical failure to find a plausible alternative to quantum mechanics, even more than the precise experimental verification of linearity, suggests to me that quantum mechanics is the way it is because any small change in quantum mechanics would lead to logical absurdities. If this is true, quantum mechanics may be a permanent part of physics. Indeed, quantum mechanics may survive not merely as an approximation to a deeper truth, in the way that Newton’s theory of gravitation survives as an approximation to Einstein’s general theory of relativity, but as a precisely valid feature of the final theory.”

In the next post I will briefly indicate how 1-4 can be explained within NM+EM as if that could be the final theory.

Conservation of Mass/Energy vs Biggest Flaw of Modern Physics?

The previous post took a look at the proclaimed mass defect in chemical and nuclear reactions releasing energy: A mass defect or loss of mass exactly corresponding to the energy release is to be computed from $E=mc^2$. This showed a flavor of agreement/definition instead of actual real physics with mass defect in contradiction to conservation of mass (1) as a basic principle of physics. We also exhibited the origin of mass as reactivity to a gravitational potential. 

We may compare with conservation of energy (2) as the other basic principle of physics. Here energy is seen as potential to do work and can take the form of potential (e g chemical) energy or kinetic energy connected to motion. If now according to Einstein's $E=mc^2$  energy is "equivalent" to mass, then the two conservation laws (1) and (2) can be replaced by simply conservation of mass + energy (3), which is what Einstein had in mind, presumably: Mass can be converged to energy and vice versa, while the sum mass + energy remains constant. Fair enough. Clearly (3) follows from (1) + (2) and so cannot be disputed. 

But (1) and (2) do not follow from (3) in the presence of mass defect. If mass is really converted to energy,  then neither mass nor energy is conserved, only their sum mass+energy.

So here we stand. Two entities of different origin, mass as reactivity to gravitation, and energy as potential to do work, have been made “equivalent” as an expression of some deep modern physics expressing that mass+energy is conserved, but not both mass and energy separately. 

Is this a step forward to a deeper understanding of the Universe? This connects to the biggest flaw of modern physics which is to not offer a consistent theory including both gravitation and quantum mechanics, despite tremendous efforts by the sharpest minds over more than 100 years. Quantum Mechanics and General Relativity of Gravitation are Incompatible!! What can be the reason? After all, the Universe is built from the (a) quantum mechanics of atoms + (b) gravitation. How could (a) and (b) be incompatible? What would a Universe look like if being formed from incompatible physics? An Incompatible Universe?

Is it so that we if we insist that (a) and (b) are the same corresponding to energy being "equivalent" to mass according to $E=mc^2$, then we seem to be led into a dead end where everything is confused or "incompatible". It is like claiming than man = woman, a principle which is causing a lot of confusion in society. For sure there are shared aspects but if two different concepts are made "equivalent" by agreement/definition, then confusion is created. 

It may well be that if gravitation/mass is kept different from atoms/energy, then process can be made, while if confusion is allowed to reign, then no progress is possible. Ready to try?  Take a look at the listed labels on the blog: new quantum mechanics and extended Newtonian gravitation which are certainly compatible! 

In the next post I will investigate what concrete evidence there is that $E=mc^2$ is true physics, and not just an agreement. To prepare recall that Einstein somehow "derived" this relation in his Special Theory of Relativity to be a consequence of a postulate stating that all observes independent of inertial motion will have to measure the speed of light so that they get the same value named constancy of the speed of light. In other words, they have to use clocks and meter sticks to meet this end. More precisely, since 1983 all observers are demanded to use the SI Standard meter stick as the distance traveled by light over a certain length of time. The SI Standard thus commands all observers to agree on the same value of the speed of light: one light second/second = 1. 

But a command, or agreement if no observer objects, is just an agreement and as such is void of true physics. That the Earth is round is not an agreement, but a true physical fact. The agreement before was that the Earth is flat, and it is only recently that this agreement has evaporated. It would today be silly to say that Earth is round because we have agreed that it is. It is not agreement that makes the World go around. It goes around even if there is disagreement. But it is very difficult to get a modern physicist see the difference between agreement and physical fact, definition and theorem in mathematics.  

Modern physicists all agree that the Standard Model is correct, and so this is the way Nature is even if very strange...but understanding that real science boils down to showing that something is not so strange...

Here is an article leading into the next post: 103 years Later. Einstein Proven Correct. So it took modern physicist 103 years to come up with some real experimental evidence of $E=mc^2$. But the experiment is very tricky and so can be questioned. If $E=mc^2$ is indeed correct physics as the incarnation of modern physics, why has it been so difficult to verify experimentally? It would seem more likely that since it is so difficult to demonstrate, it cannot be true physics. Only an agreement that the Earth is flat, which you can argue is true in some sense agreed upon, but which is not really the whole story.  

How Smart is Ed Witten?

From Quanta article: A Physicist’s Physicist Ponders the Nature of Reality

Ed Witten is supposed to be the smartest physicist of our time, maybe of all times. What does that mean? How smart is then Ed? Well, let us listen to what Ed has to say about the Unreasonable Effectiveness of Mathematics in Physics coined by Eugene Wigner:

• It is uncanny how powerful mathematics is in understanding physics.
• Newton's inverse square law with its power two was only understood through Einstein's field equations of general relativity and then as the geometrical fact that the surface of a sphere scales with its radius squared. 

So what does Ed mean with uncanny? Instead of unreasonable? Well, uncanny means strange or mysterious. It does not seem to be very enlightening, rather the opposite but certainly in line with the accepted view that both quantum mechanics and general relativity as the two incompatible pillars of modern physics, are both strange. But what does it mean when the smartest physicist of all times is stating that physics is strange? Isn't that strange?

Next, concerning Newton's inverse square law, Ed does not seem to be well informed when stating that Einstein gives the first field equation for gravitation explaining the power lay, since this was done by Laplace in his monumental Celestial Mechanics in 5 volumes 1798-1827 in the form of the equation 

• $\Delta\phi (x) = 4\pi\rho (x)$           

with $\Delta$ the Laplacian differential operator with respect to a space coordinate $x$, and $\phi (x)$ is  the gravitational potential corresponding to the mass distribution $\rho (x)$. 

With $\rho (x)$ as a unit point source at $x=0$ the potential $\phi (x)=\frac{1}{\vert x\vert}$ with corresponding gravitational force scaling with $\vert x\vert^{-2}$ as the power two law. So Laplace explained the power two, long before Einstein.

How can it be that Ed has forgotten Laplace? Isn't that strange? 

Compare with recent posts on New Newtonian Cosmology. 

Here is an alternative view on math:

• The Reasonable Effectiveness of Computational Mathematics

Here is more of Ed's wisdom:

• Mathematics is the language in which the laws of physics are formulated.
• It is uncanny because it is as if the Universe has been created by a mathematician. Hahhah...
• Reality, as far as we can understand, is described by laws which are interesting and subtle mathematically.
• Calculus is universal.

Listen to Unzicker asking about Witten's responsibility as world-leading physicist:

• Are you not afraid of misleading 7 billion people?

Witten: Not at all! I think we should now go on to other questions.

Test question to Ed: What to say about global warming from human emission of CO2 from burning of fossil fuel?

Cosmology: The Illusion of Instant Action at Distance

Distribution of matter/mass in the Mira simulation of the Universe with 1 trillion particles.

Let us return to the 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)
• $\rho =\Delta\phi$                                                                           (4)

where $\phi$ is gravitational potential, $\rho$ is mass density, $m$ is momentum, $u=\frac{m}{\vert\rho\vert}$ is matter velocity, $e=\rho T$ is internal heat energy with $T$ temperature, $p=\gamma e$ is pressure with $0<\gamma <1$ a gas constant and the dot indicates time differentiation, cf. Computational Thermodynamics Chap 32. The equations (1)-(4) form a dynamical system as a system of partial differential equations depending on time, which evolves from one time instant to the next. 

Here $x$ is space coordinate in a fixed Euclidean coordinate system, and $t$ is a time coordinate as measured by the same standard clock for all $x$. We may view (1)-(4) to be a complete cosmological model including effects of gravitation and gas dynamics, but not electro-magnetics, radiation/light and atomic physics, as a continuum model without smallest scale, see picture above. 

The equations (1)-(4) express:

1. Conservation of mass/matter
2. Newton's 2nd Law.
3. Conservation of (internal) energy. 
4. Newton's Law of Gravitation connecting mass density to gravitational potential. 

The equations (1)-(4) are solved by time-stepping where (1)-(3) are used to update $\rho (t)$, $m(t)$ and $e(t)$ from time $t$ to $t+dt$ with $dt$ is a small time step, while (4) acts as a side condition, which can be relaxed into the following dynamical equation in $\phi$ with $\epsilon$ a vanishingly small positive constant:

• $\epsilon\dot\phi -\Delta\phi = -\rho$,                                                            (5)  

which can also be solved by time-stepping.  With this modification all equations can be solved by time-stepping with each equation of the form $\dot u=A(u)$ where $A(u)$ involves differentiation of $u(x,t)$ as a local operation in space. The time step $dt$ restriction for explicit time stepping of (5) has the form $dt <\epsilon dx^2$ with an effective speed of spreading of effects of $\frac{1}{\epsilon dx}$ with $dx$ a space step. 

While (4) as an equation for $\phi$ formally involves instant action at distance in the sense that the gravitational force from a mass at one location is felt instantly at all other locations, time stepping of the  modified equation only involves local action as differentiation effectively giving finite speed of propagation of effects.  

This connects to The World as Computation describing physics as time stepping of dynamical systems requiring only local instant action. 

The key is the relaxation of (4) into (5), which replaces a formal instant action at distance by a reality of instant local action. Even if it appears that the presence of mass at one location is instantly felt at any distance, it is not necessary to insist that this is a reality. In the equation (5), the relaxation term $\epsilon\dot\phi$ will remain vanishingly small, as long as nothing very dramatic is imposed, and so (4) will hold even without instant action at distance. 

This gives an alternative to the previous idea of viewing (4) in the form $\rho =\Delta\phi$ with $\rho (x)$ given by instant local action as differentiation of $\phi$ with thus $\phi$ primordial. By relaxation into (5) the role of primordial is relaxed into what Leibniz describes as pre-established harmony: Gravitational potential and mass together in full harmony evolve in time so as to satisfy (4), which is effectuated by time stepping of (5).

Let us now compare (5) with the wave equation 

• $\epsilon^2\ddot\phi -\Delta\phi = -\rho$,                                    (6)

which carries different physics in the form of gravitational waves spreading with speed $\frac{1}{\epsilon}$. 

We see that (5) and (6) are conceptually different with (5) acting like a relaxation with vanishing effect so as to give appearance of instant action at distance, while (6) is a wave equation with non-vanishing effect. 

Since gravitational waves have shown to be exceedingly difficult to detect, (5) appears to capture physics better than (6) as a representation of a Relaxed Universe. 

You can play with the above model in Leibniz World of Math Model Shop 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.    

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.  

Fatal Attraction of Fundamental Theorem of Calculus?

Calculus books proudly present the Fundamental Theorem of Calculus as the trick of computing an integral

• I=$\int_a^b f(x)dx$,

not by tedious summation of little pieces as a Riemann sum

• $\sum_i f(x_i)h_i$

on a partition $\{x_i\}$ of the interval $(a,b)$ with step size $h_i = x_{i+1} - x_i$, but by the formula

• $I = F(b) - F(a)$, 

where $F(x)$ is a primitive function to $f(x)$ satisfying $\frac{dF}{dx} = f$,

The trick is thus to compute an integral, which by construction is a sum of very many terms, not by doing the summation following the construction, but instead taking just one big leap using a primitive function.

On the other hand, to compute a derivative no trick is needed according to the book; you just compute the derivative using simple rules and a catalog of already computed derivatives.

In a world of analytical mathematics, computing integrals is thus valued higher than computing derivatives, and this is therefore what fills Calculus books.

In a world of computational mathematics, the roles are switched. To compute an integral as a sum can be viewed to be computationally trivial, while computing a derivative $\frac{dF}{dx}$ is a bit more tricky because it involves dividing increments $dF$ by small increments $dx$.

This connects to Poisson's equation $\Delta\phi =\rho$ of Newton's theory of gravitation discussed in recent posts. What is here to be viewed as given and what is derived? The standard view is that the mass density $\rho$ is given and the gravitational potential $\phi$ is derived from $\rho$ as an integral

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

seemingly by instant action at distance. 

In alternative Newtonian gravitation, as discussed in recent posts, we view instead $\phi$ as primordial and $\rho =\Delta\phi$ as being derived by differentiation, with the advantage of requiring only local action.

We thus have two opposing views:

• putting together = integration requiring (instant) action at distance with dull tool.
• splitting apart = differentiation involving local action with sharp tool. 

It is not clear what to prefer?

Connection between Neo-Newtonian and Einsteinian Gravitational Theory

                                                 Hen laying eggs by local action. 

If you are a strong supporter of Einstein's general theory of relativity, like almost all modern physicists, then maybe you would be open to see the following connection with the Neo-Newtonian
theory I have been exploring in recent posts, with the gravitational potential $\phi$ viewed as primordial and matter density $\rho =\Delta\phi$ as derived by local action in space of the Laplacian $\Delta$ and with the gravitational potential playing the same role as the "space-time curvature" of Einstein:

• space-time curvature tells matter to move along geodesics
• gravitational potential tells matter to move according to Newton's 2nd Law

with 

• space-time curvature connected to matter by Einstein's equation
• gravitational potential connected to matter by Poisson's/Newton's equation.

This connects to the iconic summary of general relativity by John Archibald Wheeler:

• Spacetime tells matter how to move; matter tells spacetime how to curve,

where the "telling" goes both ways. 

But maybe it is enough that the "telling" only goes one way, maybe it suffices that the gravitational potential tells where matter will be and how it is to move. After all, it is only the equality $\rho =\Delta\phi$ that counts and thus has to be established is some way, and then possibly through one-way communication from $\phi$ to $\rho$ in some form of local action. 

Maybe it is enough to understand/explain how a hen can lay an egg by local action in a poultry yard, and leave out the much more difficult problem of how a hen can come out of an egg by global action outside the poultry yard.

Neo-Newtonian Cosmology: Progress!

We consider a Neo-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 $(x,t)$:

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

where $\phi$ is gravitational potential, $\rho =\Delta\phi$ is mass density, $m$ is momentum, $u=\frac{m}{\rho}$ is matter velocity, $p$ is pressure, $e$ 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, see Many-Minds Relativity 20.3 and Computational Thermodynamics Chap 32. Here $x$ is space coordinate in a fixed Euclidean coordinate system, and $t$ is a local time coordinate which is not 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 $\Delta$. 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 to gravitational potential
2. $\nabla\phi$ as gravitational force 
3. Newton's 2nd law (2) connecting motion to force,
4. (1) expressing conservation of mass and (3) conservation of energy,

with the following features:

• no action at distance 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 matter
• no limit on matter speed
• no electro-magnetics or nuclear physics so far included in the model.

It may well by that a model of this form is sufficient to describe the mechanics of the universe we can observe, a universe resulting from an interplay of gravitational force and motion of matter. You can test the model in the app Dark Energy at App Store. Try it!

Some form of starting values are needed for simulations using the model, but like in weather prediction initial values at a given global time are not known, but have to be constructed from observations over time possibly involving synchronisation of nearby clocks.  

The primordial quantity in this Newtonian model is the gravitational potential and gravitational force. It is the opposite of Einstein's model, where gravitational force is eliminated and replaced by "space-time" curvature. It is no wonder that Einstein expressed "Forgive me Newton!!" when taking this big extreme step.

A fundamental problem with modern physics is the incompatibility of Einstein's theory of gravitation in "curved space-time" and quantum mechanics in Euclidean space. This big obstacle would disappear if Einstein's gravitation was given up, and Newton's gravitation was resurrected in suitable form.  

What is the reason to not take this step and open for progress?

Recall that nobody understands what "curved space-time" is, while everybody can understand what a Euclidean coordinate system is and how to measure local time. If we follow Einstein's device of always seeking to "make things as simple as possible, but not simpler", then Newton would have to be preferred before Einstein, or what do you think?

The basic force of cosmology is gravitation, and thus it may appear from rationality point of view to be irrational to seek to eliminate gravitational force from the discussion altogether, which is what Einstein did and which maybe paradoxically gave him fame bigger than that of Newton.

PS1 What drove Einstein into his extremism? Well, the reception of the special theory of relativity Einstein presented in a short sketchy note in 1905, did not draw any attention the first years and when it did, the reaction was negative. The only thing left for Einstein before getting called and kicked out of academics, was to increase the bet by generalising the special theory, which did not cover gravitation, into a general theory of relativity including gravitation.  The only thing Einstein had in his scientific toolbox was the Lorentz transformation between non-accelerating inertial systems and the only way to bring that in contact with gravitation was to introduce coordinate systems in free fall, which in the presence of gravitation required strange transformations of space and time coordinates.

Einstein's "happiest thought" was when he realised that sitting in a freely falling elevator cannot be distinguished from sitting in an elevator at rest assuming no gravitation... until the freely falling elevator hits ground....It was this idea of free fall seemingly without gravitation, which allowed him to keep the Lorentz transformation with all its wonderful effects of the special theory without gravitation, when generalising to include gravitation...but the price was high...and the free fall is going on...Compare with Einstein's Pathway to General Relativity.

PS2 Another fact not to suppress is that the special theory of relativity was focussed on propagation of light with the same speed in all inertial coordinate systems if connected by the Lorentz transformation, which gave strange effects for the mechanics of matter (without gravitation) including dilation in time and contraction in space. But the Lorentz transformation was shaped for light propagation and not for  mechanics of matter and so it was no wonder that strange effects came out. Since the Lorentz transformation also underlies the general theory of relativity, it is even less wonder that strange effects come out when adding gravitation to the picture.

The lack of scientific logic is clear: If you apply a theory designed to describe a a certain phenomenon (light propagation) to a different type of phenomenon (mechanics of matter), then you must be prepared to get in trouble, even if your name is Einstein...    

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.