Why Newton was Replaced by Einstein, and Back

Newton's Inverse Square Law NL was until the 1960s the prime example of the power of mathematical thinking visible to everybody: All of celestial mechanics can be described and computed from NL  necessarily valid from principles of conservation in any existing Universe. 

What happened in the 1960s was that Einstein's General Theory of Relativity GR, presented already in 1915, finally was adopted to serve as one of the two pillars of modern physics (the other one Quantum Mechanics QM) and so replace Newton the icon of classical physic by Einstein as icon of modern physics. 

But this transition took place only after Einstein's death in 1955, because of the very complex mathematics of GR understood by few if any making it useless in any from of practical physics. 

However in the propaganda of the cold war it served well to strengthen the world dominance of US science formed by physicists imported from Germany during WWII inventing the atom bomb. Replacing Newton by Einstein served as a demonstration of power, and all the earlier skepticism to GR could be put under the rug. And of course the Sovjet Union followed. Einstein was put in first place, but only after his death since during his life time he appeared as an "absent-minded eccentric maybe too fond of fame". 

Today the cold war is back, Einstein is still on top of the list of fame, while the rapidly developing technology of warfare is using Newton to come to expression. 

There is only an homage to Einstein GR left as an initial offset of satellite clocks in the GPS system, which in operation is annihilated by continuous synchronisation to a master clock on Earth. 

Maybe there is reason to return to a new critical analysis of Newton vs Einstein without the fame dominating the discussion.  

The discussion could start comparing Newton's absolute space against which Einstein's relative space took stand. 

Is it true that Newton's theory of gravitation needs a notion of absolute space against which absolute velocity can be measured? Does Newton say that velocity is absolute? Certainly not! Nobody would come up with such an idea. Of course velocity of an object is always measured relative to something else.

So Newton says that velocity is relative. On the other hand Newton says that rotation as accelerated motion is detectable by forces of tension arising from acceleration. Newton's rotating bucket can thus be viewed as a form of absolute rotation which does not need any outside reference. 

Note that there is a special form of accelerated motion which is not so easily detected by presence of forces and that is free fall under gravitation where all parts of your body feel the same force and no tension arises. But that is not true for a bigger object where tidal forces occur because the gravitational force is not uniform.

So the argument that Newton says that velocity is absolute and so has to be replaced by Einstein's relativity, is not correct. The argument that Newton's theory of gravitation is a necessity in any existing Universe, is very strong. The question is if there are modern physicists willing to face this reality.

Credibility Crisis of Western Politics and Modern Physics as Miracles

The Untergang des Abendlandes (Decline of the West) predicted by Oswald Spengler in two volumes 1918-22 can now be seen as an erosion of credibility of not only political leadership but also of science and then in particular of modern physics based on Einstein's Equation EE and Schrödinger's Equations SE extending classical physics based on Newton's Equation NE and Maxwell's Equations into new physics. 

The erosion of credibility of modern physics is the essence of the crisis of modern physics of today as a basic expression of Decline of the West from a position of total success of modern physics with the atomic bomb. The crisis is rooted in an incompatibility of EE and SE, which means that modern physics in some fundamental way is "wrong" in the sense of not describing real physics, and the credibility of a physical theory which does not describe real physics cannot be maintained. 

The official picture, which is now loosing credibility, is:

• Einstein's theory of gravitation based on EE (General Relativity GR) is superior to Newton's theory based on NE and so represents an enormous advance of science. 
• NE is simply a simplified version of EE, which is the correct complete theory far superior to simple NE. 

To back this official picture some sparse evidence has been presented supposedly showing that EE describes physics better than NE, and a key such piece of evidence is the anomalous precession of the perihelion of Mercury:

• The orientation of the elliptical orbit of Mercury around the Sun is observed to turn 5574 arcseconds per century, which is 0.0043 of a full turn (small).
• 5000 out of 5574 are explained as an effect using an Earth-based reference system for  Sun+Mercury.
• Effects from the other planets according to NE gives 531.
• Analytical solution of EE for Sun+Mercury explains the missing 43 to give the observed 574.    

The message is that NE without EE including all planets is wrong (531), but when corrected by EE for Sun-Mercury (43) becomes right (574), and miraculously so since no full solution of EE with the other planets is performed, because this is computationally impossible due to the complexity of EE.

This miracle, which Einstein performed in 1915, is thus to present a solution of EE with all planets (and other effects) included which exactly fits with the observed 5574, without solving EE computationally because that was (and is) infeasible. 

The miracle is repeated today, which is evident from the fact that the EE correction to NE is still taken from the same analytical solution of Sun+Mercury used by Einstein (42.98). 

But what makes science different from black magic is that in science there are no miracles. To present science as miracle erodes credibility of science, which is what we are watching. 

Credibilty could be maintained if full solutions to EE gave 5574 in accordance with observation, but this is not what physicists deliver. Instead they offer a small EE correction to NE as the main computable model and work horse. To give EE the leading role over NE is like claiming the tail wags the dog.

Misconceptions about Newton vs Einstein: Crisis!

Modern physics in a state of deep crisis which comes to expression in the complete adoption of Einstein's Theory of Gravitation EG as replacement of Newton's Theory Gravitation NG as the most successful theory of all of classical physics. Modern physicists decided to take this step after the death of Einstein in 1955 under pressure to come up with something new after the success with the atomic bomb started to fade, based on the following arguments:   

1. NG is a "simplified version" of EG as a "limit" under low-speed and weak-field conditions. 
2. EG is thus "more fundamental" than NG. 
3. NG is "wrong" in certain extreme cases outside its (incredibly vast) area of validity, where EG appears to be "right".
4. Whatever success NG has is also a success of EG, since EG includes NG. 
5. In short: It is necessary to replace NG by EG, even if NG is used in all cases of any practical meaning. 

Let us now take a step back and see if 1-5 makes any sense. Let us start recalling that NG and EG has fundamentally different ontology or real physics:

• NG is based on Poisson's Equation based on the assumption that gravitational force is conservative (work independent of path) and conservation (no force out of nothing or into nothing). NG has a simple mathematical form and appears to cover all gravitation of some real (practical) meaning. The greatest success of mathematical modeling.  
• EG is based on a principle of curved space-time replacing gravitational force where the physics is hidden in very complicated mathematics.
• NG and EG thus have fundamentally different physical meanings, which means that NG is not a special case of EG.  

This means that the success of NG is not also a success of EG. It is necessary that EG stands on its own merits. But EG is uncomputable in all cases of practical meaning, which means that EG has very little merits of its own. 

In short: The step taken by modern physicists to replace NG by EG lacks scientific rationale and so adds  to the credibility crisis of modern physics acknowledged by prominent physicists. But there is no reason physics should be in a state of crisis, since there are so many new possibilities opened in particular by computation. A first step out of the crisis is to put NG first and view EG as fringe science without real scientific interest. This will be a relief to both educators and students giving room for real understandable physics.

If you still believe that Einstein should replace Newton, recall

• Observations of apparent instant-action-at-distance agree with a fundamental aspect of NG.
• Gravitational force with time delay as fundamental aspect of EG, requires tricky compensation/fix to agree with observations. 
• NG is computable in general. EG is uncomputable except possibly in some very special cases. 
• NG is based on fundamental physical principles of simple mathematical form. EG has very complicated mathematical form with unclear physical meaning.
• NG says nothing about possible aberration of light or gravitational lensing, since light is massless. If light is affected by gravitation, it is a matter for Maxwell's equations.  
• GPS satellite clocks are offset at launch to compensate for time dilation in EG,  but the offset is over-run by continuous synchronisation to an Earth-based master clock, and so does not show that EG is correct and NG wrong. 

Hopefully, this can start a discussion comparing the scientific merits of NG and EG. Input?

The Deep Truth about Newtonian Gravitation

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 the most remarkable mathematical model of all of classical physics allowing precise computational simulation/prediction of effects of gravitational forces in complex systems, as demonstrated in the Millennium Run tracing the evolution of the matter distribution of the Universe over 2 billion light-years using 10 billion particles interacting by (PE). There is massive evidence that (PE) captures gravitation to experimental precision in all cases of any practical interest. 

In the modern phyics of today PE has been displaced from 1st to 2nd place by Einstein's Equation EE, despite the fact that PE covers all cases of practical importance. 

This degradation of PE took a long time become the official truth of modern physics. Einstein presented EE in 1915 which was met with criticism as non-physics in a form of complicated mathematics, and it was only after Einstein's death in 1955 that EE gradually came to replace PE as the true model of gravitation of modern physics. 

The key argument used to put PE into 2nd place was that PE could be viewed to require instant-action-at-distance in the sense that a change of $\rho (x,t)$ at a certain point $x$ at time $t$ would instantly affect the value of $\phi (y,t)$ for all points $y$, since (PE) as a differential equation in space represents $\phi$ in terms of $\rho$ in terms of a global summation process according to the formula with the same $t$ on both sides, thus without time delay:

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

Since instant-action-at-distance appeared to be in conflict with Einstein's Special Theory of Relativity SR, which was accepted before his General Theory of Relativity with EE, leading physicists decided to displace PE to 2nd place in modern physics.  

Let us now take a fresh look at (PE). We see a linear partial differential operator with constant coefficients as the Laplacian $\Delta$ connecting $\phi$ and $\rho$ which can be viewed in 3 ways:

1. Differential equation $\Delta\phi =\rho$ with $\rho$ as cause and $phi$ as effect derived by global integration.
2. Specification $\rho =\Delta\phi$ with $\phi$ as cause and $\rho$ as effect derived by local differentiation. 
3. Simply a coupling of equal parts without cause-effect.   

Here 2 corresponds to local-instant-action which is compatible with SR, if that is the concern.

Here 3 connects to the Pre-Established Harmony of Leibniz as a Deep Truth. It means that $\phi$ and $\rho$ are locked to each other with the Laplacian as a linear relation of simple mathematical form the same everywhere. Such a relation can be read without causation as in 3 and then in particular without demand of instant-action-at-distance. It also makes sense from computational point of view since the computational complexity of (PE) scales linearly with number of spatial mesh points. 

In discrete form (PE) takes the following form in 1d with $dx$ a space step:

• $\phi (x+dx,t)-2\phi (x,t)+\phi (x-dx,t)=dx^{2}\rho (x,t)$ 

locking $\phi$ to $\rho$ at common time $t$ by a simple linear relation which can be read both ways. 

The basic idea of 3 is explored in many posts on New View of gravitation and New Newtonian Cosmology

Let me list virtues of (PE) and Newton motivating back to 1st with (EE) and Einstein 2nd:

• Generality.
• Simplicity.
• Minimal computational complexity.
• Compatibility with Quantum Mechanics.
• Understandable to a wide audience.

As a young patent clerk in Bern in 1905, Einstein took on a role to overthrow principles of classical Newtonian physics at the same time cubism and atonal music emerged as challenges to classical painting and music. This was the emergence of modernity at the turn of the century triggered by an explosion of new technology. Today we see a return to tonal music and figurative painting as post-modernity, and so a return of Newton may also take place after an aberration into Einstein.

New Model of Atomic Nucleus with only Coulomb Potentials

RealNucleus is model of a nucleus as a collection of $P$ non-overlapping +1 proton charge densities surrounding a kernel as a collection of $E$ non-overlapping -1 electron charge densities interacting by Coulomb potentials without presence of the strong and weak force of the Standard Model. 

We consider here a reduced model assuming spherical symmetry in the form of a central sphere of negative density of total charge $E$ surrounded by a shell system of positive density of total charge $P$. The model can be tested using this code with settings for the nucleus of 4He consisting of 2 electrons surrounded by 4 protons.  

The model corresponds to, in the setting of the Standard Model, of a nucleus with $E$ neutrons and $P-E$ protons with in the basic case $E=P-E$ with thus $P=2E$.  

The Standard Model was formed in the 1960s to explain the stability of a nucleus by introducing an attractive force overpowering the Coulomb repulsion named the strong force as a new fundamental force asking for very elaborate quark-gluon physics. The present crisis of physics is directly connected to deficiencies of the Standard Model without resolution in sight. The Standard Model is proclaimed to be the most successful theory of all of physics and as such cannot be abandoned, but then serves as a road block to progress. 

RealNucleus offers a different explanation of the stability of nuclei which does not involve any strong/weak force and so is based solely on Coulomb potentials coming with Coulomb forces. RealQM shows that the negative charges in the kernel are confined by the surrounding positive charges. More precisely RealNucleus computes a total energy of the nucleus to be negative with thus the negative potential energy from charges of different sign overpowering the positive repulsion energy from charges of the same sign, in particular from the kernel repulsion between electrons. 

RealNucleus thus shows stability of a nucleus consisting of a kernel of negative charge $E$ surrounded by a shell system of positive charge $2E$. In particular it is shown that the fact that the mass of a proton is bigger than that of an electron making an electron occupy more space than a proton, is instrumental for stability with the radius of the kernel being comparable to the radius of the nucleus. 

A nucleus thus appears as an analog of an atomic ion with electrons and protons switching roles, with the kernel of a nucleus comparably much bigger than the nucleus of an atom.  

A physicist trained with the Standard Model would say that it is impossible that a nucleus has a negative kernel consisting of electrons, because by Heisenberg's Uncertainty Principle compression of an electron to fit inside a nucleus would require 100s of MeV which are not available. RealNucleus shows that this argument may well lack real physics.  

RealNucleus is a model of a nucleus without the heavy burden of the Standard Model. Note that there is a shell model connected to the Standard Model with a nucleus as a collection of protons and neutrons swimming in a negative charge potential coming from the strong force. 

RealNucleus keeps a nucleus together as a negative kernel of electrons of charge $E$ surrounded by a shell system of positive protons of charge $2E$ interacting via Coulomb potentials only by a combination of the following circumstances: 

• The radius of the kernel is a substantial fraction of the radius of the whole nucleus,  because electron mass is much than proton assigning electrons comparatively large volume. 
• The double number of protons vs electrons allows the surrounding protons to confine the electrons in the kernel thus overcoming electron-electron repulsion. 
• The boundary/radius of the kernel is determined to make electron charge density meet proton charge density with continuity. 

If Coulomb and Newtonian gravitational potentials suffice to describe both the macroscopics of the world we can see and the microscopics of atoms and nuclei, then Einstein's dream of a unified field theory would seem to be in reach. Maybe such a theory can find an audience outside the physics community tied to the Standard Model of quarks interacting be weak and strong forces transmitted by gluons as force carriers.

Note that there are two different ways of using the concept of force:

1. Force on a particle comes from instant local in space gradient of a potential. No transmission of  force over space-time. Instant local action.
2. Force between particles transmitted over space-time by force carriers connecting particles. Action at distance. 

It is natural by Ockham's Razor to favour 1. using only the concept of potential, before 2. asking for force carriers of unknown physical nature. 

PS1 Note that the Standard Model does not describe a nucleus, only the protons and neutrons supposedly forming the nucleus, not even 2H consisting of 1 proton and 1 neutron. This is a strange short-coming.

PS2 The common understanding of well educated physicists is that a nucleus is a collection of protons and neutrons and in particular that there is not even room for a single electron.  This conviction can be challenged by the following observations: (i) a neutron outside a nucleus decays within 15 minutes into a proton and an electron and (ii) a neutron inside a nucleus can decay into a proton staying in the nucleus and an electron, which is kicked out of the nucleus ($\beta$-decay). 

Gravitational Waves as Fiction

We recall the following Newtonian model of the Universe from this recent post:

• $\rho=\Delta\phi$                        (N)             (conservation of gravitational force)
• $\dot\rho +\nabla\cdot m =0$                        (conservation of mass)
• $\dot m +\nabla\cdot (um) +\rho\nabla\phi =0$    (conservation of momentum)

describing a (zero pressure for simplicity) distribution of matter subject to gravitation, where $\rho$ is mass density, $\phi$ gravitational potential, $\nabla\phi$ gravitational force per unit mass, $m$ is momentum, and $u=\frac{m}{\rho}$ is material velocity, all depending on a Euclidean spatial coordinate $x$ plus time $t$ with the dot representing differentiation with respect to time.

We focus on the connection between gravitational potential $\phi$ and mass distribution $\rho$ expressed through Laplace/Poisson's equations (N), which formally involves infinite speed of propagation into $\phi$ from a local change of $\rho$. Let us compare with the following wave equation as a Neo-Newtonian variant of (N) with finite speed of propagation $c$

• $\frac{1}{c^2}\ddot\phi -\Delta\phi = -\rho$.            (NN)

Let us now compare (N) and (NN) in a situation where the mass distribution changes/moves with velocity $v$ which is small compared to $c$, which is the typical situation within a planetary system and collection of stars or even galaxy. This means that $\nabla\cdot m$ is small of size $v$, which means that $\dot\rho$ is small of size $v$. We conclude from (NN) that $\dot\phi$ is small of size $v$ and so also $\ddot\phi$ assuming $\nabla\cdot\dot m$ is small of size $v$. This means that the difference between (N) and (NN) is of size $\frac{v}{c^2}$ thus very small.

We conclude that even if we extend (N) (without gravitational waves) to (NN) (with gravitational waves), the difference is very small. This is in line with the LIGO experiment supposedly identifying a very very small gravitational wave from a very very large source. Indeed, very very small. More precisely from LIGO documentation:

• For physicists, a strong gravitational wave will produce displacements on the order of $10^{-18}$ meters - this is 1000 times smaller than the diameter of a proton. Waves of this strength will be produced by very massive systems undergoing large accelerations, like two orbiting black holes that are about to merge into one. Since systems like these are rare, these sources will be light-years away. Therefore, the search for gravitational waves is seeking the minute effects of some of the most energetic astrophysical systems from the depths of the universe.

We thus have theoretical and observational support of an idea that we can view gravitational waves to be fiction,  which we do not have to worry about. This makes theory simpler and also computational simulation, since (N) is much simpler to solve computationally than (NN), and so makes cosmology simpler. This is a gift from Newton.

Napoleon criticised Laplace, expert in infinitesimal Calculus, for work on infinitely small issues in his administration. Napoleon would probably similarly criticise Einstein for working with infinitely small deviations from Newton's mechanics.

Primordial Gravitational and Electric/Magnetic Potentials

Dialog between the Two Greatest World Systems with primordial potentials vs densities.  

This is a further remark to previous posts on New Newtonian Cosmology with a gravitational potential $\phi_m (x,t)$ and electric potential $\phi_c(x,t)$ with $x$ a Euclidean space coordinate and $t$ a time coordinate, viewed as primordial with mass density $\rho_m (x,t)$ and electric charge density $\rho_c(x,t)$ given by 

• $\rho_m=\Delta\phi_m$      (1)
• $\rho_c=\Delta\phi_c$      (2)

Here $\rho_m \ge 0$ while $\rho_c$ can be both positive and negative, and $\Delta$ is the second order Laplacian differential operator. 

The corresponding gravitational force $f_m\sim -\nabla\phi$ is attractive between positive mass densities and the corresponding Coulomb force $f_c\sim \nabla\phi_c$ is attractive between charge densities of opposite sign and repulsive for charge densities of the same sign. 

In principle $\rho_m<0$ is possible in (1), with then repulsion between mass densities of different sign which would separate large scales into Universa with positive and negative mass, where we happen to live in one with mass positive. It is thinkable that presence of negative mass density shows up as dark energy. It is thinkable that a very smooth $\Delta\phi_m$ corresponds to dark matter.  

The gravitational force $f_m$ acts on large masses at large distances. The electric Coulomb force $f_c$ acts on small small charges at small distances, which requires physics preventing charges of different sign to come too close, which is represented by the presence of the Laplacian in Schrödinger's equation. 

Including also a magnetic potential connected to the electric potential by Maxwell's equations and Newton's 2nd Law for mass motion subject to force, gives a model including Newton's mechanics, electromagnetics and gravitation, with potentials as primordial quantities from which mass and charge densities and forces are derived. Here Real Quantum Mechanics naturally fits in as a classical 3d continuum mechanics model. 

An important aspect of (1) and (2) is that $\rho_m$ and $\rho_c$ are derived by differentiation as an operation acting locally in space, which can be perceived to act instantly in time,  thus avoiding the hard-to-explain instant-action-at-distance coming with the standard view with mass and charge densities as primordial. 

The absence of magnetic monopoles corresponding to point charges makes magnetics different from electrics in the formation of electromagnetics.  

 

2nd Law for Cosmology

A mathematical model of the Universe can take the form of Euler's equations for a gas supplemented with Newton's law of gravitation as stated in Chap 32 Cosmology of Computational Thermodynamics.  

Computational solutions of these equations satisfy the following evolution equations as laws of thermodynamics depending on time $t$ 

• $\dot K(t)=W(t)-D(t)-\dot\Phi (t)$     (1)
• $\dot E(t)=-W(t)+D(t)$,                  (2)

where $K(t)$ is total kinetic energy, $E(t)$ total internal energy (heat energy), $W(t)$ is total work, $D(t)\ge 0$ is total turbulent dissipation, $\Phi (t)$ is total gravitational energy and the dot signifies differentiation with respect to time. Adding (1) and (2) gives the following total energy balance:

• $K(t)+E(t)-\Phi(t)= constant.$          (3)

Further (1) and (2) express an irreversible transfer of energy from kinetic to internal energy with $D(t)>0$, and so serve as a 2nd Law for Cosmology giving time a direction. Recall that the theoretical challenge is to tell/show why turbulent dissipation is unavoidable. 

Computations may start from a hot dense state at $t=0$ which is seen to expand/cool (run code) (Big Bang) to maximal size and then contract/warm back to a hot dense state (Big Crunch) (run code) in an irreversible sequence of expansions/contractions until some final stationary equilibrium state with $E(\infty )=P(\infty )$. Compare with post from 2011.

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? 

Newton vs Big Bang, Dark Energy and Dark Matter

In her latest post Sabine Hossenfelder asks if we can get energy for free e g in the form of Dark Energy as a main mystery of modern cosmology. Let us see what Newton can bring to this question starting with his law of gravitation: 

• $\Delta\phi =\rho$ or $\rho =\Delta\phi$          

connecting mass density $\rho (x,t)$ to gravitational potential $\phi (x,t)$ though the Laplacian differential operator $\Delta$ with $x$ a Euclidean space coordinate and $t$ time. 

The standard view is that mass density is non-negative $\rho (x,t)\ge 0$ for all $(x,t)$, but if we expand the scope why not allow $\rho (x,t)$ to also locally be negative, then corresponding to some form of negative mass. If we dare to take this step, we find the following remarkable facts:

1. With $\rho (x,t)$ an initial vanishingly small perturbation of an initial zero state varying very quickly in space between positive and negative values, the corresponding potential $\phi (x,t)$ will inflate to substantial size, as if gravitational potential is created out of nothing. This may correspond to a Big Bang from which a Universe filled with both positive and negative mass can evolve. 
2. Regions with negative mass density repel regions with positive mass density and so create an expansion seemingly out of nothing, which may correspond to Dark Energy, while larger regions of small positive and negative mass density can form and then locally contract by gravitational attraction into galaxies with large local density.
3. Large regions where $\phi (x,t)$ is slowly varying with $\rho (x,t)=\Delta\phi (x,t)\ge 0$ small may correspond to Dark Matter, which is not visible but still has major gravitational effect.     

In one shot, we thus open to new views on both Big Bang, Dark Energy and Dark Matter. Any comment?

More substance to such a scenario is given in blog posts on New Newtonian Cosmology. 

Motion vs Appearance or Emergence

This is a continuation of earlier posts on Zeno's paradox as an unresolved mystery of the physics of motion:

• How can an arrow move, when at each time instant it is still, that is, not moving?
• Is the arrow jumping from one position to the next in a discrete series of events in increasing time? 

No convincing resolution is offered by either classical or modern physics, and so the question is dismissed as a no-question so obvious that it does not need any explanation: Just look and see how things are moving  or shifting positions in space $x(t)$ with time $t$, with velocity $v(t)=\dot x(t)$ and the dot signifies differentiation with respect to time. 

Given a velocity $v(t)$, the corresponding motion/trajectory $x(t)$ is created by exactly solving the differential equation $\dot x(t)=v(t)$ (as if the arrow is smoothly changing position in time without jumps), or by time-stepping from one discrete time instant to a next (as if the arrow de facto is jumping).   

But a child eager to understand the World may not be satisfied with such an empty explanation, but maybe by the following argument:

Let us compare the concept of motion with that of appearance or emergence. If a certain person appears at a party, invited or not, the question may come up how the person got there, more precisely what trajectory of motion the person had followed? Today the path would be stored in the cloud, but then as a discrete sequence of still-positions just like the arrow, and the basic question would remain: How is motion possible at all? But fact is that the person did appear and so let us shift focus from motion to appearance.

We then take Newtonian mechanics to our help which describes the World by the following conservation laws in Eulerian form:

• $\dot\rho +\nabla\cdot m=0$                                (conservation of mass)                          (1)
• $\rho =\Delta\phi$                                          (conservation of gravitational force)     (2)
• $\dot m +\nabla\cdot (vm)-\rho\nabla\phi=0$        (conservation of momentum)                (3)                                                            

where $\rho (x,t)$ is mass density, $\phi (x,t)$ gravitational potential, $m$ is momentum, $v= \frac{m}{\rho}$ is velocity and $x$ a Euclidean space coordinate.

The standard way of interpreting (1)-(3) is to say that presence of mass at $(x,t)$ creates the gravitational potential $\phi (y,t)$ for all points $y$ different from $x$ by instant action at distance at time $t$, which however lacks physics explanation. Further, trajectories of motion $x(t)$ appear as solutions to $\dot x=v(x,t)$. 

I have suggested a different possibility, which is to view instead the potential $\phi (x,t)$ as primary from which mass $\rho (x,t)=\Delta\phi (x,t)$ is created by differentiation as an instant local action expressed by the Laplacian $\Delta$, which possibly is not inexplicable. The potential $\phi (x,t)$ then changes or evolves in time according to (1) with connection (2), without any need of particle trajectories of motion, 

In this view mass emerges or appears at different locations in space following the evolution of the gravitational potential, and we do not have to speak about particle/mass motion and explain exactly how the motion is realised. It connects to time-stepping corresponding to jumping from one discrete time event to the next. 

So it may be fruitful to think of appearance evolving in time rather than motion. In this perspective motion is illusionary, like a water wave appearing to move in space without corresponding motion of water.  

 

Speed of Gravity in a Static Gravitational Field?

To save General Relativity GR assuming that the speed of gravity is finite equal to the speed of light from collapse when confronted with observations apparently requiring a very much larger speed of gravity, it is commonly stated that in a static gravitational field there is no effect of time delay from finite speed of of propagation. And so common wisdom claims that there is no contradiction between GR and observations conforming to a speed of gravity much larger than the speed of light. 

It is this convincing? No problem in GR from finite speed of gravity? Let us see: A gravitational model with finite speed of propagation $c$ in a Newtonian approximation of GR takes the form

• $\frac{1}{c^2}\ddot\phi -\Delta\phi =\rho $,       (*)

where $\phi (x,t)$ is the gravitational potential corresponding to a mass distribution $\rho (x,t)$, where $x$ is a Euclidean space coordinate, $t$ a time coordinate and the dot signifies differentiation in time. 

Now a static gravitational potential is characterised by $\ddot\phi =\dot\phi =0$ and so the value of $c$ can be anything, in particular as large as desired even larger than the speed of light without changing anything. In other words it is meaningless to speak about speed of gravity in a static gravitational field. 

To state that in a static gravitational field in GR there is no effect of finite speed of gravity does not make sense. There is no speed at all. 

Further, gravitational fields are not static, not even between the Sun and Jupiter, and so this case lacks interest. 

Yet in GR the speed of gravity is viewed to be finite = speed of light c, which requires a theory of quantum gravity to explain finite speed. But no theory of quantum gravity has been found despite intense search for 100 years. Further, gravitational waves in GR are viewed to require merge of black holes to appear... 

The idea of a finite speed of gravity = speed of light is the main road block to a Theory of Everything ToE combining Newton, Maxwell and Schrödinger. What would happen if we simply remove the block by replacing Einstein by Newton? What would be missed? Nothing? And then?

 

Speed of Light vs Speed of Gravity: Maxwell vs Newton vs Aether

Propagation of light in vacuum is described by Maxwell's equations expressed in terms of an electric field $E(x,t)$ and a magnetic field $B(x,t)$ where $x=(x_1,x_2,x_3)$ is the coordinate of an Euclidean spatial coordinate system $X$ and $t$ is a time coordinate, with dot representing differentiation with respect to time:

• $\dot B + \nabla\times E =0$  and $\dot E - \nabla\times B =0$    (1)   

where $\nabla =(\frac{\partial}{\partial x_1},\frac{\partial}{\partial x_2},\frac{\partial}{\partial x_3})$, and the speed of light $c$ is normalised to 1. Observation of the speed of light in the system $X$ by an observer $O$, thus gives the value 1. Since today the meter is defined in terms of light second, $c=1$ is an agreement and not a law of physics. 

So far so good, but what about the speed of the $X$? Relative to what? 

Suppose a different observer $O^\prime$ relies on the same Maxwell's equations (1) expressed in a different coordinate system $X^\prime$ moving with relative constant speed $v$ vs $X$, as a so called inertial system. Analysis in Many-Minds Relativity Chap 18 shows that $O$ and $O^\prime$ will agree up to a precision scaling with $v^2$. For human observers this means a precision of $10^{-9}$, which may be enough for all practical purposes. This means that (1) is Galilean invariant up to a precision of $v^2$. More precisely both observers will consider the speed of light to be exactly 1, since they agree to use the same Maxwell's equations (1).

To use Maxwell's equations (1) requires specification of the coordinate system and the natural choice is to lock the coordinate system to the observation apparatus and so allow the possibility of different apparatus moving with respect to each other, with observations agreeing up to $v^2$ with $v<<1$ for human observers. Many-Minds Relativity expands the scope to $v<1$.

Sum up: Maxwell's equations requires specification of spatial coordinate system. Different observers may use different inertial coordinate systems moving with relative speed $v$ and will then agree up to  $v^2$, and exactly agree on the speed of light. The choice of a specific coordinate system effectively represents a choice of an aether, so there are as many aethers as coordinate systems. 

Let us now turn to Newtonian gravitation described by 

• $\Delta\phi =\rho$            (2)

where $\phi (x,t)$ is gravitational potential and $\rho (x,t)$ mass density, and $\Delta$ is the Laplacian in the coordinates $x$ of a Euclidean coordinate system $X$. We understand that (2) is exactly Galilean invariant since (2) reads the same independent of any motion of $X$ with constant velocity, because no time derivative is involved. All inertial coordinate systems thus give the same description of gravitation. 

In the sense of Einstein it means that (2) satisfies Einstein's definition of a (perfect fundamental) physical law, as a law of physics which takes exactly the same form in all inertial systems (as an expression of Galilean invariance). 

The speed of gravity in (2) is formally infinite if $\rho$ is viewed to be primary from which $\phi$ is created by formally instant action at distance, which is unthinkable. Viewing instead $\phi$ as primary with $\rho$ the result of differentiation replaces instant action at distance by instant local action, which is thinkable. It is also possible to view (2) as a side condition without specifying cause-effect. In the latter perspectives the notion of speed of gravity is not needed.  

Conclusion:  

1. Newton's law of gravitation (2) is Galilean invariant an so is a thinkable prefect physical law for which a notion of speed of gravity is not needed. No aether enters the discussion. 
2. Maxwell's equations is Galilean invariant up to $v^2$, where for human observers $v^2<10^{-9}$, with $c=1$ acting as an agreement. Each choice of coordinate system represents and aether. 
3. The speed of light serves a fundamental role, while a speed of gravity is not needed.
4. Massless electromagnetics and mass gravitation are fundamentally different, which contradicts Einstein. Search of gravitons as gravitational analog of photons is fruitless. 
5. There is no need to modify Newtonian mechanics, and so Einstein's relativity serves no purpose. 
6. A Grand Unified Theory as Maxwell + Newton is readily available. 

Gravitation and Continuum Models

In the CNPS talk on Febr 3 I tried to expose the virtues of a continuum as a spatial 3d Euclidean x-coordinate system without smallest scale as the reference system of continuum mechanics in Eulerian form.  As a basic example let us consider the Euler equations for incompressible flow expressing balance of momentum (Newton's 2nd Law) combined with incompressibility in the form 

• $\nabla\cdot u = 0$        (1)

stating that divergence of velocity field $u(x,t)$ vanishes for all $x$ and time $t$. Here (1) appears as a stipulation or side condition for which the Lagrange multiplier is the pressure $p$, which appears as a pressure force $\nabla p$ in the momentum equation with connection through Gauss Law:

• $\int p\nabla\cdot u\, dx = -\int \nabla p\cdot u\, dx$.

The bottom line is that $\nabla p$ appears in the momentum equation as a force effectively imposing (1) while not specifying the physical nature of the force in a pressure law. The beauty is now that solving the Euler equations computationally gives full information about incompressible flow with vanishingly small viscosity, as shown in this book and this book. The divergence zero condition (1) is in computation replaced by an effective computational pressure law of the form 

• $-\Delta p = \frac{\nabla\cdot u}{\delta}$,     (2)  

where $\delta $ is a small parameter scaling with the mesh size, for which true physics is not needed. The Euler equations as a continuum model thus in computational form constructs a pressure law imposing near incompressibility. The continuum model in computational form thus invents physics which shows to describe reality in the form of physics as computation. 

We compare the continuum model with a particle model of a fluid asking for full specification of force between particles, and understand that a computational continuum model relieves us from a very difficult if not impossible task coming with a particle model. 

We now turn to Newtonian gravitation where the analog of (1) is Newton's Law of Gravitation in the form 

• $-\Delta \phi = \rho$       (3) 

connecting gravitational potential $\phi$ to mass density $\rho$ by the Laplacian differential operator $\Delta$. The corresponding Lagrange multiplier appears in the momentum equation as 

• $\rho\nabla\phi$                (4)

interpreted as gravitational force analogous to the pressure force connected to (1). Computationally (3) may take the following form allowing time-stepping:

• $\frac{\dot\phi}{C}-\Delta \phi = \rho$           
• $\frac{\ddot\phi}{C^2}-\Delta \phi = \rho$     (5)

where $C$ is a large constant representing effective speed of propagation, and the dot signifies differentiation with respect to time. Comparing computations with observation indicates that $C$ is much larger than the speed of light. 

Recall that it is well understood by everybody, except Einstein and his followers, that (3) expresses that (i) gravitational force $F$ is conservative, thus given by a potential $\phi$ as $F=\nabla\phi$,  and that (ii) $F$ is conserved in the sense of Gauss Law with $\nabla\cdot F = 0$ where there is no mass. To question (3) lacks rationale as it would violate (i) or (ii). In fact (3) is the prime jewel of all of physics, and to seek to modify it makes no sense. 

The beauty is here that the Euler equations augmented by gravitation in the form (3) and (4) (see this book) appears to describe a very rich world on a very wide range of scales, without having to specify the exact nature of the real physics of gravitation, which is still hidden, thus following the spirit of Newton.

The beauty is enhanced by realising that also quantum mechanics can be captured as a continuum model over a 3d Euclidean coordinate system without smallest scale allowing microscopics and macroscopics to have the same seamless conceptual form as shown in Real Quantum Mechanics.  This is shocking to modern physicists educated to view microscopics beyond comprehension for humans with only macroscopic experience.

Continuum models like the Euler equations thus appear as realisations of physics as computation expressing physics in possibly new forms open to understanding. 

PS1 The total energy for incompressible flow based on (2) includes a positive contribution of the form

• $\delta\int\vert \nabla p\vert^2dx$ 

and similarly total energy balance with gravitation in the form (4) contributes (with details here)

• $\int\vert\nabla\phi\vert^2dx$     

 as a natural expression of gravitational energy (as a source of kinetic energy) and in the form (5):

• $\int\vert\nabla\phi\vert^2dx+\frac{1}{C^2}\int\dot\phi^2dx$,

where the real physics of the second term with the time derivative $\dot\phi$ is less clear, and so may be interpreted rather as computational artefact allowing time-stepping. Recall that the presence of a time derivate in an energy expression represents kinetic energy from motion of matter, which is not an aspect of $\phi (x,t)$ expressing spatial presence of gravitational potential/force.   

PS2 Multiplying (3) by $\phi$ and integrating gives:

• $\int\vert\nabla\phi\vert^2dx = \int\rho\phi dx$           

where the right hand side commonly is referred to as gravitational potential energy. We see that the left hand side includes only the gravitational potential $\phi$, which connects to viewing $\phi$ as primary, as suggested in previous posts on New Newtonian gravitation. 

PS3 We may compare (3) with a law of the form 

• $\phi = \rho$

which expresses instant local action and  connects to gas law of (isothermal) compressible flow of the form $p=\rho$ with $p$ pressure, with $\nabla\phi$ corresponding to $-\nabla p$.   

Speed of Gravity? Newton or Einstein?

In Newtonian mechanics gravitational potential $\phi (x,t)$ is connected to mass density $\rho (x,t)$ by the Laplacian differential operator $\Delta$ through the equation 

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

where $x$ is a Euclidean coordinate and $t$ a time coordinate. 

The standard way is to view the gravitational potential $\phi (x,t)$ and gravitational force $\nabla\phi (x,t)$ at some space-time coordinate $(x,t)$ as somehow being generated from the distribution of mass density $\rho (y,t)$ for all $y$ different from $x$ in an apparent instant action at distance at time $t$ as if gravitational force is propagated with infinite speed.  Solving the differential equation in terms of $\phi$ is a global operation of integration/summation. 

But instant action at distance is inexplicable and so in modern physics Newton's mechanics has been replaced by Einstein's General Theory of Relativity GR, where gravitational force is propagated with the finite speed of light c. 

On the other hand, if we in (1) view the gravitational potential $\phi (x,t)$ as primary from which mass density $\rho (x,t)$ is generated by the action of the differential operator $\Delta$ as 

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

which is a local operation at $x$ of differentiation, which is possible to view to be instant. Mass density is here generated by instant local action from gravitational potential, and then the problem of instant action at distance evaporates and it is no longer necessary to replace Newton by Einstein. 

Newtonian mechanics describes celestial dynamics in the form an initial value problem 

• $\dot x(t) = f(x(t))$ for $t>0$ with $x(0)$ given  

where $x(t)$ represents positions of celestial bodies at time $t$, and $f(x)$ is a given function of $x$ including Newton's law of gravitation. The  differential equation can be solved by explicit time stepping of the form

• $x(t) = x(t-dt) + dt*f(x(t-dt))$                      

where $dt>0$ is a time step. The position $x(t)$ at time $t$ is thus computed from previous position $x(t-dt)$ with a correction $dt*f(x(t-dt))$ determined at the previous time $t-dt$, thus with a time delay of $dt$. We can view the time delay $dt$ as an expression of finite speed of propagation $C$, and we now ask if observations can give information about the size of $C$. 

We do this in the simplest case of one small body (Earth) orbiting a big body (Sun) as expressed in this code, where we can test the effect of different time steps $dt$. By normalisation we can connect $dt$ to $\frac{1}{C}$ as the time required for a gravitational signal from the Sun to reach the Earth. The effect on position $x(T)$ at time $T>>0$ of explicit time stepping with time step $dt$ at best scales with $T*dt$ and if we ask for a precision of $\epsilon$, we have 

• $dt < \frac{\epsilon}{T}$, that is $C>\frac{T}{\epsilon}.$                              

Relevant values may be $T>10^4$ and $\epsilon <10^{-4}$, that is 

• $C>10^8$.

If we put this number in the perspective of the Sun-Earth system with the speed of the Earth about $0.0001$ times the speed of light $c$, we get $C>10^{12}$ to be compared with $c=3\times 10^8$ meter/second with thus a factor at least $10^3$. We can compare with the estimate $10^7$ made by Laplace and even $10^{10}$ from the PS below.  

In any case, observations indicate that the required speed of propagation of gravitational effects in (1) is several orders of magnitude bigger than the speed of light.

A planetary system based on (1) with a time delay from finite speed of gravity equal to the speed of light would not persist over time. 

Newtonian mechanics describes celestial/planetary motion very accurately over long time with a from observations apparent speed of gravity much bigger than the speed of light. If (1) is viewed to express Newton's law of gravitation this essentially requires instant action at distance, which is unthinkable.

But changing view to (2) replaces instant action at distance by instant local action, which is thinkable.  

Since in GR the speed of gravity is finite, one would expect to see effects in GR of time delay, but that would contradict observations where no time delay can be detected. The situation is handled by claiming that  there is a very subtle strange effect of cancellation in GR, which means that in the end the effective speed of gravity is infinite. So GR says the the speed of gravity is finite equal to the speed of light, but the effect of finite speed is cancelled and so the net effect is zero as if the speed of gravity in fact is infinite as in Newtonian gravity. Do you buy this argument? 

Recall that Einstein when claiming that Newtonian gravitation must be replaced by GR, could not refrain from expressing "Newton, forgive me." as if he had committed a scientific crime.

It is also possible consider a potential-mass connection of the form  

• $\Delta\phi - \rho = 0$                          (3)

where the cause-effect is not indicated. Here (3) appears rather as a side condition expressing a balance of potential and mass without worrying about casual connection, see this book and this computation exploring the Euler equations for fluid flow with gravitation.

Sum up: There is no reason to replace Newton by Einstein, and anyway doing so leads to a quagmire of mysteries. It is not necessary to view mass as primary from which gravitational potential/force is formed by apparent instant action at distance as in (1), which is unthinkable. We may as well turn (1) around into (2) viewing gravitational potential to be primary from which mass is formed by instant local action, which is not unthinkable. 

PS1 Tom van Flandern states in The speed of gravity - What experiments say:

• Laboratory, solar system, and astrophysical experiments for the “speed of gravity” yield a lower limit of $2\times 10^{10}c$.
• But mediation requires propagation, and finite bodies should be incapable of propagating at infinite speeds since that would require infinite energy. So instantaneous gravity seemed to have an element of magic to it.
• We will examine the explanations offered by GR for these phenomena, and conclude that in the most widely taught curved space-time interpretation of GR the acceleration of bodies through space lacks a causal connection to the source of gravity. And we will confront the dilemma that remains when we are through: whether to modify our existing interpretation of GR, or give up the principle of causality.

It thus appears that GR assuming that the speed of gravity is finite equal to the speed of light, is incompatible with experiments showing an infinite speed of gravity, which asks for modification of GR.

This modification may simply be a return to Newtonian gravitation with a law of gravitation of the form (2) with instant local action, which is not in contradiction to causality.  

The reason that mass traditionally is viewed to be primary and potential a derived quantity as in (1), is (probably) that mass may be directly visible and gravitational potential/force is not. On the other hand, all bodies directly "feel" gravitational force, and so gravitational potential/force is very present although not directly visible.   

PS2 Further evidence of speed of gravity being much larger than the speed of light from observation of satellite motion, is given here.

PS3 Maxwell's wave equations for electromagnetics describe propagation of light of all frequencies at the same speed = speed of light = c. Augmented with an Abraham-Lorentz radiation force Maxwell's wave equations also describe radiative heat transfer as Computational Blackbody Radiation as a one-way transfer of energy from warm to cold mediated by two-way electromagnetic waves, more precisely as a resonance phenomenon. The speed of radiation thus in principle can be viewed to be equal to c, even if in reality effective transfer of energy from resonance may change at a slower speed. 

Finite speed of gravity = C requires augmentation of (1) into a wave equation $-\frac{\ddot\phi}{C^2}+\Delta\phi =\rho$ with the dot signifying differentiation in time, and transfer of energy by gravitational waves requires some form of Abraham-Lorentz force. Observations show that C is much bigger than c, and so both theory and observation supporting C=c, is lacking. 

While radiative wave energy transfer is a reality, there seems to be little evidence that gravitational wave energy transfer is real. The proclaimed experimental detection of very faint gravitational wave energy transfer from distant merging black holes suffers from the difficulty of finding a needle in a haystack.  

  

The Role of Differentiation and Integration in Physics

This is a further reflection on the idea of Physics as Computation in the previous post with focus on the mystery of instant action at distance (there are many posts on this topic). 

Mathematical models of physics typically take the form of differential equations such as Poisson’s equation 

• $\rho=\Delta\Phi$        (1)
• $\Delta\Phi = \rho$      (2)

connecting gravitational/electric potential $\Phi (x,t)$, depending on a space coordinate $x$ and time coordinate $t$ coordinate, to mass/charge density $\rho (x,t)$, and $\Delta$ is the Laplacian differential operator involving second order differentiation.  

In a Hen-Egg setting $\Phi$ represents Hen and $\rho$ Egg, either as local differentiation/assignment $\rho =\Delta\Phi$ as Hen-laying-Egg,  or solution of $\Delta\Phi = \rho$ by global integration/summation as Egg-generating-Hen by instant action at distance. 

How to choose between (1) and (2)? Local differentiation or global integration/instant action at distance? 

If you are a (pure) mathematician, you would without hesitation say that there is a method for symbolic differentiation and so (1) is in a sense trivial. On the other hand there is no method for symbolic solution of (2), which is the non-trivial problem of the symbolic Calculus of Leibniz/Newton. 

Mathematicians know that if symbolic solution fails, because it has no method, it is always possible to resort to numerics as a form of trivial work-horse, which case-by-case can compute solutions by number crunching. So is mathematics split into symbolic/analytical mathematics and numerical mathematics (in descending prestige) with essentially different basic elements: symbols or numbers.

As an example, symbolic differentiation is trivial while numerical differentiation is a delicate subject because a derivative $\frac{dx}{dt}$ involves the quotient of small numbers requiring precision. In general differentiation is a delicate process because precise identification is needed. So what can be trivial in symbolic mathematics can be non-trivial in numerical mathematics. 

On the other hand, symbolic integration is non-trivial while numerical integration is trivial as it is just a form of summation. 

So the world of symbolic mathematician and numerical mathematics is very different, since what is trivial or non-trivial can be opposite. 

We now turn to real physics as something real existing in the real world (ontology). What is the relation of real physics to symbolic mathematics and to numerical mathematics? 

Since symbolic mathematics works with symbols rather than numbers it has a connection to epistemology. We now ask if numerical mathematics is closer to ontology/real physics and so if we can learn anything about real physics from numerical computation.

In particular, we seek the real physics of (1) vs (2) as the Hen-Egg question posed above. We recall that numerical solution of (2) is trivial as simply global summation, while (1) is non-trivial as delicate differentiation. 

If we believe that physics is non-trivial,  then (1) represents physics: Hen lays Egg as a delicate non-trivial local operation, but not asking for instant action at distance in a global solution process. 

On the other hand, for (2) to represent physics requires instant action at distance as instant global summation. 

We are thus led to the conclusion that (1) represents real physics as a local differentiation process. The apparent instant action at distance in (2) would then represent non-real fiction.

We thus find support of an idea that computation and real physics are closely connected, while the connection of symbolic mathematics to real physics can be difficult to assess.  

The gravitational potential generates mass by local differentiation. Mass does not (have to) generate gravitational potential by global instant action at distance. 

It seems to make sense to say that physics differentiates in the sense of evaluating force differences, while physics integrates by time stepping creating motion, which opens to physics without mysterious instant action at distance. In numerics differentiation (1) can be traded with integration (2) with fictitious instant action at distance.

Another aspect is that you can see mass but not gravitational potential itself only its effect, and you may be tempted to believe that what you can see is primary and what you cannot see is secondary. But that depends on your senses of perception and so may not tell the true story from an objective physical point of view. You see a person getting smaller receding from you, but you know it is an illusion.

PS It is also possible to give up the cause-effect aspect in the potential-mass connection and like Leibniz say that potential and mass are connected in Perfect Harmony or Best of Worlds, which has been ridiculed...maybe it is time for Leibniz to come back...in any case he laid the mathematical foundations to the digital world as a world combining ontology with epistemology... 

But of course it is possible to turn the argument around as follows: Consider Newton's 2nd Law

• $\frac{dv}{dt} = f(t)$     (3)

where $v(t)$ is velocity and $f(t)$ is force. The standard view is that the force $f(t)$ is given and causes the acceleration $\frac{dv}{dt} = f(t)$ as (2). But we may also view $v(t)$ to be given and $f(t)=\frac{dv}{dt}$ simply the force required as in (1). This describes a situation where the nature of the force is unclear, while velocity/motion is very real. This is the case with the Coriolis force and of course centrifugal force. Einstein tried to get rid of gravitational force (and aether) altogether, but did not succeed…

Numerical solution of (3) is done by time stepping $dv=f(t)*dt$ updating velocity with input from force, which has direct physical meaning as motion as summation, thus with numerics in tandem along with (2).   

 

Conclusion: We may say that numerics can connect to both (1) and (2), while the role of symbolic math in physics remains to be made precise with the physical meaning of the symbolic wave function $\Psi$ of quantum mechanics, as solution to Schrödinger’s equation, after 100 years of constant brooding, still being a complete mystery in its standard so called Copenhagen interpretation. On the other hand, RealQM offers a physical meaning in classical continuum mechanics terms with the kinetic energy of the electrons appearing as a form of elastic energy preventing the electron to fall into the kernel by an elastic force balancing kernel attraction, just like the centrifugal force of motion prevents a planet to fall into its Sun. Both forces appear as necessary conditions for maintenance of certain states (Lagrange multipliers) as virtual forces without concrete physical origin: Planets move the way they do because forces balance, and electrons assemble around the kernel because forces balance. Leibniz would agree, I am sure!

Recall that the wave function $\Psi (x)$ for the ground state of the Hydrogen atom minimises the total energy E as "kinetic" energy + potential energy: 

• $E(\psi ) = \frac{1}{2}\int\vert\nabla\psi\vert^2dx- \int\frac{\psi^2 (x)}{\vert x\vert}dx$

over all real-valued functions $\psi (x)$ with $\int\psi^2dx=1$, which can alternatively be interpreted as the state of a normalised elastic cloud subject to an elastic central force with the kinetic energy appearing as "elastic" energy.  The terminology "kinetic" energy is misleading (motivated by some deep symbolism) since no motion is involved, while "elastic" energy has a concrete physical meaning as a measure of elastic compression suggesting some form of electronic charge compression for the atom.   

Instant Action at Distance in Atom Physics/Quantum Mechanics

Instant action at distance is a fundamental element of both macro-scale gravitational mechanics and micro-scale quantum mechanics in the form of Newton’s Law of gravitation and Coulomb’s Law of electrostatics. 

The idea is that the presence of a mass/charge at one point in physical space without time delay generates a force at all other points decaying with the inverse square of distance, as the fundamental force of both classical and modern physics of Newton/Einstein and Heisenberg and Feynman as the golden boys of quantum mechanics, and of course Schrödinger.  

It also formed the foundation of the now forgotten, but once great, physicist Joseph Boscovich (1711-1787) as expressed in his monumental "A Theory of Natural Philosophy reduced to one unique Law of forces that exist in Nature" stating that the World is the result of instant action at distance of attractive and repulsive forces on both small and large scales. This a nothing but a Grand Unified Theory and what remains is to fill in details about the forces and in particular to explain how instant action at distance is realised, which has remained a fundamental mystery of physics. See the book Roger Boscovich-The Founder of Modern Science, by Stoiljkovic.

One way to summarise physics is to recall that both Newton's Law and Coulomb's Law take the form of Poissons’ equation: 

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

where $\Delta$ is the Laplacian acting in 3d space with coordinates $x$, $\phi (x)$ is  gravitational/electric potential and $\rho (x)$ is mass/charge density. This is a consequence of in the equation (1) viewing $\rho (x)$ as a locally given source generating the potential $\phi (x)$ globally as a solution to Poisson's equation which can be seen as a form of instant integration/summation process sending local source information instantly around globally as instant action at distance. Forces are generated as $\nabla\phi (x)$.

Boscovich's Theory that all force is instant action at distance contradicted the classical idea that forces are transmitted by contact, adding the explanation that there is always some little distance between different material bodies including atoms maintained by ever-present repellation thus reducing physics to one unique Law. See the book Roger Boscovich- The Founder of Modern Science by Stoiljkovich. 

It is natural to consider (1) as a limit of the following time dependent heat/wave equations:

• $\epsilon\dot\phi -\Delta \phi = -\rho$,     (2)
• $\ddot\phi -\Delta\phi = -\rho$,                  (3)

where the dot indicates differentiation with respect to time $t$, and $\epsilon >0$ is small constant formally reducing (2) and (3) to (1) when tending to zero. The expanded models require some form of heat conduction or wave propagation medium/ether giving physics to action at distance with finite speed. 

On the other hand (1) could be argued to not require any medium, since force transmission is replaced by instant action at distance, but then again without explanation. 

I have argued that that there is a way out of this dilemma by shifting the conception of the meaning of the equation (1) to a view with rather the potential $\phi (x)$ as primary source from which both force $\nabla\phi (x)$ and mass $\rho (x)=\Delta\phi (x) $ are generated through the local action of differentiation by the Laplacian differential operator. 

In this view potentials are primary from which everything (force/mass/charge) is generated by local differentiation. In particular it gives a new view on the quantum mechanics of an atom, where the primary concepts are the kernel and electron potentials, and the atom with kernel and electrons is generated by the Laplacian and then required to satisfy Schrödinger's equation. 

In physics it is natural to search for sources generating effects in a cause-effect setting, but the precise mechanism of generation may be difficult to pin down, e g exactly how differentiation generates mass from gravitational potential, or how instant action at distance comes about.

This connects to Leibniz' idea of a Pre-established Harmony beyond human inspection. The gravitational potential-mass harmony expressed by (1) may be of this kind. 

You find more under Labels.

  

 

Modified Newton Law of Gravitation

Newton's inverse square law of gravitation was by Laplace formulated as the Poisson equation

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

where $\phi (x,t)$ is gravitational potential, $\rho (x,t)$ is mass density,  $x$ is a Euclidean space coordinate, $t$ a time coordinate and $\Delta$ is the Laplacian differential operator with respect to $x$. The gravitational force is given by the potential gradient $-\nabla\phi (x,t)$. The trajectory $x(t)$ of a test particle can be computed from Newton's 2nd Law

• $\dot v (x(t),t) =-\nabla\phi (x(t),t)\equiv f(x(t),t)$        (2)

where $\dot v=\frac{dx}{dt}$ is the velocity of the particle acted upon by the gravitational force $f(x(t),t)$. This model describes the motion of a Universe subject to gravitation, and so represents a formidable achievement of mathematical physics. Test the model here.

Since the same time coordinate appears on both sides of (1), it appears that Newton's law of gravitation involves hard-to-explain instant action at distance and so invites to alternatives to (1) with finite speed of propagation of effects.              

One possibility is to relax the Poisson equation (1) into a wave equation

• $\epsilon\ddot\phi (x,t)-\Delta\phi (x,t) =-\rho (x,t)$,       (3)

supporting gravitational waves with finite speed of propagation. Another is relaxation into a heat equation

• $\epsilon\dot\phi (x,t) - \Delta\phi (x,t) =-\rho (x,t)$,        (4)

where $\epsilon$ is a small positive constant, with effectively finite speed of propagation (scaling with $\frac{1}{\epsilon}$). since only vanishingly small effects propagate with unlimited speed. 

With small $\epsilon$ solutions to (4) stay close to those of (2), while wave solutions of (3) in general do not. We therefore focus on the heat equation (4), which has received little attention in the literature. 

The relaxation in (4) corresponds to a delay of the action the effect of the gravitational force. The delay effect comes to expression in computing a particle trajectory $x(t)$ by Explicit Euler time-stepping with time step $dt$, where $x(t+dt)=x(t)+dx$ and $v(t+dt)=v(t)+dv$ are computed/predicted from $x(t)$ and $v(t)$ by Dumb Euler as position first:

• dx = v(t)*dt, 
• dv =f(x(t+dt),t)*dt, 

or Smart Euler as velocity first:

• dv = f(x(t),t)*dt, 
• dx = v(t+dt)*dt. 

Compare yourself Dumb Euler with Smart Euler and see a big difference in the delay effect. We see that in  Smart Euler velocity is updated from the force at old position, while position is updated from new velocity, and in Dumb Euler it is the other way around. 

The delay effect from replacing (2) by (4) thus comes to expression in Explicit Euler time stepping which in the form of Smart Euler is remarkably small. 

We thus find support to an idea of Modified Newton Gravitation according to the heat equation (4) with effectively a finite speed of propagation of gravitational effects, which is not critically depending on the relaxation parameter $\epsilon$. It is then natural to speculate about the possible physicality of Smart Euler with a delay effect from explicit time stepping not asking for instant action at distance. 

More posts on associated New Newtonian Gravitation with (1) updated according to (2) with Explicit Euler with effectively finite speed of propagation of gravitational effects/force/potential. Hopefully it can help to resurrect Newton's theory of gravitation and avoid the black hole of Einstein's General Relativity. 

PS We thus see a formal connection between temperature as measure of heat energy and gravitational potential as measure of gravitational energy with the connection:

• temperature $T$ $\Longleftrightarrow$  potential $\phi$ 
• heat flux $Q=-\nabla T$ $\Longleftrightarrow$ gravitational force $f=-\nabla\phi$
• heat sink $-F$ $\Longleftrightarrow$ mass density $\rho$
• heat capacity $\kappa$ $\Longleftrightarrow$ ?? $\epsilon$

both described by the same heat equation (4) expressing conservation of energy 

• $\kappa\dot T +\nabla\cdot Q = F$.

Newton's theory of gravitation is thus based on a principle of conservation of energy, which may be hard to dispute to motivate a need of Einstein's theory: It is unthinkable that Newton's inverse square law is incorrect, unless your thinking is comparable to Einstein's thinking... 

Perihelion Precession of Mercury vs Modern Physics vs Pataphysics

Science of Imaginary Solutions: Pataphysics

Einstein's General Theory of Relativity GR (1915) is viewed to be a crown jewel of modern physics replacing classical concepts of space, time and motion under gravitational force expressed in Newtonian mechanics, by an entirely new geometric world of "curved space-time" without gravitational force. 

Newton's mechanics fostered the scientific revolution in the 18th century, while GR opened to the revolution of modern physics of the 20th century. 

At least, this is what (most) modern physicists tell us: Newton's world of mechanics has to be replaced by Einstein's GR world of geometry. More precisely, Newton's mechanics has to be replaced by GR only for extreme speeds or gravitational force/curvature, while GR and Newton agree in most cases. 

The acceptance of GR has grown only slowly over the 20th century, since evidence of superiority of GR over Newton has shown to be evasive, as expressed by fact that the first Nobel Prize directly connected to GR was given only in 2020 to Roger Penrose:

•  for the discovery that black hole formation is a robust prediction of the general theory of relativity. 

The Prize is thus given to the "discovery that GR predicts" the existence black holes, which however cannot be verified. Is that evidence that GR is correct? That GR gives a prediction, the correctness of which cannot be tested? So the Prize in Physics has been awarded to the discovery of an aspect of GR as mathematical fiction regardless of any actual real truth value of GR.  It is like discovering that a certain mythological tradition admits the existence of Unicorns, regardless of the existence of any real ones. This looks like a Nobel Prize in Pataphysics as a branch of philosophy or science that examines imaginary phenomena that exist in a world beyond metaphysics. 

The first evidence of GR was presented by Einstein in a computation using GR to correct a Newton prediction of the precession of the perihelion of Mercury (very slow rotation of the elliptic orbit around the Sun) to exactly fit with observation. But the Newton prediction was made without a computer and so could not account for the full complexity of the problem involving all other planets and unknown inner motion of the Sun and more. 

So it is not clear that Newton fails as concerns Mercury. An example of the correction brought by viewing Sun-Mercury as a true two-body problem still within Newtonian mechanics, with Mercury influencing the motion of the Sun, instead of a one-body problem with fixed Sun, is given in

• The secret of planets’ perihelion between Newton and Einstein by Christian Corda. 

This shows that the correction captured by GR can also be captured by Newton. This is not surprising since the orbit of Mercury is not extreme at all, and so Newton and GR should agree. 

If then Mercury can be taken off the list of evidence of superiority of GR, what remains are extreme cases, so extreme that not even GR can be expected to work, such as black holes, so extreme that they cannot be observed, or even predicted by GR to be honest?  

Why is it important to normalise modern physics back to Newton's mechanics? Because, Newton's mechanics works very well together with quantum mechanics, where speeds are low and gravitation weak. Hopefully this can take modern physics out of its permanent crisis since 100 years caused by an unresolvable conflict between Einstein's mechanics and quantum mechanics: From pataphysics to real physics! In particular, quantum mechanics can be relieved of relativistic mechanics since speeds are low. 

PS1 The Nobel Prize to Penrose/GR is more precisely motivated as follows:

• A black hole is a supermassive compact object with a gravitational force so large that nothing, not even light, can escape from it. 
• In 1964, Roger Penrose proposed critical mathematical tools to describe black holes. 
• He showed that Einstein’s general theory of relativity means the formation of black holes must be seen as a natural process in the development of the universe. 
• He was also able to describe black holes in detail: at their farthest depths is a singularity where all known laws of nature dissolve.

Every word here triggers questions: Supermassive? Nothing can escape? Proposed? Critical mathematical tools? Must be seen? Natural process? In detail? Farthest depths? Singularity? All known laws of nature dissolve? 

Compare with Ethan Siegel: 

• This Is Why Scientists Will Never Exactly Solve General Relativity

Returning to Newton could offer a great relief from a 100 year spell. 

It is illuminating to inspect the picture presented by the Nobel Committee in its description of the scientific work of Penrose as concerns the nature of the interior of a black hole: 

Sometimes a picture tells more than 1000 words...

PS2 GR was initially met with deepest skepticism and was counted down and out by the 1950s. Then miraculously GR was revived in the 1960s and on, until the great triumph of detection in 2015 of utterly faint gravitational waves emitted by the most violent event thinkable in the from of collision of two black holes 1.3 billion years ago. Credible?  How much of the present crisis of physics can be blamed on GR?

Einstein before Newton as Root of Crisis of Modern Physics

ChatGPT can be used to get answers to scientific questions, which cannot readily be found in the scientific literature. This is because GPT has browsed a lot of text and is not yet smart enough to cover up if something is fishy and needs cover up.  Here is one example:

• Me: Did Einstein prove that Newton's inverse square law is wrong?
• GPT: No, Albert Einstein did not prove Newton's inverse square law of gravitation to be wrong. Einstein's theory of general relativity does not invalidate Newton's law of universal gravitation but provides a more comprehensive and accurate description of gravity in certain regimes.

So we are told that Einstein did not prove Newton's theory of gravitation with its inverse square law to be wrong. Einstein's general theory of relativity does not invalidate Newton's theory. This is what GPT has learned by reading text which thus is what some text says: Einstein gave more comprehensive and accurate description of gravity in certain regimes, not covered by Newton's theory. 

Newton's theory of gravitation including its spin-off of Newtonian mechanics fostered the scientific revolution of the 18th century and serves together with Maxwell's electromagnetics and quantum mechanics as the foundation of modern society. 

Nevertheless, we are told that we have to give up Newton's theory of gravitation/mechanics and replace it with Einstein's theory, even if Newton's inverse square law is not wrong, because there are regimes outside Newton's mechanics.

The prime such regime is electromagnetics described by Maxwell's equations, which does not include gravitation. 

So we are told that we have to abandon Newton's theory for Einstein's theory, because Newton's mechanics does not include electromagnetics.

Even if the logic is missing, this is what Einstein did in his special theory of relativity starting with electromagnetics without gravitation and then finding a form of relativistic mechanics without gravitation different from Newton's. Modern physicists following Einstein thus claim that 

• Newton's mechanics with gravitation but not electromagnetics, 

must be replaced by 

• relativistic mechanics with electromagnetics but not gravitation.  

The logic was missing and so Einstein went on to include gravitation in his general theory of relativity reducing to Newton's theory in regimes without electromagnetics. 

Einstein with followers are responsible for the present crisis of modern physics resulting from this unfortunate combination: 

• Newton has to be replaced by Einstein even in regimes perfectly covered by Newton.
• Einstein is not compatible with quantum mechanics, while Newton is.   

The crisis can be solved if Newton's mechanics is allowed to reign within the vast regimes it covers. This would restrict Einstein's relativity theory to concern only certain very extreme cases such as black holes, so extreme that even Einstein's theory can be questioned on very good grounds. 

Who is ready to take this step? Where are all the followers of Newton?

Sum up: 

• In Newton's mechanics gravitational force is fundamental. 
• In Einstein's special theory there is no gravitation at all. 
• In Einstein's general theory there is no gravitational force.