Creating the Universe by Plucking a String 2

The previous post outlined a possible mechanism to create a large scale Universe $U$ of non-negative mass density $\rho$ from a small scale small amplitude perturbation of a zero gravitational potential with mass density "created" from a gravitational potential $\phi$ by the instant action of the Laplacian differential operator $\Delta$ acting in a Euclidean space with coordinate $x$ according to the assignment   

•  $\rho (x,t)= \Delta\phi (x,t)$ for all $x$ and $t>0$,               (G)

where $t$ is a time coordinate starting at time $0$. The only force in $U$ so far is gravitational force $-\rho\nabla\phi$ which together with Newton's 2nd Law describes the dynamics in $U$, which is large scale.

The physics of $U$ is in a second step expanded with an electric potential $V$ with electric field $E=\nabla V$ as a small scale small amplitude perturbation of a zero electric potential with small scale large scale amplitude electric charge $q$ of variable sign created by an assignment analogous to (G):

• $q(x,t)=\Delta V(x,t)$ for all $x$ and $t>0$,                          (E)                 

with $q=q_++q_-$ split into $q_+\ge 0$ and $q_-\le 0$. A further contribution is given by connecting $E$ to a magnetic field $B$ through Maxwell's equations.

The result is a Universe including large scale large amplitude physics of gravitation and small scale large amplitude electro-magnetics created from small amplitude perturbations according to (G) and (E) through the action of $\Delta$, as a Universe created by plucking two strings. 

Note the fundamental difference: 

• Gravitational force is attractive/repulsive between masses of same/opposite sign. 
• Electric force is repulsive/attractive between masses of same/opposite sign.     

The effect is a Universe of larges scale non-negative mass density and small scale charge density of variable sign.

Creating the Universe by Plucking a String 1

Under the label New View on Gravitation I have tested the idea of letting gravitational potential $\phi (x,t)$ with $x$ a Euclidean coordinate and $t$ a time coordinate, have a primordial role from which mass density $\rho (x,t)$ is "created" by the Laplacian differential operator $\Delta$:

• $\rho (x,t) = \Delta \phi (x,t)$ for all $x$,         (G1)

assumed to act without time delay for all $t$. Mass is thus created locally for each $x$ by differentiation as an instant local operation acting at each time instant $t$.  

This is to be compared with the standard view that gravitational potential $\phi$  is created from primordial mass density as solution of the differential equation:

• $\Delta\phi (x,t)=\rho (x,t) $ for all $x$,           (G2)

represented by the integral formula

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

which requires instant action at distance and so represents a main mystery of modern physics.

We recall that the Laplacian is invariant under an orthogonal change of coordinates and so the creation process (G1) is independent of the choice of Euclidean coordinate system, which can be seen as a sign of possible physicality. 

(G1) allows "creation of something big from something small" in the following way: Let us start from a gravitational potential $\Phi =0$ satisfying $\Delta\Phi =0$ thus with corresponding zero mass density. Let $\Phi$ be subject to a small scale small amplitude oscillatory perturbation $\phi$ creating small scale large amplitude oscillatory mass density $\rho =\Delta\phi$ by the action of second order differentiation with $\rho =\rho_++\rho_-$ decomposed into positive mass density $\rho_+$ and negative mass density $\rho_-$.  

The action of gravitational force $-\nabla\phi$ will cause attraction between mass densities of the same sign and repulsion between mass densities of opposite sign, and so will segregate the small scale variation of $\rho$ into one Universe of positive mass and another of negative mass which repel each other and so recede. 

By the action of the Laplacian on a small scale small amplitude oscillating perturbation of a zero state gravitational potential $\Phi =0$, a large scale large amplitude positive Universe $U_+$ with positive mass $\rho_+$ has thus been created, which is balanced by a corresponding negative mass Universe $U_-$ at eventually large distance. 

It is thus possible to envision a scenario where a small perturbation of a zero gravitational potential by the action of differentiation + gravitational attraction/repulsion generates a large scale Universe with positive mass thus representing "creation of something very big from something very small".  

Does this mean that the scenario really starts from zero? Not quite, because starting from $\Phi =0$ satisfying $\Delta\Phi =0$ requires some spatial structure to express the Laplacian. We can see this spatial structure as a string under tension and the small scale small amplitude perturbation as a small amplitude high frequency excitation of the string. 

The creation process thus starts with a spatial structure under tension but without excitation, from which the Universe we are living in is created with "a little pluck of a string".

The scenario opens to the existence of dark matter identified by $\Delta\phi$ of small magnitude but large extension, and dark energy as influence on $U_+$ from $U_-$. 

 

Gravitation: Newton or Einstein?

Modern physics is based on an assumption that Einstein's Theory of Gravitation EG in the form of his General Theory of Relativity gives a more precise description of the true physics of gravitation than Newton's Theory of Gravitation NG. 

Is this assumption justified? What is the evidence that EG is more precise than NG? 

NG based on Newton's 2nd Law and Newton's Law of Gravitation (inverse square law) combines maximal generality with maximal formal theoretical simplicity allowing computational simulation of gravitational interaction of billions of stars/planets over billions of years. 

EG on the other hand comes with maximal theoretical complexity making computational simulation impossible for gravitational interaction already for 3 stars/planets. 

Is it then possible to verify that EG is more precise than NG? If EG is uncomputable? 

The prime evidence that EG is more precise than NG, is a back-of-an-envelope computation by Einstein in 1915 concerning the precession of Mercury showing a correction to a computation by hand using NG made in 1888 by the astronomer Simon Newcomb supposedly taking into account all the effects from the other planets, with the result of 5557 seconds of arc per century (one second of arc=1/3600 degrees). The observed precession was 5600 and Einstein's back-of-an-envelop computation came up with exactly the missing 43 arc seconds per century, which still serves as main evidence that EG is more precise than NG.

How convincing is this? Questions line up:

1. How precise is the computation by hand by Newcomb, supposed to account for all effects in the Solar system with its planets, moons and asteroids swirling around the Sun? Has the number 5557 been confirmed by best possible computation today? If so to what result? Exactly the same as Newcomb?

2. Einstein knew that 43 arc seconds were missing and so could target his correction to fit exactly. Convincing?

3. It is impossible to directly compute the precession by EG. So Einstein starts with the 5557 given by Newcomb using NG for the whole Solar system as a complex many-body system. Einstein then isolates to the two-body problem of Mercury + Sun with EG offering a correction to NG which precisely matches  the missing 43. Magic?

The weakness of Einstein's argument that EG is more precise than NG, is that direct computation with EG to this effect is impossible. It is only possible to start from a NG computation of a complex many-body problem and then isolate to a two-body problem for which EG appears as NG with an extra contribution to potential energy and use this as a correction to the many-body problem. 

It is obvious that this procedure has some weak points. Questions pose themselves:

• Is its worthwhile to spend years of study to come to at least some understanding of EG, when EG is severely uncomputable?
• Is the evidence that EG is more precise than NG convincing?
• Is it reasonable to view EG as a more precise version of NG, when only NG is computable?
• Is it reasonable to use EG as foundation of modern physics when EG is uncomputable?
• Is it reasonable to use EG only as a form of decoration, which serves no practical use?
• Is it reasonable to give up the basic concepts of space and time of Newtonian mechanics, which have served and continue to serve science and society so well?
• Is it a good idea to insist on EG when EG is incompatible with quantum mechanics? 
• Why was EG initially met with very strong skepticism?
• Why was EG accepted only after Einstein's death (and of all his original skeptics)? 

  

Normal Physics from Extreme Physics?

Modern physics is largely based on an idea to find the truth about some physics by subjecting it to extreme tests as if that will bring out the essence. This idea was introduced by Einstein in his famous "thought experiments" of his Special Theory of Relativity SR, with trains being accelerated to speeds comparable to the speed of light, and from such speculations finding revolutionary new truths about space and time today viewed as fundamentals of modern physics. 

To accelerate a 100 ton train to a speed comparable to the speed of light requires more than $10^{20}$ Joule to be compared with the energy of the total yearly production of coal of less than $10^{18}$ Joule. 

This is the idea of of finding the true physics of light by sending single photons (whatever that is) to go through a double split and finding that a photon gets confused about what slit to pass, as the basic experiment of quantum mechanics.

This is the idea in experiments at the LHC at Cern colliding protons at close the speed of light to find out the nature of protons as fundamental building stones of an atoms/molecules together with electrons and neutrons.  

The new ESS in Lund Sweden will smash high speed neutrons into different molecules to find out their essential functioning in a normal environment.. 

This idea is also present in Einstein's General Relativity as a Theory of Gravitation supposed to replace Newton's Theory of Gravitation in extreme conditions like the collision of black holes or the Universe as a whole. 

The idea of finding normal physics from extreme physics may appear strange from classic physics point of view with generality and simplicity as leading principle and not extreme particularity of extreme complication. But it has taken a prominent role in modern physics, maybe because it is now possible to perform extremely complicated experiments concerning extreme physics and finding extremely small effects. 

An example is the proclaimed detecting of exceedingly weak gravitational waves from a collision of two black holes as an event of maximal strength, awarded the Nobel Prize in Physics in 2017. 

Experts of modern physics with (i) relativity theory + (ii) quantum mechanics the essential advancements of classical physics, are eager to reveal the fact that both (i) and (ii) are very difficult to understand. An effect is that there are many conflicting efforts to simplify with little agreement since the formation of the theories 100 years ago, now manifesting itself as a crisis of modern physics.

This is to be compared with classical theories of physics formed to be maximally clear and understandable and so also being supported by a common agreement.

You find on this blog efforts to make atom physics understandable as Real Quantum Mechanics, and macroscopic mechanics as Many-Minds Relativity.

The Role of Einstein's Theory of Gravitation in Modern Physics

The general view presented to the world by the physics community is that modern physics as physics after 1900, as opposed to classical physics before 1900,  is based on 

• Einstein's Special Theory of Relativity SR and General Theory of Relativity SR replacing Newton's Theory of Mechanics NM, together with
• Quantum Mechanics QM for atomic physics as an extension of Maxwell's Theory of Electromagnetism ME and Newton's Mechanics NM to atomic scales.

The completely new aspect of modernity are Einstein contributions to physics offering a completely new insight into basic aspects of space and time with SR presented in 1905 followed by GR in 1916 after an 11 year long hard struggle. 

But the reception by the physics community of Einstein's revelations were met with skeptics or refutation or ignorance, in particular by the Nobel committee for the Physics Prize 1921, which awarded Einstein for explaining the law of photoelectricity but with the explicit mention that his relativity theory was not included in the motivation. This situation changed only after Einstein's death in 1955, but then slowly accelerated to its position today as the fundamental theory of gravitation replacing Newton's Theory of Gravitation NG included in NM, which had served humanity so incredibly well for 300 years. The history is captured in the book Building the General Relativity and Gravitation Community during the Cold War by Roberto Lalli.

Here is a typical reaction to GR (Charles Lane Poor):

• The Relativity Theory, as announced by Einstein, shatters our fundamental ideas in regard to space and time, destroys the basis upon which has been built the entire edifice of modern science, and substitutes a nebulous conception of varying standards and shifting unrealities. 
• And this radical, this destroying theory has been accepted as lightly and as easily as one accepts a correction to the estimated height of a mountain in Asia, or to the source of a river in equatorial Africa.

This is where we stand today: Physics is based on GR for gravitation and QM for atom physics, but the big trouble is that GR and QM are viewed to be incompatible, which is a catastrophe from scientific point of view.

But NG is compatible with QM, and so it is natural to ask if it is really necessary to give up Newton for Einstein? 

A modern physicist will tell you that in fact GR reduces to NG in the case of (i) weak  and (ii) static gravitation, and then admit that this covers almost everything. In fact, what is not included in (i)+(ii) is something extremely speculative, such as collision of black holes, for which the physics is unknown and so the functionality of GR. 

In other words, NG works as well today as ever before, as the most successful mathematical theory all times, and the claim that NG has to be replaced by GR appears to have little factual basis. In fact, GR is so computationally demanding that simulation of even a simple system like the Solar system is unthinkable, while with NG this is captured in a couple of lines of code and executed on a laptop in seconds. 

If NG still works fine, what was the motivation to promote GR after Einstein's death but reject it before?

Was it the result of a stalemate of modern physics in 1960s after the immense success of the atomic bomb at the end WW2? When constructive new ideas are missing, a return to some old ideas may come to rescue.

In any case GR was lifted up from obscurity to top position, but then GR as theory had to be inspected with new eyes and this was far from easy, since GR is so mathematically demanding that it can be grasped by only a few, if any. 

Today this is handled as follows: GR is fundamental and very difficult to understand and apply, but since NG works so fine it is not necessary to dig into the theory of GR in any detail. It is sufficient to know that GR has replaced NG, while NG is used in practice. GR can then be presented as a fundamental step forward as concerns fundamental aspects of space and time as an expression of the power of modern physics. 

The only trouble is that GR is incompatible with QM, and so either GR or QM must be wrong. 

  

Conundrum of Modern Physics

A modern physicist will proudly tell you that modern physics is based on two theories: 

• Einstein's Special Theory of Relltivity SR and General Theory of Relativity GR replacing Newton's Mechanics NM.
• Quantum Mechanics QM replacing Maxwell's Electromagnetics ME and NM on atomic scales.  

Modern physics is based on SR/GR + QM, while classical physics is based on NM + ME. 

A modern physicist will then inform you that SR/GR and QM are the greatest scientific achievements of all time, each with complete agreement with all observations. The only caveat well understood from the start 100 years ago, is that SR/GR and QM are incompatible, which has plunged modern physics into a credibility crisis. Real physics cannot be incompatible/contradictory and so something must be fundamentally wrong with the mathematical models. But what?

QM can be seen as a generalisation of ME and NM to atomic scales and there is no incompatibility here. So it must be SR/GR posing the problem. 

SR without gravitation introduces new relativistic mechanics based on Lorentz transformation mixing space and time into new strange effects of space contraction and time dilation, and so dismisses NM without gravitation, because it is not Lorentz invariant, as an ad hoc requirement.

GR introduces gravitation as a geometric effect of curved space-time and so dismisses NM with gravitation as a classical field theory in Euclidean space. 

NM has thus been replaced by SR (1905) and GR (1916) as a prime achievement of modern physics with very little change into our days, as if the last word was said 100 years ago. 

But NM is the most successful theory of all times encompassing a very large range of phenomena in computable form opening to a very rich world of simulations. In contrast GR is admittedly very difficult to put to work in simulations, because it is so incredibly complicated that even the largest computer and best programmer cannot make it go. The only way to put GR to use is to let it collapse to NM, while GR beyond NM is reserved for speculations on cosmological scales, or bigger.

Newton's theory of gravitation connects gravitational potential $\phi (x)$ to mass density $\rho (x)$ by the Laplacian differential operator $\Delta$ acting in a Euclidean space with coordinate $x$ by 

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

which can be viewed as an assignment creating mass by differentiation of gravitational potential and which can be motivated from conservation principles as shown in this post.  (NG) is the only possible connection between gravitational potential and matter well understood by the Creator.  You find more information on (NG) under tag New View on Gravitation.

Einstein decided to throw out (NG) as the pinnacle of mathematical thinking all times all areas. Einstein replaced (NG) by GR taking the position of Newton under the excuse "Newton, forgive me!" and so was exploited by the physics community to represent all the marvel of modern physics to the world, while his fellow physicists viewed him with pity for missing the train to modernity.

It is now time to reconsider the reasons put forward to replace NG by GR. The cost is very high, while the gains may just be fantasy. 

Galilean-Newtonian Relativity

A common view today is that Newtonian Mechanics requires the notion of absolute space, which however cannot be identified, and so is used to motivate a departure into relativistic mechanics in the form of Einstein's Special Theory of Relativity SR without gravitation and General Theory of Relativity GR with gravitation.

But Newtonian Mechanics is already relativistic in the sense that it takes the same form in all Euclidean coordinate systems moving with constant velocity with respect to each other (so called inertial systems) and so it is impossible by experiment to identify a special system to act as absolute space. This is because both Newton's 2nd Law N2 and Newton's Law of Gravitation NG take the same form in all inertial systems. This was well understood already by Galileo. 

We have seen in posts on New View on Gravitation that NG can according to Laplace be formulated as the local assignment

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

where $\rho (x)$ is mass density and $\phi (x)$ gravitational potential depending on a Euclidean space variable $x$ and $\Delta$ is the Laplacian differential operator. Viewing (NG) as assignment means that somehow matter is endowed with mass through that action of the Laplacian on the gravitational potential.

Since the Laplacian is invariant under both translation and rotation, it follows that so is NG and thus the generation of mass from gravitational potential does not depend on choice of coordinate system, which is a natural requirement. 

On the other hand, N2 is not invariant under rotation since centripetal forces from acceleration arise. It is thus possible to detect rotation by experiment such as Newton's Bucket. 

Conclusion: Newtonian Mechanics is relativistic in the Galilean sense of taking the same form in all inertial systems, and so does not require the notion of absolute space. Thus there does not seem to be any compelling reason to replace Newtonian Mechanics by SR/GR mechanics. Adding propagation of light to the mechanics picture motivates a modification on celestial scales as shown in Many-Minds Relativity. 

Remark:In Newton's Bucket experiment it can be decided that it is the bucket which rotates vs the fixed stars (and not the other way around) by considering the evolution in time from an initial state with the bucket at rest with respect to the fixed stars. With this point of view a Euclidean coordinate system determined by the fixed stars, may serve as an absolute space, but it is also possible to connect coordinate system to the Sun, Earth, or your living room. 

Gravitational Potential as Primordial without Instant Action at Distance

                                           Gravitation cannot be stopped. Why?

New Newtonian Gravitation explores a view that gravitational potential $\phi (x)$ is primordial as generator of both space with Euclidean coordinate $x$ and matter with density $\rho (x)$ by the assignment

• $\rho (x)=\Delta\phi (x)$ for all $x$ and all time       (NGnew)

as instant-local-action performed by Laplace's differential operator $\Delta$. 

This is to be compared with the standard view with instead matter density primordial and gravitational potential generated as solution to the differential equation

• $\Delta\phi (x)=\rho (x) $ for all $x$ and all time       (NGold)

with gravitational force $-\nabla\phi$ from instant-action-at-distance. (NGold) thus harbours the main unresolved mystery of Newtonian gravitation motivating Einstein's General Theory of Relativity GR supposedly without need of instant-action-at-distance. (NGold) and also GR ask for some form of graviton particle as force carrier, which has not been found, and is not needed in (NGnew). The standard view is that all atoms in the Universe are connected/attracted to each other by exchanging a stream of gravitons as force carriers, which is an absurd idea.  

The reason matter is viewed as primordial as generator of gravitational potential, is of course that matter is visible (except for dark matter) while gravitational potential/force is invisible. 

Replacing (NGold) by (NGnew) replaces unthinkable instant-action-at-distance mediated by gravitons as force carriers, by thinkable instant-local-action and so does not force giving up Newtonian gravitation as the formidable success of mathematical thinking it is. 

(NGnew) with gravitational potential as primordial without gravitons may explain why it is impossible to "stop gravitation" by some form of barrier in the same way as light can be prevented entering your bedroom by pulling the curtains.

It is impossible to prevent local action by setting up some barrier to influence from outside. We can compare with "external enemies" which can be stopped at the border, and "internal enemies" which can show up anywhere and so cannot be stopped. 

(NGnew) gives a rationale why blocking gravitation is impossible.

Summary: (NGnew) is compatible with observations that gravitation cannot be prevented from acting out. There is no space which is free from gravitation. The gravitational potential is everywhere but you cannot see it, only its action on matter like Newton's falling apple. The gravitational potential as primordial can generate both space and matter. There is no vacuum. Space is generated when being filled with gravitational potential generating matter. Is this a new revelation, or already dismissed?

Compare with fancy of Gravity with an On/Off Switch.

Poincare vs Relativity Theory

A recent post points to the fundamental difference between material objects with positive mass and immaterial objects with zero mass such as light with connection to the concept of luminiferous aether as immaterial medium for propagation of light under intense discussion as a puzzle of the modern physics emerging at the turn to the 20th century with Einstein's Special Theory of Relativity SR as one solution attempt.  

The mathematician Poincare played a key role with main idea described as follows in Einstein 1905-2005 (eds Darrigol et al):

• Poincare actually never abandoned the ether. But he refused to regard it as an ordinary kind of matter whose motion could affect observed phenomena.
• In his lectures of 1899 on Lorentz’s theory, he wrote: I consider it very probable that optical phenomena depend only on the relative motion of the material bodies present –light sources and apparatus– and this not only to first or second order but exactly.

This is precisely the idea I have pursued in Many-Minds Relativity describing propagation of light from emitter to receiver by Maxwell's equations in a coordinate system locked to the receiver, keeping full compatibility with Newtonian mechanics. 

It is natural to compare with propagation of elastic waves in an elastic (material) bar oriented along an $x$-axis of a Euclidean $xyz$-coordinate system together with a similar bar oriented along an $x^\prime$-axis moving with velocity $v$ with respect to the $x$-axis with coordinates connected by the Galilean transformation $x^\prime = x-vt$. Each bar acts as a (material) medium for wave propagation or ether with the same wave speed, and we thus have two ethers represented by the $x$-axis and the $x^\prime$-axis moving with constant velocity with respect to each other, and so being connected by a classical Galilean transformation expressing relativity.      

We can turn this into a similar situation for propagation of light by assuming that the two elastic bars now represent two (immaterial) aethers or coordinate axes connecting emitter to receiver, with the same light speed 

We understand that in both cases aethers locked to coordinate systems act like media or "universa" for wave propagation, which thus are "carried along" with the motion, which is compatible with the null result of the Michelson-Morley experiment triggering SR.

The bottom line is that there is no role for the Lorentz transformation of SR to serve to guarantee the same speed of light in different coordinate systems, since this is already the case if the medium for light propagation is carried along with the motion under Galilean transformation. Therefore SR is not needed and so can be dismissed as a curiosity, as acknowledged by Lorentz. It seems to me that this must have been the position of also Poincare, even if not clearly being spoken out under the intense pressure to come up with something spectacular at the birth of modernity. In this regard Poincare could not compete with Einstein.

More by Poincare:

• Experiment has revealed a multitude of facts which can be summed up in the following statement: It is impossible to detect the absolute motion of matter, or rather the relative motion of ponderable matter with respect to the ether; all that one can exhibit is the motion of ponderable matter with respect to ponderable matter. (1895) Compare this post.
• Laws of physical phenomena should be the same, whether for an observer fixed, or for an observer carried along in a uniform movement of translation; so that we have not, and could not have any means of discerning whether or not we are carried along in such a motion. Compare this post.
• Indeed, experience has taken on itself to ruin this interpretation of the principle of relativity; all attempts to measure the velocity of the earth in relation to the ether have led to negative results. This time experimental physics has been more faithful to the principle than mathematical physics; ... but experiment has been stubborn in confirming it And finally Michelson has pushed precision to its limit: nothing has come of it.

Newton Back! Einstein Out?

Modern physics has developed from classical physics in three steps each viewed as revolutionary:

1. Maxwell's equations 1867 describing all of electro-magnetics including light-as-wave.
2. Einstein's Special Theory of Relativity SR 1905 replacing Newton's mechanics without gravitation by relativistic mechanics.
3. Einstein's General Theory of Relativity GR 1916 replacing Newton's mechanics with gravitation by curved space-time geometry. 
4. Schrödinger's equation for atom physics.  

The present view is that a unified theory including all of mechanics + electro-magnetics + atom physics, is impossible because of severe incompatibilities between 1- 4, primarily because of SR and GR.  There is really no incompatibility between Newton, Maxwell and Schrödinger, if you do not seek incompatibility to boost your own favourite substitute.

Let us search the main reason why Newton was dismissed by Einstein and then all his followers. Newton's theory of gravitation as the main jewel of the infinitesimal Calculus created by Newton and Leibniz, describes the motion of all celestial bodies from the inverse-square law. But Newton's theory seemed to require instant-action-at-distance, which was exhibited by Newton's critics as a mystery/physical impossibility, even acknowledged by Newton himself, from a prevailing understanding that forces only can act by instant direct contact, with the history described in Newtonian Studies by Koyre. 

In GR the inverse-square-law is replaced by curved space-time geometry without instant-action-at-distance (from an ad hoc assumption that the speed of gravity is equal to the speed of light9,  thus circumventing the mystery, but at the price of an even more mysterious concept of curved space-time.

But there is a way to get around instant-action-at-distance even in Newton's theory, which is explored in posts on New View on Gravitation. The basic idea is that gravitational potential is primordial with mass secondary as the result of local instant differentiation. 

We may thus see a return of Newton, and let us then recall that Newton's world can be constructed starting with free fall of a small test particle of unit mass with velocity $v$ in a given gravitational field $\phi (x)$ depending on a Euclidean spatial coordinate $x$ described by:

•   $\dot v=-\nabla\phi$,                     (N0)

where $\dot v$ as the time derivative of $v$ is the acceleration of the test particle under the gravitational force $-\nabla \phi$. Accordingly the time of Newtonian success was named "the dot-age". 

(N0) is then generalised to small material bodies of mass $m$ as collections of $m$ particles of unit mass expressing that all small bodies fall freely the same way in a given gravitational field according to

• $m\dot v =-m\nabla\phi$.               (N1) 

We here assume the material body to be small so that $\nabla\phi$ is the same for all parts of the body. 

The next step is to transfer gravitational force $\nabla\phi$ to mechanical force $f=-\nabla\phi$ e g  hanging a unit mass in a unit linear spring and measuring its elongation under gravitation. This makes it possible to generalise (N1) to Newton's 2nd Law for a body of mass $m$ 

• $m\dot v= mf=F$                           (N2)                        

where $F$ can be gravitational or mechanical force. 

The final step is to express Newton's law of gravitation in the form given by the mathematician Laplace as the differential equation

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

where $\rho (x)$ is mass density and $\Delta$ is the Laplace differential operator, which thus connects to gravitational potential to mass density. 

The common view is that presence of a unit point mass at $\bar x$ generates a contribution $-\frac{4\Pi}{\vert x-\bar x\vert}$ to $\phi (x)$ as an apparent instant-action-at-distance. But it is possible to turn the connection around and view instead $\rho (x)$ as being generated as

• $\rho (x)=\Delta\phi (x)$       (NG2)

where the process of differentiation is local and as such can an be instant in the same way as a contact force.

It is thus not necessary to dismiss Newton's theory of gravitation as requiring mysterious instant-action-at-distance, and thus combine with Maxwell's and Schrödinger's equations into a unified model of the world as outlined in Many-Minds Relativity, Real Quantum Mechanics, Computational Thermodynamics, Computational Turbulent Incompressible Flow and Computational Black-Body Radiation.

This means that Newton, as the greatest physicist all times, is welcome come back again to constructively contribute to a unified model of the world without fundamental incompatibilities and so leave SR and GR without mission and open a way out of the current crisis of modern physics. This has been a main theme of this blog with more details to come.

In particular, with the gravitational potential as primordial with everywhere presence there is no vacuum or complete emptiness of mysterious nature.  Moreover, it opens to connect regions in space where the gravitational potential is smooth with derivatives of only moderate size, to dark matter. Further, (NG2) opens to negative mass being created subject to repulsion from positive mass as a possible source of dark energy,

After all, the world must be rational to exist at all and so must be possible to describe in rational mathematical terms like (N2) + (NG) without incompatibilities. It is impossible that the world is incompatible with itself. Only models of the world can be incompatible if incorrect is some way.