ChatGPT-4: Newton's Inverse Square Law

After a bit of discussion ChatGPT-4 delivers the following conclusion concerning Newton's inverse square gravitational law for pointlike bodies with mass/point masses: 

• Newton's law for pointlike bodies is accurate and applicable when dealing with objects that can be reasonably approximated as point masses. In such cases, where the assumption of pointlike bodies is valid, Newton's law provides accurate predictions of the gravitational interactions between those bodies.
• For example, when considering the motion of planets in our solar system or the behavior of objects in everyday scenarios, treating them as point masses yields highly accurate results using Newtonian gravity.
• So, if the objects under consideration can be effectively modeled as pointlike bodies, Newton's law is a reliable and precise description of their gravitational interactions. The law has been extensively tested and validated in numerous practical applications.
• However, it is important to note that there are situations where the assumption of pointlike bodies breaks down, such as when dealing with objects of finite size, strong gravitational fields, or high speeds. In these cases, more advanced theories, such as general relativity, are necessary to accurately describe the gravitational interactions.

GPT-4 is educated by the physics community and so can tell truths carried by the community, truths which typically may be covered up in a discussion with any member of the community. GPT-4 has not yet been educated to hide possibly inconvenient facts, so if you persist a bit with questions you can learn something.

The truth told by GPT-4 is that Newton's law for point masses is correct, and so necessarily for all bodies composed of point masses = all bodies,  and so there is no need for Einstein's General Theory of Relativity GR. 

GPT-4 seeks to cover up suggesting that bodies composed of point masses would not be covered by Newton's law, which lacks rationality, and so give room for something more advanced than Newton's law = GR as the holy grail of modern physics. The key argument used by Einstein to replace Newton's mechanics by GR, was that Newton's mechanics does not correctly predict the precession of Mercury as a slow turn of its elliptical orbit around the Sun of about 2 degrees per century since there was a 1% difference to observation, which Einstein could fix with GR to an exact fit. 

But Einstein's argument, today accepted by the physics community as the key reason to switch from Newton's mechanics to GR with all the complications coming from throwing out the best of classical physics, is weak because the precision of a Newtonian computation with all its uncertainties concerning planetary masses and the value of the gravitational is not small enough to allow a verdict that it does not match observation. This fact is acknowledged by asking GPT-4 if a Newtonian computation within its precision may fit with observation, which gives the following answer:  

• Yes, that is a possibility. If a Newtonian computation of the precession of Mercury's orbit is performed with sufficiently accurate initial conditions, precise numerical methods, and inclusion of relevant gravitational influences, it is conceivable that the computed precession could agree with the observed value up to the computational precision.
• By employing more advanced computational techniques and including additional factors such as the gravitational influences of other celestial bodies, it is possible to improve the accuracy of the Newtonian prediction. However, it is important to note that this would still be a purely Newtonian approach without considering relativistic effects.
• In practice, the precision achieved in a Newtonian computation would depend on the specific details of the calculation and the accuracy requirements of the problem. While a sufficiently precise computation could potentially yield a result that matches the observed precession, it is worth noting that the discrepancy between the observed precession and the Newtonian prediction was one of the factors that motivated the development of general relativity.
• General relativity provides a more accurate description of the precession of Mercury's orbit, including the contributions from spacetime curvature and relativistic effects. Therefore, to achieve a high level of precision that matches the observed value, it is typically necessary to consider the effects of general relativity in addition to the Newtonian computation.

GPT-4 thus states that the Mercury reason to replace Newton by GR is very weak, and that is the main reason as concerns Newton's mechanics for bodies with mass, which does not say anything about light without mass. Newton does not have to be dethroned by Einstein.  

Warning: If a theoretical/computational prediction exactly agrees with observation, such as Einstein's GR prediction of the precession of Mercury,  then an alarm signal is motivated: It requires both theory and observation to be exact, which is impossible to achieve. There is good reason to believe that Einstein knowing what observation to fit, simply adjusted his computation until exact fit, exact! Mercury is still the best argument put forward to promote GR before Newton, after more than 100 years...

PS Suppose the time step in a Newtonian computational simulation of the planetary system with objective of predicting the precession of Mercury over a century, is one second. The total number of time steps would then be about $3\times 10^9$ and to reach 1% accuracy would require a precision of size $3\times 10^{-12}$ in each time step, which seems pretty formidable...To rule out Newton's theory of gravitation by not reaching agreement with a certain observation better than 1% in a certain computation with finite time step, thus may not be called for...   

Einstein's Straw Man Argument to Cancel Newton's Mechanics

To clear the way for his 1905 special theory of relativity forming the revolution of modern physics, Einstein had to show that the classical physics of Newton/Leibniz, which had served mankind so amazingly well for more than 200 years, was in dire need of a revision. Einstein picked up a common idea (not carried by Leibniz) that Newton's mechanics required absolute space and absolute time and then in a thought experiment showed that absolute time in the sense of absolute simultaneity of events widely separated in space, was impossible to guarantee. Einstein argued that this could serve as a reason to cancel Newton's mechanics and replace it with relativistic mechanics coming with new notion time not asking for global simultaneity. 

But is Einstein's argument a valid argument, or is it just a straw man argument attaching a quality to Newton's mechanics which is not a necessary attribute? 

In recent posts I argue that global simultaneity is not needed for the World to go around according to Newton's mechanics. It is sufficient with a concept of local simultaneity/coexistence. In particular, Newton's theory of gravitation does not require instant action at distance formally asking for global simultaneity. 

One way to see this is to realise that the World can be made go around by explicit time-stepping, where the state $u(x,t+dt)$ of a system at a position $x$ at a next time instant $t+dt$ is computed/determined by the state at previous time $t$ with $dt$ a small time step, at positions close to $x$. The update can have the form 

•    $u(x,t+dt) = u(x,t) - dt*u(x-dx,t)*\frac{u(x,t)-u(x-dx,t)}{dx}$       (*)

where $dx > dt$ is a space step, which computes the solution of Burgers' equation modeling wave propagation in a compressible gas:

• $\frac{\partial u}{\partial t}+u*\frac{\partial u}{\partial x} =0$.  

You can follow the development of a shock propagating to the right in a shock tube by running this one-line code executing (*) by updating $u$ at $(x,t+dt)$ from values at $(x,t)$ and the nearby $(x\pm dx,t)$.

You see that only local simultaneity is needed and that Einstein's starting point that global simultaneity is needed, is a straw man argument. The consequences are far-reaching as concerns the role of Einstein's theories of relativity in modern physics.    

Logic Missing for Dumping Newton's Mechanics

Did Einstein seek to put the left foot in the right shoe? 

The triumphs of modern physics are presented as Quantum Mechanics QM and Einstein's Special/General Theory of Relativity SR/GR developed 100 years ago, but the success story is shadowed by the realisation that QM and SR/GR are incompatible with no reconciliation in sight. This is troublesome and contributes to the present crisis of modern physics. 

While Newton's Mechanics NM and QM are fully compatible, a modern physicist insists that NM has to be replaced/revised by SR/GR, and so it is natural to ask what is wrong with NM? An answer is given in Einstein's famous article from 1905 introducing SR with title:

• On the Electrodynamics of Moving Bodies.    

Einstein forms SR as an expression of the Lorentz Transformation LT connected to electromagnetics/light described by Maxwell's equations and compares it with the classical Galilean Transformation GT connected to NM. Einstein then declares that since NM does not fit LT, it is necessary to replace NM by SR fitted to LT. He does not say that because electromagnetics does not completely fit with GT, it has to be dumped. Only NM has to go.

But the logic is missing: LT is fitted to electromagnetics, while NM is not electromagnetics. There is no good reason to throw out NM because it does not fit with something fitted to electromagnetics. There is no good reason to throw away your right shoe because it does not fit your left foot! 

In any case this was what Einstein did and so concluded that NM had to be replaced by relativistic mechanics based on SR with all sorts of strange effects such as space contraction, time dilation, and relativistic mass increasing with speed.  

Returning to classical physics in the form of NM + electromagnetics and giving up relativistic mechanics can open to get out of the present crisis. Anybody willing to try? 

An alternative to SR compatible with NM is presented in Many-Minds Relativity.

Bear in mind that the difference between Einstein's relativistic mechanics and Newton's mechanics is viewed to be exceedingly small, so small that detection is virtually impossible. There is a heavy price to pay for replacing Newton by Einstein and the gain appears to be virtually zero. GPS does not need SR/GR to work.  

Why insist to keep SR/GR before NM arguing that Einstein was smarter than Newton, when the negative consequences are so huge and Einstein's genius can be questioned on so many good grounds? 

Local Simultaneity Saves Newton's Mechanics

Local simultaneity of galley slaves.  

In modern physics Newton's mechanics is replaced by Einstein's theory of relativity based on an argument that Newton's mechanics requires notions of absolute space and absolute time, which have shown to be difficult to give reality. 

Newton's mechanics is together with Maxwell's electromagnetics the main achievement of classical mathematical physics and to throw it out carries a large cost. It is thus natural to ask if it is really true that Newton's mechanics requires notions of absolute space en time?

Recent posts argue that Newton's mechanics only requires notions of local space as local coexistence and local time as change of local coexistence. This is seen in explicit time-stepping of Newton's equations of motion where velocities and positions are updated from one local instant of time to a next using only local information in space. In other words, the World can very well go around without notions of absolute space and time, with only local coexistence or local simultaneity in the sense of shared time locally in space necessary for locally coordinated motion. 

Local coexistence involves time synchronisation locally in space as a natural element of local physical interaction within extended bodies.  

Recall that Einstein motivated the special theory of relativity by exhibiting the difficulty of establishing a system of absolute time with global simultaneity, but did not realise that such a thing is not needed, since local simultaneity is enough.  

It may thus be possible to save Newton's mechanics from Einstein's assault, and so avoid all the complications coming with Einstein's theory of relativity. This could help modern physics out of its present relativity theory paralysis.      

   

Non-Physics of Lorentz Transformation

This post directly connects to the previous post Physics Illusion: Lorentz Transformation to which we refer for clarifying figures. 

The basic postulate of Einstein's Special theory of Relativity SR is the Lorentz transformation connecting the coordinates $(x,t)$ and $(x^\prime ,t^\prime )$ by 

• $x^\prime =\gamma (x-vt)$, $t^\prime =\gamma (t-vx)$          (L)

where $v$ with $\vert v\vert \lt 1$ is a given constant and $\gamma =\frac{1}{\sqrt{1-v^2}}$. This is a very simple linear transformation, which by the chain rule satisfies

• $\frac{\partial}{\partial x}=\gamma (\frac{\partial}{\partial x^\prime}-v\frac{1}{\partial t^\prime})$
• $\frac{\partial}{\partial t}= \gamma (\frac{\partial}{\partial t^\prime}-v\frac{\partial}{\partial x^\prime})$

which leaves a wave equation invariant in the sense that if the function $u(x,t)$ satisfies 

• $\frac{\partial u}{\partial t}-\frac{\partial u}{\partial x}=0$,         (1)

then the function $u^\prime (x^\prime ,t^\prime )=u(x,t)$ satisfies an equation of the same form:

• $\frac{\partial u^\prime}{\partial t^\prime}-\frac{\partial u^\prime}{\partial x^\prime}=0$.  (2)

This is very simple mathematics, of the same complexity as 1+1 = 2. Here $(x,t)$ may be viewed to represent Euclidean space-time coordinates with (1) expressing propagation of a wave with unit speed, while according to Lorentz $(x^\prime ,t^\prime )$ does not represent any physical reality.    

This is were Einstein stepped in with SR declaring:    

1. (L) connects the coordinates $(x,t)$ and $(x^\prime ,t^\prime )$ in two Euclidean coordinate  systems moving with respect to each other, both with the same physical meaning.
2. True physical laws must take same form in coordinate systems connected by Lorentz transformation (be Lorentz invariant)  = Principle of Relativity. 

Einstein then accepted the wave equation as a true physical law, but not Newton's 2nd Law because it is not Lorentz invariant, and so modified Newton's mechanics into a new form of relativistic mechanics with all sorts of strange effects including space contraction and time dilation by the coupling of space with time in (L).

In 2 Einstein decides over physics. In 1 Einstein contradicts Lorentz and so we ask who was right? 

An answer is given by recalling from the previous post the concept of coexistence or simultaneity, as the presence in space of a spatially extended body B represented by a range of coordinates $x$ like $0<x<1$ for which the time coordinate is the same e.g. $t=0$. If we plug in $t=0$ into (L) we get

•  $x^\prime =\gamma x$, $t^\prime =-\gamma vx$ or $x^\prime = -\frac{1}{v}t^\prime$ 

which represents a trajectory in the $(x^\prime ,t^\prime )$-system without coexistence. This is contradictory and so shows that SR is non-physics.

The bottom line is that SR does not make sense for extended bodies. But a Universe without extended bodies with coexistence is not a real Universe, only a fantasy. The consequences for modern physics based on SR are far-reaching.

Or is it possible to save SR by insisting that the Universe consists of bodies without spatial extension somehow interacting under Lorentz invariance? We are thne led to ask how different such point-like bodies may interact? So we image two point-like objects with space coordinates $x_1$ and $x_2$ in the $(x,t)$ system, and we ask about the corresponding time coordinates $t_1$ and $t_2$? With $t_1=t_2$ the interaction is instant as an expression of coexistence,  while if $t_1>t_2$ or $t_2>t_1$ there is delay excluding true interaction. We are thus led to expect that true interaction between bodies requires coexistence and so would not satisfy Lorentz invariance. 

The case for SR is weak.  

More on the non-physics of (L): Three Major Inconsistencies of the Lorentz Transformation.

      

Local Coexistence: Leibniz vs Einstein

Local coexistence.

A basic idea of modern physics is that the conceptions of space and time of classical physics formed by Leibniz/Newton have to be replaced by a concept of curved space-time of Einstein's theory of relativity. Einstein struggled through his entire scientific life to give meaning to this new concept, but today very few modern physicists claim to have a proper understanding of what Einstein wanted to tell humanity, while it is commonly agreed that Einstein was right and so that Leibniz/Newton was wrong.   

But the physics of Leibniz/Newton is the classical physics of the scientific revolution reaching full power with Maxwell's equations at the end of the 19th century thus capturing all of macroscopic mechanics and electromagnetics in differential equations in terms of functions $f(x,t)$ of classical Euclidean space coordinates $x$ and time $t$. It was a formidable success, which met the young patent clerk Einstein preparing for a scientific career at the turn to the 20th century with the demanding question: How to impress the World?

At the patent office Einstein tested his abilities on some main open problems of classical physics including black body radiation, photo-electricity and the (non-)existence of an ether as medium for the propagation of electromagnetic waves. Einstein's efforts resulted in 5 articles on these topics in 1905 named Annus Mirabilis. 

In one of the papers Einstein picked up an observation by Lorentz that wave equations take the same mathematical form under a simple linear transformation of coordinates mixing space and time, named Lorentz transformation. Lorentz cautioned that coordinates transformed from physical coordinates were absolutely not to be viewed as physical coordinates. But here young Einstein found a thread to follow, by opposing Lorentz and claiming that all coordinates subject to Lorentz transformation have a physical meaning. So was Einstein's special theory of relativity formed as the physics of Lorentz transformations, which showed a very new form of physics with space contraction and time dilation unheard of by Leibniz/Newton and classical physics. But this did not impress the physics community at all, which made Einstein double down by extending his special theory to a General Theory of Relativity GR during 10 years of hard struggle to make sense of his curved space-time. Again the physics community was not impressed and GR was not hailed as a trade mark of modern physics until Einstein had passed away and no longer could be asked to clarify his ideas, and so the physics community was free to exploit Einstein's $E=mc^2$ and nuclear power to attract funding to Big Physics, which succeeded very well until lack of results resulted in a backlash as the crisis of modern physics of today...

The mantra of modern physics is that the classical concepts of space and time by Leibniz/Newton have to be replaced by Einstein's space-time coming to expression in the Lorentz transformation mixing space and time into strange new physics. The mantra was based on a perception that Newton's physics required absolute space and time, which was hard to rationalise. 

But Leibniz viewed both space and time to be relational (and so relative) and thus not absolute in Newton's sense: 

• space is order of coexistence
• time is order of succession/change    

which can pull the rug under the necessity to give up classical concepts. It is natural to qualify Leibniz view to 

• space is measure of local coexistence
• time is measure of change of local coexistence.    

The central concept is local coexistence or simultaneity as something appearing with nearby space coordinates at the same time, like an extended body existing at a specified time. The novelty as compared with Newton's absolute space as global space is the aspect of locality. 

This aspect is expressed in explicit time stepping of the heat equation $\dot T(x,t) = \Delta T(x,t)=0$ with $T(x,t)$ temperature depending on a (1d for simplicity) Euclidean space coordinate $x$ and a time coordinate $t$, which takes the form

• $T(x, t+dt) = T(x,t)+0.5(T(x+dx,t)-2T(x,t) + T(x-dx,t))$

where $dx$ is a small space step and $dt=0.5dx^2$ a time step. We see that $T(x,t+dt)$ at new time $t+dt$ is updated from $T(y,t)$ at previous time $t$ and $y=x$ and nearby $y=x\pm dx$ expressing local coexistence (with effectively finite speed of propagation). This connects to the previous post on Newton's law of gravitation and Universe Model.

The Lorentz transformation connects space-time events without extension in space and so has no room for local coexistence/simultaneity. But without local coexistence/simultaneity the World falls apart and so must a physical theory based on the Lorentz transformation. 

Locality of space and time comes to expression by the clock of the local village church telling the villagers when to quit work and go home to eat, or a local Sunset. Global coordination is not needed to make physics without human intervention to go around. Explicit time stepping under local coexistence/simultaneity is enough. 

It is important to make a distinction between man-made physics and natural physics without human intervention. Man-made physics like the GPS system uses global time shared by satellites realised by man-made synchronisation, while natural physics only requires local time as local coexistence. 

Sum up: With a Leibnizian concept of space as local coexistence and time as change of local coexistence there is no compelling reason to replace classical concepts (which make sense) by Einstein's concept of curved space-time (which does not make sense). Leibniz thus offers a theory of relativity without the confusing aspects Einstein's theories.   

Can Newton's Universe Model be Wrong?

Newton's Model of a Universe subject to gravitational forces takes the form 

• $\rho =\Delta\phi$                      (1): Newton's Law of Gravitation
• $\frac{Du}{Dt}+\nabla\phi = 0$         (2): Newton's 2nd Law
• $\dot\rho +\nabla\cdot (\rho u) =0$      (3): Conservation of Mass

where $\rho (x,t)$ is mass density, $\phi (x,t)$ gravitational potential, $u(x,t)$ particle velocity, $x$ is a Euclidean space coordinate, $t$ is a time coordinate, the dot expresses differentiation with respect to time, and $\frac{D}{Dt}$ is the time derivative along a particle trajectory. 

Newton's Model allows prediction of gravitational motion of the Universe by time stepping after relaxing (1) into 

• $\epsilon\dot\phi = \Delta\phi -\rho$    

with $\epsilon$ a small positive coefficient. Newton's Model combines simplicity with generality in a formidable demonstration of the power of mathematical thinking in the form of the Calculus developed by Leibniz and Newton.  

Is it possible that Newton's Model does not correctly capture gravitational motion? Before Einstein this question was not posed since it (1)-(3) appeared to be rock solid as ultimately expressing conservation of mass and energy, without which the Universe could not persist over time. 

Einstein built his fame on a claim that Newton's Model is wrong and has to be replaced by Einstein's General Theory of Relativity GR, which has become a consensus of modern physics supposedly based on GR and quantum mechanics. The consensus says that there is experiments agreeing better with GR than Newton's Model, although the discrepancy is extremely small and most delicate to catch.     

So the question remains: Is it possible that Newton's Model is wrong? It would require at least one of (1)-(3) to be wrong which ultimately would require conservation of mass or energy to be wrong. Is that possible? 

Newton's Model is an expression of Leibniz view on space as expressing local coexistence and time as change/succession of local coexistence, which comes to life by explicit time stepping updating $\rho (x,t)$, $\phi (x,t)$ and $u(x,t)$ from one time instant $t$ to a next instant $t+dt$ involving only local coexistence in space.  

To question Newton Einstein thus had to leave Leibniz view with space and time separated and enter a whole new world of space-time with now time and space intertwined, which has become the trademark of modern physics. 

While Leibniz view very much makes sense, Einstein's space-time does not at all. The crisis of modern physics is rooted in this confusion. 

Note that Leibniz view only involves local coexistence and local time and so does not ask for any notion of (supposedly hard-to-explain) absolute space and time attached to Newton in advocacy for GR. 

This post connects to The World as Computation exhibiting the computational aspects of mathematical models including explicit time-stepping as fundamental technique without need of (supposedly hard-to-explain) instant action at distance again attached to Newton in advocacy of GR. 

    

Einstein vs Crisis of Modern Physics

                              Einstein Fountain in Ulm, where Einstein was born in 1880.

John D Norton describes the total confusion surrounding Einstein's theories of relativity in the often cited article from 1993:

• General covariance and the foundations of general relativity: eight decades of dispute

Summary:

• Einstein offered the principle of general covariance as the fundamental physical principle of his general theory of relativity and as responsible for extending the principle of relativity to accelerated motion. 
• This view was disputed almost immediately with the counter-claim that the principle was no relativity principle and was physically vacuous. 
• The disagreement persists today. 

Conclusion:

• The requirement of general covariance, taken by itself, is even devoid of physical content. 
• The foundations of general relativity do not lie in one or other principle advanced by Einstein.

We learn that Einstein's principle of relativity as the basic postulate of his special/general theory of relativity had no meaning accepted by the scientific community in 1993, which is still the case today. 

Nevertheless, the accepted view by the physics community is that modern physics is based on Einstein's theory of relativity and quantum mechanics. However, these two theories are inconsistent, which poses a severe obstacle to progress and has resulted in a crisis of modern physics acknowledged by the physics community. 

But here is an open road to resolution of the crisis: Get rid of Einstein and resurrect physics as science. 

Einstein was overwhelmed by a observation made by Lorentz that wave equations happen to read the same under Lorentz coordinate transformation, but did not listen when Lorentz cautioned that his transformation had no physical meaning. So Einstein proclaimed in his special theory of relativity the Lorentz transformation to a basic principle of physics demanding all physical laws to obey a dictate to be Lorentz invariant, even if it had no physical meaning and in particular ruined Newton's mechanics as the pinnacle of classical physics, because it was Galilean invariant with direct clear physical meaning but not Lorentz invariant.  Einstein then extended to general covariance in his general theory of relativity, again devoid of physical content.  

There are many physicists who understand that it would be wise to give up Einstein's relativity theories and so get out of the crisis of modern physicists, but there is also a heavy establishment invested in Einstein for which the crisis is not yet severe enough to require action.  You still have to accept Einstein's physics even if it is devoid of physical content, which makes it impossible to question on physical grounds. Einsteins was not stupid! Only his blind followers.

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... 

Einstein's Obsession: Coordinates

Einstein has become the icon of modern physics, with ever new confirmations that "Einstein was right". 

Einstein's main idea was to replace ontology of "what is" by epistemology of "how to record/what to say" with focus on the role of coordinate system or standards of measuring space and time for the description of physical reality. Einstein elevated this idea to Principle of Relativity stating that physical laws must take the same form (be invariant) under all admissible coordinate transformations, as a dictate.  

This was very different from the classical idea that physical laws (in general) take different forms in different coordinate systems, while expressing the same physical reality of "what is". The dictate appeared to give Einstein the capability to discover laws of physics by analysing standards of measuring space and time, which resulted in his Special Theory of Relativity SR in 1905 and General Theory of Relativity GR in 1915. 

In SR the according to Einstein admissible coordinate transformation boils down to the Lorentz transformation which disqualifies Newton's 2nd Law, and in GR Newton's Law of Gravitation is replaced by a geometric property of coordinates of "curved space-time".

Einstein thus challenged Newton on formalistic grounds, which was not welcomed by physicists used to prefer ontology/physics before epistemology/meta-physics. After Einstein passed away in 1955, the opinion gradually shifted to the god-like stature today under the banner "Einstein was right, Newton was wrong". 

But the skepticism during Einstein's lifetime was well founded and comes to expression in the "crisis of  modern physics" witnessed by so many today. A return to "Newton was right" with physics focussed on "what is" may open to recovery and new discovery. Many-Minds Relativity follows this line of thought. Try it!

Nature/physics does not care about coordinate systems.The World has to go around even if physicists are not measuring the progress, at least this is what you as a physicist focussed on ”what is” would say, unless you would rather prefer to think like Einstein…and not be too self-critical…

  

Equivalence Principle: Einstein vs Galileo/Newton

General Theory of Relativity is based on Einstein's Equivalence Principle EEP, to be compared with the Galileo/Newton's Equivalence Principle expressing Universality of Free Fall UFF.

UFF means that matter independent of composition reacts the same way in a gravitational field and so shows the same free fall. UFF is a prediction of Newtonian mechanics verified with very high precision starting with Galileo's free fall experiments from the Tower of Pisa. The proof that UFF is a consequence/prediction of Newton's mechanics is given in two lines here. Check it out.

Einstein formulated EEP considering two reference frames, K placed in a uniform gravitational field and K′ uniformly accelerated, as follows:

• As long as we restrict ourselves to pure mechanical processes in the realm where Newton’s mechanics holds sway, we are certain of the equivalence of the systems K and K′. 
• But this view of ours will not have any deeper significance unless the systems K and K′ are equivalent with respect to all physical processes, that is, unless the laws of nature with respect to K are in entire agreement with those with respect to K′. 
• By assuming this to be so, we arrive at a principle which, if it is really true, has great heuristic importance. 
• By theoretical consideration of processes which take place relatively to a system of reference with uniform acceleration, we obtain information as to the career of processes in a homogeneous gravitational field.

Einstein thus acknowledges UFF to be a consequence of Newton's mechanics, but finds a need to extend it beyond Newton's mechanics to all physical processes, and then things get complicated. So complicated that Einstein must leave the safe haven of Newton's mechanics with UFF secured, and resort to assuming EEP to be a postulate, now asking for experimental conformation. But that is virtually impossible, since all physical processes are involved. 

So there we stand today: UFF is secured by Galileo/Newton. EEP is waiting for confirmation. 

PS1 UFF is also referred to as the weak equivalence principle, and EEP as the strong equivalence principle.

PS2 Note that UFF does not exclude internal motion of a body subject to a non-uniform gravitational field (tidal motion). 

PS3 Einstein's bright idea was to ask what may happen to Newton's mechanics if it is extended beyond its realm of mechanics to electromagnetics with material bodies traveling at near the speed of light, however  without connection to reality. There is no reason to question Newton's mechanics, and anyway doing that is stupid.   

Why Inertial Mass = Gravitational Mass

In equilibrium the spring force = gravitational force/weight.

The Equivalence Principle EP states that: 

• Inertial Mass = Gravitational Mass.  

EP means that all material bodies react the same way to gravitational force as to other forces, or specifically that they react to inertial force the same way as to gravitational force. In Newton's mechanics this follows from the fact that the size of any force ultimately is determined by comparison with gravitational force. 

For example, the force exerted by an elastic spring under extension is determined by hanging different bodies in the spring and measuring the extension, see above picture. This means that in Newtonian mechanics the gravitational force is used as reference for all types of force and so EP is valid by definition/agreement. Gravitation is the common denominator. 

In Newtonian mechanics it is thus meaningless to seek to verify EP by an experiment, and senseless to seek to do that in a very expensive experiment, like the MICROSCOPE.  It would be the same as to seek to experimentally verify that there are 100 centimeters on a 1 meter. Most people would say that this would not be meaningful, even if very high precision could be reached.  

EP is thus a consequence of Newton's mechanics, but it is different in Einstein's mechanics, where EP in instead serves as a postulate, which then in principle can be subject to experimental testing. This is what was reported in the previous post: A very high-precision experiment confirming EP! Like testing that there are 100 centimeters on 1 meter, to very high precsision? 

On top of that the experiment is supposed to verify not only EP as postulate of Einstein's mechanics (General Theory of Relativity GR), but also serve as a confirmation of the profound revolution brought to humanity by GR. This is mind-boggling. If EP is nothing but an agreement, then GR would also simply be an agreement, and so not real science.

The very high-precision of the experimental verification of EP supports a suspicion that what is tested is in fact a definition/agreement rather than a law/theory of physics. The consequences as concerns the physicality of GR are possibly far-reaching...

Compare with the low precision of the value of the gravitational constant $G$: less than 5 decimals. 

Recall that in the new SI units standard, agreements take the role of laws of physics when defining meter in terms of prescribed speed of light, and kg in terms of prescribed equivalence with energy. However, this is not anything which modern physicists want to agree on: the prescriptions/agreements are also laws of physics! The Best of Worlds! Compare with posts on SI 2019 standard.