How to Measure/Define Mass

This is a clarification of the previous post showing gravitational force without need of force carrier.

An instrumentalist definition of a certain quantity like mass depends on how the quantity is measured by a certain specified instrument. 

According to the SI 2019 system of units, mass $m$ is determined by a Kibble balance from gravitational force $mg$ measured as electromagnetic force, where $g$ is the local gravitational constant. 

We conclude that according to SI 2019 mass is gravitational mass as reaction to gravitational force. In other words, gravitational force as gradient of a gravitational potential is primary from which mass is derived as secondary as measured by a Kibble balance. 

This suggests that the relation between mass density $\rho (x)$ and gravitational potential $\phi (x)$ should be viewed to have the form, with $x$ a Euclidean 3d space coordinate and $t$ a time coordinate: 

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

with $\phi (x)$ given and $\rho (x)$ derived by application of the differential operator $\Delta$ acting locally, rather than the standard form: 

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

with $\rho (x)$ given and $\phi (x)$ derived as solution to Poisson's equation as a differential equation.

We can view (1) as local instant action by differentiation, while (2) requires instant action at distance as integration/summation. 

By adopting (1) as the true connection between gravitational mass and gravitational potential, which fits with SI 2019 and is explored in many posts, we can thus circumvent to need of instant action at distance, which has been viewed as a stumbling stone for Newtonian mechanics, motivating Einstein's relativistic mechanics coming with many new difficulties.

Adopting (1) we thus (i) adhere to SI 2019, (ii) circumvent instant action at distance and (iii) do not need Einstein,  as a hat trick. 

In Newtonian mechanics inertial mass is defined as gravitational mass and their equivalence is not an assumption or Principle as in Einstein's mechanics. 

In the Standard Model, mass is not defined as gravitational mass as in SI 2019, but as an intrinsic quality/property carried by matter, which is formed in a very strange way by the Higg's mechanism. 

Altogether, mass as gravitational mass, appears as a very useful concept. It makes sense to define mass as gravitational mass, because gravitation is present virtually everywhere. A special feature is that free fall as motion under gravitational force, without presence of forces of other nature, is independent of the size of the mass, which defines mass as a form of sensitivity to gravitation which is additive: The mass of two particles of the same kind is twice that of one particle. 

But this is not how a modern physicist approaches the problem of defining what mass is and how to measure it. A modern physicist is trained to believe that mass according to the Standard Model is defined by QFT through the Higg's mechanism, and then connected to gravitation by General Relativity GR. The trouble is the QFT and GR are incompatible and so the mass problem is a mess problem. 

Newtonian Gravitation Does Not Require Instant Action at Distance

                                                 Instant local action.

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

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

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

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

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

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

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

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

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

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

•  $\dot\rho +\nabla\cdot m =0$ 

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Newton or Einstein or Newton?

Recall this post.

A physics student of today will to pass exams have to say that Einstein's theory of gravitation is more precise than Newton's theory of gravitation, not less precise. 

The evidence supplied by the book is that the elliptic orbit of Mercury around the Sun shows a slow turn of main axis (precession) of 5600 seconds of arc per century, while Newton predicts 5557 taking in the influence of all other planets and celestial objects, and the missing 43 is predicted by Einstein as a small correction to Newton.  

In Einstein's theory the speed of gravity is equal to the speed of light in vacuum, while in Newton's theory the speed appears as being infinite. 

In a two-body problem of one small planet and a big Sun, the planet will according to Newton follow an elliptic orbit without precession, because the gravitational force on the planet acts in the direction of the Sun without delay from finite speed of gravity. With a finite speed of gravity there would be a delay which would make the planet slowly orbit away from the Sun. Massive observations of elliptic planet orbits show perfect agreement with Newton's theory. Here Einstein's theory can only be less precise.  

Finite speed of gravity thus appears to disagree with observations, at least in a Newtonian setting. To save Einstein's theory it is necessary to show that it contains some form of non-Newtonian effect which exactly cancels the delay effect of a finite speed of gravity. It is doubtful that this is possible. 

Newton's theory appears to be perfect (except for Mercury). Einstein's theory thus appears as a less precise version of Newton's theory (except for Mercury), in contradiction to the above book answer: Newton is more precise than Einstein. 

Concerning the precession of Mercury it is natural to ask: 

• What is the accuracy of Newton's prediction of 5557? Is everything taken into account?
• What is the accuracy of Einstein's prediction of 43 as a correction to Newton? 
• In order to match the observation of 5600, it is necessary that both 5557 and 43 are correct. Possible?   

If Einstein is less precise than Newton, what does that say about modern physics vs classical physics?

What did Einstein mean by saying "Newton, forgive me?" What what so wrong that an excuse was needed? 

To correct Newton requires a full understanding of all aspects of Newton's gravitation, and then a full understanding of Einstein's gravitation as a correction. Is there any living physicist with this intellectual capacity?

The Nobel Prize in Physics 2017 was awarded "for decisive contributions to the LIGO detector and the observation of gravitational waves" in the form of an extremely weak signal from an extremely massive collision of two black holes at an extremely large distance 1.3 Billion light years away from the detector. This was viewed as extremely strong evidence that Einstein is correct.  

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.

  

 

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.      

   

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. 

    

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...