Modern vs Newtonian Physics

The concept of force carrier is central to modern physics crowned by the Standard Model of atomic/nuclear physics, with the force between two particles established by a carrier particle bouncing back and forth. The photon is identified as the carrier of the electromagnetic force and gluons as carriers of the strong nuclear force between protons and neutrons, while the graviton as carrier of the gravitational force has not been identified despite major efforts. 

In contrast, the concept of force carrier has no role to serve in Newtonian physics since forces are either transmitted by instant action by contact or by instant action at distance as gradients of electric or gravitational potentials connected to distributed charges and masses, as discussed in more detail in blog posts under tag New View on Newtonian Gravitation.

It is natural to ask if the concept of force carrier is useful, if carriers of gravitational force are missing and if electromagnetic forces more naturally arise from electric potentials rather than from photons bouncing back and forth as unnatural physics? 

But what about gluons as carriers of the strong nuclear force needed to keep protons and neutrons together in a nucleus?

In recent posts, I have tested the idea that a nucleus can be held together by electromagnetic forces, thus without need of any strong force, in basically the same way an atom is held together by Coulombic interaction between charges of different sign. Computations with RealQM suggest that this may be possible. A nucleus here consists of a central point-like negative charge density (electrons without internal repulsion) surrounded by non-overlapping positive charge densities (protons), with the "compression" of the electrons in fusion releasing massive energy.

In any case, the idea of force carrier presents severe difficulties to modern physics and one way to handle this situation is to give up the idea in a return to Newtonian physics.   

 

The Role of Differentiation and Integration in Physics

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

• $\frac{dv}{dt} = f(t)$     (3)

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

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

 

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

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

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

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

Instant Action at Distance in Atom Physics/Quantum Mechanics

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

You find more under Labels.

  

 

The Blind Space Traveler with Gravitational Potential Meter

              Hawking inside a space ship without windows with a Gravitational Potential Meter

Imagine you are a space traveler locked into a space ship without windows, or traveling through a  region of invisible dark matter. Imagine that in this difficult situation, you have access to an instrument capable of recording the gravitational potential around the space ship from near to far away, an instrument or sense which we may call a Gravitational Potential Meter. Below I discuss how such an instrument might be designed.

Would that allow you to create a normal picture of the distribution of celestial objects/matter around you including your own position, which would be the picture you could see if there were windows or dark matter somehow was made visible, a standard picture/map making it possible to navigate?

Yes, it would because the mass distribution $\rho (x)$ depending on a Euclidean space coordinate $x$ at any instant of time, is related to the gravitational potential $\phi (x)$ by Poisson's equation (in normalised form):

• $\rho = \Delta\phi$,          (*)

where $\Delta$ is the Laplacian with respect to $x$. In this setting you would naturally view the gravitational potential $\phi (x)$ as primordial, because this is what you can record/sense, and you would view the mass distribution $\rho (x)$ as a derived quantity, because this is what you can compute knowing $\phi (x)$ by applying the Laplace operator, which is a differential operator acting locally in space. 

In this new setting you would not, as in the classical setting of viewing $\rho (x)$ as primordial and $\phi = \Delta^{-1}\rho$ as derived by the inverse of the Laplacian as a non-local operator, have to explain instant action at distance, only the local action of (*), and you would thus have eliminated the question of the physics of instant action at distance, which does not seem to have an answer, and as such may be the wrong question. 

We conclude that depending on what we can see through instruments or senses, we are led to questions, which may have answers or not.  It is natural to think that questions, which may have answers, are better questions than questions which do not have answers.

As to the design of a Gravitational Potential Meter or Gravitational Force Meter, imagine a system of little satellites in free fall distributed over the space of interest and connected to a GPS system allowing tracing of the satellites, thus giving information about the Gravitational Force and from that the Gravitational Potential. It is not unthinkable that such a system could cover any space accessible for space travel and beyond. 

Instant Action at Distance and Simultaneity not Needed in New Theory of Gravitation including Dark Energy

                           Einstein won the game. But what was the game about? Simultaneity?

Einstein's theory of relativity grew out from a question of simultaneity in time of events at different locations in space, which Einstein could not answer in a non-ambiguous way and then jumped to the conclusion that a fundamental revision of our concepts of space and time was necessary. Einstein took so on the responsibility in the service of science and humanity to make the revision and thereby open the door to a modern physics of "curved space-time" with all its wondrous new effects of time dilation and space contraction, albeit too small to be detected.

It is clear that simultaneity plays an important role in our society, to set schedules and allow people to meet at the same place and for these purposes we all have clocks synchronized to a reference clock. And to decide which scientist first submitted an article reporting a certain new scientific break-through and to navigate...

But what role does simultaneity play in physics? In what sense do distant physical objects care about simultaneity? Do they all have synchronised clocks? Of course not. What they do is to react to local forces acting locally in time, and no simultaneity with the action of distant objects is involved.

Or is it? What about gravitation, isn't it supposed to act instantly over distance and thus require a form of exact simultaneity? Yes, it so seems because in Newtonian gravitation the Earth is instantly acted upon by a gravitational force from the Sun directed towards the present position of the Sun, and not towards the position where we see the Sun because of the 8 minute time delay of the light from the Sun.

The standard view on gravitation, is thus that the presence of matter instantly generates a gravitational potential/force (Newton) or "curvature of space" (Einstein) at distance. This view comes with the following questions:

1. What is the physics of the instant action at distance? Gravitons?
2. What is the physics of the simultaneity associated with instant action? 

Since no progress towards any form of answer has been made over all the centuries since Newton, it is natural to shift and instead view the gravitational potential $\phi$ as primordial from which matter density $\rho$ is obtained by the differential equation acting locally in space and time:

• $\Delta\phi =\rho$.    (*)      

With this view there is no instant action at distance to explain and no associated simultaneity, since the action of Laplacian $\Delta$ as differential operator is local is space and time. 

It may thus be that the questions 1. and 2. are not the right questions, and then also that Einstein's relativity originating from a question about simultaneity, is not the right answer to the right question.

More precisely, simultaneity does not appear to be a matter of the physics of the world, since atoms are not equipped with a man-made system of synchronised clocks, and so it is not reasonable to make a complete revision of Newtonian mechanics starting from an ad hoc idea of probably little significance.        

The equation (*) further suggests that with $\phi$ primordial there is no reason to insist that $\rho$ as a derived quantity must be non-negative, thus (*) opens to the possible existence of matter density $\rho$ of both signs, that is to both positive and negative matter. 

This idea is explored in the app Dark Energy on App Store with in particular a simulation of a universe resulting from a fluctuation of the gravitational potential with associated positive and negative matter, with the negative matter forcing a positive matter world into accelerating expansion, which may be the missing dark energy you are looking for. Try it!