Motion vs Appearance or Emergence

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

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

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

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

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

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

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

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

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

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

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

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

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

 

The Trauma of Paradoxes of Modern Physics

• How wonderful that we have met with a paradox. Now we have some hope of making progress. (Niels Bohr)

Science appears to be filled with paradoxes, which it itself is a paradox, because true science should be free of paradox. For a true scientist a scientific paradox is thus something unbearable, which requires immediate action, because one paradox is enough to kill a whole theory.

There are logical paradoxes as contradiction between words and there are physical paradoxes as contradictions between theory and observation.

One logical paradox is enough to kill a mathematical theory. Thus Russell's paradox killed set theory as the foundation of mathematics in the early 20th century.

Zeno's paradox (still unresolved) of the arrow which is moving although it is not moving at every instant, triggered the development of Calculus, but with a delay of 2000 years!

One physical paradox is enough to kill a physical theory as a mathematical theory about phenomena of physics, all according to the famous physicist Feynman:

• It doesn't matter how beautiful your theory is, it doesn't matter how smart you are. If it doesn't agree with experiment, it's wrong.

If theory does not at all fit with reality, then something is fundamentally wrong with the theory, not the other way around.

A paradox may be thus devastating to existing theory, while leading to new better theory by focussing on weak points.

Yet, the list of physical paradoxes has remained through the development of modern physics and in fact have multiplied since modern physics is loaded with many more paradoxes than classical rational physics, as if modern physics is irrational. Thus the pillars of modern physics in the form of relativity theory and quantum mechanics are both filled with paradoxes, which have remained unresolved for 100 years. This has formed the deep trauma of modern physics with no escape from ever more paradoxes.

Niels Bohr was a master of handling the many paradoxes of quantum mechanics lifting sophistry to a new level with his "complementarity principle" addressing the wave-particle contradiction with murky statements like:

• The opposite of a correct statement is a false statement. But the opposite of a profound truth may well be another profound truth.

While classical physicists had to come to grips with paradoxes, in one way or the other, modern physicists appear to welcome paradoxes as a sign of deep magical physics as opposed to shallow understandable classical physics.

The first defence line for a classical physicist is to simply deny the existence of a paradox formulated by some renegades. The next is to accept that there is indeed a paradox and then come up with an ad hoc explanation for the contradiction between theory and reality, showing that the contradiction is in fact only apparent, but not really real. If the ad hoc explanation is refuted, a new ad hoc explanation is presented and so on.

For a modern physicist, a paradox thus poses no real problem, but of course it is some kind of nuisance and so occasionally may get some attention. Like the Twin Paradox of special relativity discussed in earlier posts, unresolved since 100 years.

The prime paradox of fluid mechanics is d'Alembert's paradox comparing the prediction of zero resistance to motion through a fluid from potential flow solutions to Euler's equations of slightly viscous flow like air and water, with the observation of heavy resistance increasing quadratically with velocity.

The paradox was formulated by d'Alembert in 1755 but nobody was able to come up with a resolution until the young German fluid mechanician Ludwig Prandtl in 1904 came up with the ad hoc solution to discriminate the zero drag potential solution of Euler's equations because potential flow does not satisfy a no-slip boundary condition coming with a thin boundary layer. With the potential solution thus eliminated form the discussion, the paradox simply disappeared. But the act of discrimination of solutions of the Euler equations of course was not so glorious. Discrimination of prefect exact solutions on formal grounds carries the same weakness as discrimination of good citizens on purely formal grounds.

In 2008 we gave a different resolution of d'Alembert's paradox than Prandtl's, based on the fact that potential solutions of the Euler equations are unstable and thus turn into turbulent solutions with substantial pressure drag. This was not discrimination on formal grounds, but on real grounds; an unstable solution does not persist over time. This opened to a revolution in computational fluid dynamics freed from a perceived necessity to computationally resolve unresolvable thin boundary layers.

You find on this web site, if you are interested and make a search, resolutions of the following paradoxes:

• D'Alembert's paradox and other paradoxes of fluid mechanics.
• The Reversibility paradox of classical and quantum mechanics (Loschmidt's paradox) 
• Paradoxes of special relativity including the Twin paradox.
• Paradoxes of wave-particle and collapse of the wave function of quantum mechanics.

Yes, it is wonderful to discover a paradox and even more wonderful to resolve it!

Gravitation, Motion, Programming with Codea and Zeno's Paradox

                                     Fresh reprinting of the world at each new time instant.  

The experience with the programming platform Codea which I am reporting on Matematik-IT  connects to an earlier view exposed on The World as Computation on the connection between matter and gravitation described by Newton's gravitational equation

• $\Delta\phi =\rho$,            (1)

where $\phi (x,t)$ is gravitational potential and $\rho (x,t)$ mass density depending on a Euclidean space coordinate $x$ and a time coordinate $t$, and $\Delta$ is the Laplacian differential operator. 

The conventional way of viewing (1) is to think of mass density $\rho$ as primordial which generates a gravitational potential $\phi$, and corresponding gravitational force $\nabla\phi$, through the integral

• $\phi (x,t) =\frac{1}{4\pi}\int\frac{\rho (y,t)\, dy}{\vert x-y\vert}$      (2)

with apparent (instant) action at distance expressed by the global nature the integral. The trouble with this view is that the physics of (instant) action at distance, supposedly being transmitted by graviton particles (appearently traveling at infinite speed), is still completely open despite centuries of deep thinking by deeply thinking physicists. The idea of matter density as primordial may come from primitive thinking that what we can see/touch, must come first and what we cannot see/touch, must be secondary.

But there is another way of thinking, maybe less primitive, which is to view instead gravitational potential $\phi (x,t)$ as primordial and mass density

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

as being produced by the local operation of differentiation. With this view, there is no action at distance to explain.  What asks for understanding is the physics of (3).

Viewing the world as computation then connects to the way Codea works with the screen being freshly redrawn 60 times a second from the code under function draw()....end, without storing anything previously written on the screen.  This is the same way we perceive the world with our eyes with a fresh image at each new instant.

With this perspective we can think of (3) as the computer code which at each instant draws the world in a new configuration with new mass density $\rho (x,t)$  from a gravitational potential $\phi (x,t)$ which is changing in time according Newtonian mechanics.

In the same way as motion is exhibited by Codea by a sequence of fresh images coded under function draw()...end, the motion of matter we perceive would be a result of fresh reprinting of the world at each new instant according to the code (3).

The alternative view suggests a solution to Zeno's paradox, still unsolved after 2500 years, with the arrow being reprinted at each new instant in time giving the appearance of motion as change of position with time. Think of that!

Note that it Einstein's equations in nearly flat Minkowski space-time reduces to a wave equation variant of Newton's equation of the form

• $-\frac{1}{c^2}\frac{\partial^2h_{\alpha\beta}}{\partial t^2}+\Delta h_{\alpha\beta} =\frac{16\pi G}{c^4}T_{\alpha\beta}$,

where $h_{\alpha\beta}$ is metric perturbation and $T_{\alpha\beta}$ stress-energy. This equation allows waves traveling at the speed $c$ of light as "ripples of the fabric of space-time" in the common mysterious jargon of relativity theory, but the presence of the factor $c^4$ make such waves vanishingly weak. In short, there seems to be little reason to expect a wave equation variant of (1) to have physical significance, which can be seen as support of the alternative view.

Zeno's Arrow Paradox Still Unresolved after 2500 Years

Russell's "resolution" of Zeno's Arrow Paradox says that an arrow with position $f(t)$ changing continuously with time $t$, is moving. This not a true resolution but only an empty tautology.

Zeno's Arrow Paradox has haunted physicists and philosophers of physics since its formulation by Zeno around 450 BC:

• How can an arrow be moving in space from one point to another,  if at each instant of time the arrow is still and is not moving?

In Stanford Encyclopedia of Philosophy the paradox is described as follows:

• The third is … that the flying arrow is at rest, which result follows from the assumption that time is composed of moments … . he says that if everything when it occupies an equal space is at rest, and if that which is in locomotion is always in a now, the flying arrow is therefore motionless. (Aristotle Physics, 239b.30)
• Zeno abolishes motion, saying “What is in motion moves neither in the place it is nor in one in which it is not”. (Diogenes Laertius Lives of Famous Philosophers, ix.72)
• This argument against motion explicitly turns on a particular kind of assumption of plurality: that time is composed of moments (or ‘nows’) and nothing else. Consider an arrow, apparently in motion, at any instant. First, Zeno assumes that it travels no distance during that moment—‘it occupies an equal space’ for the whole instant. But the entire period of its motion contains only instants, all of which contain an arrow at rest, and so, Zeno concludes, the arrow cannot be moving.

Zeno was so clever in formulating his arrow paradox that it has resisted convincing resolution into our days. The commonly accepted "resolution" is the "at-at" theory put forward by Bertrand Russell, which simply says that the motion of an arrow appearing at different positions at different times can be described by a continuous function $f(t)$ where $f(t)$ is the position at time $t$. With a non-constant continuous function $f(t)$ of the real variable $t$, the arrow is thus changing position with increasing time and thus is moving. Russell writes in The Principles of Mathematics in the chapter on motion

• Motion consists merely in the occupation of different places at different times, subject to continuity as explained in Part V. There is no transition from place to place, no consecutive moment or consecutive position, no such thing as velocity except in the sense of a real number which is the limit of a certain set of quotients. The rejection of velocity and acceleration as physical facts (i.e. as properties belonging at each instant to a moving point, and not merely real numbers expressing limits of certain ratios) involves, as we shall see, some difficulties in the statement of the laws of motion; but the reform introduced by Weierstrass in the infinitesimal calculus has rendered this rejection imperative. 

The recent survey Can Continuos Motion be an Illusion? by Shan Gao (2013) points to the hollowness of Russell's view:

• According to the “at-at” theory, it is fallacious to conclude from the fact that the arrow does not travel any distance in an instant that it is at rest. Motion has nothing at all to do with what happens during instants; it has instead to do with what happens between instants. In short, motion is merely being in different locations at different times, and that is that. If an object has the same location at the instants immediately neighboring, then we say it is at rest; otherwise it is in motion. Therefore, since the arrow in flight has different positions at different instants, it is surely moving. 
• The “at-at” theory is a static theory of motion. In Henri Bergson’s cynical words, “movement is composed of immobilities.” (Bergson 1911, p.308) Continuous motion is simply the occupation, by an object, of a continuous series of places at a continuous series of times. There are no states of motion at an instant, and no instantaneous properties indicate that an object is moving or not. 

We understand that the "at-at" theory is no theory, just a tautological play with words: To say that an arrow is moving because its position is changing with time is simply a truism or tautology and thus is empty of physical content. To disguise the emptiness, a reference is often made to Calculus and the definition of a continuous function through the concept of limit, but the physics is still lacking. Another form of hand-waving is to say the motion of an arrow is explained by the theory of relativity.

The net result is that Zeno's arrow paradox still after 2500 year lacks a real physical resolution. An attempt to such resolution was made in the previous post.

New View of Motion under Gravitation without Classical Mysteries

Mysteries of Classical Mechanics

There are two main mysteries in classical Newtonian mechanics and cosmology addressed in a recent sequence of posts:

• Gravitation force created by instant action at distance from local presence of matter?
• Motion of matter (Zeno's paradox: moving arrow is still at each instant)?

Resolution of Mysteries by New View of Motion under Gravitation

Both mysteries can be resolved, or circumvented, by viewing the gravitational potential $\phi (x,t)$ depending on a Euclidean space coordinate $x$ and time $t$ as the primordial object, from which matter with density $\rho (x,t)$  and motion along trajectories $x(t)$ with acceleration $\ddot x(t)$ and velocity $\dot x(t)$ with the dot denoting differentiation with respect to time, are derived by the following equations to be satisfied for all $x$ and $t$:

• $\rho (x,t) =\Delta\phi (x,t)$                                         (Newton's law of gravitation)
• $\ddot x(t)= -\nabla\phi (x(t),t)$                                    (Newton's 2nd law).   

The gravitational potential $\phi (x,t)$ itself is governed by the evolution equation 

• $\Delta\dot\phi = - \nabla\cdot (u\Delta\phi )$               (E)

where $u(x,t)$ is trajectory velocity defined by $u(x(t),t)=\dot x(t)$, which is a reformulation of mass conservation commonly expressed in the form

• $\dot\rho +\nabla\cdot (u\rho )=0$.                                   (mass conservation)

With the gravitational potential $\phi (x,t)$ as the primordial object, matter $\rho (x,t)$ will be created locally by the operation of the Laplacian $\rho (x,t) =\Delta\phi (x,t)$ expressing Newton's law of gravitation and trajectories $x(t)$ will be defined by Newton's 2nd law, and the gravitational potential will evolve governed by the evolution equation (E) which is a form of wave equation.

Wave Motion

The gravitational potential $\phi (x,t)$ will thus transform as a wave satisfying a wave equation and matter with density $\rho (x,t)$ will be created/annihilated locally according $\rho (x,t)=\Delta\phi (x,t)$

and thereby appear to follow trajectories $x(t)$ given by Newton's 2nd law. 

Motion of matter will then be an illusion of real gravitational potential wave motion, in the same way that the illusion of horisontal motion of heaps of water across a water surface is generated by water molecules in reality in circular motion up and down, where heaps of water are continuously created/annihilated giving the illusion of progression:

Notice that the trajectories $x(t)$ are mathematical constructs, and not "real particle trajectories" since there are no "particles in motion along particle trajectories". The motion consists wave motion of the gravitational potential, which is geared to follow mathematical trajectories.

The Hen and the Egg

The gravitational potential $\phi$ thus appears as a hens farm capable of laying eggs (creating matter) anywhere which are picked up by the farmer (annihilation of matter). An illusion of eggs in motion can then be formed this way, while the reality is that each egg (like Zeno's arrow) lays still after being laid and before being picked up and a moved egg in reality is a new recreated egg. 

      

   

Summary of New View

We are thus led to a New View of Motion under Gravitation offering a resolution of the main mysteries of classical mechanics of gravitational force acting at distance and Zeno's paradox of a moving arrow which is still and not moving at instant of time.

In the New View the primordial entity is the gravitational potential $\phi (x,t)$ governed by an evolution equation expressing mass conservation under motion. In the New View all action is local and Zeno's arrow is allowed to be still at each moment of time, yet capable of giving the illusion of motion by a process of creation/annihilation resulting from real wave motion of the gravitational potential $\phi$.

In particular in the New View there is no need to introduce gravitons as some form of "force carrying particles" transmitting gravitational forces over long distance by moving with infinite or finite speed. There is no evidence of the existence of such particles.

In the New View the only motion is wave motion in the gravitational potential with velocity given by trajectories governed by Newton's 2nd law.

In the New View the equations are those of classical Newtonian mechanics, but the interpretation and cause-effect is different. In the New View the gravitational potential is the cause, matter is the effect and motion of matter an illusion.

In the New View there is no need of general relativity with motion of matter governed by the "fabric of curved space-time", which is also a mystery understood by nobody. In the New View there is no need to speculate about the existence of gravitational waves progressing at the speed of light, which lack any form of experimental support despite major efforts.

The New View offers a look behind the scene of the visible motion on the stage of our planetary system with the planets accelerating towards the Sun as if governed by a God's hand of instant action at distance.  What we can see on scene is mostly illusion of reality and not reality.

Is there then no mystery in the New View?  Yes, of course there is and it is the generation of matter from gravitational potential according to $\rho =\Delta\phi$. This is the mystery of how a hen can generate an egg. We compare with the mystery of how an egg can generate a hen. Which mystery would you prefer to explain: (i) an egg from a hen or (ii) a hen from an egg?  My choice is (i) and yours? To remove all mystery may well be beyond human capacity, but we can always seek to minimize the amount of mystery, or if your are a professional physicist to maximize the mystery in order to maximize government grants to support multiplication of string and multiverse theorists.

The New View is expanded in Newtonian Matter and Antimatter (also here).

Zeno's Paradox Unresolved in Classical Mechanics

Notice that physicists (and philosophers) are still struggling to resolve Zeno's paradox without any success declared. Often a reference to Calculus as a resolution is made with a hint that the arrow appearing at different places at different times follows a differentiable curve with a certain momentary velocity defined by the limit process of differentiation, and so must be moving. But that is a fake solution because the essence which is the physics of an arrow appearing at different places at different times, is hidden and the limit process is purely mathematical. Thus Zeno's paradox of the impossibility of motion is unresolved in classical Newtonian mechanics, which does not reveal the physics of motion simply taking it for granted by pointing to the Moon traversing the sky.

As an example, consider the nonsense explanation in Stanford Encyclopedia of Philosophy: 

• The answer is correct, but it carries the counter-intuitive implication that motion is not something that happens at any instant, but rather only over finite periods of time. Think about it this way: time, as we said, is composed only of instants. 
• No distance is traveled during any instant. So when does the arrow actually move? How does it get from one place to another at a later moment? 
• There's only one answer: the arrow gets from point X at time 1 to point Y at time 2 simply in virtue of being at successive intermediate points at successive intermediate times—the arrow never changes its position during an instant but only over intervals composed of instants, by the occupation of different positions at different times.