String theory of engineers, for physicists and mathematicians

A Special note for the Potential Employers from the Data Science field:

Recently, in April 2020, I achieved a World Rank # 5 on the MNIST problem. The initial announcement can be found here [^], and a further status update, here [^].

All my data science-related posts can always be found here [^]

1. You know the classical wave equation:

You know the classical wave equation, right?

Suppose I ask you that.

What’s there in it? Just:

u(t) = \sin( \omega t )!

or, OK, to make it more general…

u(t) = A \cos( \omega t ) + B \sin( \omega t )

Something like that might have passed in your mind, first. When someone says “wave”, people think the water waves, say the ocean waves. Or, they think of the light or sound waves, the interference experiments, even the wave-particle duality. Yet, curiously, when you say “wave equation”, people tend to think of the SHM (simple harmonic motion)—the oscillations of a point-mass, but not the waves in continua.

So, to make it clear, suppose I ask you:

How about the space part?

You might then reply:

Ah, I see what you mean. Pretty simple, too. But now it makes sense to get into a little bit of the complex algebra:

u(x,t) = A e^{i( \vec{k}\;\cdot\;\vec{x} - \omega\,t)}

You are likely to continue…

…Remember the Euler identity? The minus sign, because we want to have a wave that travels to the right? Oops, in the positive x-direction…

That might be your reply.


You know,

I would have to say at this juncture,

the wave equation? I mean, the differential equation. The linear one!

To which, you are likely to retort back

What a ridiculous question! Of course I know it!

OK, it goes like this…

You might then proceed to jot down the following equation in a hurried manner, more or less to get done and be over with my questioning:

\dfrac{\partial^2 u}{\partial x^2} = \dfrac{1}{c^2} \dfrac{\partial^2 u}{\partial t^2}

Yeah, of course, so you do seem to know it. That’s what I was saying!

You studied the topic as early as in XI or XII standard (if not in your high-school). You had mastered it—right back then. You aced your exams, always. You then went to a great engineering school, and studied waves that were a lot more complicated. Like, may be, the EM waves radiated by a radio antenna, or may be, the vibrations in the machinery and cars, whatever …. You have even mastered the simulation techniques for them. Not just FDM but also FEM, BEM, pseudo-spectral methods, and all that.

Or, may be, you weren’t driven by the lowly commercial considerations. You were really interested in the fundamentals. So, you were interested in physics.

“Fundamentals”, you remember you had said some time ago in a distant past, as if to just once again re-affirm your conviction, all in the silence of your mind. And so, obviously, it would have to be physics! It couldn’t possibly have been chemistry for you! And that’s how, you went ahead and attended a great university. Just to pursue physics.

You calculated a lot of quantum wavefunctions but only while you were in UG years—and only in order to clear those stupid exams. But you already knew that fundamental physics is where your focus really was. Real physics. Mathematical physics. Maths!

That’s why, you zipped past that ridiculously simple stage of those mere wavefunctions. You still remember that way before your formal coursework covered it, you had mastered the Dirac notation, the Heisenberg formulation (where operators are time-dependent, not the stupid wavefunction, you had announced to your stupid class-mates), the Uncertainty Principle (uh!), the Poisson brackets, and all that… You had studied it all completely on your own. Then, you had gone into the relativistic QM as well—the Klein-Gordon equation, Dirac’s equation, Feynman’s path integral formulation… All of that. Also GR. Even QFT… May be you even landed into the string theory right while you still were a high-school or UG student.

… It was long ago that you had left those idiotic wavefunctions and all way behind you. They were best left for others to look after, you just knew. That’s what you had thought, and that’s how you’d come to that conclusion.

2. Will you be able to explain its derivation, now?:

So, whether you are an engineer or a physicist, now, it indeed seems that it’s been a long time since you studied the wave equation. That’s why, if someone now asks you to explain the derivation of the wave equation, you might perhaps narrow your eyes a bit. The reason is, unless you’ve been teaching courses to UG students in the recent times, you may not be able to do it immediately. You may have to take a look at the text-book, perhaps just the Wiki? … The Wiki may not be reliable, but since your grasp has been so solid, it wouldn’t take much to mentally go on correctingt the Wiki even as you are reading through it. …Yes, it might take a little bit of time now, but not much. May be a few minutes? Half an hour at the most? May be. But that’s only because you are going to explain it to someone else…

All the same, you are super-duper-damn confident that given the derivation in the text-books (those XII standard or UG level text-books), you are going to zip through it.

Given a brilliant school-kid, you would obviously be able to explain him the derivation all the way through: each and every step of it, and all the assumptions behind them, and even the mathematical reasonability of all those assumptions, too, in turn. You could easily get it all back right in a moment—or half an hour. … “It’s high-school classical physics, damnit”—that’s what you are likely to exclaim! And, following Feynman, you think you are going to enjoy it too…

You are right, of course. After all, it’s been more than 200 years that the 1D wave equation was first formulated and solved. It has become an inseparable part of the very intuition of the physicist. The great physicists of the day like d’Alembert and Euler were involved in it—in analyzing the wave phenomena, formulating the equation and inventing the solution techniques. Their thought processes were, say, a cut above the rest. They couldn’t overlook something non-trivial, could they? especially Euler? Wasn’t he the one who had first written down that neat identity which goes by his name? one of the most beautiful equations ever?

That’s what you think.

Euler, Lagrange, Hamilton, … , Morse and Feschback, Feynman…

They all said the same thing, and they all couldn’t possibly be careless. And you had fully understood their derivations once upon a time.

So, the derivation is going to be a cake-walk for you now. Each and every part of it.

Well, someone did decide to take a second look at it—the derivation of the classical wave equation. Then, the following is what unfolded.

3. A second look at the derivation. Then the third. Then the fourth. …:

3.1. Lior Burko (University of Alabama at Huntsville, AL, USA) found some problems with the derivation of the transverse wave equation. So, he wrote a paper:

Burko, Lior M. (2010), “Energy in one-dimensional linear waves in a string,” European Journal of Physics, Volume 31, Number 5. doi: [^]. PDF pre-print here [^].

Abstract: “We consider the energy density and energy transfer in small amplitude, one-dimensional waves on a string and find that the common expressions used in textbooks for the introductory physics with calculus course give wrong results for some cases, including standing waves. We discuss the origin of the problem, and how it can be corrected in a way appropriate for the introductory calculus-based physics course.”

In this abstract and all the ones which follow, the emphasis in italicized bold is mine.

3.2. Eugene Butikov (St. Petersburg State University, St. Petersburg, Russia) found issues with Burko’s arguments. So, he wrote a paper (a communication) by way of a reply in the same journal:

Butikov, Eugene I. (2011) “Comment on `Energy in one-dimensional linear waves in a string’,” European Journal of Physics, Volume 32, Number 6. doi: [^] . PDF e-print available here [^].

Abstract: “In this communication we comment on numerous erroneous statements in a recent letter to this journal by Burko (Eur. J. Phys. 2010 31 L71–7) concerning the energy transferred by transverse waves in a stretched string.”

3.3. C. E. Repetto, A. Roatta, and R. J. Welti (Vibration and Wave Laboratory, Physics Department, Faculty of Exact Sciences, Engineering and Surveying [per Google Translate] (UNR), Rosario Santa Fe, Argentina, and Institute of Physics, Rosario, Argentina) also found issues with Burko’s paper, and so, they too wrote another paper, which appeared in the same issue as Butikov’s:

Repetto, Roatta and Welti (2011), “Energy in one-dimensional linear waves,” European Journal of Physics, Volume 32, Number 6. doi: [^] . PDF available here [^].

Abstract: “This work is based on propagation phenomena that conform to the classical wave equation. General expressions of power, the energy conservation equation in continuous media and densities of the kinetic and potential energies are presented. As an example, we study the waves in a string and focused attention on the case of standing waves. The treatment is applicable to introductory science textbooks.”

Though they didn’t mention Burko’s paper in the abstract, the opening line made it clear that this was a comment on the latter.

3.4. Burko, the original author, replied back to both these comments. All the three were published in the same issue of the same journal:

Burko, Lior M. (2011) “Reply to comments on `Energy in one-dimensional linear waves in a string’,” European Journal of Physics, Volume 31, Number 6. doi: [^]. PDF eprint available here [^].

Abstract: “In this reply we respond to comments made by Repetto et al and by Butokov on our letter (Burko 2010 Eur. J. Phys. 31 L71–7), in which we discussed two different results for the elastic potential energy of a string element. One derived from the restoring force on a stretched string element and the other from the work done to bring the string to a certain distorted configuration. We argue that one cannot prefer from fundamental principles the former over the latter (or vice versa), and therefore one may apply either expression to situations in which their use contributes to insight. The two expressions are different by a boundary term which has a clear physical interpretation. For the case of standing waves, we argue that the latter approach has conceptual clarity that may contribute to physical understanding.”

3.5. David Rowland (University of Queensland, Brisbane, Australia) also wrote a reply, which too was published in the same issue of the same journal.

Rowland, David R. (2011) “The potential energy density in transverse string waves depends critically on longitudinal motion,” European Journal of Physics, Volume 31, Number 6. doi: [^]. The author’s pre-print (pre-publication version) is available here, [^].

Abstract: “The question of the correct formula for the potential energy density in transverse waves on a taut string continues to attract attention (e.g. Burko 2010 Eur. J. Phys. 31 L71), and at least three different formulae can be found in the literature, with the classic text by Morse and Feshbach (Methods of Theoretical Physics pp 126–127) stating that the formula is inherently ambiguous. The purpose of this paper is to demonstrate that neither the standard expression nor the alternative proposed by Burko can be considered to be physically consistent, and that to obtain a formula free of physical inconsistencies and which also removes the ambiguity of Morse and Feshbach, the longitudinal motion of elements of the string needs to be taken into account,even though such motion can be neglected when deriving the linear transverse wave equation. Two derivations of the correct formula are sketched, one proceeding from a consideration of the amount of energy required to stretch a small segment of string when longitudinal displacements are considered, and the other from the full wave equation. The limits of the validity of the derived formulae are also discussed in detail.”

3.6. Butikov wrote another paper, a year later, now in Physica Scripta.

Butikov, Eugene I. (2012) ”Misconceptions about the energy of waves in a strained string,” Physica Scripta, Vol. 86, Number 3, p. 035403. doi: [^]. PDF ePrint available here [^]:

Abstract: “The localization of the elastic potential energy associated with transverse and longitudinal waves in a stretched string is discussed. Some misunderstandings about different expressions for the density of potential energy encountered in the literature are clarified. The widespread opinion regarding the inherent ambiguity of the density of elastic potential energy is criticized.

3.7. Rowland, too, seems to have continued with the topic even after the initial bout of papers. He published another paper in 2013, continuing in the same journal where earlier papers had appeared:

Rowland, David R. (2013) “Small amplitude transverse waves on taut strings: exploring the significant effects of longitudinal motion on wave energy location and propagation,” European Journal of Physics, Volume 34, Number 2. doi: [^] . PDF ePrint is available here [^].

Abstract: “Introductory discussions of energy transport due to transverse waves on taut strings universally assume that the effects of longitudinal motion can be neglected, but this assumption is not even approximately valid unless the string is idealized to have a zero relaxed length, a requirement approximately met by the slinky spring. While making this additional idealization is probably the best approach to take when discussing waves on strings at the introductory level, for intermediate to advanced undergraduate classes in continuum mechanics and general wave phenomena where somewhat more realistic models of strings can be investigated, this paper makes the following contributions. First, various approaches to deriving the general energy continuity equation are critiqued and it is argued that the standard continuum mechanics approach to deriving such equations is the best because it leads to a conceptually clear, relatively simple derivation which provides a unique answer of greatest generality. In addition, a straightforward algorithm for calculating the transverse and longitudinal waves generated when a string is driven at one end is presented and used to investigate a cos^2 transverse pulse. This example illustrates much important physics regarding energy transport in strings and allows the `attack waves’ observed when strings in musical instruments are struck or plucked to be approximately modelled and analysed algebraically. Regarding the ongoing debate as to whether the potential energy density in a string can be uniquely defined, it is shown by coupling an external energy source to a string that a suggested alternative formula for potential energy density requires an unphysical potential energy to be ascribed to the source for overall energy to be conserved and so cannot be considered to be physically valid.

3.8. Caamaño-Withall and Krysl (University of California, San Diego, CA, USA) aimed for settling everything. They brought in a computational engineer’s perspective too:

Caamaño-Withall, Zach and Krysl, Petr (2016) “Taut string model: getting the right energy versus getting the energy the right way,” World Journal of Mechanics, Volume 6, Number 2. doi: [^]. This being an open-access article, the PDF is available right from the doi.

Abstract: “The initial boundary value problem of the transverse vibration of a taut stringisa classic that can be found in many vibration and acoustics textbooks. It is often used as the basis for derivations of elementary numerical models, for instance finite element or finite difference schemes. The model of axial vibration of a prismatic elastic baralso serves in this capacity, often times side-by-side with the first model. The stored (potential) energy for these two models is derived in the literature in two distinct ways. We find the potential energy in the taut string model to be derived from a second-order expression of the change of the length of the string. This is very different in nature from the corresponding expression for the elastic bar, which is predictably based on the work of the internal forces. The two models are mathematically equivalentin that the equations of one can be obtained from the equations of the other by substitution of symbols such as the primary variable, the resisting force and the coefficient of the stiffness. The solutions also have equivalent meanings, such as propagation of waves and standing waves of free vibration. Consequently, the analogy between the two models can and should be exploited, which the present paper successfully undertakes. The potential energy of deformation of the string was attributed to the seminal work of Morse and Feshbachof 1953. This book was also the source of a misunderstanding as to the correct expression for the density of the energy of deformation. The present paper strives to settle this question.”

4. A standard reference:

Oh, BTW, for a mainstream view prevalent before Burko’s paper, check out a c. 1985 paper by Mathews, Jr. (Georgetown University):

Mathews Jr., W. N. (1985) “Energy in a one‐dimensional small amplitude mechanical wave,” American Journal of Physics, Volume 53, 974. doi: [^].

Abstract: We present a discussion of the energy associated with a one‐dimensional mechanical wave which has a small amplitude but is otherwise general. We consider the kinetic energy only briefly because the standard treatments are adequate. However, our treatment of the potential energy is substantially more general and complete than the treatments which appear in introductory and intermediate undergraduate level physics textbooks. Specifically, we present three different derivations of the potential energy density associated with a one‐dimensional, small amplitude mechanical wave. The first is based on the ‘‘virtual displacement’’ concept. The second is based on the ideas of stress and strain as they are generally used in dealing with the macroscopic elastic properties of matter. The third is based on the principle of conservation of energy, and also leads to an expression for the energy flux of the wave. We also present an intuitive and physical discussion based on the analogy between our system and a spring.

I could not access it, but it was quoted by most (all?) of the papers cited above (which I could).

5. Is it a settled matter, now?:

Have these last few papers settled all the issues that were raised?

Ummm… Why don’t you read the papers and decide by yourself?

6. Why bother?

“But why did you get into all this exasperating thing / stupidity / mess, when all engineers have anyway been using the wave equation to design everything from radios, TVs, Internet router hardware to cars, washing machines, and what not?”

Many of you are likely to phrase your question that way.

My answer is: Well, simply because I ran into these papers while thinking something else about the wave equation and waves. I got puzzled a bit about one very simple and stupid physical idea that had struck me. Far, far simpler than what’s discussed in the above papers. Even just a conceptual analysis of my stupid-simple idea seemed pretty funny to me. So, I’d googled on the related topics just in order to know if any one had thought of along the same lines. Which then led me to the above selection of papers.

What was that idea?

Not very important. Let me mention it some other time. I think there is much more than enough material already in this post!

In the meanwhile, browse through these papers and see if you get all the subtle arguments—all of them being accessible to engineers too, not just to physicists or mathematicians.

Come to think of it, it might be a good idea to post a shortened version of this entry at iMechanica too. … May be a few days later…

In the meanwhile, take care and bye for now…

A song I like:

(Western, Pop): “karma chamelion”
Band: Culture Club

[A Western song that is also hummable! … As always, I couldn’t (and still can’t!) make out words, though today I did browse the lyrics [^] and the Wiki on the song [^]. Back in the 1980s, it used to be quite popular in Pune. Also in IIT Madras. … I like this song for its “hummability” / “musicality” / ”lyricality” / melody or so. Also, the “texture” of the sound—the bass and the rhythm blends really well with the voices and other instrumentals. A pretty neat listen…]


Ontologies in physics—3: EM fields in terms of forces; space; and related ontological issues

0. Before we begin:

I wish I could have undertaken some fresh re-readings on the history of science before setting out to write this series of posts. I have most of the material, and over the years, I have gone through a lot of books on history / conceptual aspects of physics and maths. (Off-hand: Kline, Kolmogorov (ed.), Dugas, Truesdell, Hesse, Coopersmith, Encycl. Brittanica, and many, many others, not to mention the Wiki). Further, I’ve just came across a new source: “Energy, Force and Matter,” by Harman. Wish I could’ve gone through at least this one. But I just don’t have the time. …

I would like to finish this series as fast as I can, so that I can then go over to my new theorization regarding QM. So, overall, what I write here turns out to be in a rather abstract way and without specific references to the original writings. I rely too much on my memory, and on the view of these matters I have come to develop. There’s a risk here that in writing without fresh referencing, I might be mixing my own views with the original ideations. This is not at all to my liking. But I have to make do with precisely that for now, purely out of a lack of time.

A great resource for this post (and the last) is the online notes: “A visual tour of classical electrodynamics,” written for an undergraduate course at MIT [^]. There are many great books on history (some being listed above). Apart from the books, here is a relevant paper: Ernan McMullin, (2002) “The origins of the field concept in physics,” Phys. perspect, vol. 4, pp. 13–39 [(PDF) ^]. Also see a brief note at Prof. Philip C. E. Stamp’s Web pages: “The concept of the field in physics.” [(PDF) ^]. Both these resources were mentioned in a tweet I made on 04 September 2019.

1. The background object and spaces in our EM ontology:

1.1 The background object of our EM ontology:

We mentioned the last time that in our EM ontology, there are only two types of objects: (i) EC objects, and (ii) the background object.

In our EM ontology, the background object is a physically existing but a non-NM kind of an object. Our view that the background object is not an NM kind of an object makes it sharply different from Faraday, Maxwell and Newton’s view.

[As an aside, even my c. 2005 paper on QM (covering my old, PhD-time approach) had mentioned the “aether” as a physically existing but non-material object. I guess this idea has been with me for a very long time.]

1.2 The physical space, and mathematical spaces:

When you point your finger to some place in between two EC objects, i.e., if you point out the “empty space” in between them, what you are actually pointing to is an invisible background object—not space—which is present at that place. We can justify this position, though its justification will progress slowly over this and the next two posts.

So, what exists in between two EC objects is not the “empty space.” Not even in the physical sense of the term “space”. (And there is something as the physical space.)

In our view, physically, the concept of space denotes the fact that physical objects (both the EC objects and the background object) have spatial attributes or characteristics like extension, location, and also other spatial attributes like the topological ones. The concept also includes the physically existing spatial relations between all objects. The physical space is the sum total of all the spatial attributes or characteristics of all the physically existing objects, with all the interrelations between them. Mathematically, the concept of space denotes a quantitative system of measuring the sizes (or magnitudes) of those spatial attributes with which objects actually exist. [For a very detailed, in fact very long-windingly written series of posts on the philosophical ideas behind the concept of space, see my earlier posts here [^].]

Notice the logical flow:

What ultimately exist are objects—that is the most fundamental fact. In fact, it is the primary fact assumed by all of physics. A primary fact is one which cannot be analyzed as implied by or arising from other facts. So, objects exist, full-stop. Every object exists with all the attributes that it has; each object has a certain identity.

Attributes exist (as part of identity of an object) only in some specific quantities or sizes. There can be no size-less attribute. As a simple example, a pen has the spatial attributes of length, diameter, shape, etc. Each pen exists with a specific quantity or measure of length and other attributes. Thus: sizes do have a physical existence; sizes do exist in the concrete physical reality out there. However, sizes don’t exist as apart from the objects whose sizes (in different respects) they are.

Mathematics then comes into picture. Mathematics is the science that develops the methods using which physical sizes of comparable objects (i.e. objects having the same attributes but to different measures) can be quantitatively related to each other. Mathematical concepts refer to mathematical objects, not physical, even though these concepts are reached only after observing the size-wise relations among physical objects. Mathematical objects are a result of objectifying the methods invented by us for measuring the existing sizes of the physical objects. The same set of physically existing objects (or their attributes, characteristics, properties, etc.) can give rise to an indefinite number of mathematical concepts.

Coming back to the concept of space: There can be many mathematical spaces, but they all refer to the same physical space. A mathematical space is an abstract quantitative system of measuring extensions and locations of physical objects. It is a mathematical concept which is developed in reference to the facts subsumed by the concept of the physical space.

The referents of the concept of the physical space are perceived directly, but not the physical space itself. The physical space is a concept, not a directly perceivable concrete.

What you directly see, the sense you get in your direct perception that there is some “solidity” or “volume-ness” to the physical world, is not the physical space itself, but the evidential basis for that concept.

1.3 The background object

The background object is a physical substance, but having a “non-material” nature, speaking vaguely—the exact statement here is that it is not an NM object.

The physical space is just one way to characterize the background object. It is possible to spatially characterize the background object because all its parts do possess spatial attributes (like extension, location, etc.).

However, therefore identifying the background object with the physical space (or worse, with a mathematical space) is a very basic error.

Just the fact that the background object possesses spatial attributes, does not make it the same as the physical space. …Would you call an NM-object (like a ball or a table) “space” just because it has spatial attributes?

Realize, the background object, though it is invisible (in fact it is as good as non-existent for the passage of NM-object through it), is physically an active existent at all times. Force-conditions are being sustained in it at all times, which it makes it active in the same sense that a foundation stone of a building is: no overall motion, but still a transmission of forces through it, at all times.

Neglecting gravity, the background object does not interact in any way with the NM-objects. It is for this reason that no inertia or mass can be ascribed to it. This is one of the easy reasons why it can’t be regarded as an NM-object. However, as we shall see later, the background object does possess something like a state of a “stress” within itself. (The nature of these stress-strains is different from what you have in case of the purely NM-objects.) It is worth noting here that even in the NM-ontology (i.e., in solid- and fluid-mechanics), the equations defining the stress/strain fields do not have mass appearing in them. So, the point here is that even if the background object were to have mass, it still wouldn’t matter because it doesn’t enter into dynamical equations involving it.

2. A few words on the difference between the 19th century view of space and ours:

2.1 What space meant to the 19th century physicists:

The 19th century physicists (like Maxwell) took space in the same sense as Newton did—in the NM-ontological sense of the term.

Space, to them, is a directly given (i.e. directly perceptible) absolute, having an independent physical existent apart from any NM-objects there may be. They would regard objects as filling or occupying some parts (some regions) of this already given absolute space.

In our opinion, this view is somewhat Platonic in nature. Faraday and Maxwell received this view of a physical, “absolute” space from Newton.

2.2 Our view: Space as a concept derived from the spatiality shown by physical objects:

We take the view that only physical objects are what is directly given in perceptions. (Perception here also includes observations of EC objects made in controlled EM experiments.) It is true that we directly perceive something space-like—the solidity or volume-ness of objects. It is also true that we also directly perceive the particulars of configurations of objects, including their directly evident locations. But that’s about all. We don’t perceive space as such; only the volume-ness or extended-ness of objects, their spatiality.

This point is somewhat similar to saying that what we directly perceive are blue objects (of different hues). We do directly see that blue objects show blueness. But the concept “blue” is only an abstraction from these objects; it is a product of a concept formation. The blue is not out there; only blueness is, and that too, only as a characteristic or attribute of those actual objects which do possess it.

People call the “empty” region “space” simply because they are unable to think of an invisible (or untouchable etc.) object that can exist in between two visible (or perceptible) NM objects. But realize that this “emptiness” also is a part of what we directly perceive. It is not as if our perceptual fields comes to have an in-principle hole (or complete absence) in the places where there is this absolute “nothing”ness of the “empty space.” Perceptions don’t have “holes” because there is something of what we call “space.” Our perceptual field cannot contain any evidence for the existence of a literal nothing in the world out there—a nothing that supposedly is at par with the things that do exist in the world out there.

So, what we directly perceive are only objects having spatiality, but never “space” itself. Space is something which we conceive of, based on these percepts.

2.3 Filling space with objects—what it means in our view:

It is only at this point in development—after we already have the concept of space—that we are able to trace the concrete-to-abstract relationship in reverse, and say that objects can be seen as coming to occupy some region of space that was initially empty (of similar, NM-type of, objects).

In our view, this statement (that objects fill space) involves an application of a concept, a backward motion in conceptual linkages. Objects can fill the physical space only because there are other NM-objects that help you isolate the “empty” region, which can then be filled with some other NM-objects.

The reverse-tracing of conceptual linkages becomes possible only if you have it in the first place, if you start from an abstract level. Perceptual level does not permit such reverse-tracing.

2.4 Space couldn’t possibly survive literal annihilation of all objects:

To Newton, Faraday and Maxwell, if all physical objects were to be annihilated, then the absolute space would still be left behind. To us, this is an impossibility. There cannot be spatial attributes without there being objects that have spatial attributes, and there cannot be the concept of space without our grasping a certain aspects like extended-ness or volume-ness that they have, and then abstracting a concept out of such physical features.

3. Causality and interactions:

3.1 Our view of causality

A word about causality. We follow the Objectivist view of causality, put forth by Ayn Rand [^]. Thus, at the most basic level, the idea of causality has nothing to do with an orderly progression in time. Properly speaking, the nature (identity) of the objects that act is the cause, and the nature of the actions they show or undergo are the effects.

It must be understood that the concept of causality is in principle applicable to single objects as well as to interactions between two or more objects.

3.2 Characterizing all the causes and effects operative in interactions:

Ontologically, in any interaction between any two objects of any kind, both the objects participating in the interaction must be seen, simultaneously, as being agents of causal actions.

When certain actions of a given object leads to certain other actions by some other object, we say that they are interacting. Suppose one billiard ball hits a second ball that was initially stationary. We often loosely say that the motion of the first ball is the cause and the motion of the second ball is the effect. Actually, the respective natures of the balls themselves are causes—both of them; and the natures of their individual actions are the effects—both theirs. At the instant when the two balls are in contact, both their natures taken together are such that they determine both their subsequent actions (here, motions). Thus, there are two causes and two effects.

If in order to determine an effect (an action by some object) you have to consider the nature of some other object(s) too, then we say that they all are interacting with each other. Thus, in collision of two balls, the ball that is initially moving (the one that hits the other) is not the only cause. Both the objects are causes (and they both hit each other). They both produce effects, even if you typically focus on only the second ball for characterizing the effect (of that interaction).

3.3 Causality as a concept far more basic than an orderly progression in time:

To repeat, it is wrong to characterize the idea of causality in reference to an orderly progression in time. When two NM-objects remain in static equilibrium for a long period of time, they still are obeying (and exhibiting) causality even if nothing about their dynamical states is ever undergoing any changes with the passage of time. Both are interacting at all times, both are causes (they exert forces like weight and reactions), and both their stationarity (“motionless-ness”) is an effect.

4. The electric field (\vec{E}) as a mathematical generalization from Coulomb’s law of force:

4.1 Coulomb’s law as operative at only two distinct points, and nowhere else:

It is the electric charge which gives rise to the Coulomb forces between two EC objects.

Consider an EC object of a charge q_1 fixed at a position \vec{r}_1. Consider a “test charge” of magnitude q_T at some arbitrarily chosen but fixed position \vec{r}_T.

The idea behind calling the second charge a “test” charge is that it can be placed at any number of locations. However, the position of the “given” charge q_1 always stays fixed.

The force exerted by q_1 on the test charge q_T is given by Coulomb’s law:
\vec{F}_{1T} = \dfrac{1}{4\,\pi\,\epsilon_0} \dfrac{q_1\,q_T}{r^2}\;\hat{r}_{1T} \qquad\qquad r = |\vec{r}_T - \vec{r}_1| .

The equation of the law tells us that the force \vec{F}_{1T} is a function of both q_T and \vec{r}_T. If we could somehow take out the effects of these two factors (both being related to the test charge), then the field will come to depend on the first charge q_1 alone. In short, we want to isolate the action of the first (given) charge from that of the test charge.

4.2 How to isolate the actions of the first charge from that of the test charge?: The electric vector (\vec{E})

If the position of the test charge is different, we will obtain another force vector acting at that position. We can imagine having the test charge placed at different locations, successively, in different system configurations. Each \vec{r}_{T} \leftrightarrow \vec{F}_{1T} pair is unique. If we collect together all these (infinity of) unique force vectors, they form a vector field—a function that gives a vector once you plug in some specific (x,y,z) coordinates. Since the idea of the field includes all possible force vectors for all possible positions of the test charge, we don’t have to separately state a specific position—no matter what be the position, it’s already there somewhere in that infinite collection. So, by giving the entire field, we make it redundant to have to specify a single specific position for the test charge. We remove the necessity for specifying any specific position.

To factor out the effect of the magnitude q_{T} of the test charge, we now propose a new vector quantity called the electric vector, denoted as \vec{E}. It is defined as:
\vec{E}_{1} = \dfrac{\vec{F}_{1T}}{q_T}
What we have effectively done here, by dividing the force at \vec{r}_T by q_T, is to suppose that q_T always remains a unit charge. Since neither position nor charge-magnitude of the test-charge have to be specified, we have dropped the T subscript.

Whereas the electrostatic force field depends on the magnitude of the second charge, the electric vector field does not. Thus, we have found an object that captures the effect of the first charge alone. Hence the subscript _{1} to E still remains.

Actually, the second charge (T) is physically still there, but its role has been pushed back into an implicit background, using our procedure—via generalization to all space, and normalization to the unit charge, respectively.

4.3 The electric vector’s field as a mathematical device of calculations:

A field is just a simple function of the spatial coordinates. You plug in any specific position into the field-function, and you get the electric field vector (\vec{E}) that would be produced at that point—if a unit test charge were to be actually present there. Once you know the electric field vector which would be present at a point, then you can always find the Coulomb force which would be exerted by the first charge (which generates the field) on any arbitrary second charge, if it were to be actually situated at that point: you just multiply the electric field vector at that point by the magnitude of the second charge.

This is the meaning which people (the continental mathematical physicists) had in mind when they first put forth the mathematical idea of such functions. These functions later came to be regarded as fields.

5. Some comments on the MIT notes:

Since I gave a reference to the notes at MIT, a comment is in order. The MIT note defines the electric field via a limit: \vec{E} = \lim\limits_{q_0 \rightarrow 0} \dfrac{\vec{F}}{q_0}. Taking this limit is not at all necessary. In fact I fail to see even its relevance. If at all a limit has to be conducted, then it could be for a vanishing size (diameter etc.) of the test charge, so that the point of definition of \vec{E} becomes unambiguous. But taking the charge to vanishingly small charge simply does not seem to bring in anything of relevance. … For others’ comments, see, for instance, here: [^]. The answer given at a JEE-preparation site also is somewhat misleading [^].

The correct way to think about it is to think of a static situation (at least in the vicinity of the test charge). A static situation can be had either (i) by considering just one instant of time in the motion of a movable configuration of charges (EC Objects), or (ii) by introducing some imaginary support forces which keep all the charges fixed at their respective positions at all times.

We differ from the MIT notes in one more respect. They first define the electric field as a limit of the force at a point on a test charge, but without any reference to the other source of that force. Its only after thus defining the electric field that they come to relating it with the Coulomb force exerted by the first charge. Thus, their definition is, strictly speaking, half-arbitrary: it misses one of two crucial objects that are present in the empirical observations.

We regard the idea of an arbitrary field as existing at a higher level abstraction, but insist on noting that no matter how arbitrary an electric field (its pattern or distribution) might get, it still cannot come into existence without some or the EC object(s) producing it. That’s our viewpoint. We emphasize the role of the field-producing charge.

Indeed, when it comes to QM ontology, we do away completely with the idea of arbitrary fields and even arbitrary continuum charge distributions (which they demand). We *restrict* the generation of all permissible electric fields only to point charges because elementary charges are point-particles.

6. An ontological breakthrough: The entire electric vector-field seen as existing physically:

6.1 A possible reason why the continental physicists didn’t  go for a physically existing field:

Coulomb’s law states a relation for what happens at two specific points in space. The law is completely silent on what happens at any other points of space. In contrast, the electric field is mathematically defined for all points of space.

As mentioned in the last post, the continental physicists did work with the mathematical notion of fields for a long time (I off-hand suppose, for at least 70 years) but without thereby necessarily implying its physical existence.

It should not be too difficult to understand their perspective. A field would be just a mathematical device for them; using this mathematical object meshed well with their energetics program, that’s all.

Their development program primarily relied on the underlying idea of energies, and so they would certainly require the potential energy, not forces. Actually, the concept of energy was yet to be isolated as such—they simply used the mathematical functions that we today regard as defining the quantities of energy.

Strictly speaking, having a field of potential energy was not necessary for formulation of laws of physics in their program. Their laws could have been formulated with just a single number for the potential energy of the entire system. (The number would vary with positions of discrete bodies, but it would nevertheless always be a single number.) That is, as far as their laws were concerned.

However, the field idea would be convenient when it came to applying their formulation. Problems having continuous boundary conditions naturally got simplified with the working idea of a function of all possible spatial coordinates. Thus, a field came be supposed for the potential energy. They would see forces as gradients of potential function; forces were secondary or derived quantity for them. Indeed, the problems they worked on, during the development of the potential field concept, came exclusively from gravity.

Thus, all in all, the field was primarily a mathematical idea for them—just a device of calculations, and that too, only for gravity, even if electromagnetic laws also were being discovered during the same period.

6.2 Faraday makes the ontological breakthrough:

It was Faraday who vigorously advocated the idea that the force-field is not just a mathematical idea but also physically exists in the real world out there. He characterized it in terms lines of force. He believed that the space was not empty but filled with a fluid (a mechanical or NM-object like air, water, oil, etc.). The lines of force were imagined by him to be tubes formed by fluid flow. Maxwell then mathematically refined the idea.

It may be perhaps be noted here that the pattern of the magnetic field which is observed when you sprinkle some iron filings on a magnet, does not actually form enough of an evidence to prove the existence of fields. It merely suggests and supports the idea of a field.

But strictly speaking, you can always argue that a field does not therefore exist; only point-wise forces (action at a distance) do. In the context of iron-filings and a magnet, you can argue that magnetic forces are present only at the points where the iron-filings are—not in the empty spaces in between them. The picture of the field pattern produced by the iron-filings, by itself, is thus not sufficient.

The fact is that the field idea can only be introduced as based on a more general thought; it can be introduced only as a postulate, to ensure consistency in theory. We will touch on this issue later. For the time being, we will simply assume that the continuum field does indeed physically exist.

7. Our EM ontology: The electric vector field as an attribute of the background object:

Let’s now note a subtle ontological point.

7.1 An EC object is the cause of the \vec{E} field:

As the position of a given charge (the “first charge” or q_1) itself changes, the entire force-field shifts in space too. This is a direct consequence of Coulomb’s law.

The preceding sentence says “changes” and “shifts”. However, note that we don’t thereby mean an actual motion here. We are merely describing the differences which are present between two fields when they are actually produced in two different system configurations. Within each system description, everything still remains static. Taking a difference between two system descriptions does not always have to involve a continuous motion connecting them. That is what we mean here. Thus, the “shifts” here are of the variational calculus kind.

The fact that the field generated by an EC object shifts with its position further establishes the idea that the electric force field is a causal effect of that object.

7.2 The \vec{E} field is not an attribute of an EC object:

However, an important point to note here is that this fact still does not make the field an attribute of that particular EC object.

Mathematically, the field due to a point-charge is a function that is defined at all points other than its own position. Physically, therefore, the field must exist only at those spatial locations where the field-generating EC object itself is not present.

In the EM ontology, EC objects and the background object are the only two categories of object. If the field is not even present at the location of a point-charge, it cannot be an attribute of that charge. Therefore, it must be an attribute of the background object. Matter cannot act where it is not.

7.3 The existence of \vec{E} implies an interaction between two physically existing objects:

If an object is causally responsible for producing an effect, but if that effect is an attribute of some other object, clearly we have two different objects interacting here.

There is a difference in the hierarchical levels of EC objects and electric fields. EC objects are primary existents. In contrast, electric charge and mass exist only as attributes of EC objects. Similarly, the background object is a primary existent. But the electric fields are secondary—they exist only as attributes of the background object.

In calculations, we often get so much used to associating the electric field generated by a point-charge with that charge itself, that we tend to forget that the generated field is not an attribute of that charge itself. Though produced by a charge, the associated field is actually an attribute of the *other* object—the background object.

The electrostatic (and in general, all electromagnetic) phenomena can be explained in terms of the direct contact forces which arise between an EC object and the background object, or between two control volumes (of arbitrarily small size) within the background object.

8. An electric vector field as an effect produced by an EC object acting in isolation of others:

Coulomb’s law has exactly two electrostatic forces. The force exerted by any one charge acts at only one point: at the distant location of the other charge. (The support forces are taken as mechanical in nature here, not electrostatic.) Coulomb’s law thus speaks in terms of a pair of forces. It is physically impossible to have a situation in which only one of the two charges is active (exerts a force on the other).

In abstracting the idea of the electric vector field, we had to factor out the effects due to the test charge. As a result, attributes of test charge do not determine the distribution pattern of the electric field. Interpreted physically, a single charge can be taken to generate the field associated with it; a second charge (say a test-charge) is not at all necessary.

In other words, a complete electric vector field (spread over the entire infinite space) can come into being with the existence of just one charge. In contrast, Coulomb’s law requires and simultaneously relates two different charges. Ontologically, this is a significant difference.

The electric field due to several charges is simply an algebraic sum of the fields produced by each charge acting separately.

Imagine an infinitely large universe that has nothing but just a single charge. (You can’t determine its location.)

If the electric force field exists physically, and if such a field can be produced by each single charge acting singly, then actual forces will come to exist everywhere even in this universe. The field will be spherically symmetric. Faraday’s lines of forces will be straight lines that emanate from the point-charge—they will look like symmetrically distributed “spokes.”

When a second charge is added at a finite distance from the first, then the field condition actually existing in the universe is obtained simply by a linear superposition of the two fields. The effective lines of force will look distorted from their initial symmetrical shape.

A distortion of Faraday’s lines of force is implied by (and implies) the existence of the second charge. However, their very existence does not depend on the existence of the second charge.

9. A preview of the things to come:

The description in this post was mostly in terms of forces and quantities derived from them. In the next post, we will look into an alternative description, one that is couched in terms of energies and quantities derived from them. While both the approaches are physics-wise equivalent, the energy-based approach helps simplify calculations. In fact, most physicists get so thoroughly used to the energy-based approach in their PG years, that they even come to forget that ever was a force-based approach (which is quite unlike engineers—we engineers never come to forget forces, including reaction forces at supports).

We have to look at some of the basics of the energy-based approach to physics, simply because the Schrodinger equation is inductively derived using only the energies. However, since we are covering this material purely from the viewpoint of the Schrodinger picture of QM, we will try to keep the variational calculus ideas as much to the background as possible.

Thus, in the next post, we will go over the following sequence of ideas: the potential energy number of an electrostatic system; the potential energy field in the spatial region of a system; the potential energy field of a single elementary point-charge; the potential of an elementary point charge. We will also try to look into the issue of how the background object comes to support forces within itself. Finally, there also will be an issue of justifying the inclusion of the background object in an ontology—its necessity. We will try to cover it right in the next post. If not, we will do that in a subsequent post, and then we will be done with our EM ontology.

See you the next time, bye for now, and take care….

An important update on 2019.09.17 12:26 IST:

I think that the sections 7. and 8. above have come out as being somewhat misleading.

The present write-up suggests that what physically exists is the \vec{E} field. However, if the magnitude of the second charge q_2 or q_T is not 1, then it is not clear precisely what physical process occurs at \vec{r}_2 such that it results in a multiplication of \vec{E} by q_2 to produce the force \vec{F}_{12} at that point. This force, after all, must physically exist. But it is not clear what physics is there for the multiplication by q_2.

So the proper conclusion to draw seems to be that the actually existing field is always that of the force \vec{F}_{1T} and not of \vec{E}_{1}; that the latter is only a mathematical device, a convenience in formulating Maxwell’s system of equations. But what will be the implications for an isolated system having only a single charge? for bringing a second charge into it from infinity? … Need to think through.

… I will think about it for a while and then, if necessary, I will come back and update the above description appropriately.  … In the meanwhile, any thoughts / suggestions?


A song I like:

(Marathi) “hirve, pivale, ture unhaache, khovilesa kesaata ugaa kaa??”
Lyrics: Indira Sant
Music and Singer: Vasant Aajgaonkar


Ontologies in physics—2: Electromagnetic fields as understood by Faraday and Maxwell

In this post, we shall begin going through the ontology assumed in—or rather, demanded by—the physical phenomena which are covered by the classical (i.e. Maxwellian) electrodynamics. We call this ontology EM for short.

We will mostly be talking in reference to electro-statics. However, note, this ontology just as well applies also to electro-dynamics.

1. A list of the ontological objects used in the physics theory of electromagnetism (EM):

The EM objects basically are of only two types: (1) massive and electrically charged point-particles (of one of two polarities: positive or negative), and (ii) a background object (sometimes loosely identified with the fields induced in it, by calling it “field”; other times identified with an aether of a certain kind by Maxwell; more, below).

Notably, there are no separate magnetically active objects in this ontology even though magnetism has been known as a force for more than at least one millenium. Magnetism is an effect produced by the electrical charges.

2. Electrically Charged (EC) objects:

The ontology of NM (seen in the last part in this series) is basically that of the uncharged bodies. What the EM ontology now does is to further endow these same objects with an additional attribute of the electric charge. This extra attribute considerably modifies the entire dynamical behaviour of these objects, hereafter called the EC objects for short.

For the elementary charged objects (basically, here, only the electrons and protons), the phenomenon of the induced charge/polarity does not come into picture—the charge of each elementary EC object always remains with it and its quantity too remains completely unaffected by anything or any action in the universe.

The electrical charge is just as inseparable an attribute of an EC object as its mass is. As an EC object moves in space, so does its charge too. An electrical charge cannot exist at any spatial location other than that of the massive EC object which possesses it.

Just as in NM, the EC objects too can be abstractly seen as if all their mass, and now charge too, were to be concentrated at a single point. We call such EC objects the point-charges.

3. Electrostatic forces between point-charges—Coulomb’s law:

Coulomb’s law is an empirically derived quantitative relationship. There is no theoretical basis beyond the fact that such behaviour was actually observed to occur in carefully conducted experiments. The statement of the law, however, is mathematically sufficiently refined that it would be easy to suspect whether it was not derived from some other a priori basis. As a matter of fact, it was not. What does Coulomb’s law describe?

Consider an isolated system of two point-charges fixed in space at some finite distance apart. [Help yourself by drawing a free-body diagram, complete with the structural support symbols for each of the charged bodies too. No, these supports, though they look like the electrical ground, are actually mechanics symbols; they don’t discharge the charges by grounding.] It is experimentally found that both the charged bodies experience forces of certain magnitudes and directions as given by Coulomb’s law.

Let the electrical charges of the two point-charges (EC massive point-particles) be q_1 and q_2, and let their positions be \vec{r}_1 and \vec{r}_2.

Let the separation vector going from the first point-charge to the second be given by \vec{r}_{12} = \vec{r}_2 - \vec{r}_1. (There is no typo in the last equation.) Similarly, let the separation vector going from the second point-charge to the first be given as \vec{r}_{21} = \vec{r}_1 - \vec{r}_2. (Again, no typo.)

[Note, a separation vector measures the difference in the two vector positions of two different bodies at the same time, whereas a displacement vector measures the difference in the two vector positions of the same body at two different times. … There is another related idea: The variation in position is a vector that measures the difference in the two vector positions of the same body, without reference to motion (and hence time), but as imagined in two possible and different configurations within two description of the same system. We will not need it here.]

Let \vec{F}_{12} be the force that q_1 exerts on q_2 at \vec{r}_2 (and nowhere else). Similarly, let \vec{F}_{21} be the force that q_2 exerts on q_1 at \vec{r}_1 (and nowhere else).

Coulomb’s law now states that:

\vec{F}_{12} = \dfrac{1}{4\,\pi\,\epsilon_0} \dfrac{q_1\,q_2}{r^2}\;\hat{r} = - \vec{F}_{21},
r = |\vec{r}_{12}| = |\vec{r}_2 - \vec{r}_1| = |\sqrt{\vec{r}_2 \cdot \vec{r}_1}| = |\vec{r}_1 - \vec{r}_2| = |\vec{r}_{21}|, and \hat{r} = \vec{r}_{12}/r = - \vec{r}_{21}/r.

A few points are especially noteworthy:

  1. The two electrical forces so produced are always equal in magnitude, regardless of how big or small q_1 and q_2 may be with respect to each other. Even if q_1 \ll q_2, the first object still ends up exerting just as much of an electrical force on the second object as the electrically much stronger q_2 exerts on the first.
  2. The forces appear on the two EC objects even though they are not in a direct physical contact—exactly as in Newtonian gravity. They also obey an inverse-square law, once again exactly as in Newtonian gravity.
  3. The forces don’t at all depend on the respective masses of the bodies; they are only for the electrostatic interaction.In reality, the two point-charges in the fixed configuration would also experience the force of gravity, not to mention the imaginary forces exerted on them by the mechanical supports. However, we ignore gravity in this entire description, and we exclude the support forces from our system—they are regarded as at all times belonging to the environment. (If a charge moves, so does its support, and along with it, the system–environment boundary. In fact a motion of a boundary is necessary for any system to exchange energy with its environment in the form of the mechanical work. We will need to take into account the mechanical work when it comes to defining the electrical potential energy of the system.)

4. Why the big ontological issue of Action-at-a-Distance remained unresolved before EM came on the scene:

The issue slipped under the carpet the last time (while discussing NM) now once again raises its (ugly or beautiful) head: How come two charges are able to exert forces on each other even if they are separated by nothing but the empty space, i.e., when there is no direct contact of the NM-kind between them?

People had thought a lot on this question right since the time that the issue came up in the context of gravity. In fact even Newton himself had once speculated whether an invisible string might not extend between two gravitating bodies like the earth and the Moon. However, Newton was only too well aware of the limitations of the available experimental evidence. The observational data such as that by Kepler indicated very high speeds for transmission of gravitational forces; the data were not refined enough to capture any effects of a finite but high speed for the forces. So, Newton refrained from adopting any definitive position concerning either a mechanism for the transmission. (Unlike Faraday, the poor fellow could not sprinke asteroids between the earth and the moon, and thereby grow confident enough about ideas like tubes of force or space-filling fields.)

The people from the other side—from the energetics approach to formulating mechanics—were not as deeply interested in the ontological matters anyway. In the market-place of ideas, their main market-differentiator was not a superior or more refined physics but a superior method of calculating solutions when the boundary conditions became complex, e.g., too numerous, as with continuous and curved constraints. (The physics implied by the mathematics of both the approaches—Newton’s momentum-based approach and the energy-based approach—was exactly identical. What differed between the two was mathematics—the methods of calculations.)

So, there arose a feedback circle of sorts: people who didn’t care about ontology and foundations of physics, but wanted to do maths, got attracted to the energetics program; people who already were in the program directed their energy in pursuing their strong point further. So, what they kept on developing was maths. Given this feedback circle, any ontological problem concerning the action-at-a-distance couldn’t have benefitted from them anyway. In actuality, it didn’t.

That’s why even if a term for the gravitational potential energy V had appeared as early as ~1773 in Lagrange’s writing (which was devoid of not just ontology but also of even a single geometrical diagram), and even if luminaries like Laplace (~1799), Gauss (about the same time but published in 1840) and Green (1828) happily developed the potential theory for gravitation, they all were perfectly happy working with just “a mathematical function” of “coordinates” for V—not with an actually existing physical field. It was Green who described V as the “potential function”.

Of course, these physicists couldn’t have used the term “potential energy” for V. The discovery of the first law of thermodynamics and the conceptual clarity on what precisely the concept of energy itself meant, was still only in latency; the explicit identification was at least 20–30 years away.

In any case, the mathematically oriented physicists on the continent didn’t pursue the issue of the physical meaning of potential energy a lot. The quizzical end-result was this: There wasn’t just an equation for the gravitational (and later electrostatic) potential energy of a system, there also was this distinctly further development of an equation for the gravitational potential of a single gravitating body. Thus, the Continental physicists had succeeded in mathematically isolating the interacting system of two charges into components specific to single charges: the potential field due to each, taken in isolation. The potential energy of the system could be found by taking the potential of either charge and multiplying it with the other charge. They were really advanced in maths. They successfully manipulated equations and predicted results. But they didn’t know (or much care) about what ontology their concepts or procedures suggested.

That’s why the ontological issue concerning the action-at-a-distance remained unresolved.

5. Faraday’s lines of force and Maxwell’s fields:

Historically, the crucial step in developing the idea of the field was taken by Faraday. I will not go into the details simply because they are so well known: sprinkling of iron filings near magnets, tubes of force (called “lines” of force by Faraday himself), the broad laws governing their behaviour (attractive force goes with tension, repulsive with sideways pressure), etc.

What is important is to note that Faraday did explicitly advocate the idea that the lines of force physically existed; they weren’t just a device of calculations the way the gravitational potential was to the Continentals. At the same time, he also believed that even the apparently empty space in between two lines of force was also filled with more such lines, that they filled the entire 3D space.

Maxwell got impressed by Faraday’s idea. He even met Faraday, and then, set out to translate the idea into more precise mathematical terms. He arrived at his mathematical description of the electromagnetic phenomena by imagining not just the tubes of force but even many further mechanical mechanisms, invented by him, in order to give a mechanical explanation of the known EM laws (Coulomb, Biot-Savart, Ampere, Faraday, and others.) Eventually, the system of explanations of EM phenomena using mechanical means (essentially, the NM-objects) became too complex. So, in his final synthesis, he simply dropped these complex details, but retained only the differential equations.

However, importantly, he continued to keep the abstraction of the fields in his final synthesis too.

The idea of the field solved the problem of action at a distance. Maxwell put forth the idea that all of space (except for the regions where charged objects are present) is filled with a physically existing field. Charged bodies are in continuous contact with the field, and therefore, are able to induce a condition of force in space—which is the field of force. Maxwell imagined that the force field consists of non-uniform mechanical stresses and strains. (He used the term “displacements” for these mechanical strains; hence the term “displacement current,” which is still in use.) When the same field comes in contact with some other charge, it experiences a net force due to the presence of these mechanical stresses at its boundary. That’s how the forces get transmitted.

Maxwell was smart. What he started out (or invented as he went along) were rather complicated physical mechanisms (all of which were made from NM-objects). But he had a definite sense of which point-quantities to abstract away, using what kind of limiting arguments, and how. Thus showing a refined and mathematically informed judgment, he arrived at an essentialized description of all the electromagnetic phenomena in terms of point-properties of a continuum.

6. The ontological view implied by Maxwell’s ideas:

In Maxwell’s view, the entire universe could be analyzed in purely mechanical terms. This means that all the objects he employed in his synthesis were essentially only NM-objects. These came in two types: (i) point-charges, and (ii) a mechanical continuum for the field.

The two interacted (exchanged forces) using the only mode that NM-objects were allowed: using direct physical contact. The interaction proceeded both ways: from charges to the continuum and from continuum to charges.

The point-charges pressed forces on the continuum of the field at their common boundary, which resulted in there being a mechanical field of stresses and strains inside it. These stress-strain states extended everywhere in the continuum, “up to” infinity.

The continuum, in term, generated forces on the surfaces of the boundary between itself and any charged object embedded in it.

The specific stress-strain field generated by a charge was spherically symmetric around that charge. Hence, it didn’t result into any net force acting on the same charge. However, due to the inverse-square nature of Coulomb’s law, and the fact that charges were point-particles, the field they generated was necessarily non-uniform at all other points. Hence, the continuum did generate a net force on the other charges.

Ontologically, there was little difference between Faraday meant (or strived to indicate), and what Maxwell directly put forth, using mathematical concepts.

7. The basic weakness of what precisely Faraday, and also Maxwell, meant by a field—its ontology:

Maxwell’s proposal of fields had a very great virtue, and a very great conceptual (ontological) weakness.

The virtue was an advocacy of a physically existing condition in what earlier was regarded as completely empty space. This condition was identified with the mathematically defined fields. The idea of fields was not just satisfactory from the viewpoint of broad philosophic principles (we will touch on them in a short while), it also gave a reasonable-sounding solution to the physics problem of action-at-a-distance.

The weakness was that the field, even if defined very carefully (in direct reference to empirically observed electromagnetic laws, and with rigorous mathematical abstraction), still was characterized, explained, and defended as a specifically mechanical kind of a physical existent. To describe the ontology of such fields using our scheme and notation, these EM fields were “pure” NM-objects.[Professional physicists often don’t have very good ontological clarity, but they are referring to the same underlying physical fact when they say that a mechanical aether provides a reference frame that obeys the Galilean-invariance but not the Lorentz-invariance.]

This particular weakness immediately led to conceptual challenges for Maxwell. Eventually, it also led to a lasting confusion for all, a confusion that persists till date (at least in the discussion of EM and aether).

The difficulties posed by the weakness were actually insurmountable. Here is one example. If the field had to be mechanical in nature as Maxwell said, then it would have to possess seemingly impossible combinations of physical properties. It would have to be an infinitely rigid object, and yet allow other massive objects (of the NM-object kind) to pass through them without hindrance. Et cetera. For an interesting history of how creative solutions were sought, and even were supplied see [TBD]. (A candidate explanation: The field acts like a metal ball placed on a block of snow: The ball passes through the block’s thickness even while keeping the block solid everywhere else, but the ball gets reflected when thrown with a sufficiently high speed. Another example I can think of: a jet-plane (say in a tail-pin) that hits the ocean surface. When the speed is great, it first rebounds as if it had hit a solid surface, rather than sinking in the water as it eventually does when it loses speed.) However, such explanations did not bear out—no mechanism would if it produces a mechanical aether in the end.

If Maxwell’s theory were to be less successful, this weakness would be less consequential. However, this was not the case. Maxwell’s was one the most fundamental and most successful theories of his time. (It was what QM and Relativity are to us.) Since the weakness rode on a very strong theory, it irretrivably put people in the mindset that all fields, including the luminiferous aether (which transmits Hertzian waves through it), had to be necessarily mechanical in nature.

8. The consequences of the conceptual weakness of Maxwell’s idea of the field—its ontology:

The weakness had to come out sooner or later, and in the 19th century, it came out rather sooner.

In particular, if the aether was to be space-filling, then the earth would have to move through it during its elliptical motion around the Sun. If this aether was mechanical in nature, then the interaction of the earth with the aether would have to be mechanical in nature. It would be analogous to the motion of a finger through the tub-water. The aether would exert a drag force on the earth’s motion, which could be detected via light interference experiments.

However, experiments such as those by Michaelson-Morely showed that such a drag was not actually present. This momentous experimental finding led to the following situation.

The physicists and mathematicians of the energetics program (mostly on the continental Europe) had labored for almost 70 years to develop the mathematics of fields, but without regarding it as a physical entity. But they had produced excellent mathematics which greatly clarified presentation of physics and simplified calculations. Even Maxwell’s theory had its mathematics developed on the basis of these developments.

The aether-induced drag was only a deductive inference made from Maxwell’s theory. Maxwell’s theory itself was founded on very well established experimental findings. Another deductive inference, namely that light was an EM wave with the speed of c, had been borne out by experiment too. So, all these aspects had to be kept intact.

But the drag implied by Maxwell’s aether was not to be found in the experiments. So, this part of Maxwell’s theory had to be corrected for. The simplest way to do that was to drop the whole idea of the aether from the theory!

After all, in his own development, Maxwell himself had started out with a laundry list of different kinds of physical mechanisms for different aspects of electromagnetics. However, eventually, he himself had come to drop all these mechanical features, because it was hard to get all these mechanisms to work together in a simple manner. So, he had instead decided to abstract out just his mechanical field from them. This field was then identified with the luminiferous aether.

But the aether-wind gave problem. So, why not take just another step of abstraction, and entirely do away with the very idea of the aether itself? Why not regard it as just a mathematical entity? Why can’t space once again be completely empty of any physical being, just the way it had been right since Newton’s times—and even during the entire development of the potential theory and all?

If the situation is to be framed as above, then there can be only one logical way out of it. The physicists came to choose precisely that. Without challenging the specifically mechanical nature of the aether (because no one could think of any other kind of a nature for a physical aether, since none could figure out any good philosophical arguments for having a non-mechanical aether), physicists in the late 19th- and early 20-th century simply decided to remove this whole idea from physics.

In the meanwhile, Einstein was advocating a denial of the absolute space and absolute time anyway. If the space itself was not absolute but depended on the relative motion of the observer, i.e., if space itself depended on motion, then was there any point in filling it with anything?

For the rest, pick up virtually any of the hundreds of the pop-sci books on the relativity theory and/or on Einstein’s (IMO at least in part undue) glorification. … If Maxwell to be less successful as a physicist, his wrong ontological views would be much less consequential. Ditto, for Einstein.

9. A preview of the things to be covered (concerning the EM ontology):

The essential error, to repeat, was to conclude that since Maxwell was successful, and since his field was mechanical, therefore every field has to be taken as being mechanical—i.e., as if it had to follow the NM ontology. Wrong.

Next time, we will cover the correct ontological view to be taken of the Maxwellian fields. We will also look into a few issues about ideas like the electrostatic potential, the electrostatic potential energy, and their fields. We look into the details of these concepts only because they are relevant from an ontological point of view. [And yes, there is a difference between just potential and potential energy—just in case you had forgotten it.] This discussion will also help us prepare for the correct ontological view which is to be adopted when it comes to the quantum mechanical phenomena. Hopefully, the whole portion would be over in two more posts, at most three.

See you next time. Bye for now, and take care…

A song I like:

(Hindi) “o sanam, tere ho gaye hum…”
Singers: Lata Mangeshkar and Mohmmed Rafi
Music: Shankar-Jaikishen
Lyrics: Shailendra