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One Christian's Thoughts on Relativity and Quantum Mechanics

By Dr. Win Corduan

This series is currently in progress!
It will be seriously edited and revised when all the entries on the blog are done.

I started this series in the early summer of 2011, but discontinued it at the time for a number of reasons. For one thing, I was totally overcommitted and needed to keep my focus in particular on the manuscript for In the Beginning God. I really did not have the time that writing on this topic necessitated. Researching the religions of the Australian Aborigines did not much to help me with clarifying, say, the Pauli Exclusion Principle. In addition, I ran into some material that led me to rethink how I should present some of the philosophical ideas. As usual, I'm making some superficial changes in a few places, and I hope that I have corrected the mistakes that people found in a few places. If not, please remind me again.

On this site I am currently adding the various entries together without the editing. There is far more repetition and review for a stand-alone site, and some of the items should probably be sequenced better. For now, I will leave them as they come; it'll make a whole lot more sense to rearrange once it's all done.

The impetus to write on this topic has come from several places. For one thing, several people wrote to me wondering about my opinion on whether the "new" conclusions of quantum mechanics and Einsteinian relativity undercut the Christian belief that God created an orderly universe, based on cause and effect, with God being the Ultimate Cause of everything while all other beings exist and act in causal depenence on him. In fact, some non-Christian thinkers have attempted to make precisely that argument, ill-founded as it is, against theism. Simultaneously, some Christian apologists have addressed these matters with arguments that regretfully betray a lack of understanding of quantum mechanics and consequently have provided some conspicuously inadequate responses. In addition to the general concerns, I have been confronted several times with the specific contention that contemporary physics has provided empirical proof of the the existence of uncaused contingent beings. Finally, just around the time that I thought I should start on such a series, someone referred me to a really interesting article on the ongoing debate whether physics and philosophy have anything to do with each other. At that point, I started to add reflections on these issues in science to my blog, though, as I mentioned, I couldn't sustain doing so at the time.

I'm going to moderate the way that I stated my goals for this series, compared to when I first started it on the blog. When we are done, you will not be able to solve Schroedinger's equation or have achieved mastery of tensor algebra, the math that Einstein used to quantify his theory of relativity. Actually I never promised that; it's never been the goal. I am hoping that by the end you will have a basic understanding of the general principles behind quantum mechanics and relativity, and that you will find that they do not undercut a theistic view of the world, even though some people have claimed that they so. For the most part, such detrimental uses of the scientific conclusions are based on some unscientific generalizations that resort to philosophy, not scientific observation and data, and usually they do so badly. And, of course, if they do not pull the rug out from underneath theism, by implication they also do not do so for specific theistic religions, such as Judaism, Christianity, or Islam.

I'm hoping to be able to bolster my contention with a few details. However, if all you need is some positive words of reassurance, and you don't want to read a whole lot more of that "science stuff," there you go.

Let me add that, even though I am writing from a Christian perspective, unless at some point I signal a change, what follows assumes an audience grounded to a certain degree in a Christian world view. Thus, this is not an apologetic directly aimed at non-Christians, but an aid to help Christians see a little more of the relationship between their faith and science. There is, of course, an apologetic edge to the entire series simply because some of the points I am making stand in contrast to the contentions of some non-Christian thinkers, but they are not my primary target.


God created the world. God did not create science. Science is a human attempt to understand the world that God created. Science discovers the laws of nature. Did God create the laws of nature? No, not in a straightforward sense. Recognizing the "laws" are also human efforts of making sense of God's world. However, God created the world to which these laws apply, and the better our description of this world is, the closer we get to the nature of the world God has created. So, how good are our descriptions, and how close are we to understanding God's creation?

Apparently towards the end of the nineteenth century the opinion had become fairly popular that the scientific discipline of physics had attained pretty much the end point of what could be known, and there were not much chance of any new major discovery in physic. Isaac Newton had laid down the ground rules, and all that was left to be done was to fill in a few holes. Alexander Pope's intended epitaph of Newton was extremely popular, at least in the mindset that lay behind it.

Nature and Nature's laws lay hid in night:
God said, "Let Newton be!" and all was light.

If one wanted to explore new territory, one should probably go into another field other than physics, though, in the opinion of some people, other fields--such as Chemistry and Biology--were merely areas that needed to be placed under the magnifying glass of physics, and by such a reduction of all knowledge to physics the total body of scientific knowledge would constitute a completed set.

What about other areas that could not be reduced to a physical description? Remember that this is the time when Ernst Mach introduced the ideas that would eventually result in the logical positivism made famous by the so-called Vienna Circle, which decreed that all non-scientific beliefs (i.e. those not empirically verifiable) were meaningless. Also, it was around this time that philosophers such as Frege and Russell attempted to derive mathematics from a minimal set of logical principles. The idea that scientific knowledge was on the brink of being almost "finished" may seem pretty bizarre to us; but we are reading or writing about that era with a century or more of hindsight. It does not allow us to think that we are necessarily smarter than those folks.

Still, they were wrong.

The world that God has created has turned out to be far more complex and interesting than Newtonian physics could possibly describe.

Let me tell you one reason why Newton's physics could not possibly be as self-contained as many people thought in the nineteenth century. For several centuries it had become tied to some dubious, though questionable, philosophical presuppositions. We must take a quick step backwards in time to see what happened.

Whether you agree with him or not, there can hardly be a question that the integrated system erected by St. Thomas Aquinas represented the high point of scholastic thought in the Middle Ages. He gave the world a Christian scheme that relied on both Platonic and Aristotelian philosophy. But even aside from Aquinas, the philosophy of the thirteenth century was characterized by complexity and profundity. For example, long before Kant and Hegel, Aquinas modified Aristotle's philosophy in the course of analyzing the intricate nature of knowledge, which took into account the contribution of logical principles by the knowing subject.

However, towards the very end of the Middle Ages, philosophy went into a tailspin. The nominalists, headed up byWilliam of Occam, led Western philosophy into a time of skepticism, which caused grave philosophical and theological implications (and was a factor that negatively contributed to the coming of the Reformation). When a more assertive form of philosophy reemerged, it was exceedingly primitive, showing none of the sophistication one had found in Aquinas, Scotus, or Bonaventure. Unfortunately, the naive concepts of modern philosophy (from the 17th century on) have stayed with us, and for many people they have become common sense, so that they immediately read Newtonian concepts into any attempts at describing the world. Still, I must return to the thought expressed above, God created the world, but he did not create Newtonian physics.

Now, here's my first major point. A common statement in popular books is that those of us who are not physicists don't have to worry about quantum mechanics or relativity at all, and those who are working in the fields need to do so only when they are in the world of science. On the level of our everyday lives, they say, we can safely stick with a Newtonian conception of reality because the anomalies that gave rise to these theories in knowledge do not alter anything on the level on which we lead our lives. Even if we found these penomena by means of high-powered measurements, the difference between the Neutonian and the quantum-based results would be minimal, so that we can safely ignore them.

My contention is that to do so is already a mistake. I'm not talking about Newton's laws of physics; they appear to hold up quite well in most "common" situations though actually most of us don't actually live on the basis of Newtonian physics any more than on quantum physics, both of which are founded on meaurements, and we don't do all that much measuring or applying equations as we pass gently through our days. We may get a rough and ready experience of the transfer of kinetic energy when we play baseball or make use of the fact that for every action there is an equal and opposite reaction when we row a boat, but those experiences are not really "Newtonian" since we actually undertake no measurements in the process. If we did, we would discover that the Newtonian laws do, in fact, cover such actions. However, the philosophical presuppositions that undergird Newton's physics are inadequate and have always been inadequate. I don't know whether we should ever need to get worried about how quantum physics or Einsteinian relativity affect our daily lives, but the concept of causality as it has functioned in much of modern philosophy and Newtonian physics is highly problematic.

Let me illustrate what I'm trying to get at by citing from Amir D. Aczel, Entanglement: The Greatest Mystery of Physics (New York: Four Walls Eight Windows, 2001), p. 12:

Newton, building on the foundations laid by Descartes, Galileo, Kepler, and Copernicus, gave the world classical mechanics, and, through it, the concept of causality. ... Newton's laws are a statement about causality. They deal with cause and effect. If we know the initial position and velocity of a massive body, and we know the force acting on it and the force's direction, then we should be able to determine a final outcome: where will the body be at a later point in time.  (Emphasis his)

Newton did what? He gave the world causality? What a bizarre statement! I know that for many people medieval philosphy is not worth a second look, and classical Greek philosphy constitutes a collection of artifacts in the museum of archaic thought, but this assertion surely goes over the proverbial top.

Don't get me wrong. I enjoy Dr. Aczel's books, including this one, and, for what it's worth, I also recommend his Fermat's Last Theorem. Still, the assertion that Newton gave the world the concept of causality is more than a little overstated. And it is precisely the illusion that Newton's understanding of causality had become the single authoritative one that is responsible for many of the apparent conundrums posed by modern physics. Again and again I hear or read people asserting that modern physics has done away with causality. It may have buried the Newtonian view of causality, which is alright with me, but it did not bury causality.

Obviously, other people prior to Newton, including philosophers and scientists, had a concept of causality prior to Newton. In fact, it was a concept far richer and more encompassing than Newton's. What Newton gave the world, or at least reinforced with his laws, was a particular notion of causality, one that was significantly narrower than it should have been, but provided for the possibility of measurement and the establishment of mathematical formulas. Its basic nature is, to reduce it to its most elementary imagery, one of pushing and pulling. In order for there to be causal interaction between two objects, they have to be in close enough contact with each other so that they can exercise forces on each other, and that means either pulling or pushing according to quantifiable rules. Apply a few measurements, and we can express the relationship between two objects by filling in the parameters of the appropriate formula. Such a formula embodies a concept of "cause and effect," in this limited sense.

My point is this: True enough, pushing and pulling, or the interaction of forces, such as gravity or electromagnetic attraction and repulsion, are instances of causality. But where we go wrong is if we limit causality to just that expression. Causality is broader than that, as Werner Heisenberg, discoverer of the so-called Heisenberg uncertainty principle, himsef acknowledged. He argued that, even though we may not necessarily want to go to Aristotelian philosophy as a whole, it would be a good idea to go back to the notion of causality as the actualization of a potential. We need to return to that idea eventually. In the meantime, if it should be maintained that Newton really gave the world causality per se here are some quick counterexamples to his model. I can cause my wife to smile by doing or saying certain things, and I beg you, please, not to interpret such an instance along the line of molecular interactions. Even if you could do so, it would be arbitrary to say that such things on an atomic level are what was really going on. That would be an odd view of reality, one that we certainly don't live by. Furthermore, even though some people do so, it is entirely arbitrary to say that the atomic or subatomic events are what is "real," rather than the expererience of the event on the level of unaided human observation. Or, to cite another example, when we say that the downturn in the economy caused a lot of unemployment, there's certainly no pushing and pulling between objects going on.

So, as we begin to look at some things that don't seem to fit our everyday thinking about the world, a good way to start making sense of some strange phenomena is to realize that some of the concepts with which we approach the world may be too limited. That's not going to make all the strange stuff immediately comprehensible; it doesn't dissolve the paradoxes engendered by relativity or give us certainty where Heisenberg was uncertain. However, it should help us focus on the issues rather than being boxed in by unnecessary conceptual limits that are as unnecessary as they are restrictive. We'll get to those things as we move along and look at them more closely, but we can't look at them at all if we don't allow ourselves to break out of self-imposed limits.

God caused the world to exist. God caused the world to contain entities that are themselves causes, producing further effects. God caused the world to contain effects that go beyond a Newtonian understanding of causality.

Please let me clarify that I'm not trying to do anything as absurd as minimizing the great Isaac Newton's accomplishments. Insofar as the above discussion entails a critique it is directed to the philosophical concept he inherited and applied, and even more so to the idea that his use of them is ipso facto definitive for all ages.

And we're off . . .


Physics LogoLast time, as I began this series, I tried to make the point that the understanding of causality, as it emerged in modern (post-Cartesian) philosophy and as it was implemented by Isaac Newton, was a somewhat truncated one. Essentially, it consisted of "objects" pushing and pulling on each other by means of "forces." No wonder that Hume was skeptical of the empirical perceptibility of such a thing (and just being able to describe it with a mathematical formula does not make it more visible). I think it's telling for Newton's view of the world that he advocated the understanding of light as particles (photons--little pieces of material), as opposed to waves, and that, even in the development of his version of the calculus, he visualized the process of leading up to it as the movement of tiny little particles (fluxions) along the line described by a curve.

I suggested that, instead, we need to have a more open, but also more realistic understanding of causality, namely that a cause is an entity that actualizes some potential. I realize that this definition is more vague than "pushing and pulling" by means of forces, but it also does more justice to our actual use of the expression x caused  y:  was non-existent, but potentially existent. Then x brought y into existence as its cause. This definition includes applications in Newtonian physics, but does not shut the door on other situations.  As I keep saying, I don't think that we ought to try doing metaphysics without doing metaphysics, but if it seems more suitable, for now we can stay with the definition that x is a necessary condition and at least is a member of the set of sufficient conditions for y, as long as we don't limit "necessity" to logical entailment, but include factually unavoidable conditions, as based on observation or experience. In other words, we limit ourselves to alpha, the actual world. Neither logically possible worlds in which the laws of the universe are different from the ones we know, nor so-called alternative universes fit into this definition.

I'm still speaking in general philosophical terms. I will explain the following physical phenomena later on, but I really want you to see the difference between a Newtonian understanding of causality and a broader view. Many people are aware that Albert Einstein distanced himself from quantum physics, more specifically from the "Copenhagen School" led by Niels Bohr. Did you know that Einstein actually received his Nobel prize, not for either theory of relativity, but for his contribution to quantum mechanics in which he posited light as consisting of particles (good old Newton's photons rediviva)? Subsequently he abjured the newer trends in quantum theory, especially as espoused by Niels Bohr et. al., based on his judgment that their theory was incomplete. But, it only appeared to be incomplete for him, I can say with confidence, because it did not meet Newtonian criteria of causality. In short, and it is really bizarre to say this, When it came to atomic physics, Einstein was Newtonion at heart.

Before giving you the reasons for my claim, let me re-emphasize that the point that I'm trying to make is simply to highlight the difference between coming at certain phenomena with a Newtonian world view or with a broader one. There is no further polemic intended, and I'm not even making any particular truth claims with regard to the physical phenomena at this point

So, now let me try to explain what I mean by Einstein having taken a Newtonian view. I'm going to refer to the phenomenon known as "entanglement," the subject of the book by Amir Aczel that I mentioned above. "Entanglement" is the physics behind the imaginary idea of the "quantum computer," which plays a significant role in Michael Crichton's Timeline (New York: Ballentine, 2003), as well as my little piece of fiction The Absence of the Bloggist. IBM announced recently that they think they will have a functional quantum computer ready in ten years. We'll see.

The fundamental idea is this: Imagine that you have an electronic gun that shoots out one pair of sub-atomic particles. Because they leave together, they are "entangled" with each other.  Each particle could have certain properties out of a set {{A or B} & {C or D}}. Thus, a particle could have the properties: A & C, A & D, B & C, or B & D.


However, when first emitted, both of these particles will be in the state that is called "superposition," which means that until someone has actually measured the properties of a particle, it acts as though it had all of the available properties, even though they may be mutually exclusive. This is weird stuff, and, as I said, I'll try to describe it better later. For now, we just need to realize that, when we use our particle gun to shoot out a pair of particles, both of them are in the state of superposition; both demonstrate all four subsets of properties. So, now we select particle 1 and measure it. We keep track of its properties and check particle 2. Its properties will immediately show up to be equal and opposite to particle 1.

Entangled Particles

Clearly, under a Newtonian paradigm our action of analyzing particle 1 must have released some kind of force that affected particle 2 and told it which properties it should adopt. Well, we can test that: Let us say that we create such a large distance between them that we can rule out any communication by any conceivable Entanglementforce, gravitational, electromagnetic, nuclear, whatever, even to the point where the proposed communication between the two particles would have to be faster than the speed of light. We set up our apparatus so that at precisely the time we measure the properties of one particle, the other one is a football field's length away. The same phenomenon still occurs. Once we measure the properties of one particle, the properties of the other particle immediately become its equal and opposite. So, how can the properties of one particle influence the properties of another particle a hundred yards away? That would be a trick question because quantum mechanics, according to Bohr's interpretation recognizes no "influence" and, aside from personal interpretations, an influence with a velocity in excess of the speed of light is impossible anyway. It just seems to be the nature of particles to come out that way. If one is A, the other is B. If one has a clockwise spin, the other one's spin is counterclockwise. Even though neither one had either property prior to the measurement (or perhaps both), once you've determined the properties of one, the other one instantaneously must have the opposite properties.

Nevertheless, seen from a Newtonian point of view, there must have been some way in which, say, particle 1, communicated with particle 2, so that particle 2 could know which properties particle 1 possesses and take on the opposite properties. But no such factor is known.

Albert Einstein and Niels BohrI've been writing about this phenomenon as though it were based on experimental observation. Actually, that's not how it first came up. (See Aczel, Entanglement, pp. 111-121, which is also the continuing source for some of the information below). Albert Einstein brought up entanglement in an article published in 1934 (co-authored with Nathan Rosen and Boris Podolsky) merely as a thought experiment involving quite complex and apparently flawless mathematics on how the wave functions of the two particles would become entangled at the outset prior to observations, thus laying claim to the presence of a communication factor (the wave functions) at the outset. On the other hand, he asserted that that, if the Copenhagen version of quantum mechanics were true, then entanglement would be a definite result. However, entanglement is not physically possible because it would involve what he called "a spooky action at a distance," which is to say a causal influence of one particle on another particle without any physical force between the two particles. This was clearly impossible, and so Einstein, along with some other notable physicists, wound up parting theoretical ways with the main stream of quantum mechanics. He considered the definitive ("Copenhagen") version of quantum mechanics, if not quantum mechanics altogether, to bear the seeds of its own refutation. 

The big names in quantum physics, Werner Heisenberg, Erwin Schroedinger, Wolfgang Pauli, and others were furious at Einstein--probably more at the fact that he wrote against their theory than at what he wrote--though they couldn't really refute it. Niels Bohr went into a tizzy trying to find a way to prove Einstein false, but struggled helplessly. He eventually announced that the article was irrelevant since it had no experimental application.

Einstein was right. Entanglement turned out to be a definite consequence of quantum theory. But Einstein was also wrong because it did not falsify quantum theory. Decades after he and Bohr debated the issue, entanglement was experimentally verified at distances of several miles. Quantum mechanics was vindicated because what Einstein thought of as a reductio ad absurdum turned out to be physically real.

When it came to science, Einstein was a materialistic determinist. Notwithstanding occasional references to God (such as "God does not play dice"), his theology was a deistic one at best: God was the creator who started the clock of the universe running. Now, I'm not saying that this world view is worse or better than the agnosticism and skepticism expressed by many of his colleagues, but we need to realize that such was his approach, and that it figured in his response to later quantum mechanics.

And then we need to make sure that we don't buy into the same paradigm. The idea that all there is to the universe is a collection of objects and the forces they generate, which push and pull at each other, just doesn't exhaust all that happens in the universe.

I'm reminded of an argument made by Kai Nielsen, a leading atheist of the previous generation. I guess that, inconrast to the current "new atheists," he should be considered an "old atheist." His over-all case didn't hold in the final analysis, but, in contrast to the contemporary version, he was a rigorous philosopher who presented real arguments with which one could actually interact. Nielsen (Introduction to Philosophy of Religion, New York: St. Martin's, 1982, pp. 17-42), following the canons of analytic philosophy, tried to show that the concept of God is not meaningful. His basic argument was that God is described as both incorporeal and as acting in the world. But how can a non-corporeal being possibly carry out actions in the material world? According to Nielsen, the only experience of action that human beings have is that of one material being acting ("pushing or pulling") on another material being. Actions by a non-corporeal being on material objects are beyond our experience and are, therefore, incomprehensible and self-refuting. But, since the concept of God is intimately tied to his acting in the world, the concept of God must also be intrinsically incoherent.

A response to Nielsen on this point is fairly easy and straightforward. His idea of the unintelligibility of actions by an immaterial being, to which he appeals, is obviously gratuitous. He is not observing that it is unintelligible, but decreeing that it should be so. Millions of people seem to find it comprehensible, for example when they pray. Thus, such an argument based on the meaningfulness of words is based on an assumption that cannot be sustained.

Now, I'm certainly not saying that quantum mechanics proposes divine action as an alternative explanation for entanglement. But the phenomena it presents to us challenge us to back off from an unexamined materialistic and deterministic understanding of the world.

Next time: So, what is a quantum anyway?


Physics Logo

How can one have specifically "Christian" thoughts on twentieth-century physics? Isn't science supposed to be value-neutral? --- Oh please! --- Science can never be separated from the world view of the people pursuing it. It makes all the difference whether one believes that there is a Creator behind the phenomena of the universe or not. In the latter case the exploration of the universe can only be an exercise in futility when all is over. To quote from Bertrand Russell's A Free Man's Worship:

    Brief and powerless is man’s life; on him and all his race the slow, sure doom falls pitiless and dark.  Blind to good and evil, reckless of destruction, omnipotent matter rolls on its relentless way; for man, condemned today to lose his dearest, tomorrow himself to pass through the gate of darkness, it remains only to cherish, ere yet the blow fall, the lofty thoughts that ennoble his little day . . .  (emphasis mine)

You can find this essay in many collections and anthologies. If you don't have time to look for it and check up the context of the quote, let me assure you that it does not get any more cheerful either before or after this excerpt.

So, I promised you that I would tell you in the next entry what a "quantum" is, and this would be the next entry. Now, I need to ask you to forget for a moment all of the paradoxes and mysteries that usually come to mind if you're familiar with anything about quantum physics at all. Those are merely items that followed eventually from a very straight-forward phenomenon. So, to misquote Mrs. Misaprop, "Illiterate them from your mind, I say." At least for the moment.

Please think of the difference between going up a ramp or a stair case. If you go up a ramp, there is going to be a direct correlation between the amount of energy you expend and how far up you have gotten, and you can stop anywhere along the way. On the other hand, if you walk up a stair case, you have to make progress in increments, depending on the height of the steps. If one step is ten inches above the previous one, and you want to get to the top, your next step has to consume enough energy to go up another ten inches. Any less effort will be futile, and you will remain on your present step. The same thing is true if we contrast walking down a ramp or a stairway. On a ramp, you can shuffle down by moving forward one inch at a time. If you try that on a staircase, you're most likely going to wind up making a much quicker and less orderly descent than you had planned on.

Ramp and Stair Case

Now, I really get impatient with books that are punctuated by words like "clearly," "obviously," or "simply." Nevertheless, I'm going to risk it this time: The fundamental pointpoint of quantum mechanics is fairly simple. On the atomic level certain processes can only occur in steps or stages. In this context (at least) when energy is added to a system or a system releases energy, the process works more like a staircase than a ramp. Energy comes in stair steps, and one step is one quantum. So, a quantum is the minimum amount of energy necessary to raise a system from one step to the next higher step; in reverse it is the amount of energy released when a system moves down in energy by one step. Another way of looking at this is to think of the energy released when a change occurs on the subatomic level (either adding or emitting), the energy has to be bundled in packets of adequate size. Let us say that it takes one packet (= 1 quantum) to move from step 1 to step 2. Half a packet will not do it. A full packet will take us there. A packet and a half won't take us any further than the single packet would have.

Theoretically, if the radiation of energy worked like a ramp, there would be no limit to how much radiation you can create. Let's say that you're sitting cozily in front of your fire place, watching the red, orange, and yellow flames. The colors that you see are an indication that electromagnetic waves are being emitted by the fire, and some of them are in the visible range. From the heat that you feel you infer correctly that there is also radiation in the infrared range. But what about the other side of the spectrum?

Skip this paragraph if you hate numbers. A clarification: Electromagnetic waves come in different sizes, called wave lengths and most of them are not visible to us. The magnitude of a wavelength stands in an inverse proportion to its frequency of vibration; that is to say, the longer the wavelength, the slower the frequency and vice versa. If we arrange different types of electromagnetic radiation by the size of wavelength starting with the longest, at the top of the list we find the electric energy that we get out of our outlets in the U.S., which has a frequency of a mere 60 cycles per second and a wave length of approximately 5000 km or 3125 miles, a number that is close to the radius of the earth. At the very bottom are gamma-rays whose frequency can get at least as fast as 5 x 1024 cycles per second with a wave length possibly as short as 6 x 10-15 centimeters. That would be a 5 followed by 24 zeroes for the frequency and a 6 preceded by 14 zeroes after the decimal point for the wavelength. (5,000,000,000,000,000,000,000,000 cycles/sec and 0.0000000000000006 centimeters. You may convert the latter into inches if you care to.) 

Here is the hierarchy of wavelengths (ignoring some areas of overlap or imprecision in official nomenclature):

 Household Electricity > AM Radio > TV and FM Radio > Microwaves > Infrared > Visible Light > Ultra-violet > X-rays > Gamma-rays. [1]

So, now, you are thinking that, as of right now, there probably are no X-rays or gamma rays coming out of your fireplace, and it is questionable whether there would be any ultra-violet radiation. But, since you haven't studied contemporary physics, you think that if you were to increase the total energy, the radiation would move further and further to the right of the spectrum. So, you add a few logs to the fire. It gets hotter, and the flames get more intense. After letting the fire work itself up a little more, you see areas that are white hot alongside some blue flames. You add some more fuel; the fire gets increasingly blue. As you continue, colors in the blue and violet range begin to dominate, and you're starting to cross over into the ultra-violet zone. More logs! More energy! Now the ultraviolet is dominating; your fire place works like a "black light." But you don't stop. Lots more wood to increase the energy! Pile it on and watch it crawl further up the spectrum. Another huge load of logs later you have added enough energy that your fire place is beginning to give off x-rays. Your ambition knows not limit, and you don't stop. More wood! Lots more fuel! Finally, you have increased the energy to the point that your fire place radiates gamma rays. You have created a powerful weapon and are ready to use it to take over the earth and become king or queen of the world.

Max PlanckSorry for all the potential Dr. No's who may be reading this. It doesn't work that way. It is not possible to inject sufficient energy into a fire of, say, wooden logs, to generate more and more powerful forms of electromagnetic radiation. To be sure, it was a great puzzle at one time why a sufficiently heated body does not cross over the threshold so that--with a whole lot of more sophisticated equipment--the fire would phase over into ultra-violet radiation (the "ultra-violet catastrophe"). In that case something along the line of the fireplace scenario would have be theoretically conceivable. 

We have a German physicist named Max Planck to thank for recognizing around 1900 A.D. the nature of the limitation. It was he who discovered the boundaries imposed by the quanta. The amount of energy that is possible to derive from any particular system (your fireplace, an x-ray tube, a light bulb, etc.) is based on the nature of the system under consideration multiplied by a constant number, called, appropriately, Planck's constant.

    Skip this paragraph is you hate equations! We can create a formula in which the nature of the system is represented by the letter v and Planck's constant h, so that one quantum level for that system is described by multiplying h and v. Then you can write something along the line of Eq= vh. In the case of burning logs of wood in your fireplace, the quantum step between blue flames ultra-violet is  too large to be reached by burning wood, as expressed by the variable v.

The value of Planck's constant, by the way, is 6.6262 x 10
-34 joule-seconds (the number 6.6etc. preceded by 33 zeroes prior to the decimal place). That's a tiny little number for sure. How can that make a real difference? It can because the phenomenon we're talking about here takes place on the atomic level where everything is extremely small.

Now please notice that we're talking about something that surprised Max Planck himself as well as presumably all other physicists. Radiation is organized in quantum steps or packets. Very well.  But there doesn't seem to be any paradox here, just a new regularity that we have to take into account. Of course, now the question is what is happening on the atomic or subatomic level to cause the quantum delineation. This is where the mystery starts.

[1]  Kenneth R. Atkins, Physics, 2nd. ed. (New York: Wiley, 1970), p. 393. This is an ever-so-slightly newer version of the physics textbook I had in college, so some numbers may have changed since then.


Physics LogoAs promised, I am now going to pick up with the series on Christian thought and the so-called new developments in physics, many of which are well over 100 years old by now. In my last entry I provided a link to the first three installments in combined and somewhat revised form, and I highly recommend that, if you are interested in this topic and my thoughts on it, you may want to go back and read that material.

Still, in case you're not so inclined, here is a very brief and insufficient summary of the three main points I attempted to make up to now.

  1. We need to be aware of the rather limited understanding of causality, derived from modern philosophers such as René Descartes, which is intrinsic to Newtonian physics, in contrast to the broader, more inclusive, views of causality that arose in classical Greek philosophy and were expanded in the Middle Ages, particularly by St. Thomas Aquinas. The fact of the matter is that the transition from medieval philosophy to so-called modern philosophy during the time period that we usually refer to as the Renaissance constituted a reversion to a rather primitive understanding of knowledge and of human nature. As an example, the Cartesian notion of causality as "pushing and pulling," which was also utilized by Isaac Newton, is simply not adequate to describe the various phenomena that we would classify under the heading of "cause and effect." In short, if quantum mechanics were to imply the abrogation of the philosophy underlying the Newtonian vision of the universe (and I'm inclined to think that it has), I'm not sure that we have lost all that much.

  2. EntanglementIn the second entry, I used none other than Albert Einstein to demonstrate how pervasive the Newtonian view of causality actually is. Despite the fact that Einstein sparked a revolution in the way in which a scientifically minded person now needs to look at the world and how it functions, he still clung to the Newtonian view that a physical event must have a "pushing" or "pulling" cause. This mindset of his became clear in his disagreement with the leading exponents of quantum mechanics, such as Niels Bohr. Einstein contended that, if Bohr's version of quantum mechanics were true, "impossible" consequences would follow. Specifically he pointed to the phenomenon called "entanglement," which posits two subatomic particles emitted simultaneously but now outside of the range of imcommunication due to the limit imposed by the speed of light. Nevertheless, they still act as though they were communicating with each other. Einstein argued that such an uncaused event would be impossible, and that, therefore, the "Copenhagen" interpretation of quantum mechanics must be false. We'll get back to the example. My point for now is simply to show how even someone like Einstein, who certainly had no problem "thinking outside of the box," was still tied to a Newtonian understanding of cause and effect. He would not countenance the concept of what he called "spooky action at a distance," which he believed would be the logical outcome of quantum mechanics. At this point in time, entanglement has actually been demonstrated experimentally and has thereby confirmed the value of quantum mechanics as an appropriate theory.

  3. In the third installment I addressed the all-important question of "What is a quantum?" The idea of a quantum was first discovered by the German physicist Max Planck. He recognized that there was no one-to-one correspondence between the amount of energy one might infuse into a physical system and the increase in the systems level of energy. Raising the system's energy required injecting a clearly defined amount of energy, and raising it to an even higher level, would demand another full full dose. Amounts in between would have no effect, and amounts higher than what was required for the next step, but below what would constitute a two-step advance, would have no further effect than an increase of exactly one level. Furthermore, the same thing would be true if one sought to lower the level of energy within a system. The system would release a certain amount of energy, but it would also do so within the precise limits of the packets. The increase or decrease of the energy within a system must be viewed as a stairway, rather than as a ramp. The "step" is what is called a "quantum."

I need to, and I intend to, revisit most of what I have said so far with more detailed and hopefully more helpful explanations. However, I feel a great amount of urgency in addressing one particular claim that goes beyond the specifics of quantum mechanics, namely that modern physics has shown that there can be uncaused subatomic particles. I'm going to attempt to give a specific explanation without losing ourselves in details. Let's see if that will work. 

The two most common examples of this alleged possibility are the phenomenon described by the Heisenberg Uncertainty Principle and the observations made by scientists by means of particle accelerators (also known as "cyclotrons" or "atom smashers"). For now let's stick with the latter since getting involved with Heisenberg is going to take a whole lot longer than I can give to this entry today. [Speaking of cyclotrons, I recall the "atom smasher" in the lobby of the physics building at Rice University. It consisted of a small board and a wooden mallet. The instructions that came with it specified that one should take one good-sized atom, place it in the center of the board, and then come down on it vigorously with the hammer — or words to that effect.]

Particle AcceleratorWhat, you may ask, does a particle accelerator do? It accelerates the speed at which subatomic particles travel by constantly feeding them more energy. When I took physics in my undergraduate years at the University of Maryland, the new cyclotron on campus had just been finished, but was not yet operational, and so we were given a tour of it. Its main structure was a big hollow ring of sufficient size for a person to be able  walk in it upright. Electrons or other subatomic particles are shot into this ring, and they follow the path through this circular tunnel.

How do you move a subatomic particle? A lame answer is that whenever you move an object, you're moving all of the atoms and all of the particles that make it up, but that's not very interesting. Let me rephrase: How do you move subatomic particles without moving the entire atom? Actually, chances are good that you've seen one way of doing so hundreds, if not thousands, of times. All it takes is a magnet. If you don't have one, you can make one. Take an iron screwdriver and store it for a while lying in the north-south direction, and if you pick it up some time later, it will have become magnetized. Then, say, an iron nail lying next to it adheres to it. A magnet works by causing the electrons in an object to face in one particular direction. So, you have moved the electrons independently of the nuclei of the iron atoms, though not for a great distance.

Now imagine that the electrons in the particle accelerator are no longer attached to an atomic nucleus, and that they have full freedom of movement. The big round tunnel of our particle accelerator has incredibly powerful magnets stationed strategically all the way around the circle. As the particle travels along it is constantly attracted by more and more magnets, which slingshot it at an increasingly faster pace around the structure. Since we are looking at a circular construction, there is no end to the magnets, and so the particle's speed continues to get faster and faster. The scientist keeps track of what the particles are doing by means of photographic plates or other instruments that record the events in the chamber; nobody sees what's happening with the naked eye. Some accelerators are now so powerful that particles have actually been measured as racing close to the speed of light. The results are intriguing. There are two items in particular that are of great interest. 1) What happens per se when atoms and particles are pumped up to such high levels of energy. 2) What happens as the result of the inevitable collisions that occur when they race around at these incredible speeds. The latter point is, of course, responsible for the monicker "atom smasher." Many new particles have been observed to come into existence as the original atoms and particles are transformed in this process. However, there are some particles that appear all of a sudden without any precursor particles or atoms. They show up, continue to exist for a very, very short time, and then they disappear again just as mysteriously as they showed up. Thus, it has been contended that these particles are, in fact, uncaused. We cannot identify any physical item that produced them, nor, for that matter, any physical item that's destroyed them again. They came; the scientist saw them; they went.

So, are these particles "uncaused"?

Since this topic is intended to be a part of an ongoing series, I don't feel too bad about stopping at this point and promising more the next time.


Physics Logo So, you have spent millions of dollars building a particle accelerator in which you are using incredibly huge amounts of energy in order to speed up the velocity of subatomic particles to levels that they do not usually attain. Finally, some formerly unknown kinds of particles emerge in a metamorphosis of previously existing ones, and then suddenly some tiny bits of matter appear seemingly out of nothing, only to disappear again just as rapidly. So, it would appear, as some people mistakenly argue, that here we have a situation that traditional philosophy would have considered impossible, namely that here we have contingent, but uncaused entities.  These particles that blitz on- and off-stage are alleged to be contingent in so far as their existence is not logically necessary. They can exist; sometimes they do exist, but at other times they do not exist. On the other hand, they seem to be "uncaused." We cannot point to any specific event in this system that brought about their existence. In broader terms, they supposedly challenge the popular idea that "whatever begins to exist must have had a cause of existence." 

There are a couple of points worthy of comment here. It's not that long ago that I posted an entry concerning this claim and showed that an uncaused, contingent being is in a league with a square circle. But, aside from the logical problems for this claim, I am amazed (see above) that anyone would think that on the factual level these particles were actually uncaused.  As I said last time, I'm happy to concede that these phenomena do not conform to a Newtonian version of causality. But that understanding, even if correct, would only show that the Newtonian definition of causality, along with the philosophical roots on which it is based, is simply inadequate. Even without going back into the Middle Ages we can come up with a better notion of what constitutes a cause than the Cartesian understanding that a cause is a thing that influences some other thing by pushing or pulling it. If we can wean ourselves even a little bit of a purely physicalist understanding of causality we can say that a cause is a necessary and sufficient condition for the existence of some entity. "Necessary condition" means that without the entity that we call a "cause" the effect would not exist. A "sufficient condition" means that once the appropriate entity or, in many cases, the appropriate entities are in place, the effect will definitely come into being. When working on complex situations, inside or outside of science, it may not always be easy to identify the exact sufficient conditions causing an effect, and even the general necessary conditions may not be obvious. However, even though it may not always be possible to identify the precise cause of a phenomenon, if it never appears in the absence of certain conditions, we can at least be sure that there is something--perhaps unknown to us--that causes the phenomenon. 

Take the following example. Let us say that it is a heavily clouded day, and it is raining. We may safely conclude that, together with some other factors, the clouds are the primary cause of the rain. When the meteorological parameters are just right, we have the sufficient condition for there to be rain. Clouds are clearly a necessary condition. However, in and of themselves clouds in the sky are not a sufficient condition for there to be rain. There are many days when the entire sky is covered with clouds, yet it doesn't rain. Still, and this is the points that I'm trying to get to here, if there are no clouds we know that it is not going to rain. So, clouds are a necessary condition for rain, and in conjunction with some other factors, they become a part of the sufficient condition for rain. We can make the following statement with full assurance. The clouds together with the appropriate factors must be the cause of the rain. We can be certain that rain is not uncaused because it is easy for us to know a set of conditions (such as a cloudless day) when it will not rain.

Now going back to our particles being driven to greater and greater frantic activity by our huge cyclotron, we can identify some necessary conditions. The experimental data so far allows us to go no further than to say that a particle accelerator with a sufficient amount of energy is a necessary condition, and a conjunction of various items, not all of which may have been identified at this point, constitute the sufficient conditions for the appearance of the strange temporary particles. If we were not looking for the sufficient conditions for these bizarre particles, we would no longer be doing science, and we would be wasting an enormous amount of money and equipment just to amuse ourselves. I do not believe that the scientists engaged in this research are simply running these tests so that they can laugh and clap their hands with glee when they see something for which they have no explanation yet.

But the last point is really neither here nor there. The point is that, just as we can say that rain must be caused by various conditions because we can be certain of it not raining under other conditions, the emergence of these particles must be caused because it is clear that under different conditions, they do not show up. Take away the large circular tunnel, the magnets, the "gun" that emits particles, and we get no evidence of self-existent particles. Perhaps we will have the facility someday to secure evidence of their existence outside of such conditions, but even then we cannot say that they are "uncaused." As I have kept saying and will continue to elaborate, what we mean by "cause" in this context may be radically different from what a Newtonian physicist may have called a "cause," but that fact simply shows up a deficiency in his philosophical understanding of what should be meant by "cause." These particles that seem to appear spontaneously do not qualify for being labelled "uncaused."

What captures my interest even more than the physical side of asserting whether there are uncaused subatomic particles, is the philosophical matter of invoking the notion of uncaused contingent entities. Doing so is to take recourse to a self-contradictory phrase akin to a "square circle" or a "married bachelor." Whatever the dedicated philosopher may want to express, it cannot be that. Being "contingent" in conventional philosophical vocabulary entails being dependent on other entities, which is to say to be caused in some way. An entity that is not "contingent" is "necessary," and that means that it exists without a cause for its existence. On the other hand, if an entity is uncaused, then it cannot be called "contingent."

Even in philosophy, in order for communication to remain sensible, words must retain their meaning. Unfortunately, we live in a world governed by the principle ascribed by Lewis Carroll (Charles Lutwidge Dodgeson) to Humpty Dumpty in Through the Looking Glass:

Humpty Dumpty

I don't know what a philosopher in such a position would have to call these particles if they really met the conditions of being both contingent and uncaused. Nor do I know what someone would call a square circle. Fortunately, neither exists and so we need not worry about it.

The Heisenberg Principle is a very different matter.


physicslogoI can't imagine that too many people reading this entry are not familiar with the fact that, according to physics, light has the properties of both particles and waves. The question of whether it is one of the other was debated for a long time. Isaac Newton thought of light as small particles, and, given his success in other regards, it is not surprising that his judgment became authoritative for quite a while, especially since it seems to fit in rather nicely with a “common sense” view of the issue. There are some properties of light, easily noticed without any scientific apparatus, which might lead us to think of light as consisting of particles, viz. teensy-weensy little balls. 


For example, as anyone who has ever played miniature golf has had to take into account, if you roll a sphere against a firm barrier, the angle at which it meets the wall (angle of incidence) is the same as the angle at which it bounces off (angle of reflection).The same thing is true for a beam of light that encounters a mirror; its angle of incidence is also equal to its angle of reflection. So, the corpuscular theory was easily embraced.

Then along came Thomas Young (1773–1829), whose double-slit experiment, as described below, brought about a rather large-scale conversion to understanding light as waves. Waves, as I'm sure we all know, are up-and-down pulses that move through a medium, such as water or air, without the medium actually making forward progress. The medium by itself does not actually travel, except up and down, but the wave's linear progression travels through it. A surfer catching a ride on a big wave will move in the direction of the beach, but the portion of water on which he sets out will only move up and down and for the most part stay where it is. As the wave proceeds, our eyes tend to focus on the upper portion, but actually the pulse consists of an up-and-down path. It is only when the wave encounters the beach that it causes to flow the water back and forth. With regard to light, the idea was that there is an “ether” that serves as the medium through which light waves propagate.

Thomas Young's Double Slit ExperimentNow, consider two waves coming into contact with each other. They will create an interference pattern. Where two peaks join, their heights will add up, where two troughs come together, they will be deeper. In some places peaks and troughs will cancel out each other, while in still other locations we will see the original heights and depths. A typical interference pattern is going to show alternating bars of light and dark, slowly diminishing from the center of the image to its two sides.

For a time, it was thought that Thomas Young demonstrated definitively that light consisted of waves. He created a set-up in which a light beam was aimed at a barrier with two very narrow slits. The result was captured on a screen a distance away from it. Encountering the two apertures, the light splits into two beams. As they spread out, they generate the expected interference pattern consistent with the notion that light is waves. As a result, the interpretation of light as waves gained superiority.

But then Planck and those who followed his lead initiated a change in the other direction again. Light, along with other emissions of energy, comes and goes in packets. The quantum effect, viz. the observation that energy, including light, can only be added and subtracted in defined stages (Planck units), is not compatible with the wave theory. Albert Einstein contributed to this research, which earned him his Nobel Prize.

An important question became to what extent the wave effect was true of individual particles. Let's go back to Young's box. We will set up a “particle gun” in the form of a small mass of a radioactive isotope of some element, which will release tiny amounts of subatomic particles into the box, so that we can observe them on the screen. Which kind of particle is being emitted, whether photons, electrons, or some others, apparently does not make a difference. Reducing the number of particles does not eliminate the interference effect. In fact, imagine that we can “shoot” a mere two particles through the two slits, one left and one right. (I say "imagine," but these are actual experimental results.) If the particles were like bullets, we would expect two simple marks on our screen. But even on the level of just one pair of particles, the wave interference pattern still shows up. Two particles are manifesting the properties of waves, a rather confounding notion!

Particle InterferenceBut wait, the fun is just beginning!

What if we were to shoot just one particle into our setup? Surely the particle is limited to going through just one slot or the other, and, thus, there could be no interference pattern. This is where it gets good. The interference pattern is still there. The single particle, it appears, went through both slots and the particle seemingly interfered with itself.

If this little experiment and its result are new to you, and you're now expecting me to show you a resolution, I'm sorry to disappoint you. There is no way out of this paradox. And I'm not saying "no known way." It's a part of the way God created the world. I can give you this much comfort: Once the scientist with his equipment investigates the properties of the particle (including which slit it passed through), everything comes out nice and tidy. The particle will then, in retrospect, show us only one side of the various mutually exclusive properties of particles, including having traveled through just one slit. But prior to direct observation (by way of machinery, of course), the particle indicates that it went through both slits.

It would be easier to grasp such occurrences if the particle initially gave no evidence of its path. Then the whole question would merely be a matter of something unknown becoming known, and there would be no big deal to the matter. I have seen any number of arguments along this line from Christian apologists who feel that there is a problem here for a Christian world view and that they need to defeat it somehow. However, the problem (if it is one) is not that there is at first no indication of which set of properties the particle has. It gives evidence of having gone through both slits, not through neither. Thus, such arguments miss the point. Also, I cannot see the threat to a Christian world view in this matter, as I'll try to show later.

Before going on, I need to clarify some terminology.

• We can refer to this phenomenon as the principle of indeterminacy. It is not the same as the so-called Heisenberg Uncertainty Principle. We'll get to that eventually.

• The state of the particle in which it demonstrates properties of having gone through both slits is called the state of superposition 

I'm going to let the cat out of the bag to this extent: I have said earlier that quantum phenomena call for a broader conception of causality than many people bring to it. Now, I will add that it also requires a more thorough understanding of logic than many commentators come up with to deal with superposition. (And no, I'm not going to advocate trivalent logic.)

In the meantime, speaking of closely confined felines, let us amuse ourselves momentarily by mentioning Erwin Schrödinger's hypothetical cat. Schrödinger (1887-1961) was one of the important developers of quantum mechanics, and this illustration stems from him. Since he brought up this thought experiment, it has been adopted and adapted, and the version I'm giving is not exactly how he presented it, but the point is the same, namely a potential problem that could be engendered by indeterminacy and superposition. Let us imagine that particles have a property we call its “spin.” Furthermore, when we observe the particle directly we notice 1) that the spin of the particle is either clockwise or counterclockwise and 2) that the direction of the spin reflects which slit the particle has used: clockwise for the right one and counterclockwise for the left one.

Now we are going to attach some wires to each of the slits and let the slits serve as switches. The wires are attached to an enclosure that holds a cat together with a canister of some extremely lethal cyanide gas. If the particle spins clockwise and goes through the right slit, the canister is sealed permanently and the cat stays alive. However, if it should spin counterclockwise and take the left slit, some cyanide gas is released instantaneously and the cat dies immediately. We shall refrain from observing the particle and leave it in its superposition. Thus, for all that we know (and we cannot know differently), the particle travels through both slits, both switches are tripped, and the cat is both dead and alive.

Schrodinger's Cat

We've got a lot of stuff to sort out here.


physics logo

January continues to show why it belongs into the winter months. Once again our temperatures here in Indiana are below 0°F. Nick and Meghan's plumbing issues have finally been resolved, and last night they moved back to their home.

Poor Schrödinger's cat! There it is, stuck in a state that leaves it both dead and alive at the same time. The cat itself is, of course, not in the quantum superposition, but its sad fate is due to the superposition of a particle that acts as though it were two particles, thereby triggering the "dead" and "alive" switch simultaneously. Still, before helping the cat out (if that will even be possible), let's complicate things a little more.

The problem of the indeterminacy caused by the superposition of a particle is often confused with Heisenberg's Uncertainty Principle, but these are two very different things. The Heisenberg principle is not just the result of the two-slit experiment; it obtains in any attempt to specify both the momentum and location of a particle. In plain English, the principle reads roughly:

At any given time it is impossible to ascertain both the momentum and the position of a particle.

Schroedinger's CatWhy, apart from certain philosophical extrapolations, should this apparent deficiency be a problem?

One of the important aspects of modern science has been the principle that, given sufficient information about any state of affairs, it becomes possible to predict what may happen in its immediate future. For example, we can predict that combining certain amounts of oxygen and hydrogen and applying a certain amount of energy, water will result. We can safely say that if a billiard ball with a certain momentum collides with another billiard ball at rest, its momentum will transfer to the second ball, and, if we know the exact angle of impact, we can even predict the path the second ball will take. On the level of subatomic particles, we lose such an all-around predictability. Since it's not possible to specify with sufficient exactitude both the location and momentum of a particle, we cannot predict where it will head next. I suppose that we could predict that we cannot ascertain its next location, but that's not very helpful.

"Well," someone may reply. "What we need here is a little bit of patience. We may not be able to do so right now, but surely in the future we will have much better equipment, and then we'll be able to make the precise measurements we want."

Werner HeisenbergUnfortunately--and you knew this was coming--at this level better equipment could not help. It is not even possible to envision what "better equipment" could mean here.

A fairly popular conception of Heisenberg's principle concludes that the very act of observing the particle already interferes with the particle's properties. Take its location, for example. By "looking" at it, you're introducing more photons , which will collide with the object of your investigation, thereby changing its location. This summary is not as accurate as people thought it was a while ago, but it is close enough that it suffices if you don't want to deal with the mathematics involved. I would, however, like to go a little further because the math demonstrates why the uncertainty is intrinsic to the phenomenon and not just a deficiency in our capacity for observation.

Werner Heisenberg will most likely remain a controversial figure in 20th-century history. He was a leading figure in the Nazi's attempt to build an atomic bomb, which, of course, did not succeed. His own personal commitment to Nazi ideology has been debated, and I will leave the matter there (at least for now). Regardless, his insights into quantum physics are subject to neither ideology nor flag.

I would be hypocritical if I pretended to know or understand all of the math involved, and I am bringing it up only in extremely broad strokes to make it clear that it is the math that forbids an easy solution--or even any solution at all, if by "solution" we mean circumventing the principle. Let us work up slowly to what is involved.

Let's stipulate an equation in which the product of two variables (a and b), always come out as equal to a certain constant (c). As we learned in algebra, we'll write ab for (a x b), and, for the sake of convenience, we'll limit ourselves to natural numbers (positive integers). So, we'll say


Let's have c =36. Then


There are many numbers for a and b that would satisfy this equation. Still remaining with natural numbers, a could be 1, 2, 3, 4, 6, 9, 12, 18, or 36, in which case b would have to equal 36, 18, 12, 9, 6, 4, 3, 2, or 1 respectively. The point I'm making is that, if you increase the value of one of the variables you have to decrease the other one in order to retain the set value of the constant.

This is the idea behind Heisenberg's Uncertainty Principle. The more precise our measurement of one variable is (either the momentum or position of a particle), the less precise the measurement of the other one will be. Expressed as a formula, it can be represented by

ΔpΔx ≥ h

Δp stands for the precision with which we can ascertain the momentum of a particle, and Δx is the precision of measurement for the position of a particle. The symbol h is none other than our previous acquaintance, Planck's constant. I pointed out when I brought it up before that its value is a mere 6.6262x10-34, though it's a significant number on the level of particles and atoms. The "greater than" aspect of the formula must be understood as a tiny fraction of h. So, just as with our example equation above, if the precision for one value (momentum or location) increases, the other one must decrease.

Now, someone with a just a little knowledge of algebra might ask "Where's the problem?" The most basic rearrangement of variables in an algebraic formula should allow us to conclude that

Δx ≥ h/Δp

and that

Δp ≥ h/Δx.

Then if we can measure one value with precision, we can get the second one simply by dividing Planck's constant by it.

But it's not that easy (or even possible) because we are now outside of the realm of traditional algebra. Even apart from the uncertainty principle, Heisenberg made a significant contribution to quantum theory by creating a model in which the properties of a particle are described by matrices. There are some significant differences between "regular" algebra and matrix algebra. More specifically, whereas in simple algebra we learned that multiplication of numbers is commutative, viz. ab = ba, multiplication of a matrix by another is not commutative. In Heisenberg's mathematical analysis Δp and Δx are both matrices and they cannot be substituted for each other in the simple algebraic manner we suggested above.  

A matrix is an array of numbers written in table form. Eg., here is a matrix that we shall call A. This is just your ordinary everyday matrix, which bears no relationship to the ones set up by Heisenberg. I just want you to see how ornery matrices can be.

1 2
3 4

Let's provide it with a companion, which we'll call B.

2 3
4 5

Heisenberg's matrices obviously looked different, but these samples will show up the same peculiarity that emerged in his equation from the fact that it includes the multiplication product of two matrices.

For anyone interested or willing to help me out if I made a mistake, I'm going to demonstrate the procedure involved in matrix multiplication and show my calculations below in an "addendum." Here I will simply give you the conclusions. Multiplying matrix A with matrix B yields a totally different result than multiplying matrix B with matrix A.

Matrix AB is:

10 13
22 29

while matrix BA comes out as:

11 16
19 28

And that kind of inequality cannot be overcome, no matter how much harder we try to be precise in our observations. Unless you deliberately choose the numbers beforehand to make the calculation come out as you want it to be, the order in which you multiply the matrices makes a difference in the resultant product.

Now, Heisenberg's math seems to tell us differently at first when he states that the product of Δp and Δx will always come out to Planck's constant, h. How can that be? If multiplying matrices is not commutative, Δx multiplied by Δp should come out to a different result. But it doesn't. The result is still h.  That result is only possible if at least one of the matrices, Δp or Δx, takes on a different value when the multiplication goes into one direction rather than the other. And, thus we are locked into the situation that we cannot determine a precise value for the momentum and the location of the particle simultaneously.

There are other ways of constructing the math. Schrödinger did it with a different-looking formula based on the wave function of particles (and I'll try to come back to his efforts in the future). But the bottom line is the same; the uncertainty remains. And my point is this: Let's forget about the more or less ingenious ways in which people try to get around the fact that we cannot ascertain the momentum and location of a particle at the same time. Whether you go with Heisenberg's matrix analysis or Schrödinger's wave function/probability scheme, we cannot rewrite math--or logic for that matter, in order to liberate us from a conundrum created by math. What we need to focus on is whether there is a need to react to this phenomenon from a Christian philosophical perspective at all, and, if so, how.

Addendum: Multiplying Matrices

You can find a very nice introduction to matrix algebra at the "Math is Fun" website. The multiplication of matrices is not difficult, but it can be extremely tedious. The "Purple Math" website calls it "a royal pain."

If we want to multiply matrix A by matrix B, first of all we need to be sure that the number of columns of A is equal to the number of rows of B.

Then we multiply the first element of the first row of A with the first element of the first column of B and the second element of the first row of A with the second element of the first column of B and add up the two products. Doing so gives us the first number for the product matrix AB, assuming we have a small 2x2 matrix, as in this example.  

Then we do the same thing for the first row of A and the second column of B, the second row of A and the first column of B, and finally the second row of A and the column of B. An example may clarify the procedure more than further attempts at verbal explanations. Let's set A and B side by side.  And calculate their product: AB

A                B                     AB

1  2          2  3                   a  b
3  4          4  5                   c  d

In order to make sure you can see what I'm doing, I'm going to violate protocol and enclose the multiplications with unnecessary parentheses.

a = (1x2) + (2x4) = 10
b = (1x3) + (2x5) = 13
c = (3x2) + (4x4) = 22
d = (3x3) + (4x5) = 29

Then the matrix AB looks like this:

10  13
22  29

And now for the whole point of this. In the earlier simple example, it didn't matter whether we wrote ab=c or ba=c. We say that the multiplication of numbers is commutative. But this is not the case for matrices. Let us calculate the values if we reverse the position of the two matrices and find the product matrix BA. Once again, for the sake of convenience, let's set them side by side.

B             A                  BA

2  3         1  2                a  b
4  5         3  4                c  d

We calculate:

a = (2x1) + (3x3) = 11
b = (2x2) + (3x4) = 16
c = (4x1) + (5x3) = 19
d = (4x2) + (5x4) = 28

And so the result for BA is this:

11  16
19  28




Physics LogoBefore getting further into various scientific aspects of quantum physical phenomena, I think I should make sure that we don't lose sight of the philosophical issues. The indeterminacy demonstrated by the two-slit experiment and Heisenberg's uncertainty principle certainly present a challenge to our understanding of how the world works. The question is how big that challenge is. Do we need to revise everything we thought we knew or believed in light of those discoveries, or is this just one of those minor annoyances that we can ignore?

Several of the outstanding figures in the development of quantum physics had some formal exposure to philosophy, but it affected them in different ways. According to Amir Aczel (Entanglement, p. 88), Niels Bohr had a friend named Harald Høffding who was a philosopher. One evening Høffding invited Bohr and another quantum physicist to his home in order to discuss the superposition phenomenon in general and the problems of the existence of the particle during the experiment in particular. Apparently Bohr had more questions than answers in this regard because, the story goes, as the evening wore on Bohr was overheard asking himself in an undertone: 

"To be ... to be ... what does it mean to be?"

The other physicist invited for that evening, whose name I shall tell you in a little while, had a strong background in philosophy and has written on physics and philosophy. His easy command of the important figures and ideas in the history of philosophy, including competent interaction with the ancient Greek philosophers, will surprise anyone who might expect that he was just another scientist stepping outside of his field to make toe-curling assertions. My faithful readers are aware that it takes a lot for me to make such a laudatory statement; I fear that the latter group is still in the vast majority.

I believe it is fair to say that both Bohr and the so-far-unnamed figure agreed that one cannot create a theory of reality by starting with what is not there or a theory of knowledge beginning with what we cannot know. In fact, it appears to me that there is no a priori reason to believe that whatever science teaches us about the nature of the world is the one and true reality, and that our life-worlds are consequently illusory. If I place my cup on the little table next to the couch, I know that it will stay there because the wooden boards of the table are solid. Now, ever since Rutherford shot particles through gold foil, we have known that solids, including boards, consist of atoms, whose main mass is represented by tiny little nuclei, surrounded by even tinier electrons. (The ratio of the mas of a proton to an electron is 1836 to 1.) In other words, an atom consists mostly of empty space, and thus the boards are also overwhelmingly empty space. Is this a description of reality? Certainly. Does it supersede my uninformed judgment that the board is solid? Of course not. They are both aspects of reality, and there is no "really real."

However, I'm pretty sure that for most of us there is little or no experience of the quantum world to take into account. Our experience of the life-world, assuming that our epistemic functions are not impeded in some way, is pretty much synonymous with the experience of reality. As physicist X reminds us, even quantum physicists, in conveying their thoughts concerning matters inside or outside of the quantum world, will do so in the language of classical physics. I was tempted to say something like, "they must 'resort' to the language of classical physics," but that would have been a wanton overstatement. There is no "resort" here. "Classical" language is all that we have. Schrödinger did not speak to Heisenberg in wave-probability equations, and Heisenberg did not respond to him by multiplying matrices. Physicist X asserts:

"The concepts of classical physics are just a refinement of the concepts of daily life and are an essential part of the language which forms the basis of all natural science. Our actual situation in science is such that we do use classical concepts for the description of the experiments, and it was the problem of quantum theory to find theoretical interpretations of the experiments on this basis."

Take the notion of "prediction" in science, on which I touched in the last post in this series. I stated that quantum mechanics does not allow us to predict the position of a particle, and I shall stick by that statement. However, the quantum scientist still proceeds along the same line as any other scientist. There are theories that will predict certain outcomes by the rules of quantum mechanics, and--possibly after a fairly long time--the theories are confirmed or disconfirmed by experiments which tell us if the predicted results came about. For example although the direction of the spin of two entangled particles in the superposition is both up and down, we can predict that once we have observed the value for one particle, the other particle's spin will have been in the opposite direction. Einstein predicted this result as a defeater of Bohr's theory. As it turned out, the prediction came true, as confirmed by much later experiments, and, paradoxically, strengthened the plausibility of Bohr' point of view.

I'm not saying that the naive world of our everyday lives is always the best measure of reality. To ignore the findings of science carries a great liability. For example, even though I do not see the electrons in an electric outlet, to ignore what I have been taught about such things might lead me to suffer electrocution. The results of modern science have important implications beyond their own realm and, thus, must be taken into account in a full picture of the world. My point is that they are not necessarily more paradigmatic than the conclusion of direct experience or even the particularism of phenomenology.

Furthermore, this reaffirmation of the life-world as the starting point for epistemology and metaphysics does not imply a simultaneous affirmation of bad epistemologies or bad metaphysics. So, our still-anonymous celebrity physicist pointed out the deficiencies of modern philosophy which influenced classical (Newtonian) philosophy. Specifically, physicist X blamed Descartes' distinction between a res cogitans (a "thinking thing") and a res extensa (a "material object", literally "a thing that has spatial dimensions"). This division created a conundrum for Descartes in trying to understand how the human soul and the body can interact causally and led him to consider animals to be nothing but machines. It also left us with the rather meager understanding of causality as pushing and pulling of one res extensa on another. Physicist X wrote that

" has always seemed difficult to deny the existence of some kind of soul in the animals, and it seems to us that the older concept of soul for instance in the philosophy of Thomas Aquinas was more natural and less forced than the Cartesian concept of the 'res cogitans,' even if we are convinced that the laws of physics and chemistry are strictly valid in living organisms."

Physicist X  suggested a direct parallel between the conclusions of quantum mechanics and Aristotle's metaphysics. With reference to the particles that seem to appear and disappear without benefit of a Newtonian-style cause, he did not debunk the existence or importance of causality, but suggested that an Aristotelian model might be more appropriate.

"Therefore, we have here actually the final proof for the unity of matter. All the elementary particles are made of the same substance, which we may call energy or universal matter; they are just different forms in which matter can appear. If we compare this situation with the Aristotelian concepts of matter and form [--which is the basis of causality in Aristotle and Aquinas--] we can say that the matter of Aristotle, which is mere "potentia," should be compared to our concept of energy, which gets into "actuality" by means of the form, when the elementary particle is created." [The comment in brackets is mine.]

May I admit that I was startled (in a good way) when I read those words for the first time? At the time I was preparing a refutation of the philosophical conclusions attributed to physicist X as I had come to understand them from secondary sources.  I intended to show that an Aristotelian model of causality fits where a Newtonian one does not. Of course, I needed to read the primary material, and, after reading what X actually said himself, that specific project of mine very quickly came to an end. I think that the reason why his proposal has not become more widely known is the fact that he took his commentators into unfamiliar territory. I'm thinking of other scientists, but even more so of contemporary professional philosophers who would not likely be sufficiently familiar with, let alone accepting of, Aristotle's metaphysics. 

To continue with the description, a little later in his book X returned to the same idea in the course of discussing the language physicists use when they refer to the expected outcome of an experiment. The example he had chosen was the quantum theory of thermodynamics. 

"Again, as in classical thermodynamics, it is difficult to call the expectation objective. One might perhaps call it an objective tendency or possibility in the Aristotelian sense of 'potentia.'"

I started out by intimating that the conclusions of quantum mechanics have left many a person floundering in a philosophical quagmire. At least some of that confusion can be avoided by going beyond the breezy simplistic concepts of modern philosophy, which are obviously inadequate to do justice to the phenomena of quantum mechanics. If Cartesian causality fails, the rational alternative is not the absence of causality, but a more sophisticated understanding of causality. 

The mystery physicist in question is none other than Werner Heisenberg. The quotations come from his book Physics and Philosophy: The Revolution in Modern Science (Amherst, NY: Prometheus, 1999; orig. 1958). Heisenberg's father was a professor of philosophy, specializing in the history of philosophy, particularly classical Greek thought. Werner grew up in a household where philosophy was a common subject of discussion.

We're not done yet with Heisenberg. He also discussed the logical implications of the two-slit experiment, where his interpretations are right on target as well--until the very last moment. We'll leave that topic for the next installment.


Physics LogoFirst, a medical update from the veterinary department. I need to confess that I was goofing around. I like cats, and I'm beginning to feel bad for Schrödinger's hypothetical zombie cat. It is undoubtedly dead by now, but the cause of death may as likely have been old age as being hooked up to a double-slit device. It really doesn't matter what conceptual play one engages in, a cat cannot be both dead and alive. As a matter of fact, the very principles of quantum mechanics prohibit an object the size of a cat to be in the state of superposition, which the thought experiment, my own earlier statements notwithstanding, would have to be the case for the cat. Earlier I mentioned that one cannot produce higher and higher states of radiation just by burning more and more fuel. I cannot rule out that there might be cosmic gamma radiation in your fireplace, but if so, it did not come from your burning logs. Similarly, for an object to be subject to quantum phenomena, its size may not exceed its wave function, which means that only items of the size of atoms and particles qualify.

I have a number of nephews (some of them by marriage) who are geniuses, and one of them (D.W.) is in graduate school actually working on the cutting edge of this subject matter. He pointed out to me that, since the superposition only obtains when the particle is not being observed, the cat would suffice as an observer, and, thus, the whole experiment would collapse. He's right. If a lifeless photographic plate is sufficient to count as an observer, then surely an animate being can fulfill that role.

Please note that I am continuing to compile the installments of this series on its own web page, so you don't have to try to navigate the blog in order to find the predecessors to this one.


The last entry on this topic was Werner Heisenberg's philosophical interpretation of quantum mechanics. I mentioned that he had a better grasp of the history of philosophy than some professional philosophers I have run across, and that his response to the idea of "uncaused" particles was not to ditch the concept of causality altogether, but to return to an Aristotelian-like conception of causality as the actualization of a potential. Also, specifically with regard to the two-slit experiment, Newtonian causality cannot provide any categories to understand the superposition and its dissolution upon observation of a particle, but, as Heisenberg pointed out, the Aristotelian categories of potential and actuality can. Of course, I'm not talking about "explaining" the phenomenon. I wouldn't even know what an "explanation" could be in this case; as far as I can tell, what you see is what you get. The point is that one doesn't have to fall back on silly notions such as "uncaused contingent entities."

But what about the logic of having a particle exist in two mutually exclusive states at the same time? Giving it a name, i.e. "superposition," does not do anything to alleviate the apparent contradiction. The descriptions of how the particle presents itself are a perfect fit for the category of contradiction. Before addressing the quantum aspects, let me give you a quick bit of background.

Let's think a little bit about logic in general. In really rough terms, logic has to do with correct thinking, and you don't learn that from taking a course in logic any more than you will get healthy by taking a course in physiology or become a better person by studying ethics. Take some simple valid arguments:

All birds have feathers.
All penguins are birds.
Therefore, all penguins have feathers.

If Nairobi is in Kenya, then it is in Africa.
Nairobi is in Kenya.
Therefore, Nairobi is in Africa.

Are these inferences sound or not? You need not have studied logic to recognize that both of these arguments are valid. Also, the premises are true, and so the conclusions are true as well (as long as we don't engage verbal sleight of hand). Both are based on the inclusion of items in various classes: 

Euler Circles

In the first one, the class of things that have feathers includes the class of birds, and the class of birds includes penguins. So, in order to find any penguins, you have to look through the class of feathery things. In the second argument, the class of places in Africa includes Kenya, and the class of places in Kenya includes Nairobi. Consequently, if you are looking for Nairobi, you must look for it in Africa But I didn't need to explain that to you, did I?

Feel free to skip this paragraph; my point is that it is eminently skippable. We could get even more technical. Formally, the first one is a classical AAA-1 syllogism, in which "having feathers" is the major term and the predicate of the conclusion. "Birds" is the minor term, which appears in both premises and is distributed exactly once. "Penguins" is the minor term and the subject of the conclusion. Furthermore, the minor term is distributed in the conclusion, and in the premise in which it appears. The second argument is a conditional argument (if...then) called a modus ponens, in which affirming the antecedent of the first premise (the "if" part) necessitates that we must also affirm the consequent (the "then" part).

You didn't have to know all that technical stuff to know whether the arguments were valid or not. Of course, arguments can become a whole lot more complex. Then the rules can also become more and more demanding, and some people will be better at applying them than others. But on this basic level, drawing a rational inference does not require any formal training. The starting point for logic as a discipline is not a set of rules that we must learn, but rather systematizing (and perhaps explaining) what we already know to be valid.

There has been general agreement that there are three fundamental principles of logic, which in English we call "laws." There is the law of identity, which says that a thing is identical to itself.

a = a

The second one is the law of contradiction (sometimes called the "law of non-contradiction"), which states that nothing can be what it is and its contradiction (at the same time in the same sense).

not (a and not-a)
~(a & ~a)

And finally, we have the law of excluded middle. It tells us that anything in the universe either has a certain property or it does not. This principle seems to follow from the law of contradiction. If something cannot be both what it is and its contradiction, then it must be one or the other.

either a or not-a
a V ~a

Choose any property you want: You either have it or you do not. Something either is an aardvark or it is not. My car is either larger than a breadbox or not. 2437 either is a prime number or it is not. I either am Overlord of the Galaxy or I am not. (Hm. We may have to come back to that last one).

The law of excluded middle tells us that there is no "in-between" when we are looking at contradictory, viz. mutually exclusive properties. The statement that something either is an aardvark or some other kind of mammal is not necessarily true because it could be something else. The law of excluded middle is applicable only when there is a radical disjunction between the two options, as expressed by the law of contradiction. (Note that in customary rational thinking, contradiction and excluded middle are easily transformed into each other as expressed in DeMorgan's theorem.)

What impressed me about Heisenberg's reference to the law of excluded middle is the fact that he recognized that it is not dependent on epistemology (the order of knowing). We may not know what something is, but we do know that it must be what it is and not its contradiction.

The law of excluded middle is not in the least perturbed by matters such as light being both particles (photons) and waves (even without resorting to the ingenious idea of "wave packets," developed by DeBroglie). This is where the qualification of "at the same time in the same sense" comes into play. Light acts as waves under certain conditions and as photons under others. The same thing applies to electrons, protons, and so forth. One can specify when they will act as waves and when they act as particles.

The chestnut is the superposition, as brought out by the two-slit experiment. We are looking at a single particle acting as if it is two particles with mutually exclusive properties. What we see is that it traverses both slits and produces a two-wave interference pattern. Did it take the right slit? Yes. Did it take the left one? Yes. Is its "spin" up or down? Both. Once the observation (measurement) has taken place, we're back in business with the usual principles of logic, but the particle in the superposition does not seem to want to fit. So, one frequently hears that the law of excluded middle does not apply to this case, and, since it takes only one case to the contrary to falsify a principle, we must question the principle.

Let me emphasize again that the law of excluded middle does not say that we know for everything what it is and, therefore, that it is not its contradiction. In an elementary course in truth-functional logic students (should) learn to evaluate the truth of various formulas with unknowns truth values. For example, take a simple conjunction,

(T & T) = 1

We can say that this conjunction is true only if both sentences are true, where 1 stands for true and 0 would stand for false, such as when at least one of the two statements is false.

(T & F) = 0

If we know that one statement is true, but do not know the truth value of the other one, we have to suspend judgment.

(T & ?) = ?

However, if we know that one statement to be false it doesn't matter what the other one may be since both have to be true in order for the conjunction to be true. Thus:

(F & ?) = 0

So, if it were just a matter of not knowing the state of the particle in the superposition, we're certainly equipped on the logic side of things to deal with it. We know what to do when we do not have sufficient information. The problem is that the particle thumbs its nose at us (so to speak) and gives us too much information. What to do?

My answer is: as little as possible. The reason why I brought up the fact that we are rational by nature is that I can now say that the bizarre case of the superposition could not falsify the law of excluded middle if it wanted to. It is still the case that the particle is either in the superposition or it is not. It is still true that the particle in the superposition manifests the properties of two particles, and the contradiction of that statement is false. In fact, I'm going to go out on a fairly short limb and declare that we wouldn't be bothered by the whole phenomenon if excluded middle were not a universal principle without which we cannot think.

Let me go on with two further thoughts, one nothing more than an undeveloped hunch, the other one hopefully helpful. For the first one, I'm not sure where to go with it right now. It is simply that we should keep in mind that not all "opposites" are contradictories. Take the two statements "All people celebrate Christmas," and "No people celebrate Christmas." It is a fact that if one of those sentences were true, the other one would have to be false. However, they could both be false, as indeed they are. Sentence pairs such as those are called "contraries." I would like to see if there is a way to apply this idea to the particle in superposition, but I'm not sure that can be done, or, if so, how.

I have tried my best to present the logical problem caused by the phenomenon of the superposition of particles as clearly as I could. I also stated that there is a common perception that a principle can be falsified by a single fact to the contrary. But whether that's really true depends on the nature of the principle and the alleged counter example. And I'm not convinced that the superposition should be considered a direct counter example to the law of excluded middle. I already made reference to the fact that, even if we tried to chase it away, it stubbornly persists to direct our speech and thoughts.

As to the second thought. I also made the point that, under the law of excluded middle, we may not know what a thing is, but we still know that it has to be one or the other of a set of properties. However, I think that I have to clarify the scope of its application a little bit. The ascription of a certain property or its contradictory to a thing has to make sense. Here are some examples that would try to combine items from incompatible categories:

Goodness either does or does not weigh eight pounds.

George either is or is not a prime number.

The Bill of Rights either does or does not sing soprano.

You get the picture. When the subject under consideration combined with the alleged property amount to incoherence either way you go, we simply can't apply the law of excluded middle. That doesn't mean that it is no longer in force, but that in cases like this its application is meaningless.

I would like to suggest that the phenomenon of the superposition of particles is a similar case. By it very meaning (it's "definition" if you will), it is a state in which it does not make sense to apply excluded middle to it. The state of superposition is not green, it does not play the oboe, and the consequences of excluded middle simply do not apply to it. Otherwise, we are left with either a contradiction or an empty tautology. 

"The state that is characterized by its apparent defiance of the law of excluded middle appears to defy the law of excluded middle." That's not helpful. 

Neither is the other side of the coin: "That state that is characterized by its apparent defiance of the law of excluded middle is subject to excluded middle in some hidden, unknown way."

We cannot characterize the state one way or the other and say anything edifying. But that's intrinsic to the state. I mean, it either has to be intrinsic to the state or not. Regardless of whether the state is bound to excluded middle, we are.

We didn't learn about excluded middle by inductive logic as a generalization from specific instances, and so it cannot be shot down by a single enigmatic observation. It is an integral part of the rational universe that God created. God also created the particles that people can send through two slits and be amazed at its contradictory manifestation. However, the experiment cannot falsify a basic principle of thought. No one, neither Heisenberg nor Bohr nor any other quantum physicist gave up thinking in a way that circumvented the basic laws of thought, including excluded middle. Doing so would be impossible. My recommendation, as alluded to above, is to marvel at the universe with its mind boggling complexity. 

Does the fact that the universe contain phenomena that cannot be crunched into our finite understanding give testimony to the existence of a Creator whose thoughts must have gone much farther than what makes sense to us? I don't think I'm ready to say so as an argument, but as a Christian I can most definitely affirm it as an observation.


Physics Logo

We need to come back to Erwin Schrödinger, the infamous cat owner for another moment. His approach to quantum mechanics was actually slightly different than that of some of the other leading figures, such as Bohr or Heisenberg. As I mentioned earlier, Schrödinger came up with a formula for assessing the state of a particle based on wave functions rather than on matrices, as Heisenberg did. The upshot of his work was that, even though we cannot know both the position and momentum of a particle at any given time, we can estimate those values in terms of probability. Take, for example, the picture of an atom that may have had to deal with in first semester beginning Chemistry. Look at the illustration, linked to from Wikimedia. The orbits of the electrons surrounding the nucleus do not resemble a miniaturized solar system, but they seem to take on highly bizarre pathways. I hope that they don't bring back too many bad memories of a required chemistry class to anyone reading this.


However, we must get away from the idea of "pathways" altogether. Note that the "orbitals" shown in the diagram are three-dimensional, which would certainly not make any sense if we were tracking the course of electrons. What they show is the probability of finding the electrons in those areas with about 95% probability. The more electrons an atom has, the more orbitals will be utilized, but the probability map can be changed by raising or lowering the energy of the atom.

Thus, Schrödinger took an approach that was in some ways more realistic than that of the so-called Copenhagen school. By "more realistic" I mean a greater commitment to the idea that the idea of measurement of a truly existing physical particle was still possible, though only on the basis of probability.

Albert Einstein, who had won his Nobel Prize for his early contributions to quantum mechanics, was no happier with the probability interpretation of quantum mechanics than with the Copenhagen school. He was convinced that the theory as it stood was incomplete, viz. that there was some factor or parameter that would remove the inexplicability or even probability from particle physics. -- I really don't understand why some Christian writers intuitively tend to side with Einstein on this matter. What Einstein advocated was a materialist deterministic understanding of the universe.

Einstein and Bohr were on friendly terms, though that fact did not keep them from some heated debates. The nature of the relationship was emphasized particularly whenever they met face to face, particularly at the "Solvay Conferences." These were meetings sponsored by Ernest Solvay, a wealthy Belgian industrialist who was convinced in his mind that he was a great physicist. In order to get a hearing for his ideas, he invited the premier physicists of his day to come together and discuss the latest discoveries. At these meetings, it became a little bit of a routine for Einstein to greet Bohr, as well as, say, Heisenberg and Schrödinger in the morning at breakfast by presenting them with a theoretical counterexample that would defeat the new quantum mechanics. Bohr took these little challenges very seriously, and usually he came up with some way to refute Einstein's objection in some way.

At one point, Bohr almost got stuck for an answer to one of Einstein's thought experiments. Significantly, when he finally solved it, he wound up using Einstein's own innovations to refute him.

I really wanted to finish this anecdote tonight, but I'm getting way too sleepy. So, more on this debate between these two intellectual heavyweights next time.


physics logo

In the light of a rather unfortunate set of comments in Facebook, please let me reiterate what I said at the beginning of this series, which I'm assembling at a single site as I'm moving along. My announced intention from the outset has not been to provide an exposition of quantum mechanics and relativity per se. I clearly declared my Christian perspective and that "I'm addressing an audience grounded to a certain degree in a Christian world view. Thus, this is not an apologetic directly aimed at non-Christians, but an aid to help Christians see a little more of the relationship between their faith and science." As much as anything I have two particular aims:

  • To help Christians who are under the mistaken impression that the "new" developments in physics present a risk to the truth of Christianity;
  • To try to provide a more accurate description of quantum physics and relativity than I have run across from some Christian apologists whose responses are beside the point as a result, and possibly a little embarrassing.

Even though I'm trying as hard as I can to get the physics right, the point is primarily philosophical, and, thereby, in my case, also theological. Non-Christians and atheists are certainly invited to follow, but this series is not an apologetic directed at them. I realize that contemporary atheists' feelings get hurt when they are not the center of attention, but I'm afraid that's how it is.

I had to leave off rather abruptly last night and hopefully left you in suspense concerning the debate between Einstein and Bohr at the Solvay Conferences of 1927, 1930, and 1933. A good summary is provided by the aforementioned book Entanglement by Amir D. Aczel on pp. 110-116. Einstein demonstrated a somewhat monomaniacal attitude during the time before and after sessions in coming up with thought experiments to cut defeat both the Copenhagen and the probability versions of quantum mechanics. As mentioned, typically he would come up with a potentially defeating argument; Niels Bohr would work on it all day, frequently consulting with some of the other celebrity physicists there, and come up with a sound response by dinner. But there was one time when it looked as though he was stumped.

Here's the basic principle: How much does your cat weigh? Don't try to place it on your bathroom scale; cats don't usually cooperate with such maneuvers. Instead, first weigh yourself without the cat, and then do so with it. Or vice versa. Subtract the weight-without-cat from the weight-with-cat, and you have the weight of the cat. Usually physicists make a distinction between the weight of a thing and its mass, though in this case it does not make a difference.

Einstein's Thought ExperimentEinstein imagined an experiment that would supposedly disprove the uncertainty principle along similar lines. Let me try to describe it. The picture on the side was obviously made after Einstein's initial sketch, and is being circulated without attribution these days. The point of the uncertainty principle, you will remember that one cannot ascertain with certainty the momentum and location of a particle at a particular time. Keep in mind that in classical physics the momentum of an entity is understood as the product of its mass and its velocity. So,in order to ascertain the momentum, you need to know its mass, and for the velocity, you need to know the distance it covered over time. Thus, one needs an incredibly accurate clock as well. In the case of this experiment, having an accurate measure of the time and the energy are sufficient to satisfy the variables needed to satisfy a potential refutation of the uncertainty.

So Einstein posited a box filled with "excited" particles that would give off photons as they returned to a lower state of energy. At the outset, the box is weighed on an extremely precise scale. The box has a small aperture that opens briefly and immediately closes after a single photon has escaped. Now we can weigh the box again, and subtract the weight-without-photon from the weight-with-photon, and we presumably know the mass of the photon [m].

Assuming that we're working in a vacuum, we know that the photon traveled at the speed of light [c]. So we can calculate the energy of the photon with Einstein's formula E=mc2. Our super-precise clock gives us the exact time when the photon escaped. Thus we get the equivalent of what we need for the other side of the uncertainty teeter-totter.

From the contemporary accounts, Bohr was beside himself. He went from colleague to colleague trying to get their help, but no one came up with a problem in Einstein's scenario. That evening at dinner he did not have an answer. Einstein looked smug; Bohr looked harried.

However, on the next morning, instead of Einstein handing Bohr another puzzle, Bohr presented Einstein with a refutation of his thought experiment. It is based on Einstein's own general theory of relativity, which we'll come back to. Let me summarize the gist of Bohr's reasoning in my own words. How much did the box weigh before the release of the photon? Call the number x. How much does it weigh after the release? Presumably, there is a reduction in the total mass, which we can call x-Δx. The point that Bohr raised was that x did not carry the same value before and after weighing. According to Einstein's general theory of relativity, the difference in mass implies a displacement in the gravitational field, which affects the velocity of the particle (keeping in mind that velocity is a vector). And that phenomenon also entails that the clock will slow down. Consequently, the uncertainty is still there. Einstein had been defeated by use of his own theory.

Obviously, some of the numbers in this context are infinitessimal. On the other hand, the speed of light squared is a rather large quantity. Thus, what we have here is a meeting of quantum mechanics and relativity, providing a nice segue from one area to the other. We'll pursue the theories and implications of relativity as we continue the series.

A good comment from a reader came up on the blog: "One cannot measure speed/weight of the box without shooting billions of photons on it." I assume that he was referring to the short-hand, non-quantitative version of the Heisenberg uncertainty principle: "You cannot observe the mass or momentum of a particle because in the process, the photons you are shooting at it will distort your measurement." That's a good point, once you assume the Heisenberg principle to be true. However, Bohr could hardly use it in this case because the principle was the item in question. It would be circular reasoning to use the Heisenberg principle in order to defend the Heisenberg principle.


physics logo Link to camel post

Student Bodies and Babies. Relativity is not a new idea. Nor is the idea that what we see is not necessarily what we get. Kenneth R. Atkins, the author of my undergraduate physics book, provides an amusing illustration of how we adapt to otherwise misleading perceptions(Physics, 2nd ed., New York: Wiley, 1970; orig. 1965; 459-60). He imagines that he walks into his lecture hall and realizes that 1) the students in the front row of the auditorium are about ten times the size of the ones in the back row, and 2) the university has been considerate enough to provide just the right-sized chairs, ranging from larger ones for the bigger students in the front to smaller ones for the tiny students in back. How thoughtful!

Professor Atkins tries to confirm his observation by walking to the back of the lecture hall, but realizes that his earlier observation was incomplete. The further back he moves, the more the students in the last row increase in size, while the ones in front start to shrink. He congratulates himself on having special powers that allow him to control the size of the students in his lecture room. While still in back, contemplating this unique ability, another professor walks in and lingers at the front of the auditorium. He repeats Professor Atkins' earlier observation that the students in front are larger than the ones in back. This is curious. 

As the two scholars move forwards and backwards in the room and communicate with each other, Dr. Atkins has to concede that apparently his colleague has the same superpower to manipulate the size of the students. A fascinating additional consideration is that, when one of them stands in front and the other one in back, their observations should cancel out each other, but they don't. The matter is truly baffling.

Babies in SpaceAfter discussing the phenomenon further, the two scholars agree that the different perceptions of the students' sizes is just a matter of appearance. They are actually all roughly the same size, and if they were to establish a standard of measurement, this latter interpretation would be confirmed. In fact, given such an invention, they could express the spatial relationships with mathematical ratios and even build rules of geometry on them. However, Atkins reports, "Upon later referring to the library, we are disappointed to find that we must concede prior publication to a certain Dr. Euclid" (460).

Atkins' point is that our minds learned to adjust for visual perspective at a very early age, and so, it has become intuitive for us. The same automatic compensation could theoretically become true for people growing up with direct exposure to the results of relativity. He concludes with a dry sense of humor:

    "Unfortunately, the upbringing of our children is such that they are confined to frames of reference whose relative velocities are small compared with the velocity of light. If we could take our babies and propel them through space with large variable velocities, we should presumably raise a generation to whom the results of the theory of relativity would be intuitively obvious."

I considered what that might look like, as pictured on the right, and decided that it is not likely to happen, and, thus, we will have to continue to strain our minds to comprehend some of the phenomena of relativity.

Galileo's Ark.  Galileo did more than drop weights off the Tower of Pisa and look for planets around Jupiter. He also came up with the first principle of relativity, which was subsequently amplified by Isaac Newton. In the Dialogue on the Two Chief World Systems (1632), Galileo's main character, Salviati asks a group of friends to join him inside of a ship in a room with no windows (portholes). As they walk in, they notice that there are mosquitoes and butterflies in the air, and that there is a tank with some fish. He goes on to request that they would hang a leaky bucket of water from the ceiling so that its drops would fall directly into a container underneath it with a very small opening. The last favor Salviati asked for was that his visitors should jump both forward and backward. The nice thing about thought experiments or philosophy written in dialog form is that people always say and do exactly the thing that the author thinks would be appropriate. So, Salviati's guests did as requested.

Now Galileo has Salviati put the ship in motion and attain a steady cruising velocity. He and his guests once again visit the same room and nothing is changed. The insects fly as before; the fish swim around the tank with no additional trouble; the bucket still drips into the same small-mouthed container, and the people can jump back and forth the identical distances. As long as the ship maintains its velocity, everything is the same as it was without moving. The fact that the ship moves in one direction does not either increase or decrease the distances that people are able to jump. The ship represents an inertial frame of reference. Whether we use ships, automobiles, or planets revolving around the sun, the laws of physics hold true in any location from a given observer's vantage point.

What makes Galilean relativity interesting is that someone in one frame of reference may observe something different from an observer in a different frame. For example, if I sit in a moving railroad car and toss a tennis ball straight up in the air, it will come straight down to me again. But if someone were to stand at a vantage point to the side of the train and tracked the course of the ball from there, they would see it traversing a parabolic course.


This is the basic meaning of relativity. It has nothing to do with epistemological relativism except for the etymology. In fact, even in its revised form by Einstein, the whole point was to clear up the mathematical description of objects moving through space. Remember that Einstein was a Newtonian at heart, and he did not believe in any events in the universe actually occurring outside of a rigid uniformity. They might be weird, but they would obey the letter of his laws. 

Let me put in a quick mention of a book and a website here. If you're fairly serious about this subject, though not necessarily intending to become a professional physicist, you may want to pick up Peter Collier, A Most Incomprehensible Thing: Notes towards a (very) gentle introduction to the mathematics of relativity. (Published by Incomprehensible Books, 2012; the location of this publisher is not given.) I think I need to state that this introduction is probably not quite as gentle as Collier hopes, but it works as long you're patient with yourself. If you are allergic to math, please stay away from it. If you do use it, or if you're working with college level math for other reasons, there are numerous websites that will be of assistance to you. Collier recommends the WolframAlpha Calculus and Analysis Calculator. I have found it to be really astounding in the range of calculations it does for you.

Ether Or Not

"Space, the final frontier ... " What exactly is "space"? I don't mean what is "outer space," but space in and of itself. Well, one might say that space is that big huge open area in which all the stuff of the universe is located, a humongous living room with virtually uncountable pieces of furniture. John Locke contended that space was infinite and paralleled eternity in God's attributes. Immanuel Kant promoted the idea that space and time are a priori forms that our minds impose on an otherwise chaotic sensory intuition. Still, for most people, space is something outside of themselves, the empty areas not occupied by other things.

But how empty can space actually be? Can you have water waves without water? Of course not. Can there be sound waves without air or some other gaseous or liquid medium? I don't think so. Can you have light waves without water, air, or some other discernible medium? Yes, apparently you can. If it were possible to construct a space with an absolute vacuum, you could still send a beam of light through it. But that doesn't seem right since waves are up-and-down pulses that move through some sort medium.

Michelson-Morley AnimationSo, for centuries, right into the early twentieth century, physicists stipulated that there was a rarefied medium, called the "ether," which fills the empty spaces in the universe. Light waves are possible in the absence of any other factors because they propagate through the ether. We can't observe it or even notice it directly. It is present in all locations where there is no matter. Among various theories, the one that received the most acceptance was that the ether does not move. However, due to the constant motion of the earth (rotation around its axis and revolutions around the sun), objects moving on the earth should have to adapt to an ether wind. This stream, be it ever so light, makes a positive contribution to the speed of objects pointed in its own direction, and constitutes resistance to objects moving against it. 

But how real is the ether? As solidly as its reality was embraced, it seemed to be immune from experimental testing for a long time, but by the end of the nineteenth century, experiments were forthcoming that questioned its existence. As a matter of fact, at least two other scholars were a hair's breadth away from discovering Einstein's special theory of relativity [Henri Poincaré (1854-1912) and Hendrik Lorentz (1853-1928)], but it was the fact that Einstein had liberated himself entirely from the idea of an ether that made it possible for him to make the breakthrough in a precise, non-gimmicky way. In other words he could do so without introducing some ad hoc property of the alleged ether.

The most famous experiment to refute the existence of an ether was carried out by Albert Michelson and Edward Morley in 1887. It involved splitting a beam of light and having its two halves come back together again. Since each of the two halves was now a beam in its own right, when they converged they were expected to form an interference pattern, as we mentioned earlier in connection with the double-slit experiment, and they did. If the two beams traveled along two different routes, in which one had to go counter to the direction of the ether, it should drag down its velocity somewhat and produce a different pattern than if they were both unimpeded by any resistance from the ether. No such difference was detected. I have drawn my own diagram, but please, if you're interested at all, go to the webpage where there is a professional animation put together by faculty and grad students at the University of Virginia. You can even change the settings in numerous ways and see meaningful results, including the data (here expressed in pixels per frame) you would get if a) either the speed of light were changeable, b) there were an ether moving with variable speeds, or c) if the orientation of the apparatus would make the light move in different reactions.  The last part is important a) in case neither direction in the initial set-up goes directly with or against the direction of the ether, and b) since theoretically, the direction of the ether's resistance should alternate every six months. 

The experiment was repeated under many different circumstances with various light sources, and the speed of light always came out identical regardless of which way the motion of the photons went. Poincaré himself later declared that the idea of the ether was a useless fiction since it contributed nothing to experimental data or predictions. Michelson, however, did not trust the experiment for which he became famous and continued to believe in the ether until the day he died.

Still, if the speed of light is truly constant, then other problems need to be addressed.


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I’ve been doing a lot of writing and erasing trying to present the theory of relativity in such a way that it is both understandable and correct. Please let me know (nicely) if you see any mistakes. And please remember that this is not an essay in physics per se, but that I’m stating my take on developments in physics and whatever impact they may or may not have on a Christian world view.

It seems as though much, if not most, of what follows hangs on the constancy of the speed of light, 3 x 1010 cm/sec in empty space. You may wonder why the speed of light is so important. That’s because this is not a course in physics, and so we haven’t dealt with much of what would precede this material in a physics textbook. Let me put it this way: If you read casual articles on science, you may see statements along the line of: Quantum mechanics becomes crucial on the level of the tiniest entities, such as atoms and subatomic particles, while relativity becomes an important factor when investigating enormous speeds and long distances. From there your mind may run to the cosmic level of distant galaxies and space ships propelled by warp drives on a mission "to go where no one has gone before.” However, the speed of light is also important on the atomic and subatomic level. Remember that visible light is only one small section of the spectrum of electromagnetic radiation, and that ideally electrons or other particles also move at the speed of light. So, in the process of investigating the relationship between electricity and magnetism, first Michael Faraday (1791-1867) and then James Clerk Maxwell (1831-1879) developed some important equations, in which the speed of light plays an important role.

Actually, Maxwell’s equations already started to undermine some of Isaac Newton’s theories. Newton, as I have mentioned several times already, basically understood the interaction between entities in the Cartesian manner as instantaneous pushing and pulling on each other. However, Faraday and Maxwell brought a new idea into the mix: the idea that, at least in electromagnetism, there is no such thing as simultaneous interaction; any contact is “slowed down” by the speed of light. So, for example, in Newton’s system, the kinetic energy of billiard ball A is immediately transferred to billiard ball B when they collide, and there is no time lag between them. When it comes to electric charges, things don’t work that way. Let’s look at what happens when charged particles interact. The effect of one electromagnetic charge on another can be no faster than the event is communicated at the speed of light. That's pretty fast, to be sure, but it's not instantaneous.

As an illustration, imagine that someone is walking through the forest, stumbles over a root on the path, falls down, and let's out a yell for help. Fortunately, there is a boy scout nearby who can respond to the accident immediately. Well, not quite immediately. First the shout must have reached his ears, which would have taken the time necessary for it to reach him at the speed of sound. The box below is a little more accurate, and the speed of passing on a message is the speed of light.



I am going to follow Aktins, (Physics, 438) very closely here because I really don't want to make a mistake,  and I am reproducing one of his diagrams. (The anthropomorphism are mine, and I don't really believe that particles have consciousness and communicate with each other in language.) Look at the illustration below:

charge interaction 1

Find the black ball on the left, and right above it you'll see the designation "q1." q1 represents an electric charge intending to move from location A to location C by way of B. However, at location B a neutral particle is resting, and q1 slams into it. We will call the time at which this occurred t0. The path of q1 is diverted, so that q1 is now heading toward C'. Also q1 releases some electromagnetic radiation, which will affected other charges in its vicinity.

In the meantime, another charge, q2, is tooling along its own way from location D to F. Let us say that at the crucial moment t0, when q1 collides with the neutral particle, q2 is at a location that we can call E.

q2 is going to be impacted by q1's adventure, but not until the electromagnetic energy given up by q1 reaches q2. And the speed of that "communication" is c, the speed of light.

We'll give the label R to the distance that q2 traveled before receiving the message from q1. Then the time at which q2 learns of q1's collision is calculated by adding the time it took for something moving at the speed of light (c) to cover the distance R. In a formulaic expression we get:  ti=t0+R/c.

So, to return to the previous topic, it became important to know to what extent the hypothetical ether might impede or accelerate the velocity of particles. The Michelson (pronounced Mikkleson)-Morley experiment seemed to demonstrate that there was no ether through which light waves needed to propagate, but physicists were slow to accept that conclusion. As mentioned in the last entry, Michelson himself never came to terms with it. Hendrik Lorentz was swayed by it after Einstein’s paper of 1905. Henri Poincaré was toying with the idea of giving it up, but there was just too much temptation to adjust other parameters rather than concede the absence of any ether, and it took him a couple of years or so to abandon it altogether.

According to a biographer, [Albrecht Fölsing, Albert Einstein (New York: Penguin, 1997)] Einstein had been wanting to find a way of getting rid of the undetectable and unproductive ether for about ten years prior to his celebrated 1905 paper “On the Electrodynamics of Moving Bodies,” which you can find in English translation as well as the original German in multiple sites on the web. If you do take the time to look at it, note how little math is involved in the first two sections that actually set up the theory. The further sections on consequences and implementations are a different story.

The problem with giving up the ether was that without it there was no longer any theoretical privileged reference point for observation of motion. As I showed in the last entry, Galilean relativity already stated that the laws of physics hold true in any inertial frame of reference, regardless of its velocity. But with the presupposition of an ether, at least one had a theoretical way of deciding the truth. Things might appear differently to different observers, but in the final analysis there was an absolute standard, as it were. If one could get a glimpse of the observed phenomenon in the ether, one would have an unimpeded picture of true reality behind the different observations. If one yanked the ether, anyone’s observations are as true as any others.

Does that sound like a relativistic epistemology to you? “Anyone’s beliefs are as true as anyone else’s.” “Nobody has a privileged vantage point from which he or she can gain absolute truth.” Well, it may sound like it, but it isn’t. I didn't say "beliefs." The discussion here is very specifically about the observation of bodies in motion, not about beliefs in general. For example, either Einstein’s theory of relativity is true, or it is not. Both points of view on this issue cannot be true. Very few people even have significant beliefs with regard to the electromagnetic properties of bodies moving at or close to the speed of light, do they? The fact that one cannot designate one of multiple observations of motion as the "true" one, does not mean that how people observe such phenomena is arbitrary or due to cultural conditioning. There is nothing capricious about how people observe things; the only limitation is that in connection with physical motion and timing they can only do so from their frame of reference.

Of course, God has an absolute frame of reference, but that’s of little help to us since we cannot share it or, for that matter, use God as our absolute frame of reference when it comes to observations in space and time. God does not have dimensions. He is without time or space (eternal and omnipresent); time and space are a part of God’s creation.

With the speed of light as a constant, certain interesting phenomena occur. Let’s say that two baseball players are riding in a convertible at 60 miles per hour (ca. 97 kilometers per hour). Just to let you know, in case you’re not familiar with baseball, a really strong pitcher may be able to throw the ball as fast as 100 mph (ca. 161 kph), but that would be a rarity. Let’s say that one of players in the car boasts that his throwing speed has been clocked at 101 mph. The other one chuckles and says that he can throw the ball a lot faster than that. He rises as much as he can in the moving vehicle and throws a ball forward in line with the direction of the car. Given the odd conditions, the throw from his arm actually only travels 85 mph, but a nearby policeman with a radar gun registers it at 145 mph. So, the ball was indeed moving a much faster velocity than that of his rival. But the pitcher really cannot take credit for it. The reason is, of course, that from the external observer’s perspective, the velocity of the ball is added to the velocity of the car: 85 mph + 60 mph = 145 mph.

combined velocities

The same addition of velocities will not work with speeds in the vicinity of the speed of light, however. Let us say that, instead of throwing a ball, someone in the car shines a strong flashlight ahead of the vehicle. Would that mean that an outside observer would see the flashlight emitting light at the speed of the car plus the speed of light? That is not possible. The velocity of the light beam is not c + 60 mph. It is c for an external observer just as it is for the person in the car holding the flashlight. It can get no faster than the speed of light.

speed of light

Next time we’ll look at one of the first consequence of the theory of relativity: the idea that relativity supposedly makes some clocks run faster than others.


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How does relativity slow down clocks?

I’ve phrased the question in a way that actually already embeds any number of misconceptions. However, judging by some Q&A sites on the web, it’s apparently the way that any number of folks have been led to think of the phenomenon. Here are some necessary qualifications.

•   Relativity doesn’t do anything. It’s not some kind of entity exerting causal power, but a concept that arises out of an analysis of the manner in which we need to interpret certain observations of the world.

•   The interpretation does not say anything about clocks in general. It’s about observing one or more clocks outside of one’s own frame of reference. Your own clock, presuming that it’s not defective, is always correct. (The German Uhr is a little bit more helpful in this case because it does not distinguish between “watch” and “clock.” Any accurate time-keeping device is fine, though we'll use a really unusual one in a few moments.)

•   Speaking of being picky about the language we use, let me guard what I’m saying a little bit more. Strictly speaking, Einstein did not say that there could not be an ether or other absolute frame of reference, but he said that it was superfluous and rejected it as playing any role in our understanding of the motion of bodies. (“On the Electrodynamics of Moving Bodies,” 1).

Up to now I have left the distinction between Galilean relativity and Einstein’s special theory of relativity somewhat blurred. Let me state the difference now succinctly. Atkins, (Physics, 450) defines the classical (i.e. Galilean and Newtonian) principle of relativity as

The fundamental laws of physics are independent of the velocity of the observer.

For Einstein’s special theory of relativity we can go to the man himself (“On the Relativity,” 4). His version includes two principles. The first one is pretty much the same as the Galilean version, just expressed with more technical language.  The second one is the addition of the constancy of the speed of light.

1)  The laws by which the states of physical systems undergo change are not affected, whether these changes of state be referred to the one or the other of two systems of co-ordinates in uniform translatory motion. [“Uniform translatory motion” refers to the supposition that the systems do not change velocity. If they accelerate the principle becomes a whole lot more complex. That eventuality will be encompassed by the “general theory of relativity.]

2)  Any ray of light moves in the “stationary” system of co-ordinates with the determined velocity c, whether the ray be emitted by a stationary or by a moving body. Hencespan>

Velocity Formula

So let us illustrate this idea with an example, based on Einstein’s own paper, but with your creative bloggist’s slight reworking of the details. First of all, I’m going to just “talk” you through this thought experiment. Then, for those who are interested, I’ll try to specify the math involved in describing the result.

Imagine that there are two scientists, call them Jeff and Tony, at the controls of two spaceships. They are able to communicate with each other. For purposes of this example it is not necessary that they would be flying anywhere near the speed of light. Each of them has a clock on board, and each of them can see his own clock as well as the other’s clock.

Light clockNow, when I say clock, I’m not thinking of the apparatus that we usually call a clock (e.g., a watch, a grandfather clock, a wall clock, etc.), though, again, any of them would theoretically work as long as they’re not defective, but would not be as suitable for a direct explanation of this topics. The clocks we have in mind are rods, each one exactly one meter long, and each one of them sporting a light-emitting source on the bottom end as well as a mirror on the top and one on the bottom. The source emits a beam of light that hits the upper mirror and reflects it down to the lower mirror. Since the speed of light is a constant, this back-and-forth oscillation will always take exactly the same time in any inertial system. We'll call the amount of time it takes for the light to make one up-and-down trip a “tick.” Quantitatively, a tick is the time that it takes for light to move up and down one meter for a total of two meters or 200 cm. 

Now, let’s say that Tony remains stationary and observes his own clock. Throughout this experiment his clock never changes. The light beam travels up and down consistently following its vertical path of 2 meters, with each up-and-down stroke constituting a tick.

Then he looks over at Jeff’s rocket. Jeff’s clock functions in the same way: The time of one up and down sequence by the beam along the rod constitutes one tick. However, because Jeff's system is in motion, Tony does not see the light pursuing a neat up-and-down path on Jeff’s pole. Keep in mind the earlier illustration of what an outside observer sees when a ball is tossed up in the air on a moving train. Similarly, the movement of the light takes place in a more complex manner. As it moves up and touches the upper mirror, Jeff’s spaceship has already moved a certain amount, and the same will be true when the light returns to the bottom mirror. At each interesting juncture, Jeff's frame of reference has moved along a short distance. So, what Tony sees is a triangle-like configuration for the light beam. Just by looking at a drawing of the scenario, we can see that Tony’s light has had to travel farther than Jeff’s within the time span of one tick. Even if the angle at the midpoint is a whole lot steeper than I have drawn it, an angle is there.

Spaceship relativity

Consequently, one or the other must be true:

Either Jeff's light beam traveled faster than Tony’s in order to complete its movement, or Jeff’s tick was longer than Tony’s in order for the light beam to complete its motion in the span of one tick.

Well, the speed of light is constant, and so we have to discard the option of the light speeding up. Thus, we are left with the other alternative. Since the light had to travel a longer distance on Jeff’s space ship than on Tony’s in one tick, Jeff’s tick must last longer than Jeff’s. Therefore, Jeff’s clock runs somewhat slower. How much longer Jeff’s tick is, compared to Tony’s, obviously depends on how fast he is moving with respect to Tony.

At least that’s what Tony observes. If the roles were reversed, and Jeff’s frame of reference were stationary while Tony was zipping along, Jeff would observe that Tony’s clock was slower than his for the same reason. It comes down to who examines the other one’s clock. In each person’s own frame of reference the tick is going to be 200 cm divided by the speed of light, while in the observation of the other’s it’s going to be a little longer than that.

To summarize, since the beam of light is observed to go through a greater length in one system than in the other, and since the length of time is defined by the time it takes for the beam of light to complete its path, the clock outside of the observer’s frame of reference will be found to be slower than the observer's in order to accommodate the additional distance the beam of light had to traverse.

Okay, let’s look at the math, which in this section of Einstein’s paper is comparatively manageable, though my math never comes with a guarantee of being mistake-free. Once again I’m following Atkins (467-69) pretty closely.  

Tony’s system is stationary, his rod is exactly 100 cm long, and the speed of light is about 3x1010 cm/sec. We apply the equation

t=l/c = 200cm/c.

So the up and down journey of the light is 2x102 cm divided by 3x1010 cm/sec, which works out to 2/3x10-8 seconds, or better 6.7x10-9 seconds.  That’s the length of the tick in Tony’s immediate environment: 0.0000000067 seconds. Now let’s look again at what Tony sees on Jeff’s clock. Jeff’s spaceship is moving along at a velocity, for which we can’t give it a number, but just simply call v.

For our present purposes, light travels in straight lines. Let’s call the point where the beam originates A, the point where it makes contact with the upper mirror B, and the point where it makes contact with the lower mirror C. We’ll refer to a “tick” simply as time, abbreviated as t.

We want to know how much longer Jeff’s tick is compared to Tony’s, and to arrive at that conclusion, we'll leave behind the pictures of rockets and just think of the geometry of the path of light as a triangle.

Triangle 1

There is no justification for supposing that any of the angles are right angles, but we can bisect the triangle at point B and refer to the midpoint between A and C as D. Then the angle at D is a right angle, and we can do some good old trigonometry.

Triangle 2

We know that the line BD is exactly 100 centimeters long. The hypotenuse is line AB. So, we can formulate the other two lengths algebraically, since we know that in each case the length it took the light to move along that line is ½ t. So, the length of AB is one half of a tick multiplied by the speed of light.


The ship’s velocity is not the speed of light; we can only give it the generic variable v. We can say that the length of AD is one half tick multiplied by the velocity of the rocket.


Now we can bring out the Pythagorean Theorem according to which


and substitute the more complex formulas we just came up with:

(½ tc)2 = (½ tv)2 + 1002

Subtracting (½tv)2 from both sides we get

(½ tc)2 – (½ tv)2= 1002

A fundamental rule of algebra is that whatever you do to one side, you have to do to the other. But that applies only if we do something that changes the values. We can rearrange one side of an algebraic equation as long as it still says the same thing. So, we can factor out ½ t from the left side and derive

¼ t2(c2-v2) = 1002

Let’s get rid of that silly fraction by multiplying both sides by 4.

t2(c2-v2) = 4x1002

Next, we’ll divide both sides by (c2- v2):

t2 = 4 x 1002
       (c2- v2)

We happen to know that (4x100)2 = 2002, so we can write:

t2 =     2002
         (c2- v2)

And if we factor out c2 from the bottom of the fraction we get:

t2 =        2002           
             c2 (1 – v2/c2)


Now, all we need to do is take the square root of both sides and we arrive at our goal (more or less):

t =        200         
     c √(1-v2/c2)

(The square root sign is supposed to cover the entire expression (1-v2/c2). Now if we want to express the length of a tick on Jeff's clock as seen by Tony, we can express it as a ratio, in which we can ignore (or set as "1") the vertical length and get:


That's the length of a tick on Jeff's clock according to Tony's frame of reference. We don’t know the number for v, but we are allowed to assume that there are no imaginary numbers involved. Therefore the denominator of that fraction is less than 1, and the total value of the fraction is larger than 1. Therefore, time has expanded or, to put it the other way around, Jeff's clock appears to be slow.


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Who's on Second; What's on First. One of the fascinating consequences of the special theory of relativity concerns the sequence of events. More accurately it leads us to the conclusion that under certain circumstances, observers may see some events in reverse order. Did A come before B, or did B come before A?

Let me state right now that the theory of relativity does not allow time travel, visions into the future, or an effect preceding its cause. We may have to come back to those points if necessary.

What the theory does lead us to conclude is that different observers may see certain events in different sequences. There's nothing particularly spooky about that. We are used to the fact that people may hear sounds at different times because sound has a definite speed. For example, if you were to set up a sound system for a large theater, you would have to take into account that there will be a lag in the time between the speech an actor makes on stage and when the people in the last row hear it.

Once again, the matter hinges on the two givens of relativity theory: there is no privileged point for absolute observations, and the speed of light is constant.

Again, even though I can think of lots of my own illustrations (and you probably can as well, at least when we're done here), I'm going to borrow one from Atkins, Physics (471-74), so that I can feel secure that I'm not just running away with my imagination. Needless to say, I'm still telling the story itself in my own way.

Let's bring out once again our intrepid rocket scientists, Jeff and Tony. Today Jeff has decided to do an experiment involving a sequence of flashing lights. To this end, he acquired a long metal girder, made sure that he had identified its exact center, and suspended it with precision below his spaceship. When you're dealing with the speed of light, accuracy is imperative. At each end he mounted a packet of explosive material that will, upon ignition, give up an extremely bright momentary light. In the meantime, Tony is hovering in his own space ship a good distance away.

light sequence setup

In the diagram, J stands for Jeff, T for Tony, M indicates the middle of the pole, and the lights are installed at points P and Q. When Jeff pushes a button, he triggers two rays of light that move in opposite directions from the center M to the ends at P and Q. When they reach the packets, they will set off the explosions, and both of our adventurers will keep track of when they see the lights.  

Since Jeff sits equidistant from the packets, he will notice the two lights flashing simultaneously.

simultaneous lights

But not so Tony. Due to his position in relationship to P and Q, he is going to see the explosion at P before the one at Q. Here is how it will appear to him:

sequential lights

Now Jeff decides to make things more interesting. He adds a third exploding packet to the beam at point R, a little closer to the center than Q. The same impulse will trigger the new flash to go off.

additional setup


What Jeff will see is the explosion at R, followed by the simultaneous explosions at P and Q.

light sequence 3

What will Tony see? If he is stationed properly, which he must since this is a thought experiment, his sequence will be: first P, then R, and then Q. So, like Jeff, he sees R before Q. But he sees P before R, while Jeff saw R before P. Thus, the frame of reference determines which event has been observed prior to another one.

light sequence 4

By my constantly qualifying the subject matter to the effect that it is all about observations and perceptions, hopefully you see that there is nothing illogical here. You have a contradiction or inconsistency when two statements cannot both be true at the same time in the same sense. Clearly, statements by or about the perceptions of Tony and Jeff do not fulfill the requirement for having the same sense because they come from different frames of reference. Furthermore, "at the same time" has taken on a whole new meaning in this context.

But, just for fun, let me add another part of this scenario as Atkins actually presented it and amplify it somewhat. In his version, the explosions at the ends of the beam are triggered by the rays of light caused by a small explosion at M.

Now, I am going to add the stipulation that this explosion is also visible to both investigators, but I'm not going to insert animations for this part because what I'm going to do with it is not possible. On this hypothesis, Jeff's sequence of observations would be M, R, and then both P and Q simultaneously. So far, so good. Tony's sequence, on the other hand, would be P, M, R, Q.  Aha! There's the hiccup: The event at M caused the event at P. But then, if Tony saw P before M, he actually would have observed an effect before its cause.

However, that cannot happen, and not just on the grounds of philosophical dogma. If the light from an explosion at M caused the explosion at P, then we cannot forget about the time that the light took to travel from M to P where it set off P's explosion. Thus, there is a built-in time lag that cannot be overcome. Necessarily, since we're talking about the speed of light, Jeff's steel girder must be extremely long so that all of the explosions don't appear to have occurred simultaneously to any observer. And, if that's the scale we're having to work with, it should be evident that the distance between M and P must be long enough so that Tony cannot see M before P. (In the pure abstract reality, the scale does not matter.)

The point is clear, I hope. The principle of relativity describes the universe in such a way that different people have different perceptions about some hypothetical sequences of events. But the fundamental order of existence, in this case the necessary succession of causes and effects, is left intact.


physics logo

[Friday afternoon: I have added some pictures and diagrams as promised. I really wanted to get this information up late last night and continue the series without too much of a hiatus. I think that you could probably follow it without the additional help, but I may be giving my explanatory powers more credit than they deserve. Regardless, below are some pictures, and thanks for your patience.]

Let me begin by stating where I am heading with this entry. If you’ve heard anything about Einstein’s general theory of relativity, you may know that it involves gravity. Traditionally, gravity has been considered a force, and a force has been defined as the mass of an object multiplied by its acceleration. And it is actually acceleration that the theory addresses. Gravity happens to be an all-pervasive instance of acceleration, and we need to work with it, but it’s acceleration that we need to nail down a little bit first.

And so we move on from the special theory of relativity. It is called “special” because it only applies to a certain set of circumstances, namely when the frames of reference involved are moving at steady velocities, which includes being at rest. But what would happen if we extended the same principles to frames of reference that are not at rest or not moving at a constant velocity? We call in Einstein’s general theory of relativity, which he finally finished to his satisfaction in 1915.

I dare say that many, if not most, material bodies that we are likely to encounter as we trudge through this weary world are frequently changing their velocity. In the language of physics, they are “accelerating.” However, that does not mean that everything is speeding up.

Acceleration as used in physics is not a tricky concept, but we need to be clear on its meaning and be able to distinguish it from its ordinary use. For instance, when we say that a car is accelerating, we most likely mean that it is increasing its speed. If a vehicle decreases its speed, we might just say that it is slowing down, or we could also get a little fancier and announce that it is “decelerating.” Actually, "deceleration” is not a very useful expression for our purposes since we want to be able to compare apples to apples, and so we’ll talk about “negative acceleration” instead of “deceleration.”

I just used the word “speed” several times. As an auto racing fan, it’s a “purr” word (as opposed to “snarl” word) to me. Let’s, however, use the word “velocity” instead for the moment. Scientists make a distinction between “speed” and “velocity.” Speed is simply the rate at which you are moving regardless of your direction. In your car, that’s what your speedometer shows. If you drive a newer model car, it may also have a directional indicator or compass. When you combine the speed with the direction, you are expressing the velocity. For example, your speedometer may tell you that you are going 70 mph. Your compass reveals that you are heading straight northeast. Put the two together you have your velocity, i.e. 70 mph northeast. Thus, velocity is represented by an arrow of a certain length and direction, which is called a vector. The number that determines a vector’s length is often called its “scalar.”

We can represent a vector by indicating its length and direction in a coordinate system or by drawing it really nicely with a protractor, ruler, and a sharpened number 2 pencil (or some other graphic apparatus). The important matter for us is that any change in the vector is considered an acceleration. The vector can get shorter or longer and it can shift its direction; any such alterations are considered accelerations. An interesting point in this connection is that an object moving in a circle is always accelerating because it is always changing directions, even when it maintains a constant speed.

Please forgive me for stating the obvious in the next few paragraphs; I’m just trying to make sure that we will be on the same page as we move along. An object’s velocity is measured by dividing the distance it has moved by the time it took to traverse that distance.


I know that this is old stuff. You probably learned that in fourth grade (or were supposed to, but maybe you had to miss that week with measles--[Oh no! I wrote that before Nephew Michael's FB post called my attention this article]). But please bear with me. Let’s make things a little bit more exciting by inserting much higher speeds. Buckle your seat belt.

Let’s say that I drove exactly 150 miles, and it took me exactly one hour. During that time I departed from neither my exact speed nor from my direction. Then my velocity was 150 miles/1 hour --- and, again, you knew that. Now please permit me to change to the metric units I’ve been using all along (the cgs system: centimeter/gram/second). 150 mph is 242 kilometers per hour (kph) rounded off, which I looked up on one of the many conversion utilities on the web.





10 million

10,00,00,000 (1 crore)


1 million

10 lakh


1 hundred thousand

1 lakh


10 thousand



1 thousand



1 hundred







0.1 = 1/10  


0.01 = 1/100  


0.001 = 1/1000  


0.0001= 1/10,000  


0.00001 = 1/100,000

1/1,00,000 = 1 divided by 1 lakh

I’ve used scientific notation before without giving a long explanation. It comes in rather handy, particularly when you’re up against both the colossal and infinitesimal numbers that come into play in relativity and quantum mechanics. Look at the table on the right for a list of equivalents. So, instead of saying 242 kph, I’m going to write it as 2.42x102 kph. A kilometer consists of a thousand meters (103 meters), and a meter has 100 (102) cm. Then one kilometer equals 100,000 (105) centimeters, so my distance for that hour would have been 242 x 105 centimeters. We prefer to have a first-order number before the power of ten, so we’ll say that our distance was 2.42x107 centimeters. If you insist that I should use plain English when I could say it in a more complicated, less intelligible way, I can tell you that I covered a little over 24 million centimeters during the allotted time of one hour. (Actually, once you’re used to it, scientific notation is easier to manage than writing out numerous zeros before or after a decimal point, just as symbols in logic or math may scare you away at first, but later on become handy short cuts.)

Speaking of time, we know that an hour has 60 minutes, and a minute contains 3,600 seconds. So, the time in which I covered that distance is 3,600 seconds or, 3.6 x 103 seconds.

Consequently, if I want to express my speed in terms of cm/sec, I need to divide 2.42x107 cm by 3.6 x 103 sec, which turns out to be 6.72 x 103 cm/second. So 150 mph roughly equals 6,720 cm/sec.

Now, we said that I drove the car for the entire hour at exactly 150 mph in a straight line. Thus there was no acceleration to worry about. However, if I changed speeds (either faster or slower) or direction, then my car has undergone acceleration (either positively or negatively), and we can calculate what the acceleration was. We do so by dividing the change in the velocity by the amount of time that the change took. The delta sign is frequently used to stand for “change.” Thus, we can shorthand a change in velocity as Δv. We get its value by subtracting the original velocity (v0) from the later one (v1). Thus,

                                                        Δv=(v1 - v0)

Similarly, in order to arrive at the amount for the time, we subtract the final time from the time at the outset of the change.

Δt=(t1 - t0).

Then the acceleration is

Δv / Δt     or

( v1 - v0)  /  ( t1 - t0)

If there was no change in velocity, then Δv =0, and there clearly was no acceleration. If no time was used up, there is nothing to calculate since we cannot divide by zero. However if we applied a mathematical device found in calculus and some other areas of math, we could say that

“the limit of (t1 - t0) approached 0,”

and then the change in velocity would approach infinity/sec, which doesn't mean anything since a change of 18 kph cannot be an infinite quantity. Furthermore, I don't drive that fast.

So, to return to the example, if I was moving at, say, 242 kph and then sped up to 260 kph within 5 seconds, the equation or my acceleration would be

(260 kph - 242 kph)/5 secs =

18kph/5sec =

18x105 cm/sec/5sec .

Let’s divide 18 by 5, and we get 3.6. The acceleration was

3.6 x 105 cm/sec/sec.

This business of making the units be cm/sec/sec is a little awkward. Instead, we can write that the acceleration was

3.6 x 105/sec2.

"360,000 centimeters per second squared"

If you prefer to write it all in one line without trying to display a lengthy divisor line from your word processing program you can also write it out as

3.6 x 105 sec-2.

A negative exponent signifies that it is the denominator of a fraction.

Thus, the car increased its speed at a rate of 360,000 cm/second for each second. That’s a lot, but we should keep in mind that the actual speed of the car was higher by two orders of magnitude: tens of millions compared to hundreds of thousands. (For my South Asian readers: crores vs. lakhs.)

Now, let’s talk about gravity, one of the most ubiquitous and yet elusive forces of nature. The sun is restricted in its motion by the gravitational forces exerted by other stars in our galaxy, the Milky Way. The earth is maintained in its revolution around the sun by the sun’s gravity. The earth’s gravity keeps me sitting on the couch, and my laptop rests on my thighs due to—no, not my gravity, but the earth’s as well. Even though there are gravitational forces between any two masses such as me and my computer, they are far too small to be significant. The simple Newtonian formula for a force is

F = ma,

where F stands for the force, m for the mass of the object, and a for its acceleration.

Pound WeightWe need to refine our understanding of "mass" some more. If you took high school (or perhaps even college) physics or chemistry, you may have had it drilled into your head that there is a difference between an object’s mass and its weight, and that scientists are more interested in mass than in weight. The difference is that mass is a property of the object itself, which can be ascertained in comparison with other objects, whereas weight is the interaction between an object and gravity. Mass is ascertained by scales. Weight is actually measured rather infrequently. I remember from my childhood days way back in Germany that the gentleman who came around from time to time collecting rags and old paper would place the materials in a sack that was attached to a measuring stick by a hook at the bottom of a spring. The spring, whose strength would presumably have to have been calibrated, pulled down an indicator on the ruler. The more objects were in the sack, the lower the pointer would go, the greater the weight it registered, and the more pennies one would receive for the contribution. So, in that case, it actually was the effect of gravity on the filled sack that measured its weight.

Now wait! (No pun intended.) Didn’t Galileo prove that all objects, regardless of their size fall to the ground with the same acceleration? Yes, he did. Ignoring wind resistance and other interferences, a pea and a bowling ball would drop to the earth with the constant acceleration of gravity. But remember, the acceleration is only one part of the force. Assuming you were standing underneath Galileo’s perch on top of the Tower of Pisa, and you had to choose between either the bowling ball or the pea to land down on your head, I recommend that you choose the pea. The pea's rapid and constant acceleration, which is equal to that of the bowling ball, will still do a lot less damage. It has far less mass than the bowling ball, giving it a far smaller force, resulting in less damage to my skull. The mass of an object may not make a difference with respect to the acceleration caused by gravity, but it certainly makes a huge difference in the impact when it lands somewhere.

But wait again! Since we’re talking about Galileo cleaning out the attic of the Tower of Pisa by dropping stuff from its top,  aren't we actually referring to the objects’ weights and not their masses. What gives?

Here we have one of the first planks of Einstein’s general theory of relativity. He called it the “principle of equivalence,” and it came down to the idea that mass and weight, though not identical, had equivalent functions in the universe and would be indiscernible in a non-rotating frame of reference. Let’s be a little more precise and engage in another thought experiment. (I am adapting the scenario describe by Atkins in Physics, 506-508.) By means of our imagination we place ourselves into a space ship. At first, it stands at rest on the ground (the rocket on the left in the picture below). If I drop a ball (literally, not figuratively) it will fall with the ever-increasing speed brought about by the acceleration of earth’s gravity, which, for now we can call g.

rocket in gravitational field

Rocket no acceleration, no gravity

rocket acceleration resembles gravity

Having measured the acceleration and speed of the ball, we take our spaceship to a distant place, far enough from any celestial bodies so that there is no gravitational influence on the ball or the space ship (the rocket in the center). If we are moving at a steady velocity, then I cannot drop the ball. If I were to release it, it would simply hang in the air where my hand had held it. Neither the ball nor the space ship would have enough mass to exercise significant attraction on each other.

Let’s accelerate the space ship at exactly the rate equal to the acceleration due gravity on earth (the rocket on the right). The acceleration (a) has the value of -g just to remind us that, if there had been a gravitational attraction, our acceleration vector would have to be negative with respect to g. In this case there actually isn't any, and so the only thing that actually counts is g's absolute value.  Now if I hold up the ball and let go of it, the spaceship will move away from it, and things will look to me exactly as though I had dropped the ball in earth’s gravitational field. Thus, according to this insight, which became known as the “principle of equivalence,” an observer would not be able to tell whether he is in a gravitational field or if his frame of reference is accelerating at precisely the value of the gravitational force.

That insight brings with it a number of implications, which we will look at briefly before we're done.