Norton’s Dome (Column 687)

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With God’s help: Norton’s Kippah

A few days ago I was sent, for the Q&A section on the Paradox website, something I had not previously known, known in the trade as ‘Norton’s dome’. It challenges the deterministic conception of physics, and really of Newtonian mechanics. This is a fascinating case that stirred many thoughts in me, and at the request of several readers I will try to share them with you. I should say at the outset that this column is a bit technical, and I use mathematical symbolism in it. I will try to explain it here as best I can, but I am not sure it will be clear to someone who is altogether unfamiliar with it.

A brief look at determinism in physics. I have often dealt here with the question of determinism. This is the view according to which the state up to the present uniquely determines the future, that is, that it is impossible for the same history to yield two different futures. This question bears on our attitude toward the human being, and on his relation to the rest of nature: the inanimate, plant, and animal realms. It is commonly assumed that nature is certainly deterministic, and the question concerns only the human being: is he exceptional, possessing free will, or not? From the beginning of the twentieth century, quantum theory began to develop, demonstrating non-deterministic domains within physics itself. Later came chaos, which is also often presented as a deviation from physical determinism. I have written more than once that with respect to chaos this is a mistake. There is nothing non-deterministic there. What there is, is our computational difficulty in predicting the future, but the future itself is uniquely defined by the history. In quantum theory, by contrast, at least in the accepted interpretations, there indeed seems to be a departure from determinism. But regarding classical physics, at least after ruling out the matter of chaos, there is broad agreement that it is a completely deterministic theory.

The properties of Newtonian mechanics: determinism and reversibility. The laws of physics, and in particular Newton’s mechanics, are described by differential equations that determine the state at the next moment on the basis of the state that prevails in the present. This means that one can, in principle, determine the future categorically on the basis of complete knowledge of the present. Of course, if we lack information about the present, we cannot know the future (this is exactly the situation in chaos), but that is only a technical problem. If we had full information (and sufficient computational ability), we would know everything that is going to happen. Let me preface this with a brief explanation of the concept of the derivative for those unfamiliar with it. If I have a function that depends on another variable, for example the position of a body, X, which depends on time, t, I may be interested in its rate of change over time. Does it change quickly or slowly, and by how much? Newton and Leibniz formulated the concept of the derivative, which is a mathematical concept representing the magnitude of the change in X

with time and its direction (change upward or downward, that is, whether the position increases or decreases). The derivative of position with respect to time is defined as: (1) 𝑋̇ = ∆𝑥 ∆𝑡

where ∆x is the change in position and ∆t

is the change in time. Thus, if the body underwent a change in position of 10

meters over a time interval of two seconds, the rate of change of the position is 5

meters per second. That is defined as the body’s velocity. Velocity is the rate of change of position over time. Of course, we are familiar with motion at constant velocity (the rate of change is the same), but it may also happen that the body continually changes its velocity. In such a case we have to consider very small intervals of time (∆𝑡→0) and ask what the rate of change is in each tiny interval of time. The result may be different at each point in time along the path. If we perform the same trick on the velocity itself, that is, on 𝑋̇, we obtain the rate of change of velocity with time, which is the body’s acceleration. This is the second derivative of position, and you will surely not be surprised to hear that we denote it by 𝑋̈

We can now understand the second law in Newtonian mechanics, which determines a body’s position through the following equation: (2) m𝑋̈ = F(X,t)

where X is the body’s position, and m

is its mass, and F

is the force acting on it, which itself may depend on position and time. The two dots above X denote, as stated, the second derivative with respect to time.

This is a differential equation, that is, an equation whose unknown is a function (in our case

X(t)), and it also contains its derivatives. Solving this equation means finding the body’s trajectory, that is, its position at every point in time, in other words, the function X(t).

It is not hard to see that this equation describes only the change in position (that is, the position after some time has elapsed

in terms of the position at time t). Therefore, if we want to write the position X

itself at every point in time t, this depends on where we started (the initial position) and what the velocity was at the beginning of the path. From that point onward, the subsequent dynamics are described by the above equation of Newton’s second law. It follows that in order to determine uniquely the trajectory describing a body’s position over the entire time axis, we must know its initial position and velocity, and then the differential equation of the second law will carry us forward. If we do not have the initial position and velocity (or the position and velocity at any other point in time), then we do not have complete information. In such a case we will not be able to know the body’s position uniquely, but only because of lack of information. Given the initial conditions, this equation uniquely determines everything that will happen thereafter.

Another property of Newtonian mechanics is symmetry on the time axis, that is, reversibility. Any legitimate mechanical process that we observe, if we run the film backward we obtain another legitimate process. In other words, mechanics is indifferent to the direction of time. As far as it is concerned, time could just as well flow backward and nothing would change. Indeed, the puzzle of the directionality of the time axis (why it flows from past to future rather than the reverse) occupies quite a few philosophers and physicists. The reason for the reversibility of mechanical processes is that the differential equation describing motion in Newtonian mechanics is of second order, that is, the position is differentiated twice with respect to time. This is roughly like dividing by t squared, a factor unaffected by reversing the direction of time, since for positive or negative t one always gets a positive result. Therefore, whether time flows forward or backward, the result is the same. Think of a small ball standing on the top of a mountain. At some point it is released and begins to roll downward. Its speed increases with time until it reaches the bottom, and it continues moving on the plain. If we run the film backward, we will see a body moving quickly on the plain, reaching the foot of the mountain and beginning to climb. Its speed decreases with time until it reaches the top of the mountain and comes to rest there. This is literally a mirror image, on the time axis, of the path I described earlier. This is the meaning of time reversibility in Newtonian mechanics. The conclusion is that according to Newtonian mechanics, if we have a body’s position and velocity at a certain moment, we can know its position and velocity at any future moment (and also in the past), that is, the entire path uniquely. This is the essence of determinism in Newtonian mechanics. A body’s path, insofar as it is governed by Newtonian mechanics, is both uniquely determined by its present state and reversible in time.

Norton’s kippah. Norton was a well-known designer of kippot in Meah Shearim. He had been given his name in memory of his two forebears, Nora and Tony, of blessed and holy memory, prominent members of the local Satmar community. One day, as he sat and daydreamed in his shop, his grandmother Nora, of blessed memory, appeared to him in a dream and instructed him to design a new kippah with cylindrical symmetry, whose shape is as follows (this is, of course, a cross-section, and it is the same cross-section in every direction):

A cross-section of Norton’s kippah, where the vertical axis is Z (the height, denoted here by h) and the other is the horizontal axis. These two distances are measured in units of 2𝑔2 3𝑏4

This same Nora even gave Norton the formula for the profile of the kippah, which was naturally in cylindrical coordinates (as befits a kippah whose shape looks the same in every direction):

(3) 𝑍= 2𝑏2 3𝑔 𝑟3/2 ; 0 < r < 𝑔2 𝑏4

where b is some constant, and g

is the acceleration due to gravity (a number whose value is about 9.8).1 Nora explained to him that r is not the horizontal distance from the Z-axis but the geodesic distance, that is, the distance one travels from the top of the kippah down along the kippah itself. The geodesic distance from the top of the kippah to the bottom is r = 𝑔2 𝑏4

(Note that in the drawing the horizontal axis is X and not r, that is, this is a cross-section. We are referring to it as a three-dimensional dome, where r is the geodesic distance from its summit.) You can now understand that the width of the kippah is, of course, in the units described at the bottom of the figure.

Norton, of course, knew the material thoroughly (in his Toldaot Avraham Yitzhak school, formulas in cylindrical coordinates were core kindergarten material), and he immediately built the kippah in question and set it in his shop window. At that very moment Elijah the Prophet happened into Norton’s shop. He placed a point-particle exactly at the apex of the kippah. Norton, who was very well versed in Newtonian mechanics, expected one of two things: either the little ball would remain standing there forever at the top of the kippah (for those who hold that exact precision is possible in the hands of Heaven

see Bekhorot 17b), or it would immediately begin to slide down the side of the kippah (for those who hold that exact precision is impossible in the hands of Heaven). And indeed the ball stood there for about an hour, and a heavenly voice went forth and said: exact precision is possible in the hands of Heaven. But Norton had a difficulty with this, because there is friction on this kippah, and Elijah at once pulled from his pocket a wonder-paper and polished the entire kippah until there was no friction on it at all (as the saying goes: Tishbi will polish kippot and frictions). After that he again placed the ball on top, and behold, a wonder: the ball continued to stand there. In the meantime they were almost ready to conclude that exact precision is possible in the hands of Heaven, when suddenly, without any prior warning, the ball slowly began to roll downward.

The whole study hall was in an uproar. Mighty ones and humble cliff-dwellers contended, yet the holy ark was not captured; the heavens bent down, the truth of the waters took wing, and the inhabitants of the study hall raged. That little ball had acted in accordance with no one’s view: according to the opinion that exact precision is possible, it should have stood forever, and according to the view that exact precision is impossible, it should have rolled immediately. Thus the matter has remained wondrous to this very day. The bewildered Norton announced throughout the holy city of Jerusalem that whoever solved the riddle would receive his daughter in marriage and inherit his shop (unless bandits should burn it down for the sin of engaging in outside wisdoms, Heaven forfend). And the kippah remained with its maker and was sold to no one, until it was forgotten by all. Many years later, a certain sage happened into the shop of old Norton and found the solution to the matter (and ever since he has been known as Peter the Great). He pulled from his bag a mechanics textbook (an appendix to the Yiddish translation of Eyal Meshulash by a disciple of the Vilna Gaon, from whom no secret was hidden), and wrote Newton’s second law for this case. Taking into account the fact that the only forces acting on the body are gravity and the normal force of the kippah (recall: there is no friction), the motion is of course in the tangential direction to the kippah (and it is accepted that the force in that direction comes out to 𝑏2√𝑟, for obvious reasons). Naturally, this equation has cylindrical symmetry (there is no dependence on the angle). We therefore arrive at the following rule:

(4) 𝑟̈ = 𝑏2√𝑟 This is a differential equation whose solution gives the geodesic distance of the little ball from the top of the kippah as a function of time. As noted, to define the trajectory at every time we must add initial conditions. In our case, the little ball begins by standing at the top of the kippah, and therefore the initial conditions are: r(t=0) = 0 ; 𝑟̇(𝑡= 0) = 0 It turns out that this equation has two solutions. The first is:

(5a) r(t) = 0 This is the solution in which the particle remains standing at the top of the kippah the entire time and does not move.

But it also has a second solution: (5b) r(t) = { 0 ; 𝑡≤𝑇 144 [𝑏(𝑡−𝑇)]4 ; 𝑡≥𝑇

One could have written simply some constant K here. It is more convenient to express it with g, and for that we have to introduce a proportionality constant b.

This solution says that the particle waits, standing at the top of the kippah, for some time T, and from that point onward it begins to roll downward. The time T

is arbitrary, that is, it can be anything. In other words, there are really infinitely many solutions here (for every value of T, and the first solution is simply the second when T → ∞).

That is, the little ball can decide to roll at any moment that comes to mind, and there is nothing that determines for it when to do so, if at all. Moreover, because of the cylindrical symmetry, it can roll downward in any direction that comes to mind, that is, at whatever angle it wishes. One can see the multiplicity of solutions through the reversibility of Newtonian mechanics. Suppose the particle is moving at some speed from below toward the kippah, and it climbs it while slowing down. One can tune its initial speed so that when it reaches the summit it stops there exactly. This will of course happen from any direction from which it approaches the kippah at that speed, because of the angular symmetry. If we return to the reversibility of mechanics, this means that if we run the film backward, we get an equivalent solution to the problem, namely infinitely many solutions in which the little ball rolls in every possible direction and begins to do so at any time T

it wishes. This also allows you to understand why this paradox cannot be formulated for an ordinary spherical dome. In a spherical dome, if the little ball reaches it with a speed that causes it to stop exactly at the top, it must continue to climb and slow down more and more until it stops on the dome. It turns out that on a spherical dome this process takes infinite time, and therefore its reversal cannot begin. A little ball that stands at the top of a spherical dome will never begin to move downward, because the beginning of the motion would be after an infinite time (this parallels the first solution to our problem: T → ∞). What distinguishes Norton’s dome is that the ascent time of the little ball from below until it stops at the top of the dome is finite, and therefore the reverse solutions are admissible and occur in finite time. This is the reason we had to define the shape in precisely the special way given in formula (3).

And when Norton beheld this wonder, he took the magical kippah and beheld another wonder: the author of Chiddushei Ha-Rim had already foreseen all of this through his holy spirit (and some say he received it by tradition from the Kotzker), and thus was born the Gur kippah with the bump in the middle.

To the problematic point itself: a first look. Up to this point, this has been a mathematical calculation. But Norton’s dome is considered a paradox, not merely a strange case. The reason is that it contradicts the assumption of determinism in Newtonian mechanics. If one looks at these solutions, one sees that in a given state with well-defined initial conditions, several solutions emerge. The little ball can begin to roll at any time and in any direction it wants (or not roll at all). What, then, causes it to choose a time and direction? Nothing. This is determined arbitrarily, a kind of lottery. It follows that the path of the little ball’s motion has no reason, and neither does the very beginning of the motion. It simply suddenly decides to move without anything having changed in its environment, and it chooses a direction without there being any reason to prefer that direction over others. It seems as though it has free will, or perhaps merely a randomizing mechanism. This contradicts the deterministic character of Newtonian mechanics that we encountered above.

I have pointed out that when we encounter a paradox, three possibilities stand before us: to find a mistake in the logical course of the argument; to give up one of our assumptions; or to give up the conclusion contradicted by the paradox (that is, to see in it a proof by contradiction that we were mistaken).

I noted there that there is also a fourth possibility: to remain with the matter unresolved, that is, not to give up either the assumptions or the position contradicted by the paradox, and to assume that there is some flaw in the argument even though for the time being we are not finding it. How does all this apply in our case?

The first possibility is probably not correct. This is a mathematical calculation, and it appears entirely correct. The second possibility means giving up Newton’s laws in mechanics. But that is not really an option, because even if Newton’s laws are not correct, the paradox remains. Note that the problem did not arise from an experiment that ostensibly shows us that our theory is incorrect. The problem arose from applying the theory itself. In other words, the theory itself is paradoxical, not the physics. It is supposed to be deterministic, and it is not. The third possibility is to give up the assumption of determinism. After all, we derived determinism from Newton’s laws (because they are described by a differential equation), and this example shows us that we derived it incorrectly. It turns out that there are cases in which Newton’s laws are not deterministic, even though they are nothing but a differential equation. The fourth possibility does not seem relevant here, since not only have we not found a solution, but we have proved that there is not and cannot be any

solution. Seemingly there is no escape but to give up the assumption of determinism. Our mistake was in thinking that Newton’s laws are deterministic. It turns out that they are not.

Is determinism really a product of Newton’s laws? I think that what is so troubling about Norton’s paradox is that determinism is not merely a product of examining Newton’s laws. If that were the case, then indeed it would be called for to give it up: we would simply have been mistaken, since it turns out that determinism does not follow from Newton’s laws. Our problem here is that the principle of causality states that nothing happens without a cause, and that does not come only from Newton’s laws. Determinism flows from this a priori principle and not from Newton’s laws. Newton’s laws merely express it, among other things. Even if it turned out that Newton’s laws are not correct, I would still expect the correct laws to be deterministic. The same therefore applies here. Even if I were willing to accept the conclusion that Newton’s laws are not deterministic, the conclusion should then be that we can no longer see them as a reasonable and acceptable description even of the behavior of physical objects, since that behavior is required to be causal and deterministic. Things do not happen without a cause, and therefore the direction and the time at which our little ball ‘chooses’ to roll ought to be the result of some cause and not be chosen arbitrarily. In other words, it seems that what this thought experiment shows us is that Newton’s laws are not a correct description of physical nature, since that nature ought to be deterministic.

One can of course argue that quantum theory teaches us that this is not so. It turns out that there are indeed physical occurrences without a cause. If so, the principle of causality is not really universal and binding, and there can be physical situations in which things happen without a cause. And if that is so in quantum theory, why should the same not be true in classical mechanics? Even so, the feeling remains that something here is problematic. The principle of causality really ought to hold at least on the classical plane. Somehow this is not an a priori conclusion that cannot be tested empirically (after all, quantum theory undermines it), yet we still are not prepared to give it up in classical contexts. Why? It is a kind of intuition, and even if it is not universal, we have not lost our confidence in it. At the margins of my remarks I will note that even with respect to quantum theory there is no agreement that it is indeed non-causal. There are various interpretations (for example, that it is causal but nonlocal, or that there are hidden variables that govern quantum phenomena, only they are not measurable and therefore we do not see their causal influence, and so on). Our causal-deterministic intuition is very strong, and if with respect to quantum theory we do not relinquish it so quickly, then all the more so with respect to classical mechanics. Very well, then what more can be done? Seemingly everything here is closed off and there is no way out. The necessary conclusion is that if we do not give up the principle of causality, then Newton’s laws necessarily do not fully describe nature.

The accepted solutions. In the Wikipedia entry to which I linked above, there is a claim that the force is not differentiable at r=0, and when Lipschitz continuity is absent, the rule of uniqueness of the solution to the differential equation does not hold. Of course, this does not solve the problem; it only says that Newton’s laws are indeed not always deterministic. But above we already saw that this is not a solution to our real problem. At most, it says that Newton’s laws do not describe nature, since they are not deterministic.

Later in the thread in which the problem came up, Aryeh raised several points that try to solve it. In one of them he argued that the analysis we saw does not indicate that something happens without a physical cause, but only that in this system such a thing could happen at any moment. He even added that this could open the door to the claim that there are gaps in nature (contrary to what I usually write), and therefore there is room for free will or divine involvement even within the framework of the laws of nature. But I replied there that this is not a solution, because the thought experiment shows that something like this can happen, and that alone is enough to undermine the determinism of the laws of mechanics. It does not have to happen in practice. Besides, why assume that it will not happen in practice if in principle it can happen?

He also brought there that Norton himself claimed there is no paradox if one formulates Newton’s second law differently: at every moment when no force acts, there is no acceleration, and when there is force, there is acceleration. If we look at our little ball, up to the time the body begins to move (t<T), its acceleration is indeed zero (and its velocity is also zero, and then indeed no force at all acts on it). At the moment t=T its acceleration is still zero (because of the shape of the curve), but still no force acts on it because it is still at the top of the kippah. But afterward (when t>T) there is acceleration, except that by then a force is already acting on it. If so, there is no point in time

at which the law is violated, and therefore there is no paradox. But this formulation does not explain how the body moves from the top of the kippah to the nearby point at which a force is already acting on it. Something here still does not sit right.

My solution: there is no square from the six days of Creation. From the moment I saw the paradox, my initial thought was that the problem lies in the shape of the dome. Such a dome simply cannot exist in nature. True, the shape of the kippah is continuous and differentiable, but only once. There is no defined second derivative there. The meaning is that if you draw the height as a function of the horizontal distance X

you will get a sharp point at 0. That means the slopes to the right and left of the point are different, and the second derivative will be discontinuous (it has no single value at that point).

The claim is that there are no discontinuities and no sharp points in nature. Therefore such a process cannot occur in our reality. In other words, one may say that in order to speak of an object with such a shape, one would have to treat distances at a very high, indeed absolute, resolution, and that cannot be done. In reality, when one goes down to absolute resolution, continuity has no meaning at all, since we live in a world with discrete atoms. Moreover, at such resolutions quantum theory rules and not classical mechanics, and therefore the entire analysis here is incorrect. In other words, classical mechanics does not deal with this kind of shape. Perhaps that is what Norton himself (the real one, not the one from Meah Shearim) meant when he said that force and acceleration always correspond. I was asking there how he would explain the transition of the little ball from the top of the kippah to the point on the side where the force already exists. The answer is that there is no such transition, because such a shape cannot exist. Nature is continuous. Hence the mathematicians have taught us that it is not right to describe a continuum as a dense collection of points standing side by side (that is the density property of the continuum). And from this it follows that one cannot speak of a transition from the point at the top of the kippah to the closest point beside it. The density property teaches us that there is no closest point. But that is exactly what it means to say that such a shape, in which we speak of one defined discrete mathematical point, cannot be found in reality. In reality there is only a continuum. As I remarked to Aryeh, Norton’s explanation is really an expression of the non-smoothness of the shape. It is interesting that this matter is already mentioned in the Jerusalem Talmud (Ma’asrot 5:3; see also Tosefta Ma’asrot 3): Rabban Shimon ben Gamliel says: there is no square from the six days of Creation.

To be sure, the midrash in the Mechilta, Beshalach 14:1, says that among foods there is squareness. The plain meaning is that a human being can make square shapes; only nature does not make ‘sharp points’. In nature things are rounded (that is, continuous and differentiable). This may be connected to the fact that exact precision is impossible in the hands of Heaven, even if by human hands it may perhaps be possible. Things that come into being by happenstance and not by intention are always rounded. Such is the way of nature. To create a sharp point requires the deliberate intention of a human being. The deeds of flesh and blood are finer, as our friend Norton from Meah Shearim should know. To be sure, there is room to discuss this, since the shape itself is continuous and differentiable. Does nature also not make ‘sharpness’ in the second derivative? That is, could there not arise in nature a shape that is indeed rounded, but whose second derivative does not exist? My intuition is that the answer is no. In nature everything ought to be continuous: the functions and the derivatives. The variables we choose to use are not important to nature. That is only our choice. From its point of view, once there is some non-continuous quantity, that is impossible (after all, we could have chosen דווקא to use it in our mathematical formulation, and then the discontinuity would already have appeared in the first derivative).

I think that the solution that speaks about Lipschitz continuity is really aiming at this. The claim is not that the force is not differentiable, but that the shape is not twice differentiable. The natural entity here is the shape and not the force generated by it.

Returning to the paradox, the bottom line seems to be that one can retain the causal-deterministic assumption in classical mechanics and, together with that, not give up Newton’s laws as a description of reality. Newton’s laws describe classical reality accurately, and this does not contradict the determinism that prevails within it, although those very same laws, when applied to situations that cannot occur in reality, are not necessarily deterministic. That too is no small novelty, but it is one that can be digested. That is the point.

Admittedly, the shape above is defined by the dependence of Z on r, and I did not examine the dependence of Z on X. It may be that there the non-smoothness already appears in the first derivative. I do not think so, but in any event that is not important. If there is no discontinuous second derivative in nature, then certainly there is no such discontinuity that with respect to one variable appears in the first derivative and with respect to another in the second. Nature does not care which variables I use. It likes continuity and differentiability, and that is all.

Basically, this is a mathematical rather than a physical novelty (I think mathematicians will not be excited by this at all. It is obvious that if the conditions are not met, there is no uniqueness of the solution. The problem arose from the outset only because of the physical implications of the issue).