Some Metaphysics of
Angular Momentum and Gravity

It is a postulate of Einstein's General Theory of Relavitity that inertial and gravitating mass are identical. This is the "Equivalence Principle." The strong interpretation of this is that weight, whether induced by the application of a force, e.g. a rocket engine, or found on the surface of a gravitating body is metaphysically identical, i.e. the weights must occur because of the same phenomenon at the ontological level. The nature of this point is not well understood. Karl Popper says that, according to Hegel, "Newton's theories of inertia and of gravity contradict each other (of course, he could not foresee that Einstein would show the identity of inert and gravitating mass)" [The Open Society and Its Enemies, Volume II, Hegel and Marx, Princeton, 1971, p.27]. However, Einstein didn't "show" anything of the sort, if this means that he proved it. The postulate was part of the theory, in fact foundational to the theory, and the success of the theory vindicates it, to the extent that this is possible, as Popper himself was well aware.

I have argued elsewhere at this site that this identity occurs because of the relative, accelerated motion of the body with weight against the background of space itself. In the case of the rocket engine, the body with weight, namely the spaceship and what is in it, will describe a curved path against a background grid of spatial (Cartesian) coordinates. For there to be an ontological equivalence between this and the case of a body that has weight sitting on the Earth, it must be the background grid of spatial coordinates that itself described the curved path across the postion of the body with weight. We have an elegant mirror image. In the one, the curved path (which maps acceleration) is described by the moving body; and in the other the curved path is described, against a fixed surface and a fixed body, by the motion of space itself. This means that inertia, against which the rocket works, and gravity, induced by the mass of the planet, are simple inverse functions of each other, like addition and substraction, multiplication and division, or differentiation and integration.

But what is inverted in this case, and what makes it a metaphysical issue, is the role of space itself. In one case space is a passive background, and we see motion against it, but in the other we do not see motion at all, as a body sits on a planet, and the acceleraton is in the invisible background of space. Of course, if a body is not on the surface of the planet, but nearby in space, then we see induced motion, as the body falls towards the planet. What is then noteworthy about this case is that the body in free fall has no weight. In this, it is the equivalent of a body in empty space. Again, we see situations that are ontologically identical, and it is the recognition and reflection on this identity that are supposed to have started Einstein off on the theory. As time passes, space itself accelerates in one case but not in the other, as the body in free fall experiences identical conditions of weightlessness. Both bodies are in free fall, although we only think of one of them as actually falling -- the one that is dangerously headed for a collision with the surface of the planet, while in the other it harmlessly drifts in the void.

I have not seen these considerations elsewhere; and I expect that the metaphysical nature of the explanation, with a role for physical space, will not be agreeable to many, perhaps most, physicists and philosophers. After all, Einstein proved that Absolute Space doesn't exist. All space is Relative; and Leibniz was decisively vindicated against Isaac Newton. The very idea of the coordinate grid of space moving, although this no different from the common talk of the "curvature of space" in explanations of General Relativity, might be taken to posit a form of Absolute Space. However, none of this is quite right. Even in Einstein, there is a feature of space that in fact continues to be Absolute.

I recently came across a treatment of the issue in a discussion of Ernst Mach (1838-1916) by Paul Halpern in Einstein's Dice and Schrödinger's Cat [Basic Book, 2013]. Mach had a problem with Newton:

Mach's argument against Newton's definition of inertia referred to a thought experiment involving a bucket that Newton had concocted to demonstrate the need for absolute space. Here's the gist of the argument:  Imagine hanging a bucket filled almost to the brim with water on a rope tied to a tree. Now twirl the bucket carefully round and around until the rope is all twisted up. Hold the bucket, wait until the water within it has settled and has a flat surface, then let it go. The bucket will start to spin around on its own. If you look down within it, you'll see the water slosh around too as it forms a vortex, its surface becoming increasingly concave. That's because inertia makes the water try to escape. Since it can't leave the bucket, its outer edge rises. If you look at the inside of the bucket itself, ignoring its exterior, you might wonder why the water had a concave surface. Relative to the bucket, the water would seem to be perfectly still. Only by reference to an outside framework -- which Newton called absolute space -- would the concavity make sense. The water's rotation relative to absolute space, Newton asserted, remolded its surface.

Mach begged to differ, arguing that there was no empirical evidence for absolute space. More likely, he said, there was a pull on the water from unaccounted sources, such as the aggregate influence of distant stars. Just as the moon's tug causes the tides, perhaps the combined pull of the stars somehow causes intertia. Einstein's would later dub this "Mach's principle." It would inspire him as he developed relativity. [pp.23-24, color added]

Paul Halpern is a physicist and a popular historian of science, but there are many strange things about this passage. First of all, the bucket experiment may have been "concocted," whatever that means (I think it means that Halpern doesn't want us to take it seriously), by Newton, but it is attested in the correspondence between Samuel Clarke, defending Newton, and Leibniz. Clarke wasn't arguing against Mach or Einstein, who had not yet been born, but against the thesis of Leibniz that space does not exist. So the issue was not "absolute space," but the existence of space at all. It is not clear to me whether people like Ernst Mach understood, as Halpern clearly does not, the context of the Newtonian argument and what the metaphysical theory of Leibniz happened to be.

Next, the rotating bucket is not, strictly speaking, a "thought experiment," since it is easily done as an actual experiment; and, in the absence of an actual experiment, it is not clear that one could necessarily predict what the result would be. Either way, we discover that the spinning water climbs up the sides of the bucket. In ordinary life, we easily explain the difference between the water with the flat surface and the water with the concave surface by noticing that the bucket is spinning. The concave surface is itself "empirical evidence" that the bucket is spinning even if we could not detect the motion of the bucket by reference to external landmarks. Leibniz wanted to argue that the motion of such a rotating bucket could not be detected except by reference to such landmarks. This was clearly not true, and Clarke, but not Halpern, wins the point.

I think we should ask what was Mach's problem. Since he must allow that Clarke was right in the argument, and that the rotation of the bucket can be detected from an internal inspection alone, he was required to fall back on a hypothesis that the mere existence of external objects, even distant stars, is responsible for the effects of inertia as the bucket rotates. Why Mach would have thought this explanation "more likely" than the commonsense existence of space should leave us puzzled. And the language used by Halpern, that this is the influence of "unaccounted sources" that "somehow" institute inertia, does not cohere very well with the portrait of Mach as a hard-headed Empiricist and Positivist. The bucket -- any real bucket, and not just a thought experiment -- displays the incontrovertable empirical evidence, while a theory about the gravity of "distant stars" is not so much a theory as a wish list. Indeed, if "distant stars" pull up the water the way the Moon pulls us the tides, why does the bucket need to be rotating? What has rotation got to do with it? And if the influence of the "distant stars" somehow imposes an orientation, in reference to which a rotation can be detected, how can it do that when the "aggregate" gravity of the universe is "more likely" to be symmetrical (in an isotropic and homogeneous universe) and thus unable to establish a preferred direction on anything? After all, Leibniz needed to argue that the rotation of the bucket can only be detected in reference to external landmarks which, as such, must be distinguishable from each other. A uniform background of gravity would be an indistinguishable continuum [note].

But even if we suppose that Special Relativity has destroyed "absolute space," and we can "somehow" discount the meaning of Clarke's Newtonian argument against Leibniz, Halpern and many others have overlooked something important. Special Relativity preserved the absolute nature of the velocity of light through the "Lorentz Transformations," with which length, time, mass, and velocity vary in relation to the velocity of light. We will observe that length contracts, time slows, mass increases, and velocity can approach, but cannot equal, the velocity of light.

What is always overlooked in all of this is that length contracts only in the direction of motion. Velocity has two physical components, (1) the magnitude or speed, and (2) the direction of the vector of motion. The vector determines the dimension of length that will contract. Thus, the vector, which is a direction in space, presupposes the existence of space and the three dimensional coordinates that give direction a physical meaning. Thus, while the magnitude of velocity is relativized in Special Relativity, the vector and the direciton of velocity are not. Direction posits and requires Halpern's "absolute space," even as it continues to refute Leibniz's thesis of the non-existence of space [note].

This happens to give special meaning to the "bucket" experiment. Along with the acceleration of the rocket, or the surface of a gravitating body, rotation also generates the phenomenon of weight. Again, according to the Equivalence Principle, all these must physically be the same thing. It is easier to see this if, for the bucket, we substitute a centrifuge, especially one in space in free fall. From science fiction movies like 2001: A Space Odyssey [1968], Interstellar [2014], and The Martian [2015] we see how "artificial gravity" can be generated on a spaceship just by spinning it. But there is nothing artificial about the weight experienced by the astronauts [note].

Compared to the rocket and the planet there are peculiarities involved with this phenomenon. One is when we notice that, having spun up our centrifuge or spaceship, it will, in the absence of friction, simply continue to spin. The Earth has been spinning for billions of years. In other words, the spin enjoys interial motion. This is odd, because inertia attends constant velocity; but while the spinning object may experience constant speed, it does not have constant velocity. Instead, it is constantly accelerated. But that seems to be the trick. The speed of the rotation does not change, and the inertia here seems to be a phenomenon of the speed. But the rotation is an acceleration entirely because the direction of motion is always changing. The acceleration is entirely a phenomenon of the change in the vector. Our astronaunt is pulled around in a circle.

This makes our centrifuge exactly the opposite of the case with the accelerating rocket, as long as the rocket does not change direction. Thus, acceleration and weight in the rocket is due entirely to its change in speed. Direction, which is constant, is no factor at all. On the other hand, acceleration and weight in the centrifuge is due entirely to its change in direction. Thus, the two cases form a logical pair, which means that in all such discussions of these issues, they should be treated together. But from what I have seen, at least in popular and general presentations of science, that is never done. This is especially odd when the centrifuge, with its angular momentum, enjoys the benefits of inertial motion. Despite its remarkable features, the significance of weight in the centrifuge is ignored.

But the fun doesn't stop there. If weight in the centrifuge suggests comparisons with weight in the rocket, it does so also with weight on the surface of the planet. In both cases, weight is felt on a surface whose postition, periodically, does not change. Thus, the planet may or may not rotate, but its rotation is irrelevant to the experience of weight (unless the planet is actually a small asteroid, whose spin may toss an astronaut off its surface). The surface of the centrifuge does indeed move, but it does so in a periodic way, which returns our astronaut to where he started. This comparison may be too loose, but there is a deeper connection. In each case a force is exerted. We press down on the planet or on the wall of the centrifuge. A force, ordinarily, expends energy. In each of these cases, we might wonder where the energy comes from. With gravity, we get extensive explanations. The energy comes from the gravitational "field." At any point in the field, an object has "potential energy" because of its position. If we drop something towards the planet, the object accelerates and the potential energy is turned into "kinetic energy," the energy of motion. This energy becomes manifest as a falling object may heat up and glow from friction with the atmosphere; and then, if we get an impact and the object doesn't simply burn up in the air, we get a massive release of energy in the shock, with a great release of heat and light. On the other hand, if we are simply standing on the surface of the planet, we still have all that potential energy, and it is continually converted, not into motion, but into the force that we feel as we are pressed down on the surface. Given the principle of the Conservation of Energy, how much is available to fuel that force? How long can it go on? Well, unless this somehow reduces the mass of the gravitating body (since E=mc2), it looks like that could go on forever. But doesn't that violate the Conservation of Energy?

Even better, something rather like this goes on with the centrifuge. The astronaut feels weight when pressed against the wall, or, for him, the deck, of the spaceship. This force comes from an acceleration, which is the continuous change in direction of the wall of the ship. In terms of forces, this is a curious and often confusing case. The astronaut presses down with a "centrifugal" force, as his body would rather fly off into space; but then, as it correctly is always explained, this "force" is a fiction. There is no centrifugal force. His body only wants to continue in the straight line of its inertial velocity. No force. Just momentum. The real force is the centripetal force of the rotating wall pulling his body away from its straight path. Where does that force come from? Well, it all comes from the rigidity of the material with which the wall is made. If the centrifuge spins fast enough, the soundness of the material may fail, the wall may fracture, and the astronaut would continue on the vector of his intertial motion. Otherwise, the centrifuge keeps on spinning and the astronaut keeps experiencing weight for, what, forever? The astronaut may walk one way, and add force to the spin, or walk the other way, and substract, force, but chances are these motions will cancel out, and the angular momentum of the centrifuge will maintain its motion.

So in the case of the centrifuge and the case of weight on the surface of a planet we seem to have a paradox, where force and energy come out of the Blue, without exhausting themselves. Perhaps gravity leaches mass out of the planet -- although I've never seen anyone say that. But with the centrifuge, the situation is even stranger. The astronaut experiences weight thanks, on the one hand, to the inertial speed and angular momentum of the spinning centrifuge, and, on the other hand, to the acceleration that is due to the change in the vector of whole rotating body. This is rather like having your cake (weight) and eating it too -- although the trick is that you don't eat it, namely you don't expend any of the energy that would have been required, for instance, to run the rocket engine. And it's got to be a trick. No roar. No vibration. No heat or light or flame. Just, well, the Blue Danube playing in 2001: A Space Odyssey.

Thus, both Leibniz and Paul Halpern, and perhaps even Newton and Einstein, were faced with something a lot stranger than any of them appreciated. In terms of the Equivalence Principle, it is no problem. The spinning centrifuge describes curved lines in space-time -- as we see in the diagram at right -- crossing the grid of spatio-temporal coordinates, just as weight does elsewhere in General Relativity. The strangeness is that weight occurs for an entirely different reason than with the rocket or the planet. Like the rocket, we can say that weight occurs because of acceleration, as it does; but this acceleration involves no change in the magnitude of the velocity, of the speed, and the result of this pecularity is that the inertia of the moving mass is not impaired. Momentum keeps the centrifuge spinning, just as momentum keeps any body in inertial motion moving at a constant velocity. The constant and periodic change in the vector does not affect this, even as it results in the phenomenon of weight [note]. So it is acceleration without forfeiting the advantages of inertia. People should really ask how it gets away with this, when the poor rocket must expend large and continuous amounts of energy.

With such questions left hanging, let me finish with the reminder why Clarke refuted Leibniz. According to Leibniz, rotation could only be detected by reference to external objects. However, since we find weight occurring in the centrifuge, or the rotating bucket, we know it is spinning without reference to any external objects. So Leibniz was wrong to hold that space doesn't exist. Leibniz was also refuted by Kant's argument about handedness and by, of all things, the existence of geometry, which, after all is about the structure of space. Leibniz, like Hegel, would have thought that the axioms of geometry were analytic and self-evident, which means you don't really need to have space for them to be true. Kant argued that the axioms of geometry are synthetic, which means that they can be denied without contradiction and that some ground outside of them would need to be responsible for their truth or falsehood. Why Ernst Mach simply didn't like any of this is a good question, and he may not have even understood that Leibniz denied the existence of space. As for Mach and Einstein rejecting "absolute space," it is obviously still not understood that Special Relativity did not relativize the vector component of velocity, which has meaning, not only in relation to the existence of space, but in the absolute terms of direction in space. Rotation is the Joker in this business since it involves an acceleration where, extraordinarily, only the vector component changes.

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Einstein's Equivalence Principle

The Clarke-Leibniz Debate (1715-1716)

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Angular Momentum and Gravity, Note 1

Mach's problem, of course, was his philosophy of science, not anything about the empirical evidence. While Halpern wants to present Mach as some sort of hero of science and inspiration to Albert Einstein, much of what he says about the man would serve to discredit him. Thus, Mach most famously denied the value or meaning of the atomic theory of the chemical elements that had been introduced early in the 19th century by John Dalton (1766-1844). Halpern incautiously quotes Mach to this effect:

If belief in the reality of atoms is so important, I cut myself off from the physicist's mode of thinking, I do not wish to be a true physicist, I renounce all scientific respect -- in short: I decline with thanks the communion of the faithful. I prefer freedom of thought. [Halpern, op.cit., p.23]

Mach, of course, has ignored the empirical evidence for the atomic hypothesis, which, as it happens, is really in chemistry, not in physics. Perhaps that is part of his problem. Dalton discovered that common compound substances appeared to be derived from whole integer quantities of the atomic weight of the primary elements. Pretty much the only explanation for this that has ever existed is that the basic particle of each compound consists of atoms of each element, e.g. two hydrogen and one oxygen atom in each molecule of water. It is not like Mach had an alternative theory. Instead, we are back to the necessity that "somehow" there be another explanation. There never was. This is not "freedom of thought." It is prejudice.

The nature of the prejudice is of interest. It is purely philosophical. Mach was an extreme Empiricist who could not believe in any entitites that could not be directly observed. This is also what we saw in Hume, who forthrightly asserted:

Our senses inform us of the colour, weight, and consistence of bread; but neither sense nor reason can ever inform us of those qualities which fit it for the nourishment and support of a human body. [Enquiry Concerning Human Understanding, Shelby-Bigge edition, Oxford, 1902, 1972, p. 33, color added]

This sort of embarrassing claim, which is little discussed or publicized by modern Empiricists, is starkly revealing about the limits of Hume's ideas about knowledge. He did not believe that whatever hidden qualities of bread made it suitable for nutrition could ever be known. What you see is all that we can know; and all we see is "the colour, weight, and consistence of bread." Mach is little different from this.

But, when Mach must ignore the evidence that had accumulated in science since Hume, this makes him, not an Empircist, but a Dogmatist. And the irony is that Mach is also supposed to be Positivist, following the doctrine of Auguste Comte (1798-1857) that the only kind of knowledge is scientific knowledge. But his doctrine motivated Mach to reject scientific knowledge on the basis of an epistemological theory that was not part of science. All later Positivists, including Ludwig Wittgenstein, would continue to be plagued by a similar incoherence and paradox. Moreover, we can accuse Mach's ideology of perhaps leading to a particuarly ugly consequence. Ludwig Boltzmann (1844-1906), who was a far greater scientist than Mach, would be so unsettled by this totally gratuitious controversy over atomism, that he committed suicide. I am not aware that Mach ever apologized for his harrassment, even when, in 1905, while Boltzmann was still alive, Einstein argued that Brownian Motion itself was evidence for the existence of atoms. Perhaps Mach didn't believe Einstein's argument either.

Nevertheless, since Paul Halpern presents Mach as some sort of hero of science, we should take this opportunity to credit Halpern with the principle that purely philosophical considerations, such as Mach's, can override a quite solid consensus about scientific results, which means that Hume's Empiricism and Comte's Positivism are both wrong. If he can admit this, then perhaps Ludwig Boltzmann did not die in vain.

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Angular Momentum and Gravity, Note 2

We might say that General Relativity subsequently Relativized direction also, because of the curvature of space. But since the "straight lines," the geodesics of gravitational paths, nevertheless simply look curved to us in Euclidean space, this does not produce the kinds of paradoxes that contractions do in Special Relativity. That is, everyone already thought that the orbits of the planets looked curved. Einstein didn't change that. He just explained it differently. Special Relativity not only explained things differently, it predicted bizarre effects.

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Angular Momentum and Gravity, Note 3

The spaceship in The Martian is beautifully constructed with fixed and rotating parts. Since the experience of space flight is that prolonged weightlessness results in bone loss and other unwanted physiologically problems, it is now obvious that something needs to be done to prevent this if flights to Mars or anywhere beyond the Moon are going to be possible. In Interstellar, the whole ship rotates, which means that crew on the command deck are watching the whole sky spin. This makes one crew member dizzy and nauseous. So Interstellar probably does not have the best arrangement if we want officers to be watching what the ship is doing. In the ship Hermes of The Martian, the rotating crew quarters are separate from the stationary command deck, power plant, etc. Thus, within their spinning quarters, the crew does not need to be strapped down to sleep and does not need to be strapped down to exercise in order to delay the physiological degeneration. The movie features nice shots of crew members floating down the weightless axis of the ship and entering the spinning quarters at the hub of motion.

It is nice to imagine artificial gravity, as in Star Trek, Firefly, or Star Wars, but there is no basis for this in current physics or prospect for it. It may be because of this absence that recent movies have begun returning to what used to be common in science fiction in the '50's and '60's, which is the use of rotation for "artificial" gravity, which physically is real gravity, according to General Relativity.

Another aspect of gravity induced by spin is found in The Expanse books by James S.A. Corey (the combined pen name of Daniel Abraham and Ty Franck, evidently proud residents of New Mexico), beginning with Leviathan Wakes in 2011. Now, if what is spun up is not a cylinder but a sphere, the result on the interior does not result in a uniform phenomenon of weight. Indeed, if we imagine the situation at the axes of rotation, there will be no weight at all. The sphere will simply be rotating under the weightless astronaut, as at the hub of the rotation section in The Martian. Between the axes and the equator of the sphere, however, there will some combination of weight and spin, with full weight at the equator and with less weight and more spin as one moves towards one of the axes. As it happens, the spin phenomenon in the intermediate locations is evident on the surface of the Earth itself. It is the "Coriolis Effect," by which straight inertial paths, as of the wind, are deflected.

Now, there might sometimes be a good reason to built a spaceship as a sphere, but in generally this will be unnecessary and, in this case, a bad idea. However, Abraham and Franck realize that a Coriolis Effect could become an issue elsewhere. If humans colonize asteroids, they are all too small, even the largest, to have significant gravity. Also, human habitation will be built underground (as in Heinlein's The Moon is a Harsh Mistress), mainly to use the surface as shielding from radiation, both cosmic and solar (something Heinlein might not have appreciated). These two circumstances lead Abraham and Franck to anticipate that the asteroid could be put into a spin in order to create spin gravity in the interior. But then, in roughly spherical bodies, such as Ceres, this could mean that a Coriolis Effect could arise in some parts of the colony. Residents might just need to get used to it.

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Angular Momentum and Gravity, Note 4

Note that the "vector" referenced like this is not the well defined "angular momentum vector" of the whole spinning body. That vector is given by the "right-handed rule," whereby if you curl the fingers of your right hand in the direction of the rotation, you thumb points in the direction of the angular momentum vector. This is another unique attribute of rotations and is reponsible for some bizarre properties of rotation, such as that a force applied against the vector results in motion that is displaced 90 degrees. This is what happens with gyroscopes and, consequently, something as humble as bicycle wheels. A moving bicycle will not fall over because the force of gravity applied against the wheel is displaced away from the vertical. As a child learning to ride a bicycle, my father did not think, or know, to tell me that my moving bicycle could not fall over if I steered a steady course. I did not understand what was happening until learning physics years later.

Instead, the reference to the "vector" above is to the ordinary vector associated with the inertia of any parts of a rotating body. As the body spins, this vector is continually changing direction.

If we reduce space to two dimensions, a rotating body in space-time will describe a spiral in the temporal dimension. Our graphs here are space in one dimension, so that we see rotation describing a waving line in space-time. This periodicity should remind us of the mathematics of periodicity, which involves imaginary numbers. The value of imaginary numbers is evident in this circumstance alone. The strangeness of imaginary numbers matches, no less, the strangeness we see in rotation as an acceleration that enjoys interial momentum, which means that the weight induced by spin requires no expenditure of energy.

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