Symmetries and Persistent Identities

By the way, I have never discussed your heuristic point of view with my boss. It is possible that there are difference of opinion between him and me on this question. In any case, every paper in which probability considerations are applied to the vacuum looks dubious to me.

                                                                                Max von Laue, 1906

The idea of inertial coordinate systems was not formalized until the 17th century, but such systems of reference had always been the implicit basis for the intuitive concepts of space and time. The assumption that inertia is homogeneous and isotropic serves as the organizing principle for conceiving the extrinsic relations between the components of our experience. In this context, the principle of inertial relativity asserts that for any particle in any state of translatory motion there exists an inertial coordinate system in terms of which that particle is instantaneously at rest. We can then ask whether the set of all (or all possible) inertial coordinate systems is perfectly symmetrical, or whether (for example) one particular system of inertial coordinates is to be considered the “center” of the set, representing what might be called the state of absolute rest.

This is sometimes compared with the corresponding question for the set directions in space. We ordinarily regard those directions as inherently symmetrical, and physicists have rarely felt a need to single out a particular direction or orientation in space as the “absolute” or “central” direction or orientation. In the same spirit, we are inclined to think that no particular state of motion is distinguished over any other (at least not locally, although in a cosmological sense there may be a state of motion at each point in terms of which the large-scale structure of the universe is isotropic). The tendency to make the analogy between spatial orientations and inertial coordinate systems is even stronger in the context of special relativity, because of the formal similarity between Lorentz boosts (hyperbolic rotations) and spatial rotations.

However, there’s an interesting difference between the symmetry of inertial coordinate systems and the symmetry of spatial orientations. These two sets may be considered as analogous respectively to the symmetries of an infinite straight line and a circle. In both cases we can say that each point on the locus stands in symmetrical relation to all the other points, viz., no point is intrinsically distinguished. Nevertheless, the two sets are different in the sense that there exists a uniform probability distribution over the set of points on the circumference of a circle, but there does not exist a uniform distribution over the set of points on an infinite straight line.

We should note that in both cases it’s questionable whether we can really select a single element of the set “at random” from a uniform distribution. This obviously can’t be done for the points of an infinite continuous straight line (i.e., the real numbers), because there is no uniform distribution over that set, but in a sense it is not even possible to select a point from the continuous perimeter of a circle, or, equivalently from the unit interval from 0 to 1. Granted we can define a uniform probability density over such a locus, and such a density enables us to determine the probability of selecting a value from any given sub-interval of positive extent, but the probability of any specific element of the set is zero. Indeed if we were to actually attempt to select a real number “at random” from the unit interval, we would find ourselves unable to do this, because even to specify an arbitrary real number requires (almost always) an infinite amount of information. The set of computable (or finitely specifiable) real numbers on the unit interval has measure zero over the set of all real numbers on that interval. To specify a real number from a uniform distribution over all the reals, we would essentially need to specify infinitely many digits to the left of the decimal point, which is impossible, but to specify a real number on the unit interval requires us to specify an infinite number of digits to the right of the decimal point, which is equally impossible (except in special cases when the digits are computable from a finite algorithm).

Thus the claim that we can choose a spatial orientation “at random” is somewhat misleading. At best we can select an approximate orientation at random, with any finite degree of precision (in proportion to the circumference of a circle or the surface area of a globe). Still, this is a meaningful capability for a compact set, whereas even in this sense it is not possible to select a “point” on an infinite line (or plane) at random – basically because an infinite line (or plane) has no natural scale. We can “approach” a similar problem in a compact space by increasing the size – which is to say, the resolution – of the space, while measuring the precision in terms of some fixed unit. Indeed a circle or sphere approaches an infinite line or plane in the limit as their radii go to infinity, so the “compactness” of those loci is ambiguous in that limit.

Whenever we have a probability distribution over a set of elements, such as the points of a continuous locus, for which the probability of any individual element is zero, the selection of one individual element “at random” is problematic. At best we can only specify an element with some finite precision, which amounts to splitting up the locus into segments of non-zero length, each of which has some non-zero probability, and then select one of those. This is meaningful for any finite characteristic size (e.g., circumference) of a compact locus like a circle, sphere, or set of orientations, but apparently not for an infinite locus such as an infinite line or plane, because in such cases there is no finite characteristic size. In this sense, the symmetry of the points of a circle is different from the symmetry of the points of an infinite line. We can define a uniform probability distribution with some finite precision on the former, but not on the latter.

Likewise the symmetry of spatial orientations differs from the symmetry of Lorentzian or Galilean systems of inertial coordinates, because although we can define a uniform probability distribution over the orientations, and thereby select one “at random” (albeit with only finite precision), we cannot do the same for the systems of inertial coordinates. If we “weight” every small arc of directions equally, we can easily conceive of a well-defined uniform result over all the directions in space, but if we try to “weight” all the systems of inertial coordinates equally (in either the Galilean or Lorentzian senses) we find that essentially all the probability must be placed at ±infinity or ±c. This tends to undermine the idea that all frames of reference could be “equally probable”, even from a purely mathematical perspective. If we imagine a pre-existing Minkowskian manifold, it is not possible – even in principle – to introduce a material particle with a definite state of motion selected “at random” from a uniform distribution of all possible states of motion, because mathematically there does not exist a uniform distribution over the set of states of motion.

Of course, this presupposes some kind of substantialist status for the initially empty manifold. We could, as an alternative, assert that the manifold has no independent existence, and is merely induced by the relations between substantial entities. Thus the “first” particle has no definite state of motion, because there is no pre-existing independent spacetime. But on this basis we encounter a problem when we introduce the second particle. What is the distribution of the differences between the states of motion of the first and second particles? At what point do we assume the full Minkowski spacetime (with uniform distribution) takes shape? It could be argued that only a complete Lorentz-covariant universe of fields and/or particles could entail the full Minkowskian structure, but it’s difficult to imagine such a universe, because (again) essentially all the substance would have to be at the infinite extremes in order to be Lorentz covariant. In other words, in order for the universe to “look the same” with respect to every state of motion, every state of motion would need to be equally occupied, which implies a uniform distribution where such a distribution is not mathematically possible.

This can be taken as an argument against the neo-Lorentzian idea of a substantial ether underlying spacetime. If spacetime is truly Minkowskian, then it cannot be embodied as a substantial entity with definite state of motion (or rest). Einstein stressed this point when he spoke about how the spacetime of general relativity could be regarded as a kind of ether, in the sense that it possesses physically meaningful properties, but that it cannot be consistently assigned a state of motion.

Incidentally, in some circles Einstein’s remarks about “no state of motion” and “cannot be tracked through time” are often misinterpreted as signifying that the ether must be stationary, or that the ether must have no parts, or else as signifying some kind of epistemological limitation on our knowledge of the ether’s state of motion. It’s interesting that the actual meaning of his remarks, while simple and fairly obvious, is somewhat difficult to express verbally in a way that is totally free of ambiguity. This seems to be because our natural languages have no terms for describing an entity with no velocity as distinct from an entity whose velocity is zero. This is evidence of how deeply ingrained are the concepts of persistent entities and temporal progression. Only rarely have people expressed ideas outside of this conceptual framework, such as when St. Augustine remarked that the world was made, not in time, but along with time. This was his attempt to point out that time is part of “the world”, and it obviously makes no sense to ask when the creation of time took place, since the concept of “when” is associated with time.

The non-compactness of the possible states of motion is closely related to the ambiguity in the interpretations of special relativity. With changes in spatial orientation we can “turn all the way around” and arrive back at our original orientation without ever changing our direction of rotation, whereas we cannot do this for Lorentz boosts. To this extent the formal similarly between spatial rotations and frame boosts is incomplete, and some people take this, along with the non-existence of a uniform distribution over reference frames, to support the view that a unique temporal ordering of events is a more useful principle to take as the basis for arranging our understanding. Combined with the cosmological arguments (giving primacy to the local frame in terms of which the universe appears maximally isotropic), this gives a somewhat plausible argument in favor of the absolutist interpretation. Closely related to the cosmological isotropy argument is the fact that any posited closed dimension (capable of circumnavigation by a light pulse) must be globally absolute, in the sense that there is a unique state of motion for which light pulses circumnavigating the dimension in both directions give equal frequency shifts. This, in turn strongly suggests that, if such a dimension exists, then space must be an entity with its own independent existence. Of course, this consideration can just as well be put forward as an argument against the existence of any such closed and circumnavigatable dimensions (which are problematic for other reasons as well, related to self-intersecting null intervals).

Incidentally, the story of Einstein’s inaugural lecture at Leiden on the subject of “Relativity and the Ether” has several interesting historical sidelights. In January (1920) Einstein wrote to Ehrenfest

For my inaugural lecture I would like to treat “The Ether and Relativity Theory” because during my last visit to Leiden Lorentz already expressed the wish that I air my position on this problem publicly when the occasion arose.

The same day Einstein wrote to Lorentz saying “I shall hold the inaugural lecture you mentioned on the ether. It is a fine opportunity to make the clarification you suggested.” The subject had obviously come up between the two men previously. In fact, just two months before, on 15 Nov 1919, Einstein had written to Lorentz

I will lay out in detail my position on the ether question as soon as the occasion presents itself. I would have been more accurate in my earlier publications if I had limited myself to emphasizing the non-reality of the ether’s velocity rather than the non-existence of the ether in general. For I see that with the very word ether one is saying no more than that space must be conceived as a carrier of physical properties.

Ehrenfest, however, was apparently not privy to these discussions. Hearing only the title of the proposed lecture, Ehrenfest naturally anticipated that Einstein was planning to lambaste the idea of ether. He took Einstein’s original (1905) stance against the ether as one of the definitive features of Einstein’s work – which of course it was not. In preparing a lecture of his own on relativity theory Ehrenfest had written to Einstein

What a monstrous torture you subjected me to with the etherlessness of the special theory of relativity! Your inaugural address will sell like hotcakes! Make sure to have a large enough print run!!

And in a subsequent letter (April 1920) he wrote

I know that many hundreds of people will fall on your ether speech like a hungry pack of wolves. [In a public talk before 4-500 people I said in response to endless “ether questions”: “I am informed that Einstein will soon be publishing on this question” – Emotional stir in the auditorium!] So if it does not appear in the following day we shall receive 50 letters of request per day.

To help Einstein with the formalities of the lecture, Ehrenfest provided a sample of the expected format, and gave as the hypothetical title: “Down with the Ether Superstition!” However, as happened more than once during their long association, Ehrenfest was to learn that he had misunderstood the subtlety of Einstein’s views on this subject. In the actual lecture, Einstein stated that the spacetime of general relativity could be called an “ether”, because it is characterized at each point with certain invariant properties. In fact, he went so far as to say that spacetime without ether was unthinkable, but only with the understanding that the word “ether” referred to the metric tensor representing the gravitational and inertial field. He stressed that this “ether” was devoid of any of the mechanical properties of the ether imagined by 19th century physicists. Lorentz had already deprived the ether of all mechanical properties other than immobility, and Einstein’s view was that this last property must also be abandoned, because it is not possible to regard the ether as consisting of identifiable parts with positions in time. As discussed previously, there is no well-defined Lorentz covariant distribution over all possible states of motion, so the parts of spacetime can only exist at events, not as identifiable entities persisting over time.

Ehrenfest’s reaction to Einstein’s lecture doesn’t seem to have been recorded in their correspondence, but Besso’s wrote in December:

Today I read your speech of May 20 in Leiden – on the ether. It was yet another one of those serene moments for me that you have brought down from the stars. I do think, though, that you in fact gave that word the only possible meaning in the new domain, so that people who cling to it, Lorentz in particular, are not intimidated even further by apparent divergences – hence something humane. Yet also something humanely beautiful.

Oddly enough, this humane beauty seemed to have vanished by the time Infeld and Einstein produced the popular book “The Evolution of Physics” in 1938. There we read, after a summary of the 19th century attempts to isolate the ether, that

Nothing remained of all the properties of the ether except that for which it was invented, i.e., its ability to transmit electromagnetic waves. Our attempts to discover the properties of the ether led to difficulties and contradictions. After such bad experiences, this is the moment to forget the ether completely and to try never to mention its name…

Admittedly this passage occurs in the section on special relativity, but the ether is not thereafter mentioned again, except for two places where it is written as “e___r”, so as to avoid even writing the word. This seems rather silly, and reinforces the suspicion that Einstein had little to do with the actual writing of this volume. (It seems to have been written for the express purpose of making money to enable Infeld to remain in the United States after his research grant ran out. Einstein agreed to let his name be listed as co-author to increase the sales.) It should also be noted that, even in the context of the special theory, Einstein conceded (to Lorentz) that he should not have insisted so strongly on the non-existence of the ether, but only on the fact that it has no identifiable state of motion.

It’s interesting to compare this with E. T. Whittaker’s summary of how the ether was viewed at the end of the 19th century. He wrote (in 1952)

It came to be generally recognized that the aether is an immaterial medium, sui generis, not composed of identifiable elements having definite locations in absolute space.

This point of view, which Whittaker attributed to Larmor, sounds similar to Einstein’s, although if interpreted literally it is actually more radical, because Einstein merely denies to the parts of the ether any identifiable state of motion, whereas Larmor (according to Whittaker) was ready to deny them even an identifiable position. The meaning of this is not clear to me, because it seems that we can always identify the ether at any location with that location. I suspect this was simply Whittaker’s version of Einstein’s prescription for a relativistic ether, but garbled so as to make it plausibly attributable to Larmor. (This also shows, again, the difficulty of verbally conveying the idea of atemporal entities.)

Granted that the spacetime of general relativity does not consist of identifiable parts that can be traced through time, we might ask why we cannot (for instance) trace through time the location in space where the invariant curvature scalar has a certain value. The answer is that the curvature invariant at a given time and place does not represent localizable energy or information. In fact, a change in the physical conditions can result in the instantaneous appearance of that same value of the curvature invariant at separate locations (possibly even over entire regions), and the disappearance of that value from still other locations. Thus although the attributes of spacetime (according to general relativity) can be defined at each time and place, they are not entities with persistent identities through time.

It’s worth noting that the same lack of persistent identity through time is explicitly present in quantum mechanics, in which the wave functions of all particles overlap (at least to some degree). As a result, it is not possible to unambiguously identify one particular electron (for example) at a certain interaction with an electron at some later interaction. The identities of particles can be exchanged, and this exchange symmetry has very significant and measurable consequences. Furthermore, massless particles (such as photons) are not even conserved, so they certainly do not possess persistent identities through time. This is reminiscent of some remarks made by Lorentz in his 1909 Theory of Electrons. He discussed how

Poynting's theorem [for the energy flow in an electromagnetic field] throws a clear light on many questions. Indeed, its importance can hardly be overestimated, and it is now difficult to recall the state of electromagnetic theory of some thirty years ago, when we had to do without this beautiful theorem... [However], I will call attention to the question, as to how far we can attach a definite meaning to a flow of energy. It must, I believe, be admitted that, as soon as we know the mutual action between two particles or elements of volume, we shall be able to make a definite statement as to the energy given by one of them to the other. Hence, a theory which explains things by making definite assumptions as to the mutual action of the parts of a System must at the same time admit a transfer of energy, concerning whose intensity there can be no doubt. Yet, even if this be granted, we can easily see that in general it will not be possible to trace the paths of parts or elements of energy in the same sense in which we can follow in their course the ultimate particles of which matter is made up...

Lorentz assumed that we can trace the “ultimate particles” of matter, even though they too represent flows of energy and hence are, strictly speaking, subject to the same reservations about the persistence of identity that he expressed for energy. To support his claim that energy cannot be traced through time, he observed that we may add a certain quantity of energy A to an object at a certain time (say, by accelerating the object), and at a later time we may add another quantity of energy B to the object (by accelerating it some more), and subsequently some quantities of energy may be extracted from the object (causing it to decelerate), but we have no way of identifying the extracted quantities of energy with the quantities A and B that were previously added to the object.

For this reason, the flow of energy can, in my opinion, never have quite the same distinct meaning as a flow of material particles, which, by our Imagination at least, we can distinguish from each other and follow in their motion. It might even be questioned whether, in electromagnetic phenomena, the transfer of energy really takes place in the way indicated by Poynting's law, whether, for example, the heat developed in the wire of an incandescent lamp is really due to energy which it receives from the surrounding medium, as the theorem teaches us, and not to a flow of energy along the wire itself.

Here Lorentz was referring to the remarkable fact that Poynting’s formula implies that the flow of energy that heats up a current-carrying wire is not along the direction of the wire but rather perpendicular to the wire, i.e., the energy flows into the wire from the electric and magnetic fields in the surrounding space. This was considered so counter-intuitive to physicists of Lorentz’s generation that it led to various kinds of speculation for how the conclusion might be avoided. For example, Lorentz went on to say

All depends upon the hypotheses which we make concerning the internal forces in the system, and it may very well be, that a change in these hypotheses would materially alter our ideas about the path along which the energy is carried from one part of the system to another.

Nevertheless, he had to admit that Poynting’s conclusion was inescapable unless we posit that some other – presently unknown – aethereal forces (besides the electric and magnetic forces) are at work in the conduction of current through a wire. As he wrote,

There is no longer room for any doubt, so soon as we admit that the phenomena going on in some part of the ether are entirely determined by the electric and magnetic force existing in that part. No one will deny that there is a flow of energy in a beam of light; therefore, if all depends on the electric and magnetic force, there must also be one near the surface of a wire carrying a current, because here, as well as in the beam of light, the two forces exist at the same time and are perpendicular to each other.

At the same time (1909) when Lorentz was contemplating these issues, Lewis and Tolman were raising a challenge to Lorentz’s previous explanation of the Trouton-Noble experiment, in the idealized form of the so-called right angle lever paradox. They claimed that a net torque must exist on the lever (or on the charged capacitor) unless the transformation of electromagnetic force was different from what Lorentz had derived in 1904. In one sense Lewis and Tolman had simply made an elementary error, failing to recognize that the internal forces maintaining the shape of the lever (or capacitor) must transform in exactly the same way as the electromagnetic forces, thereby canceling the couple represented by the external forces. However, in a deeper sense, the ability of internal forces to oppose an external couple seems to demand a more meaningful explanation. This was provided in 1911 by Max von Laue, who analyzed the system in terms of the energy flow in accord with Poynting’s theorem, and the corresponding momentum flow, in accord with the equivalence of mass and energy. He found that the flow of energy-momentum in the transformed frame was entirely consistent with the applied external torque. Still, some people have expressed reservations about this explanation, essentially appealing to Lorentz’s argument against treating energy as something whose movements can be explicitly traced through time. Laue’s explanation is similar to an existence proof in mathematics, as opposed to a constructive proof, in the sense that neither the amount of energy nor the speed of its propagation are determined, but the product of those two is determined, and this is sufficient to account for the lack of rotation of the lever (or capacitor). However, it is possible to reduce the “right angle lever” to just three charged particles, and to use Poynting’s formula to explicitly compute the flow of energy and momentum. When this is done, the results agree with Laue’s analysis, and there is no reason to doubt that the same holds true for more complicated arrangements of charged particles. Moreover, even though, at some level, we can no longer regard energy as consisting of parts with persistent identities existing through time, the same is ultimately true of all substance, so this doesn’t undermine the validity of Laue’s analysis.

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