Historically we are moving backwards, because the next case was not only discovered, but in practical terms, more nearly ``understood'', by the time Planck's constant was even discovered to be as universal as it is. The next case is easier to ``see'' because it is manifest even in big people-sized systems, unlike the Planck constant, whose effects tend to become irresovable because of their exceeding smallness. But it may have been a little less obvious without the precedent set by the Planck constant, that it too is a dimensionful constant associated with repeatability. That constant is the speed of light.
By the beginning of the 20th century it had been known for some time that light had a finite speed (cite Weinberg for complete presn), but even that discovery had required the observation of the moons of Jupiter, because they are objects that travel large enough distances and yet themselves are far enough away that delay-effects that could be associated with a finite speed of light were noticeable. Thus the speed of light is much too fast to make it an easily measureable quantity, and if there were no particular search for dimensional constants, it could have been overlooked for a long time that light even has a repeatable speed (much less that this speed was special and independent of how the light was created or what the observer was doing), had not that result been expected for other reasons. While the remarkable observer-independence of the speed of light is the aspect of it that we will use most to see later why the theory of special relativity is necessary, the original understanding of this issue that led Einstein to create this theory is more obviously connected to the essence of this speed as a repeatable quantity. It was only after realizing that the arguments that necessitated this repeatability made no reference to how the light was seen, that the issues surrounding relativity were even understood to exist.
To understand how the speed of light was predicted to be a constant, even long before its value was measured with any substantial accuracy, we need to mention some observations about electric currents, electricity and magnetism.
Electric currents are familiar; they are what travels in wires and provides the power to do work, make heat and light, and so forth. But it is also clear that the general phenomena associated with ``electricity'' extend beyond the currents in wires to other effects, which will concern us more in later chapters. The additional relevant property that was recognized as the essential feature of electricity was that it does something to empty space, which can effect objects that are not in direct contact with each other in any sense of currents. The quintessential example is that electric charge, which we have noted is like an ``accumulation of current'', can be trapped on the surface of a balloon or at the front of a television screen, for example, and then causes other things to be attracted or repelled as a result. The balloon can cling to a wall or ceiling, and the television screen or the balloon can cause the hairs on your arm or head to pull toward it, and even cause a sensation in the skin. All of these things happen ``because'' there is charge trapped on these surfaces, but they happen to distant things that are not in any direct contact with the surfaces or the charges trapped on them. We will give names to these effects later, when there is a reason to do so in a particular way.
This space-affecting property of electricity is actually not so readily associated with the currents as with the charge which is their accumulation. And in fact, it is familiar that currents can flow in wires without giving rise to any of the effects of charged balloons. But there is another very similar effect which, symmetrically, is sensitive to currents and not to charges. That effect is magnetism. Since long before electricity was understood, magnetism had been recognized as something that affects objects across empty space, because magnetic compasses could make each other turn in this way. It was not until relatively recently, though, with the work of (?) Galvani (when?) that it was recognized that currents traveling in wires produced the same kinds of effects as other magnets like pieces of magnetized ore (lodestones), and that in fact that two ways of producing these magnetic effects were completely interchangeable. Once this connection was realized, people started to make careful measurements and determine the relations that described the kinds of magnetic, as well as electric, effects that could be produced and how these related to respectively the currents and charges that were used to create them.
The important thing to understand is that in this realm, as everywhere, there is a notion of scale associated with these effects. It is easy to recognize that there is a difference between a strong magnet and a weaker one of the same shape and qualitative character. Similarly there are strong electric attractions and weak ones, that can be produced by accumulated charges. And these differences of degree can be measured quantitatively and made precise by comparing them to standards, and in that way giving numbers to characterize their strengths. It was immediately recognized that ``more charge'' on the same balloon (as one can obtain, to a point, by rubbing it against cat fur (which can be left on the cat) more times, for example) leads to ``stronger'' electric effects, and similarly that ``more current'' makes stronger magnets. If this were the end of the story, perhaps the strength of electric and magnetic effects would have been used as the relevant characteristic by which to quantify what was meant by ``more'' charge or current.
But the strengths of their empty-space properties are not all we know about the amounts of charge or current. Because recall, we also have a knowledge of chemistry. And chemistry involves measureable and countable quantities of different elements, in an absolute sense like one counts marbles. And chemistry and electricity are intimately related, as is clear from the fact that it is a chemical reaction in a battery that is responsible for its ability to provide currents and thus also charges. Thus there is an absolute sense of ``how much current'' has flowed through a wire, which can be related to how much lead was consumed in the battery that made it, and converted into some kind of chemically different salt. And since charge can be accumulated and stored on objects (such as balloons, but there are also much better ways), it is possible to relate the amounts of charge stored to the amounts of lead that would have been consumed to move them there. So from these considerations of electric and magnetic influences and of chemistry, we have three separate quantitative properties with senses of scale that we can measure.
The particular relations among the different effects were found to be very simple. The ``strength'' of electric effects (which could be discussed, for example, in terms of how many identically charged ``standard'' ballons would have been needed to produce the same effects) was found to be just proportional to the charge that made them, as it would be measured in the chemical sense of how much lead was reacted to produce it. Magnetic effects, in addition to the amount of charge, hinge on the amount of time involved, because some amount of stored charge, allowed to pass through a loop of wire in a short time, produces a larger magnetic effect than the same amount passing through more slowly, though in the latter case the weaker magnetic effect is maintained for a longer period. But again, comparing either a number of reference magnets for a given amount of time, or asking how long a magnetic effect of a particular amount can be sustained, one finds that the answer is proportional to the charge as measured chemically. Since the whole notion of a current comes to be understood through these investigations as an amount of charge flowing through someplace in a given time, it becomes easy to state that the strength of the magnet is again linearly proportional to the amount of current used to create it, when other things are kept the same.
So now, not only is chemistry repeatable, but the electric and magnetic strengths that can be produced with the aid of chemical processes like batteries (as well as in other ways) are repeatable, and they have a mutual scaling relation. But this scaling relation inevitably involves some new constants of proportionality, of the form:
much like the constant that appeared when we related a spring's strength of resistence to how far it was pushed. But these ``spring constants'' are very strange, because they seem to relate the strength of a response of empty space to currents and charges that ``live in'' or move through it at some place. These constants of proportionality could be and were measured very accurately, but for some time they were regarded as aspects of independent phenomena and simply catelogued.
But just as charges are related to currents as their accumulations, there turns out to be an intimate relation between the two empty-space effects, electricity and magnetism, which was understood a piece at a time by several people, with the last piece of the puzzle discovered, and all of the pieces assembled into a single idea, by Maxwell in (?). (We will deal with the details in a later chapter; they distract from the development here.) In light of his unified understanding of electric and magnetic effects as two aspects of a single thing, Maxwell was able to guess that what we see as light is a wavelike disturbance of these empty-space effects that sloshes back and forth between electric and magnetic expressions of this unified ``thing'' as it travels along. (Very allegorically, it would be like having seen ocean waves somehow, without having been able to see the ocean that they ``are waves in''. Maxwell identified light as the waves, and then to make sense of them ``guessed'' the existence of the ocean, in one of the most insightful and useful leaps of intellect perhaps ever performed.) And since light is a property of electricity and magnetism that exists in empty space, it would be expected that it draws its properties from the empty-space properties of electricity and magnetism. In particular, its speed could have been inferred from dimensional analysis even before Maxwell was able to predict it from his equations, because the dimensions of these constants were known, as they had to be to give a relation between the essentially ``dimensionless'' counting numbers that describe charges in chemistry and the dimensionful forces that they exerted on those charges.
The traditional dimensions for these proportionality constants are somewhat beside the point in this discussion, especially because their important dimensional content can be seen directly and independently of the path in which it came to be understood historically. Experience with balloons and magnets, even of fixed strengths, tells us that their effects are not equally important at every distance. For instance, people in China are almost certainly doing things with magnets at this moment of which I have no awareness as I sit halfway around the world from them. Thus even if we do the most convenient thing and always compare electric effects only to one another in terms of the amounts produced by some standard (chemically ``dimensionless'', because it is simply counted) amount of charge, we find that there is some characteristic distance involved in the proportionality relation. Making the measurements reveals that it is the same relation as we found for gravity in the last chapter: Specifically,
Thus the electric proportionality constant, expressed in this way, has dimensions of . In similar terms for magnets:
Here it was most convenient to separate the amount of charge, which can be thought of through chemistry as a dimensionless counting number, from the speed at which is is flowing, to form the description of the current. The most natural way to form the standard is, of course, to use some pre-determined amount of charge moved at some predetermined speed through a particular-length loop of wire held at a particular distance. In this way all of our dimensionless numbers are explicitly used in the same way. From this we see that the factors of length from the length of the wire and the speed of the current cancel the factors from the , so that the magnetic proportionality constant, expressed in this way, has units of , because it must both cancel the factor of time from the speed of the current and add another factor for how long the magnetic effect can be held by that amount of charge flowing at that speed.
(This is not the traditional way to represent the electric and magnetic constants, because while the ``counting'' of atoms is a very intuitive language in which to discuss charge, the very large number of atoms involved in actually doing anything big enough to work with easily makes this impractical, so charges, currents, and also their associated electric and magnetic effects, came historically to be measured in terms of chemically defined, more convenient but also arbitrary units. The choice of units is in this case distracting, because it intertwines the real essence of the electric and magnetic constants of proportionality with the historical accidents by which they came to be discussed in terms of certain chemical units instead of others. By using the understanding of the atomic theory as much as possible to remove extraneous dimensions from the effects we are discussing, we obtain a representation of the constants that is much more completely a description of the physical objects of relevance, but we step out of the historical context in the process).
In this language the natural guess for the speed of an electric and magnetic disturbance, which we will denote with the letter c is immediate:
as usual up to dimensionless factors that cannot be guessed in this way. From the already-measured values of the electric and magnetic constant (in their somewhat different-looking forms, but with the same combined dimensional content), the speed predicted was very fast relative to most experiences, but very plausible as the correct speed for light. Since there has now been a reason to look carefully, many improvements have been made in measuring this speed accurately, and the agreement of this relation is exactly as Maxwell predicted (which goes beyond dimensional analysis to provide the numerical factors as well) to within all of the precisions of the measurements. (Clean up previous magnetic example with better one to get around issues of dipole geometry etc.)
Einstein's remarkable realization was that, since both of these constants appear as properties of empty space, and since that conclusion is even more strongly reinforced by the fact that light, which is purely a ``property of empty space'' depends on them, there is no place in the dimensional analysis that predicts the speed of light, in which to account for how the light was created. This is very different from the case of cannonballs, for instance, which themselves have no intrinsic property that predicts a characteristic speed, and which can have seemingly any speed desired, depending on how they are launched. Even though the electric and magnetic proportionality constants were measured under particular circumstances, once they took their place in the description as properties of space and the affects that propagate through it, those circumstances disappeared. He was left, even more remarkably than that there was a familiar ``substance'' that happened to travel at this speed, with the recognition that there seemed to be a characteristic speed associated with empty space, at least as measured in electric and magnetic phenomena, including light. Thus physics was given its second seemingly fundamental dimensionful constant in c. For the record, in terms of familiar units, this speed is roughly
While the significance of this constant as a fundamental dimensionful scale was not appreciated until well into the development of relativity, at least its appearance in physical systems, and its relation to measured properties of electricity and magnetism, were understood with the work of Maxwell. In this sense that speed of light was more ``familiar'' than Planck's constant, and in some sense part of its significance was already understood.
The one reference that we have made to the speed of light as a dimensionful reference-constant was our mention of a different ``addition'' scheme that must be used if speeds of objects are to be ``added'' in a way that respects what we have learned through relativity. What we mentioned in the chapter on language, and will come to understand properly in the chapter on relativistic kinematics, was that for ``very slow'' speeds, the ``relativistic addition'' is much like familiar addition, with 1/1000 + 1/1000 nearly = 2/1000, with the speeds represented in some appropriate units. But for much ``faster'' speeds, in these units identified as the speeds approaching 1, we find that the relativistic addition becomes much different from familiar addition, culminating in the rule 1 + 1 = 1. But for such an addition to even be defined, we must know what we mean by ``fast'' or ``slow'' speeds, specifically fast or slow relative to what? The answer is, relative to the speed of light. In other words, if we want to use an ordinary number, like 1/1000, to refer to a speed, but also want the addition rule to treat that number correctly in the context of relativity, the number we must use to refer to the speed is its ratio as a fraction of the speed of light. In other words, it is for those objects moving at 1/1000 the speed of light that we use the number 1/1000 as the label for the speed in the addition rule. And we see that it is precisely the speed of light (a ratio 1 to itself) which is the special scale at which the relation 1 + 1 = 1 is obtained.
In other words, if I am standing on a roadside, and a car drives toward me at a speed I say is very close to c, and its headlights produce light that the driver says move away from him toward me at speed c, I nonetheless find that the light from his car that shines on me is moving relative to me with speed c. Hence 1 + 1 = 1.