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DIMENSIONS AND SCALING REVISITED

The repeatability we have just seen would be utterly unimaginable out of context of what is generally called ``the atomic theory of matter'', which is the claim that all ``chemicals'' are made of a certain set of comparatively simple building blocks called atoms. Thus while chemistry is itself full of variety and complexity, as long as all of the rules that determine the forms of the chemical species are derivable from the structures of the atoms themselves, then the reproducibility of structure of these more fundamental building blocks ensures the reproducibility of all of the much greater sophistication of chemistry. It is useful to say that all ``chemicals'' are made of atoms, which is a little different from the original language of the theory, to avoid some problems with the original language that make it confusing in light of what we have learned since. When the ``atomic theory'' was first advanced, for instance, it was not known that atoms were themsevles composed of more fundamental parts (hence the name ``atom'', meaning ``indivisible'' is ``wrong'' from this later perspective). Yet fairly ``common'' objects like stars, or even burning things in the middle of their chemical reactions, consist more of the torn-apart pieces of atoms than they do of atoms, and so while they have the same ratios of the various constituents, they are not strictly composed of ``atoms'' as an intermediate level of structure. These ``torn-apart atom'' states, called plasmas, are awkward to describe with the original terms. Even more extreme problems with the original ideas are the modern realizations that the constituents of atoms themselves, such as the nuclear material at their cores, can combine to form whole objects, like nuetron stars, which not only are not composed of atoms as intermediate levels of structure, but do not even have the same ratios of the various constituents as do atomic matter and ``regular'' stars.

But these can be considered minor modifications relative to a certain notion of the atomic theory that has survived and is represented in chemical repeatability. That notion is that the repeatability is a repeatability of building blocks, which can be interchanged one for another but leave the form of what they make unchanged. The only modification to the original language of the ``atomic theory'' is a technical one, that the building blocks cannot be considered fundamental at the level of atoms, but require at least to be subdivided to a few more levels beneath that (i.e., those which we have found so far), and possibly that this subdivision goes on many levels further or even indefinitely. That essential kind of building-block repeatability is what was intended in the original language, that all ``matter'' was composed of atoms. Thus a division was assumed between what it means for something to be ``matter'' and the other notions that are still needed to describe nature, such as whatever characterizes the way these clumps of ``matter'' interact, gravitationally or through electric or magnetic effects, or however. In permitting the building blocks to be parts smaller than atoms, we have updated our language to acknowledge that ``matter'' also comes in plasmas or even forms like nuetron-star matter, but we have retained the essential perception of ``matter'' as the name for actors on a stage, which is thus only a part of the complete language necessary to describe ``what matter does''.

This original perception, that ``matter'' is somehow separate from ``what matter does'', persisted for a long time, but it is a distinction which loses meaning at more and more levels as we progress in our understanding of nature, and as we find that an even better understanding of what we should mean by ``matter'' incorporates more and more of ``what matter does'' as intrinsically a part of the description of ``what matter is''. In keeping with this loss of distinction, we look again at the notions of dimensions and scaling relations, to see that in fact they encode many of the same aspects of repeatability as those we have seen for chemistry, though not so obviously in the language of ``building blocks''. Much of the rest of this book could be taken as a further exploration of the meaning and importance of the objects we will introduce in this chapter, and pursuing that understanding takes us all the way to the forefronts of physical research and speculation. Part of what people would like to accomplish is an incorporation of all of ``what matter does'' as part of an improved and better language to describe ``what matter is'', at which point the distinction will be gone altogether.

(Need to go back and mention in chapter on language that in physics, the notion of ``why'' is taken to have meaning only in the sense of a notion of ``sub-what''s.)

We look more carefully at what a ``dimension'' is. We discover them when we find that there is an interchangeability of the parts of nature. For instance, the important function of a ``ruler'' is to fill the space between objects that are sitting somewhere. The fact that any object which fills the intervening gap can be replaced with any other object whose ends rest at the same places, independently of how otherwise-different the objects are, leads us to abstract the notion of ``length'' as the character of ``having ends that rest at the same places''. Thus rulers either ``have the same length'' or they don't. The abstraction of ``taking the same time'' from clocks that fill in the interval between ``events that happen'' is basically the same kind of idea. And, to the extent that one looks at ``masses'' as things that can be used to balance a scale, or to make springs oscillate with a certain characteristic time, the notion of ``having the same mass'' (or not) arises as the same kind of abstraction. Thus it is interchangeability of one whole object for another whole object that brings the notion of a dimension into existence. Obviously the notion would not be useful unless if two rulers are interchangeable to do one job, they also turn out to be interchangeable to do other jobs, and all rulers that are interchangeable one with another are interchangeable in any combination. That is why the notion of ``length'' survives as something independent of the rulers that are used to measure it.

But a dimension is useful for more than specifying either that two things are ``the same'' or not when doing a certain job. That is because not only can whole rulers of the same length be exchanged, but also many little rulers can be laid end to end to do the same space-filling job as a single big ruler. Similarly a sequence of ticks from different stopwatches can be substituted for ticks from a single watch, or many little lumps can be used to balance a scale in the same way as a single large lump. Thus the exchangeability that characterizes a dimension is also an exchangeability of many ``parts'' for a ``whole''.

The ultimate refinement of this notion that collections of parts are interchangeable for whole objects is the whole concept of ``scaling'', which says that one ``whole'' can be ``equivalent'' to another except for how many parts can be interchanged for it. In a language of building blocks, this is natural. A big volume of water is qualitatively ``the same'' as a little volume, except that there are more water building-blocks (molecules) in the big volume. Otherwise all the important properties that make it identifyable as ``water'' are no different. Similarly, two rulers can be qualitatively ``the same'' except that one is ``longer'' than the other.

These two properties are what we recognize as special about what we call the ``dimensions''. Each dimension represents a kind of interchangeability with respect to which two things either are ``the same'' or are not. Further, that interchangeability, which extends to make collections of things ``the same'' as other ``whole objects'' makes possible the description that two things are ``the same except for a rescaling'', and makes the concept of a dimension a useable one to describe changes that are only changes of scale and leave other qualitative ``properties'' of objects the same.

If, as we considered conceivable above, we had found the world to be a chaos of independent dimensions that we could discover but could not relate, this interchangeability and scalability might have been nothing more than evidence of the corresponding interchangeability of the building blocks. In a sense, discoveries much like this led to the discovery and acceptance of the modern theory of atoms and elements from chemical properties. There are many interchangeable chemicals which have repeatedly recognizable properties, such as ``ability to neutralize acid'', or ``ability to consume oxygen when they burn'', and literally endless others. And these properties are to a degree scalable, which makes it possible to say that, in equal amounts, one substance is ``a stronger acid'' than another, by a meaningful numerical amount. But the informational content of these repeatable properties has been completely extracted when we realize that they arise from the interaction-properties of atoms created by their own internal structures. And once we understand the structures of constituent atoms well enough to predict a given chemical property of molecules made from it, the scaling of that property with the amount of the chemical tells us nothing new.

But this is not what we found in our discovery that dimensional analysis is possible. Whereas in chemistry it was found that all chemical properties could be reduced to consequent properties of about a hundred kinds of atoms, each with a somewhat intricate internal structure, we have found that every dimension we discover to measure is reducible to a set of only three. If this hints at building blocks in a literally analogous way, the number of such blocks suggested seems very small.

More importantly, in chemistry, the molecular compositions of the various chemicals is assumed to provide a complete description of everything one wishes to consider about them and their reactions. Therefore, even within the long-standing distinction of ``matter'' from ``what matter does'', all of the behavior of chemical species is intelligible fairly directly in terms of the structure of the matter in question. However, if one maintains this distinction, the very existence of the scaling relations we have seen in the process of dimensional analysis seems intrinsically to contain relations between ``what matter is'' and ``what it does''. We have even gone so far as to remark that the central premise of dimensional analysis is that ``it does what it does because it is the way it is''.

But even preserving the distinction seems an odd thing to do when considering dimensional scaling relations. For instance, consider the spring oscillating with a massive lump on the end. The fact that changing the mass without changing the spring has the effect of changing the time of oscillation led us to even define the notion of ``mass'' as ``the dimension that characterizes the resistance of an object to being accellerated, or shaken back and forth''. Yet the very fact that the dimension exists permits us to replace the big lump with any collection of little lumps ``with collectively the same mass'', and the oscillations will happen in the same way. Yet even if we understand the mass as made up of building blocks of ``matter'', the existence of the scaling relation tells us something more. This is because, while a single little lump may have one characteristic time for oscillation, several such lumps stuck together all become characterized by a new oscillation time that is different from the individual time for any of them. Thus even something intrinsic to what, about the interaction of lumps of ``matter'' with empty space, causes them to resist ``being shaken'', or causes something else like the end of the spring to ``push on them'' to try to shake them, is represented in this scaling relation.

This scaling relation is somewhat peculiar, because it is not obvious that there is anything special about ``empty space'' that interacts with ``matter'' ``in it'' and causes it to resist being shaken. Even less obvious is that this relation has anything to do with gravity, as we shall see in later chapters (maybe, if it seems good to talk about Mach's problem). But unfortunately, in the most obvious scaling relation for gravity, the one thing that never appears is the mass, or number of lumps, so it is difficult to make the point with that example. Similarly, since we have been somewhat crude and not worried about just ``how'' a spring ``pushes'' on anything, it is difficult to describe how that ``push'' gets distributed among a bunch of lumps stuck together, so that each of them takes on a different characteristic period for motion than it would have alone. Thus unfortunately, the particular way in which these scaling relations do tell us something about the interactions of ``matter'' with things like ``empty space'' or ``the electromagnetic field'' (which is responsible for the pushing of the spring) will have to wait for development in later chapters to become more clear.

The point to remark is that the whole notion of a dimension as repeatable and scalable is very much the same sort of idea as the notion of repeatability and interchangeability-of-parts of the atomic theory. But the fact that dimensional analysis works, that is, that scaling relations of all sorts exist and can be reduced to the same three dimensional interchanges, seems to greatly extend the building-block interchangeability of ``matter'' to all of the ways it can behave. Thus, much as various chemical, and then later nuclear and various particle properties hinted at the forms of the building blocks of ``matter'', the presence of dimensionful scaling relations that are always found to be reducible to the same three dimensions seems to provide a similar description of all the kinds of behevior that that ``matter'' can have. Yet unlike the case for the ``structures'' of matter, we do not have a similar ``building-block'' language (at least not a complete one) from which we can derive the existence of the interchangability characterized by the various dimensions.

The fact that these descriptions are indeed so similar, and the never-violated ability to reduce all dimensional comparisons to the same absolutely repeatable set of three, suggests for one thing that the ``behavior of matter'', or whatever it is that ``mediates'' its interactions, such as the gravity that affects things falling toward earth, may have a repeatability and scalability, and hence perhaps a substructure, somewhat like the atomic substructure that accounts for the chemical properties. That this point of view is fruitful is born out more and more as we find that light, and then various nuclear ``forces'', that ``control'' the behavior and properties of ``matter'' can indeed be cast in a language no different from that which describes what we formerly regarded as the ``matter'' itself. But, if such a total building-block theory is possible, we have not succeeded in finding it, and precisely such suggestions as are provided by the successes that have been accomplished, along with the existence of the scaling relations, are what direct modern research to look for further developments toward the same reduction.

The similarity of the forms of scaling relations to the consequences of building-block descriptions constitutes much of the incentive for the latest stages of work, and is good to keep in mind as a kind of longer-range order that physicists are trying to discover, because it establishes a certain perspective for the ways modern discoveries are organized and presented. But much before this, and still a potent motivation in its own right, is the most basic description that can account for the repeatability of structures that we see. The first major progress in this direction was the atomic theory of matter, which reduced the repeatability of chemistry to the redundancy of atoms. But even when it was thought that atoms were in some sense ``fundamental'' or at least indivisible, there remained the problem of accounting for the repeatability of the particular structure of each kind of atom, and for the the fact that while individual atoms of a kind were repeatable, there were still so many different kinds that could be formed.

The fact that what we will discover in accounting for these observations also tells us something about the ``forces'' through which different objects interact is a bonus to the original question. It becomes natural from the modern perspective, that atoms are themselves collections of objects which occupy space, that the details of their structure tell us about space and about the forces that hold the parts together, as well as about the nature of the parts themselves. But our concern is less with particular applications of this repeatability than with finding any way to account for it at all, however it is expressed. The subject of this chapter is what has been discovered that relates to the repeatability of the various parts of nature that we examine, and to scaling relations themselves.

This section has been one of those ``looking ahead'' sections that refers to several ideas, which the reader cannot possibly be expected to interpret at this point, but rather to watch as they unfold before him in the succeeding chapters. Most of the ``un-understandable'' ideas relate to the way in which modern physics is coming to describe ``matter'' and ``the forces that affect matter'' as really two versions of the same thing. This removal of the disctinction is primarily the accomplishment of particle physics and of its implications for General Relativity. The understandable points relate to an interpretation that seems well-motivated by nature's high degree of repeatibility. They are:



Next: THE CONSTANT h/ Up: THE FUNDAMENTAL DIMENSIONFUL Previous: STARLIGHT AND LIFE


desmith@
Thu Aug 31 12:01:42 CDT 1995