ACHILLES AND THE TORTOISE

From a philosophy a science is derived; from a science, an engineering system can be deduced. From engineering come the blueprints, and the machine-shop directions. And the philosophy of present-day science, summated in Einstein's relativity, can-not lead to the interstellar cruiser. But another philosophy º ?

In the pioneer days of wireless telegraphy the wife of one of the scientists engaged in developing the new invention was asked by her friends whether she could explain to them what her husband was doing. "Of course," she said, "my husband explained it to me just yesterday. Imagine a very long dog. His forelegs are in Washington when his hindlegs are still in New York. If you pinch this dog in New York, he will bark in Washington. Wireless telegraphy is exactly the same - but without dog."

Something similar could be said about interstellar space-travel. It will be space-travel all right - but without space. This seems to be a rather asinine statement. Nevertheless, we shall see later on that it expresses, in a some-what cryptic fashion, the very secret of a possible space-travel technique.

We learned in Part 2 of "Achilles and the Tortoise" that neither Space nor Time are absolute data of our scientific experience, and that they exist only in an interdependent rela-tion with Matter. But so far we have not yet discussed Matter. Well, what is Matter? Ancient Greek philosophy had an answer. One of its outstanding representatives (Democritus) said that Matter is the accumulation of tiny indivisible particles of an eternal and indestructible substance. All objects and phenomena of Nature can be explained by quantitative changes in the accumulations and configurations of these particles. This is the early theory of atoms. It has influenced scientific development for more than two thousand years. Recently, how-ever, this theory has been modified, and modified to such an extent that the original idea is hardly recognizable. There are elementary particles, yes, but - there is something else as well. And, if somebody tells you that these particles are no particles at all, you won't be able to contradict him.

At first sight the situation looks rather confusing, but this is only due to the fact that we are momentarily caught in a transition period where we are passing from Democritus' idea of absolute atoms and absolute Matter to the very novel discovery that Matter is as relative as Space, Time, color, motion et cetera. Just the same, in order to find out something about the modern concept of Matter the theory of elementary particles is still a good starting point. The following table enumerates the particles which were known to physics up to the year 1953, and also shows, us some of. their properties:


Name	Charge	Mass	Lifetime	Decay Scheme	Spin
Graviton	0	0	stable	no	2
Photon	0	0	stable	no	1
Neutrino	0	0	stable	no	½
Electron	-	1	stable	no	½
Positron	+	1	stable	no	½
Meson					mostly
Group	0, -, +	210-1400	not stable	yes	unknown
Proton	+	1836	stable	no	½
Neutron	0	1838.5	750 sec	yes	½
Neut.V-Part.	0	2190	not stable	yes	?
Pos. V-Part.	+	2200	not stable	yes	?

We are rather certain that this table is not exhaustive. According to present theory there should be an "anti-proton". It is also improbable that the Meson-group is complete with its eleven members. If the present system of counting particles is retained, a final table might enumerate twenty-seven or twenty-nine particles. But this is just a hypothesis.

However, these particles do not have the properties Democritus ascribed to them. Perhaps the best way to define them is to say, that each particle represents a "localized" manifestation of a quantum field. Each quantum field, on the other hand, fills the whole of Space and Time of our present Universe. But there is a differ-ence between these fields. Only the gravitational and the electromagnetic field are genuine long-range fields. All the others are extremely short-ranged in their observable effects. It might be said with a certain modicum of truth that the first two fields represent - -partly at least - the long-range be-havior of the short-range quantum fields. The gravitational and the electromagnetic field are, therefore, called the classical fields.

This abolishes completely the con-cept of empty Space. It looks as if Space is physically real only as an extension of the quantum fields be-yond the existence of "solid" matter. This also implies that distance is a quantized property, produced by gradual accumulation of space-quants.

And what goes for Space should have its analogy in Time. There is no ex-perimental proof yet, but it is per-fectly safe to predict that there is also a smallest physical unit length of Time, an irreducible time-quant. The concept of the quant, first discovered to be a property of Matter in its fluid state as energy (Planck), is undoubt-edly the general criterion of whether something is measurable or not. Space, Time and Matter - as far as they have measurable properties - are quantized. But, as far as they do not represent an accumulation of quants, they are not measurable. And what is not measur-able does not exist-scientifically speaking. Or does it? Well, the equiv-alence between physical existence and measurability is certainly true for classical physics, but we shall obtain a very different - and most unexpected answer if we couch the available information in terms of quantum physics.

However, before I attempt to for-mulate the quantum-physical answer - which, incidentally, will lead us straight down the road to the problem of space-travel - permit me to recapit-ulate the salient features of our problem:

2) We also know from the solution of Zeno's paradox that the phenom-enon of motion is independent of the number of physical - denumerable - space-quants a moving body transverses. Achilles passes in the same time as many real points, but more space-quants than the tortoise.

With the help of arguments (1) and (2) the theoretical feasibility of inter-stellar and even intergalactic space-travel can be demonstrated. I shall show this by first analyzing the significance of these arguments. Achilles is capable of traveling, within the same time, a longer distance than the animal because by doing so he passes no more and no less real - non-denumerable - points than his compet-itor. Zeno assumed for the sake of simplicity that our hero travels exactly twice as far as the animal. But from what we know about the absolute equivalence of shorter and longer dis-tances in terms of the transfinite num-ber c [cf. argument (1)] Zeno's point would also be valid if his fast runner would travel a quintillion times as far as the slow one. The ratio of the distances is irrelevant. We therefore ask: What makes Achilles overtake the tortoise? The answer is trivial: He uses longer legs, and by doing so, he compensates for the fact that he has to pass more space-quants than his opponent.

I'm afraid we should need awfully long legs to step from here to the Andromeda Nebula. Achilles´ personal method is not very practical for inter-stellar distances. But his example demonstrates a general principle. Dis-tance per se does not mean a thing! The traversing of distance is "purely a matter of operational method" (Campbell). We have not yet reached the stars, because we use an extremely limited operational procedure for loco-motion - a general procedure, by the way, which includes everything from the crawling of a toddler to the flight of a jet plane or a rocket.

If you have various forms of loco-motion, some slower and some faster, but all unsatisfactory for a certain purpose, you may ask: What is their common characteristic which makes them all so inadequate? The answer in our case is simple. The toddler, the tortoise, Achilles or the jet plane, they all try to cover distances by passing space-quants. But in terms of space-quants there are always shorter and longer distances, and no matter how good your operational method is, there always comes a point where the num-ber of space-quants becomes too much for your technique. And viewed from the order of magnitude of galactic distances there is hardly any difference in speed between the toddler and the jet. Both are equally outclassed in the race for the stars.

The point, therefore, is, can we envisage an operational method of locomotion which does not try to cover distances in terms of space--quants? Because as long as we do so we have to distinguish between shorter and longer distances. Arguments (3) and (4) suggest that there should be such novel method. Let us find out, therefore, what these arguments really signify concerning the possibility of interstellar travel. According to (3) absolute empty Space does not exist in a strict physical sense. Conse-quently, absolute distances do not exist either. What does exist is "spa-tiality" (Einstein) as a measurable property of matter-in-general, i.e. of all physical states of nature. So the "absolute" distance between us and, let us say, the Andromeda Nebula, which has so far made it impossible for us to visit that distant galaxy, is not absolutely real. It is only relatively real with regard to our toddler-locomotion, and it, would disappear at once if we were to discover the proper operational technique to deal with such distances.

At this point I imagine I hear some of my readers mutter: "This sounds fishy! Even apart from the problem of space-flight we know there is a stupendous remoteness between us and a distant galaxy." Surely, there is! But the point is: Do we have to interpret it in terms of spatial distance? A trip to the Andromeda Nebula per-formed with the velocity of light, would take approximately two million years - terrestrial time. There is noth-ing which could prevent our saying: There is a temporal interval between us and this galaxy. Our space-vessel rests motionless in Space, but a time interval of two million years will affect the immediate proximity of the Andro-meda galaxy. Motion relative to empty Space is not observable! Motion relative to eventless Time is not ob-servable either. It follows with inex-orable logic that the two statements: I travel from here to the next, galaxy through Space or - exclusively - through Time are absolutely equiva-lent. For, if I travel through Space with the velocity of light, my docks will slow down to an absolute stand-still.

Obviously, Space and Time are interchangeable entities. But they are interchangeable only on the basis of Matter. When I said, we travel from here to the next galaxy two million years through Time but not through Space, I measured time outside the spaceship. When I stated that we travel the same way exclusively through space and not through Time the latter was measured by the clocks of the ship. The difference between the two sets of clocks is a material condition! But this means there are three inter-connected interpretations by which the mutual relations of Earth and Andromeda Nebula may be defined. We may say:

We discovered that (T) and (S) were interchangeable, provided the mate-rial gradient (M) was represented by a constant. This constant is the veloc-ity of light. It was assumed that our ship was traveling with that velocity.

In fact, we are quite accustomed to the interchangeability of the three cosmic components of the Universe in our daily lives! But we never think about it, and we fail, therefore, to notice the general significance of the most familiar phenomena.

In the preceding article I compared the Michelson-Morley ether-drift ex-periment with the situation on two cars traveling on a highway, and I stated that the difference in results as to the relative velocity of moving objects was due to the fact that in the case of the two cars there existed an "absolute" constant: the highway. Now, everybody who has ever driven a car, knows that his locomotion is subject to the following laws:

;	;

In the case of our highway travel (S) is always a constant, let us say, the distance between New York and Chi-cago. (S) is basic and cannot be changed. But (T) and (M) are variable, and to a certain degree interchange-able. We know from our experience that we can reduce the value of (T) by increasing the value of (M). In plain language: If we spend more gasoline, rubber, oil et cetera (M) we can save on time (T). But, if we want, to be economical with our engine, tires and fuel, we must pay for it in terms of longer duration. There are, of course, practical as well as theoretical limits to this experiment. Don't forget, not all traffic cops read Astounding, and are ready to accept our argument that we are only testing space-travel the-ories as a novel excuse for speeding.

There is, in fact, a very fundamental limitation to the operational procedure which affects the interchange between (T) and (M). This classical procedure can never be good enough to make one of the two components disappear completely, because in this case the value of the other would become in-finite. But just the same, even the primitive level of terrestrial locomo-tion permits us to study the basic interchange-relation of (T) and (M) with (S) being an "absolute" constant. -

In the case of the Michelson-Morley experiment the interchange-relation of the cosmic components is calculated on a different basis. This time the absolute constant is (M), the material factor, and (M) is represented by the most general property of Matter, the velocity of electromagnetic waves. Consequently we are entitled to ex-pect in this case an interchange-rela-tion between (T) and (S). This is indeed confirmed by the transformation equations of H. A. Lorentz:

	and

If we have two systems K and K´ being in rectilinear uniform motion relative to each other, then the following relation exists between temporal intervals (T) and spatial distances (S) according to Einstein's interpreta-tion of the formulas /2/: The faster K moves relative to K´, the longer will be the time interval transferred from one to the other system, and the shorter will be the spatial distance in the same transfer. At the same ratio as time grows, Space tends to disappear. And vice versa. The basis of this Space-Time interchange-relation is mass or Matter, because at the velocity of light the increase in mass for any moving body becomes infinite.

T ´ M	(1)

S ´ T	(2)

M ´ S	(3)

because the mutual interdependence of (S), (M) and (T) is such that the first two interchange relations could never exist unless a third between (M) and (S) would balance them. (We shall later demonstrate this in terms of symbolic logic.)

Interchange-relation (3) is the most interesting for us. We know (1) only to the limited extent of our daily experience in terrestrial travel. (2) Is at the moment a purely metric problem between the two continua Space and Time. But (3) represents the very core of modern quantum-mechanical phys-ics. It is becoming more and more difficult nowadays to draw a distinc-tive line between what is purely "spa-tial" and what is "material" in any physical datum. The borderline be-tween Space and Matter shows a sharp demarcation in classical physics only. It tends to disappear under the "boundary" conditions of microphys-ics as well as of astrophysics. It is only in the intermediate field of "terres-trial" macrophysics that we seem to know with some certainty what the difference between empty Space and solid Matter amounts to. In popular language: It is the difference between something and nothing. Matter is the sum total of "all things", and empty Space is the total absence of things. Every child can understand this. But not too long ago a most important formula was discovered: (Einstein)

E = mÞc2

(where E = the energy of a body at rest, m = its mass and c = velocity of light). Since then it has become general knowledge that "solid matter" (mass) may be transformed into energy (atom bomb), and that it is at least theoretically possible to re-transform radiant energy (light) into Matter. Does our distinction between Something and Nothing still hold? Is energy a thing? We are told that Matter is an electromagnetic phenomenon. But the electromagnetic field extends through all of Space. Space itself ap-pears to be a field-phenomenon. That means, microphysically speaking, it becomes more and more impossible to draw a sharp line of demarcation be-tween a thing and the space that surrounds it.

An analogous intimate relation be-tween Space and Matter exists in macrocosmic physics. A preliminary remark: Imagine you have a box partly filled with marbles. You take one after the other of these marbles out of the container till there is nothing left inside. Nobody will doubt that there is one thing left. That is the empty box. Its capacity of being a container has not been affected by the removal of the contents. This idea of the rela-tion between Space and its contents is that of classical macrophysics. But now let us proceed to macrocosmic physics.

Imagine yourself to be an entity with divine power, and located outside the Universe. You reach into the Universe and remove from it one galaxy and one nebula after the other, and you continue to do so till nothing material, not even the smallest mete-orite or the most tenuous wisp of cosmic dust is left in the Universe. According to common sense two objects should then be left: the empty Space waiting to be filled again with things, and empty Time waiting for the event of a new creation. Common sense tells us further that the dimen-sions of our Universe should not be affected by the removal of Matter and of events. But common sense has failed us once before when we dealt with Cantor's theory of transfinite number, and it will fail us again in macrocosmic physics.

We shall throw only a fleeting glance at what happens to Time in a Universe deprived of all Matter. In our present Universe Time has two directions. It stretches toward the past as well as toward the future. But in an empty Universe Time would have only one direction - toward the future. A past would not exist. The possibility of "passing" Time demands the presence of Matter.

But what about Space? Some time ago the English physicist Sir Arthur Eddington tried to define the inter-dependency of Space and Matter in certain equations - in a somewhat simi-lar way as the Maxwell-Hertz field theory connects the electromagnetic fields with charges or poles. Eddington´s equations permitted two inter-pretations which became known as, the Einstein- and the de Sitter-uni-verse. Einstein's universe is "static"; de Sitter's is in constant expansion - and contains no Matter! Consider what that means! A space that "con-tains no matter" but is in constant expansion is Matter in some state of radiant energy. It was later discovered (Friedmann, Lemaitre, Robertson) that Eddington's equations allow of a series of solutions which define a connection between the extremes of the Einstein- and the de Sitter-universe.

The gist of the theory can be de-scribed as follows: If you put some Matter into the de Sitter-universe, then its gravitational energy will start to counteract the expansion. The ex-panding world will start to slow up. If you add more and more material, you will finally arrive at a point where expansional and gravitational forces balance each other. This is the static world of Einstein. But if you add even more Matter, then gravitation be-comes stronger than expansion, and the Universe will start to shrink.

It seems as if Matter has a "con-tracting" effect on Space, and a corresponding influence on the struc-ture of Time. As far as Space is con-cerned one of the ways to describe its relation to Matter is represented by the formula:

	(5)

As (5) is at present a highly speculative formula I shall not deal in detail with it. But, even if it is only a rough approximation of the truth, it clearly implies that distance is a field-effect. By increasing or decreasing this effect spatial distances can be shortened or made longer. This seems probable anyhow. But distances in our universe are so enormous that even a considerable reduction of spatial dimensions would not help us much. What would be the use of pulling a galaxy which is about one billion light-years away into our immediate "neighborhood" of fifty million parsecs. Moreover, we would need the energy of millions or billions of galaxies to do it. The very idea is the height of absurdity.

Fortunately, our basic interchange-relations (1), (2) and (3) suggest some-thing else. Some way back in these articles I stated that Space and Time are continua; but Matter has a dis-continuous structure it is quantized. Then a very confusing thing happened: We found out that Space was quan-tized too, when we traveled through it. Zeno's paradox was produced by the conflict between denumerable (quan-tized) and nondenumerable numbers. How did that happen? The answer is simple: All classical forms of motion demand a partial "materialization" of Space. We travel through Space by means of a highway, for instance. And a highway, representing distance, as a functional part of an act of locomotion, is materialized space. And dis-tance in form of a highway is, of course, quantized. For the airplane our atmosphere plays the corresponding part. And even the principle of rocket propulsion - which may suffice for the short hop to, the Moon - is still based on the idea of quantized motion. It is only a little more sophisticated - you take your "highway" along with you. This time it is your rocket tubes.

But this convertibility of the spatial continuum into quantized material ex-istence is only possible if there is a reversed process by which quantized material distance can be converted into a non-quantized state of con-tinuity. We have at the moment not yet the slightest idea how this can be managed. Nevertheless we know two things with absolute certainty. We know that there can be no doubt that this inverse convertibility from quan-tized into non-quantized existence does exist. Because our three inter-change-relations,

T ´ M
S ´ T
M ´ S

would not be possible if only: Space could turn up as a quantized form of Matter, but Matter never as a spatial continuum Matter has spatial exten-sion in quantized form nobody ever doubted it. Our whole technology is based on that knowledge. But, if the inverse relation between Matter and Space did not exist we would have to assume that Matter - and with it Time - would constitute absolute data of nature. No modern scientist is ready to make this concession.

The other thing we know with equal certainty is that in a technology no longer based on the discreteness of Matter but on the continuity of Space and Time distance, goes by the boards. There is no longer any question of shorter or longer distances. They dis-appear completely. A line segment one billionth part of a millimeter long is numerically equivalent to a line segment of any trans-cosmic length.

Only a "little" question is now left. How can we develop a technique of locomotion which does no more use Space in its quantized material aspect, but which utilizes Matter in its, non-quantized spatial version? We are fortunate in so far as we are not totally ignorant in this respect. We have at least some negative knowledge - we are aware of the data that are missing. First: We shall have to dis-cover the law which describes the physical interaction between the elec-tromagnetic and the gravitational field. We need, further, detailed information about the cosmic "glue" that holds the atomic nucleus together (Meson theory).

One scientist (H. Bethe) has re-cently expressed the opinion that we need more powerful mathematical tools to tackle the problem of the nucleus. It may well be that Cantor's theory of non-denumerable sets will finally provide the answer. It should not be forgotten that Cantor only dis-covered the existence of transfinite numbers. How to use them in physical science is still a mystery to us. In the preceding article I could only show that non-denumerable sets do apply to the problem of motion in Space, and that their application demon-strates that distance is a property of quantized Matter but not of a con-tinuum like Space or Time.

Thus these articles on the problem of interstellar and intergalactic space-flight would have to conclude on a very unsatisfactory negative note, if it were not for a recent discovery in the field of applied symbolic logic. We are now well acquainted with the fact that symbolic logic can be used in analyzing electrical circuits and power patterns. I shall use this technique in order to demonstrate the basic power pattern all interstellar spaceships will have to apply. I know, of course, nothing about the details which will depend entirely on as yet unknown operational procedures of electromag-netic, graviton, and messianic character, but I know by logical analysis that they will have to conform to the fol-lowing structural pattern.

"M," "S" and "T" shall again be the symbols representing our three cosmic components of the Universe. Their interchange-relation shall be represented by ´´, ´´´ and ´´´´ respectively. We then obtain three elementary principles of inter-change:

´´		´´´		´´´´
M	S		S	T	T	M
S	M		T	S	M	T

We know, on the other hand, that any physical event involves all three cosmic components; because, it is physical, and it happens in Space as well as in Time. That means: No isolated interchange relation of ´, ´´, or ´´´ ever happens in the Universe. We, therefore, introduce two operational procedures:

and

with the intent to combine all three interchange-relations. The choice of these two procedures is by no means arbitrary. In fact they are the only possible operations, if we want to combine all three interchange-rela-tions. We shall define

as the combination of the upper line of all three relations of ´; that means in this case: of M to S, S to T and T to M.

In order to establish our opera-tional procedures

and

we arrange the cosmic components M, S and T in all possible combinations in two horizontal lines X and Y. We then look for the values of

in the upper lines of our ´´, ´´´ and ´´´´ tables, and we always take the second value. Thus, if our two horizontal lines contain the combination M S or we choose the value S from S M ´´ table. If the combination is

we choose again the second value of the upper line, this time of ´´´. Accordingly our value must be T, and so on. For

we select the second value of the upper line, this time of the ´ tables. We thus obtain as definitions of the operations

and

the following table:

X	M M M S S S T T T
Y	M S T M S T M S T
X Y	M S M S S T M T T
X Y	M M T M S S T S T

This arrangement may seem redundant, to the uninitiated. But it is not. As a basis for further calculations we need all logically possible combinations, even such seemingly redundant ones as between M and M, S and S, T and T.

One thing is absolutely certain - no matter how interstellar and intergalactic space-flight achieved, and what detailed technical arrangement may be used - all issues space-ships capable of traveling cosmic distances will have a switchboard based on the two operational patterns of -

and

. The interesting point, of course, is what could be done with it. Well, many things. It is impossible to predict which of the individual technical procedure that fall into the pattern of

and

will finally be used. This depends entirely on what future discoveries will be made in nuclear and astrophysics. But we might as well - for the sake of the practical demonstration of our "switch board" - assume that the description of interstellar flight, given by one of our outstanding science fiction authors is approximately correct. Permit me to quote this description from one of Isaac Asimov´s novels, where one of the officers of a spaceship explains the principles of interstellar flight to the pasengers /4/:

"Ladies and gentlemen! We are ready for our first Jump º The Jump is exactly what the name implies. In the fabric of space-time itself, it is impossible to travel faster than the speed of light º Therefore one leaves the space-time fabric to enter the little-known realm of hyperspace, where time and distance have no meaning. It is like traveling across a narrow isthmus to pass from one ocean to another, rather than remain-ing at sea and circling a continent to accomplish the same distance º Great amounts of energy are re-quired, of course, to enter this 'space within space' as some call it, and a. great deal of ingenious calculation must be made to insure re-entry into ordinary space-time at the proper point. The result of the expenditure of this energy and intelligence is that immense distances can be transversed in zero time. It is only the Jump which makes interstellar travel possible."

Asimov like many other science fiction writers sees quite clearly that the possibility of space-travel depends on the elimination of spatial distance. We know now that distance must be interpreted as an accumulation of space-quants, and we travel distances by passing space-quants. I remarked in Part 2 of this series: No matter how good our travel methods might become, there is always a critical point where the accumulated number of space-quants becomes too much for our operational methods. The question, therefore, is: Will it be possible to devise a tech-nique of locomotion where we "jump," as Asimov says. In other words:

We know that what is quantized is the M-factor in the Universe. Neither Time nor Space, being genuine con-tinua, are quantized per se. It is only their matter-aspect that gives them a quantized structure. It all boils down to the problem: Can we get rid of the M-factor in our operations

and

. Our highway experience has already taught us that M and T might be traded against each other - at least to a limited degree. The same applies to S and T. But if there is general con-vertibility of all three cosmic com-ponents into each other, it should indeed be possible to eliminate one of the components operationally. Asi-mov's description suggests that the M-component should disappear, be-cause it is the matter-factor that produces distance.