Robert Lionel, page 10
ments involving various coincidences. And the same remarks apply to theorems on congruence. Euclidean geometry is entirely dependent, at least in the form in which it is handed down to your generation, on the idea that a straight line is the shortest connection between two points.. I am now going to outline briefly how the basis of Euclidean geometry may be gained from the concepts of distance. We will start with the equality of distances. Suppose of two unequal distances one is always greater than the other, the same axiom must hold for the inequality of distances as holds for the inequality of numbers. We will do it simply. There are three distances: AB, BC, and CA.
Now if CA can be suitably chosen, you will have the marks AAl, BBl, CCl superimposed on one another in such a way that a triangle ABC results. The distance CAl has an upper limit for which this construction is stilI just possible.
The points A(BBl) and C then lie in a straight line. This leads to the following concepts, first producing a distance by an amount equal to itself; second, dividing a distance into equal parts; and third, expressing a distance in terms of a number by means of a measuring rod, or in other words, definition of the space interval between two points.
When the concept of the interval between two points or the length of a distance has been gained in this way, all we need to do is to follow Pythagoras' theorem, in order to arrive at Euclidean geometry analytically. To every point of space, or body of reference, three numbers of three co-orrlinates X Y Z may be assigned, and conversely in such a way for each pair of points A (Xl, YI, Zl) and B (X2, Y2, Z2) the theorem holds.
"Measure number AB-the square root of (X2-Xl)2
{Y2-Yl)2 plus (Z2-ZI)2.
"I think 1 must have now said enough to remind you of the fundamental principles of the Euclidean geometry with which, as an educated man of your time, you should have been familiar. But I must now go on to tell you the serious difficulties encountered in those representations and inter-
pretations of geometry, insofar as the rigid body of experience does not correspcnd exactly with the geometrical body. There are no absolutely definite marks, and moreover, we must always remember that temperature, pressure, and other circumstances of the environment trust modify the laws relating to position. It is also to be recollected that the structural constituents of matter-such as atoms and the electron-assumed by modem physics, are not in principle commensurate with rigid bodies, for they are, in actual fact, electrical discharges Nevertheless the concepts of geometry are applied to them and to their parts. For this reason modem thinkers havc been disinclined to allow real contents and facts, or as, the Germans would put it, Teale tatsachenbestande. These facts they make out do not correspond to geometry alone.
In fact, they consider it preferable to allow that the concept of experience corresponds to physics and geometry conjointly.
"Now, if we go on to apply the theorem of Pythagoras to infinitely near points, it reads like this:-DS2_DX2 plus DY2 plus DZ2. Where DS denotes the measurable interval between them, for an empirically given DS the coor-dinate system is not yet fully determined for every combination of points by this equation, and I rather doubt if it ever will be. In applying Euclidean geometry to pre-relativistic mechanics, a further indeterminateness is bound to come in when we use the co-ordinate system.
The state of motion of the co-ordinate system is arbitrary to a certmn degree, namely, in the substitutions of the co-ordinates of the form:-
"XI-X-VT
"YI-Y
"Zl-Z"
Bathurst, brilliant man that he was, was experiencing a great deal of difficulty in keeping up with these expositions of space, time and relativity. The other, the tall dark stranger, was speaking slowly and clearly, as a patient teacher speaks to a backWard child. And yet he was beginning to wonder how much of his explanation was taking root in this fertile mind that had been tom out of its native century. . . .
"Let us consider now," he went on, "the appearance of time itself. Every event that happens in the world is deter mined by the space co-ordinates of XYZ and the time c0-ordinate T. Thus the physical description must be four dimensional right from the beginning. But the four dinIen-sional continuum seems to resolve itself iJlto a three dimensional continuum of space, and a one .dimensional continuum of time. Now this apparent resolution owes its origin to the illusion that the meaning of the concept of simultaneity is self-evident. This illusion arises from the fact that we receive news of a new event almost simultaneously, owing to the agency of light. This faith in the absolute significance of simultaneity was destroyed by the law regulating the propagation of light in empty space.
Or, if my memory serves me rightly, in the dim days of scientific history, by the Maxwell~Lorentz electro-dynamics, two infinite near points can be connected by means of a light signal, and in that case the relationship may be mathematically expressed like this: DS2
C2D'f2-DX2- -DY2 - DZ2 - O. It must further follow from this, as I am sure you will agree, that the expression DS has a value which for chosen points, infiIiitely near in space time, is indepeDrlent of the particular inertial system that is selected. In agreement with this, we fiIid that passing from one inertial system to another, when the equa-tions of transformation hold, which they do not in general, leaves the time values of the events unchanged. So surely we must now realize," his voice became very firm as he hammered home the points of his argument, "it must thus become quite clear, that a four-dimensional continuum of space cannot be split up into a time continuum on the one hand, and a space continuum on the other, for the invariable quantity DS may be measured by means of measuring rods and by clocks."
Bathurst shook his head as though to clear his mind of the new ideas and concepts which were being presented to bin} by this brilliant 24th century scientist. He realized that he was attempting something well nigh impossible, as he strove to gain a fundamental mastery of the rudiments of this technology, five hundred years ahead of his oWn generation. The stranger smiled sympathetically, "Let me just go on now and give you a word about four dimensional geometry," he said softly. "On that invariable DS
that we've been talking about, it is possible to build up a complete four dimensional geometry, which is analagous to Euclidean geometry, in three dimensions. When we think of it like that, physics becomes a sort of statics, in a four-dimensional continuum, because apart from the difference in the number of dimensions, the latter continuum can be distinguished from the old Euclidean geometry be--
cause DS2 may be greater' or less than zero. Corresponding to this we can separate and distinguish time-like and space--like line element. The boundary between them is marked out by what we may call the light cone. The light cone is simply DS2_0, and that cone starts out from every point." He paused to let that piece of basic mathematical information sink in before going on again. "When we think of all the elements which belong to the same time value, we can put forward this mathematical formula DS2-:-DS2_DX2 plus DY2 plus DZ2-and then we find that the elements DS have real counterparts in distances at rest, and the Euclidean geometry will still hold good for these elements. With regard to the effects of Relativity, special and generalized, we come to a modification of the whole idea of space and time, a modification which the idea has undergone through the restricted theories and concepts of relativity. The whole idea of space has been still further modified by the general theory of Relativity, simply because this theory insists on denying that the three-dimensional space perception of the space-time continuum is Euclidean in character. And it goes on to assert, and to assert most strongly, that the old flat surface Euclidean geometry cannot, does not and will not hold for the relative positions of bodies that are con-tinuously in contact. Let us regard the empirical law of the equality of inertial and gravitational mass. You will find that we are led to interpret the state of the continuum as a gravitational field and to treat non-inertial systems as equivalent to inertial systems. A paradox, my friend, a
parad6x that has no basis in ultimate reality. . . .
"There was a great early scientist of considerable ge.-
nius,' who lived well over three hundred years ago, a man who was not very much after your own time. His name was Riemann, and he held up a system of geometry which holds good for a space of N dimensions, and his geometry bears the same relation to Euclidean geometry, as does the general geometry of curved surfaces to the geometry of the pure and I must add imaginary flat plane. There is a local co-ordinate system for the infinitesimal neighborhood of a point on a curved surface, in which the distance DS between two infinitely near points is given by this equation:-DS2_DX2 plus DY2, Eor other co-ordinate systems, however, we may use a different expression. The expression is DS2-Gll~ DX2 plus2G12, DXl, DX2 plus G22, DX22, and that holds good in a finite, bounded and limited region of the curves surface. Just as the old-fashioned Euclidean geometry space concept that we have been thinking about refers to the position possibilities of.
rigid bodies, of firm, unmoving bodies, so the generally accepted and understood theory of Relativity, that we may delineate as the space-time concept. refers to the general behavior of matter, energy, space and time and of course is measured by our old friends the clocks in part, at least.
Now to the really important parts of what I have been saying. I want us to consider time itself. If what I have said now has been complex, I trust you will forgive me; I have endeavored to simplify it insofar as my limited verbal powers will permit. . . . The physical time concept answers to the time concept of the extra scientifically inclined mind. We experience the moment 'now,' or to express it very accurately, what may be called the present sense experience. Or since we are stilI in Perleberg, we will use the motheT tongue of the proVince and say the sinnen Erlebais in combination with the recollection of earlier sense experience. That is why our sense experiences seem to form a series. In other words, the time series which we indicate by the simple words, earlieT and later, before and after.
Now and then, or in reverse, then and now. The experience series is regarded as a one-dimensional continuum; although we have already seen that a one-dimensional time continuum cannot be separated from the three-dimensional space continuum. We must realize that iliere is a fallacy in the argument up to this point, and that ilie natural everyday experiences and the primary sense in which they are accepted differ very vastly from the true state of affairs in the space-time continuum of the universe in which we exist.
"The experience series can repeat itself, and can of course be recognized and understood and comprehended.
It can also be repeated without necessary exactness, and in that case, some events are replaced by others without the character of the repetition being completely lost. In this way we form in our minds a time concept as a one-dimensional frame which can be fi11ed in by experiences in various ways. The experiences themselves are really subjective time intervals. When we move from subjective time, or as the mother tongue of this location would have it, the ich-Zeit, when we turn from that to the time concept of pre-scientific thought, we find that it is connected
with the creation fomlation, the manufacture, or the idea that there is a real external world, independent of the subject. In this sense, the objective event, which we have just been think;ng of, corresponds with the subjective experience, and they are attributed to the subjective time of the experience and real time of the corresponding objec~
tive event. When we then come to contrast experiences of outward events and their order of precession and recession in time, then we find that we have at last reached a concept which has a validity for all subjects. We find that we have arrived, if I may use an old cliche, at a moment of truth.
"Then, of course, there is the process of objectivism. . .
which doesn't encounter difficulties except for the fact that the time order of the experience corresponding to a series of outward events is not the same for all individuals.
Let us look at something straightforward from everyday life. When we see something, when we make a visual perception in our daily lives, then the two things correspond exactly, and it is because they correspond exactly in normal straightforward, uncomplicated everyday life that there is an object time order, which has beoo established to a remarkable degree and to a very wide extent. But when we come to work out the idea of an objective world of external events, and when we try to take this working out in-to a system demanding the utmost detail, then we find it very necessary to make both the experiences and the events depend on each other; to make them interdependent and interrelated in a much more complicated way. The. original attempts to bring this about were made by means of rules and methods and laws of thought, if you like, which were gained by man instinctively, as part of the evolutionary process and the uprising of the human mind. By these instinctive rules and modes of thought, the conception of space plays a particularly prominent part. H we go on to define the process, to condense and to analyze it down to its ultimate conclusio~ then we find that we have led ultimately to the goal of natural science. We know pedectly well that we measure time with the clock Or the watch or the chronometer, but let us ask ourselves exactly what a clock does. Let us ask ourselves for a dictionary definition of this common everyday object, for in both your centwy and mine a clock was as familiar as a house, or clothing or food. The purpose of the clock is automatically to pass in succession through an equal series of events. The number of those periods, which we will call clock time, which has elasped, serves as a simple and not terribly accurate measure of time.
"Let us think of a simple time observation experiment.
You will see immediately what I am driving at. If an event occurs in the neighborhood of one clock in the same space as the occurrence of the event, everybody watching the event observes the same clock time simultaneously with the event. They observe by means of the eye, and they observe independently of their position." He sighed deeply. "How wonderfully simple it would have been if everything had been as direct as that. . . ." There was just the slightest trace of bitterness in his voice. "Until the theory of Relativity was propounded, it was assumed that the conception of simultaneity had an absolute objective
meaning for events that were separated in space, just as it had a meaning for events that took place in the same area of space. That simple happy, homely little idea was shattered by the discovery of the law of the propagation of light. You see the velocity of light in empty space is a quantity that is independent of the choice of the inertial system to which it is referred. So no definite absolute or invariable meaning can be assigned to the conception of the simultaneity of events that occur at points separated by a distance. in space. A whole system has had to be allocated to every inertial system. And If no co-ordinate system or inertial system was used as a body of reference, there was no sense in asserting that events that took place at different places in space occurred at the same time.
"Before I can really come to the climax of the explanation that I am trying to give, I must just put you in the picture as regards the measurement of time, which has taken place between your day and ours. You see, the whole problem of the measurement of time means that we must refer first of all to its psychophysical basis, unless we are going to be completely simple and give it a mere empiric treatment because of its practical necessity. When experiences are repeated closely enough to each other, then we associate these with ideas of things and indeed with ideas of reality. Indeed, the whole psychological history of the construction of our world, of our universe and of our solar system, of the entire galaxy and the cosmos would appear to have its basis here. We look, then, over a measurement of time, toward some process or other which has recognizable repetitions-repetitions which are easy for us to count. This idea replaces the impalpable idea of duration. And it thus takes us into the realm of measured quantities. Let us think of a few simple examples; the beat of the pulse, the alternations of day and night, the appearances associated with definite spectral lives. These things, of course, are all natural processes. They have actually served this purpose, aiid then of course we have all our man-made instruments: the balance watch, the pen-dulum clock, the modem electrode chronometer, the radio-active clock, and so on, almost to infinity. We have turned the measurement of time into an extremely fine art.
Let us think then for a few moments of what the absolutely essential features of an ideal time measurement instrument would be, and we find that our ideal instrument would be a gyroscope, mounted without friction inside a case, which mayor may not rotate itself and is held absolutely friction free at the common axis. Or more abstractly still, let us imagine two particles which we will call A and B revolving at different angular speeds at a common center which we will call O. Each new passage of A past B is a repetition, giving account of a step of time. An apparatus like that might be used for measurements of time, as precisely as a pair of compasses can be used for measuring space. But if we do not know that the revolution of A with respect to B
is completely constant and unvaried, that it is, in other words, an invariant, then, using our old analogy of a pair of compasses, we have a pair of compasses that are not known to be stiff at the joint. The measure can still be made for what it is worth, and it may, in fact, be the only one possible. H the interval of time measured falls between
two interval counts, we have a problem in which we must subdivide the standard unit by constructing a smaller unit, which is standardized by comparing it with the standard.
