The organisation of thought, educational and scientific · Alfred North Whitehead
SPACE, TIME, AND RELATIVITY
(_Paper read to Section A at the Manchester Meeting of the British Association, 1915, and later before the Aristotelian Society_) FUNDAMENTAL Problems concerning space and time have been considered from the standpoints created by many different sciences. The object of this paper is the humble one of bringing some of these standpoints into relation with each other. This necessitates a very cursory treatment of each point of view. Mathematical physicists have evolved their theory of relativity to explain the negative results of the Morley-Michelson experiment and of the Trouton experiment. Experimental psychologists have considered the evolution of spatial ideas from the crude sense-data of experience. Metaphysicians have considered the majestic uniformity of space and time, without beginning and without end, without boundaries, and without exception in the truths concerning them; all these qualities the more arresting to our attention from the confused accidental nature of the empirical universe which is conditioned by them. Mathematicians have studied the axioms of geometry, and can now deduce all that is believed to be universally true of space and of time by the strictest logic from a limited number of assumptions. These various lines of thought have been evolved with surprisingly little interconnection. Perhaps it is as well. The results of science are never quite true. By a healthy independence of thought perhaps we sometimes avoid adding other people's errors to our own. But there can be no doubt that the normal method of cross-fertilising thought is by considering the same, or allied problems to our own, in the form which they assume in other sciences.
Here I do not propose to enter into a systematic study of these various chapters of science. I have neither the knowledge nor the time. First, let us take the ultimate basis of any theory of relativity. All space measurement is from stuff in space to stuff in space. The geometrical entities of empty space never appear. The only geometrical properties of which we have any direct knowledge are properties of those shifting, changeable appearances which we call things in space. It is the sun which is distant, and the ball which is round, and the lamp-posts which are in linear order. Wherever mankind may have got its idea of an infinite unchangeable space from, it is safe to say that it is not an immediate deliverance of direct observation. There are two antagonistic philosophical ways of recognising this conclusion. One is to affirm that space and time are conditions for sensible experience, that without projection into space and time sensible experience would not exist. Thus, although it may be true to say that our knowledge of space and time is given in experience, it is not true to say that it is deduced from experience in the same sense that the Law of Gravitation is so deduced. It is not deduced, because in the act of experiencing we are necessarily made aware of space as an infinite given whole, and of time as an unending uniform succession. This philosophical position is expressed by saying that space and time are _a priori_ forms of sensibility.
The opposed philosophical method of dealing with the question is to affirm that our concepts of time and space are deductions from experience, in exactly the same way as the Law of Gravitation is such a deduction. If we form exact concepts of points, lines and surfaces, and of successive instants of time, and assume them to be related as expressed by the axioms of geometry and the axioms for time, then we find that we have framed a concept which, with all the exactness of which our observations are capable, expresses the facts of experience. These two philosophic positions are each designed to explain a certain difficulty. The _a priori_ theory explains the absolute universality ascribed to the laws of space and time, a universality not ascribed to any deduction from experience. The experiential theory explains the derivation of the space-time concepts without introducing any other factors beyond those which are admittedly present in framing the other concepts of physical science. But we have not yet done with the distinctions which in any discussion of space or time must essentially be kept in mind. Put aside the above question as to how these space-time concepts are related to experience--What are they when they are formed? We may conceive of the points of space as self-subsistent entities which have the indefinable relation of being occupied by the ultimate stuff (matter, I will call it) which is there. Thus, to say that the sun is _there_ (wherever it is) is to affirm the relation of occupation between the set of positive and negative electrons which we call the sun and a certain set of points, the points having an existence essentially independent of the sun. This is the absolute theory of space. The absolute theory is not popular just now, but it has very respectable authority on its side--Newton, for one--so treat it tenderly.
The other theory is associated with Leibniz. Our space concepts are concepts of relations between things in space. Thus there is no such entity as a self-subsistent point. A point is merely the name for some peculiarity of the relations between the matter which is, in common language, said to be in space. It follows from the relative theory that a point should be definable in terms of the relations between material things. So far as I am aware, this outcome of the theory has escaped the notice of mathematicians, who have invariably assumed the point as the ultimate starting ground of their reasoning. Many years ago I explained some types of ways in which we might achieve such a definition, and more recently have added some others. Similar explanations apply to time. Before the theories of space and time have been carried to a satisfactory conclusion on the relational basis, a long and careful scrutiny of the definitions of points of space and instants of time will have to be undertaken, and many ways of effecting these definitions will have to be tried and compared. This is an unwritten chapter of mathematics, in much the same state as was the theory of parallels in the eighteenth century. In this connection I should like to draw attention to the analogy between time and space. In analysing our experience we distinguish events, and we also distinguish things whose changing relations form the events. If I had time it would be interesting to consider more closely these concepts of events and of things. It must suffice now to point out that things have certain relations to each other which we consider as relations between the space extensions of the things; for example, one space can contain the other, or exclude it, or overlap it. A point in space is nothing else than a certain set of relations between spatial extensions.
Analogously, there are certain relations between events which we express by saying that they are relations between the temporal durations of these events, that is, between the temporal extensions of the events. [The durations of two events A and B may one precede the other, or may partially overlap, or may one contain the other, giving in all six possibilities.] The properties of the extension of an event in time are largely analogous to the extension of an object in space. Spatial extensions are expressed by relations between objects, temporal extensions by relations between events. The point in time is a set of relations between temporal extensions. It needs very little reflection to convince us that a point in time is no direct deliverance of experience. We live in durations, and not in points. But what community, beyond the mere name, is there between extension in time and extension in space? In view of the intimate connection between time and space revealed by the modern theory of relativity, this question has taken on a new importance. I have not thought out an answer to this question. I suggest, however, that time and space embody those relations between objects on which depends our judgment of their externality to ourselves. Namely, location in space and location in time both embody and perhaps necessitate a judgment of externality. This suggestion is very vague, and I must leave it in this crude form.
_Diverse Euclidean Measure Systems_ Turning now to the mathematical investigations on the axioms of geometry, the outcome, which is most important for us to remember, is the great separation which it discloses between non-metrical projective geometry, and metrical geometry. Non-metrical projective geometry is by far the more fundamental. Starting with the concepts of points, straight lines, and planes (of which not all three need be taken as indefinable), and with certain very simple non-metrical properties of these entities--such as, for instance, that two points uniquely determine a straight line--nearly the whole of geometry can be constructed. Even quantitative coordinates can be introduced, to facilitate the reasoning. But no mention of distance, area, or volume, need have been introduced. Points will have an order on the line, but order does not imply any settled distance.
X = β_x_, Y = γ_y_, Z = δ_z_. [β ≠ γ ≠ δ.] They proceed to adjust their differences, and first take the _x_-coordinates. Obviously they have taken different units of length along O_x_. The length OU_{1}, which A calls one unit, B calls β units. B changes his unit length to OU_{1}, from its original length OV_{1}, and obtains X = _x_. But now, as he must use the same unit for all his measurements, his other coordinates are altered in the same ratio. Thus we now have X = _x_, Y = γ_y_/β, Z = δ_z_/β. The fundamental divergence is now evident. A and B agree as to their units along O_x_. They settled that by taking along that axis a given segment OU_{1} as having the unit length. But they cannot agree as to what segment along O_y_ is equal to OU_{1}. A says it is OU_{2}, and B that it is OU_{2}′. Similarly for lengths along OZ. The result is that A's spheres _x_^2 + _y_^2 + _z_^2 = _r_^2, are B's ellipsoids, X^2 + β^2Y^2/γ^2 + β^2Z^2/δ^2 = _r_^2, _i. e._ X^2/β^2 + Y^2/γ^2 + Z^2/δ^2 = _r_^2/β^2. Thus the measurement of angles by the two is hopelessly at variance. If β ≠ γ ≠ δ, there is one, and only one, set of common rectangular axes at O, namely that from which they started. If γ = δ, but β ≠ γ, then there are a singly infinite number of common rectangular axes found by rotating the axes round O_x_. This is, for us, the interesting case. The same phenomena are reproduced by transferring to any parallel axes.
The root of the difficulty is, that A's measuring rod, which for him is a rigid invariable body, appears to B as changing in length when turned in different directions. Similarly all measuring rods, satisfactory to A, violate B's immediate judgment of invariability, and change according to the same law. There is no way out of the difficulty. Two rods ρ and σ coincide whenever laid one on the other; ρ is held still, and both men agree that it does not change. But σ is turned round. A says it is invariable, B says it changes. To test the matter, ρ is turned round to measure it, and exactly fits it. But while A is satisfied, B declares that ρ has changed in exactly the same way as did σ. Meanwhile B has procured two material rods satisfactory to him as invariable, and A makes exactly the same objections. We shall say that A and B employ diverse Euclidean metrical systems. The most extraordinary fact of human life is that all beings seem to form their judgments of spatial quantity according to the same metrical system. _Relativity in Modern Physics_ Owing to the fact that points of space are incapable of direct recognition, there is a difficulty--apart from any abstract question of the nature of space--in deciding on the motion to be ascribed to any body. Even if there be such a thing as absolute position, it is impossible in practice to decide directly whether a body's absolute position has changed. All spatial measurement is relative to matter.
Newton's laws of motion in their modern dress evade this difficulty by asserting that a framework of axes of coordinates can be defined by their relations to matter such that, assuming these axes to be at rest, and all velocities to be measured relatively to them, the laws hold. The same expedient has to be employed for time, namely, the laws hold when the measurement of the flow of time is made by the proper reference to periodic events. Thus the laws assert that the framework and the natural clock adapted for their use have been successfully found. But, if one framework will do, an infinity of others serve equally well; namely, not only--as is of course the case--all those at rest relatively to the first framework, but also all those which move without relative rotation with uniform velocity relatively to the first. This whole set of frameworks is on a level in respect to Newton's laws. We will call them Dynamical frameworks. Now, suppose there are two observers, A and B. They agree in their non-metrical projective geometry, _e.g._, what A calls a straight line so does B. They also both apply a Euclidean metrical system of measurement to this space. Their two metrical systems also agree in having the same plane at infinity, that is, lines which are parallel for A are also parallel for B. Furthermore, they have both successfully applied Newton's laws to the movement of matter, and agree in having the same sets of dynamical axes. But the framework (among these sets) which A chooses to regard as at rest is different from the frame (among the same sets) which B so regards.
Without alteration of their respective judgment of rest, they choose their co-ordinate axes so that the origins (O for A, and O′ for B) are in relative motion along OO′, which is the axis of _x_ for both. Further, since OO′ is the line of symmetry of their diverse Euclidean systems, we assume that the two measure-systems agree for planes perpendicular to OO′, _i.e._, we assume a symmetry round OO′. Then if, for A at O, the distance OO′ be ξ, the relations at any instant between A's coordinates (_x_, _y_, _z_) and B's coordinates (_x′_, _y′_, _z′_) for the same point P are given by _x′_ = β(_x_ - ξ), _y′_ = _y_, _z′_ = _z_. Also, according to A's clock, O′ is moving forward with a uniform velocity _v_, and we measure A's time from the instant of the coincidence of O and O′. Thus ξ = _vt_, and _x′_ = β(_x_ - _vt_), _y′_ = _y_, _z′_ = _z_. We now consider B's clock, and ask for the most general supposition which is consistent with the fact that their judgments as to the fact of uniform motion are in agreement. We do not assume that events in various parts of space which A considers to be simultaneous are so considered by B. But we assume that at any point P, with coordinates (_x_, _y_, _z_) for A, there is a determinate relation between B's time T and _x_, _y_, _z_, _t_.
Text from Project Gutenberg, public domain in the United States.