Albert Einstein

Mein Weltbild, Amsterdam: Querido Verlag, I934.

Scientific thought is a development of pre-scientific thought. As the concept of space was already fundamental in the latter, we must begin with the concept of space in pre-scientific thought. There are two ways of regarding concepts, both of which are indispensable to understanding. The first is that of logical analysis. It answers the question, How do concepts and judgments depend on each other. In answering it we are on comparatively safe ground. It is the certainty by which we are so much impressed in mathematics. But this certainty is purchased at the price of emptiness of content. Concepts can only acquire content when they are connected, however indirectly, with sensible experience. But no logical investigation can reveal this connection; it can only be experienced. And yet it is this connection that determines the cognitive value of systems of concepts.

Take an example. Suppose an archeologist belonging to a later culture finds a textbook of Euclidean geometry without diagrams. He will discover how the words "point;" "straight-line;" "plane" are used in the propositions. He will also recognize how the latter are deduced from each other. He will even be able to frame new propositions according to the rules he recognized. but the framing of these propositions will remain an empty play with word for him as long as "point;" "straight-line;" "plane;" etc., convey nothing to him. Only when they do convey something will geometry possess any real content for him. The same will be true of analytical mechanics, and indeed of any exposition of a logically deductive science.

What does it mean that "straight-line;" "point;" "intersection;" etc., convey something? It means that one can point to the sensible experiences to which those words refer. This extra-logical problem is the problem of the nature of geometry, which the archaeologist will only be able to solve intuitively by examining his experience for anything he can discover which corresponds to those primary terms of the theory and the axioms laid down for them. Only in this sense can the question of the nature of a conceptually presented entity be reasonably raised.

With our pre-scientific concepts we are very much in the position of our archaeologist in regard to the ontological problem. We have, so to speak, forgotten what features in the world of experience caused us to frame those concepts, and we have great difficulty in calling to mind the world of experience without the spectacles of the old-established conceptual interpretation. There is the further difficulty that our language is compelled to work with words which are inseparably connected with those primitive concepts. These are the obstacles which confront us when we try to describe the essential nature of the pre-scientific concept of space.

One remark about concepts in general, before we turn to the problem of space: concepts have reference to sensible experience, but they are never, in a logical sense, deducible from them. For this reason I have never been able to understand the quest of the a priori in the Kantian sense. In any ontological question, our concern can only be to seek out those characteristics in the complex of sense experiences to which the concepts refer.

Now as regards the concept of space: this seems to presuppose the concept of the solid body. The nature of the complexes and sense-impressions which are probably responsible for that concept has often been described. The correspondence between certain visual and tactile impressions, the fact that they can be continuously followed through time, and that the impressions can be repeated at any moment (touch, sight), are some of those characteristics. Once the concept of the solid body is formed in connection with the experiences just mentioned-which concept by no means presupposes that of space or spatial relation the desire to get an intellectual grasp of the relations of such solid bodies is bound to give rise to concepts which correspond to their spatial relations. Two solid bodies may touch one another or be distant from one another. In the latter case, a third body can be inserted between them without altering them in any way; in the former, not. These spatial relations are obviously real in the same sense as the bodies themselves. If two bodies are equivalent with respect to filling out one such interval, they will also prove equivalent for other intervals.

The interval is thus shown to be independent of the selection of any special body to fill it; the same is universally true of spatial relations. It is evident that this independence, which is a principal condition of the usefulness of framing purely geometrical concepts, is not necessary a priori. In my opinion, this concept of the interval, detached as it is from the selection of any special body to occupy it, is the starting point of the whole concept of space.

Considered, then, from the point of view of sense experience, the development of the concept of space seems, after these brief indications, to conform to the following schema solid-body; spatial relations of solid bodies; interval; space. Looked at in this way, space appears as something real in the same sense as solid bodies.

It is clear that the concept of space as a real thing already existed in the extra-scientific conceptual world. Euclids mathematics, however, knew nothing of this concept as such; it confined itself to the concepts of the object, and the spatial relations between objects. The point, the plane, the straight line, the segment are solid objects idealized. All spatial relations are reduced to those of contact (the intersection of straight lines and planes, points lying on straight lines, etc.). Space as a continuum does not figure in the conceptual system at all. This concept was first introduced by Descartes, when he described the point-in-space by its coordinates. Here for the first time geometrical figures appear, in a way, as parts of infinite space, which is conceived as a three-dimensional continuum.

The great superiority of the Cartesian treatment of space is by no means confined to the fact that it applies analysis to the purposes of geometry. The main point seems rather to be this: the Greeks favor in their geometrical descriptions particular object (the straight line, the plane); other objects (e.g., the ellipse) are only accessible to this description by a construction or definition with the help of the point, the straight line, and the plane In the Cartesian treatment, on the other hand, all surfaces, for example, appear, in principle, on equal footing, without any arbitrary preference or linear structures in building up geometry.

In so far as geometry is conceived as the science of laws governing the mutual spatial relations of practically rigid bodies, it is to be regarded as the oldest branch of physics. This science was able, as I have already observed, to get along without the concept of space as such, the ideal corporeal forms-point, straight line, plane, segment-being sufficient for its needs. On the other hand, space as a whole, as conceived by Descartes, was absolutely necessary to Newtonian physics. For dynamics cannot manage with the concepts of the mass point and the (temporally variable) distance between mass points alone. In Newtons equations of motion, t.he concept of acceleration plays a fundamental part, which cannot be denied by the temporally variable intervals between points alone. Newtons acceleration is only conceivable or definable in relation to space as a whole.

Thus to the geometrical reality of the concept of space a new inertia-determining function of space was added. When Newton described space as absolute, he no doubt meant this real significance of space, which made it necessary for him to attribute to it a quite definite state of motion, which yet did not appear to be fully determined by the phenomena of mechanics. This space was conceived as absolute in another sense also; its inertia-determining effect was conceived as autonomous, i.e., not to be influenced by any physical circumstance whatever; it affected masses, but nothing affected it.

And yet in the minds of physicists space remained until the most recent time simply the passive container of all events, without taking any part in physical occurrences. Thought only began to take a new turn with the wave-theory of light and the theory of the electromagnetic field of Faraday and Maxwell. It became clear that there existed in free space states which propagated themselves in waves, as well as localized fields which were able to exert forces on electrical masses or magnetic poles brought to the spot. Since it would have seemed utterly absurd to the physicists of the nineteenth century, to attribute physical functions or states to space itself, they invented a medium pervading the whole of space, on the model of ponderable matter the ether, which was supposed to act as a vehicle for electromagnetic phenomena, and hence for those of light also.

The states of this medium, imagined as constituting the electromagnetic fields, were at first thought of mechanically, on the model of the elastic deformations of solid bodies. But this mechanical theory of the ether was never quite successful so that gradually a more detailed interpretation of the nature of etheric fields was given up. The ether thus became a kind of matter whose only function was to act as a substratum for electrical fields which were by their very nature not further analyzable.

The picture was, then, as follows: space is filled by the ether, in which the material corpuscles or atoms of ponderable matter swim around; the atomic structure of the latter had been securely established by the turn of the century.

Since the interaction of bodies was supposed to be accomplished through fields, there had also to be a gravitational field in the ether, whose field-law had, however, assumed no clear form at that time. The ether was only supposed to be the seat of all forces acting across space. Since it had been realized that electrical masses in motion produce a magnetic field, whose energy provided a model for inertia, inertia also appeared as a field-action localized in the ether.

The mechanical properties of the ether were at first a mystery. Then came H. A. Lorentz great discovery. All the phenomena of electromagnetism then known could be explained on the basis of two assumptions: that the ether is firmly fixed in space- that is to say, unable to move at all, and that electricity is firmly lodged in the mobile elementary particles.

Today his discovery may be expressed as follows: physical space and the ether are only different terms for the same thing; fields are physical states of space. or if no particular state of motion can be ascribed to the ether, there does not seem to be any ground for introducing it as an entity of a special sort alongside of space.

But the physicists were still far removed from such a way of thinking; space was still, for them, a rigid, homogeneous something, incapable of changing or assuming various states. Only the genius of Riemann, solitary and uncomprehending, had already won its way by the middle of the last century to a new conception of space, in which space was deprived of its rigidity, and the possibility of its partaking in physical events was recognized. his intellectual achievement commands our admiration all the more for having preceded Faraday's and Maxwell's field theory of electricity.

Then came the special theory of relativity with its recognition of the physical equivalence of all inertial systems. The inseparability of time and space emerged in connection with electrodynamics, or the law of the propagation of light. Hitherto it had been silently assumed that the four-dimensional continuum of events could be split up into time and space in an objective manner-i.e., that an absolute significance attached to the "now" in the world of events.

With the discovery of the relativity of simultaneity, space and time were merged in a single continuum in a way similar to that in which the three dimensions of space had previously been merged into a single continuum. Physical space was thus extended to a four-dimensional space which also included the dimension of time. The four-dimensional space of the special theory of relativity is just as rigid and absolute as Newton's space.

The theory of relativity is a fine example of the fundamental character of the modern development of theoretical science. The initial hypotheses become steadily more abstract and remote from experience. On the other hand, it gets nearer to the grand aim of all science, which is to cover the greatest possible number of empirical facts by logical deduction from the smallest possible number of hypotheses or axioms. Meanwhile, the train of thought leading from the axioms to the empirical facts or verifiable consequences gets steadily longer and more subtle.

The theoretical scientist is compelled in an increasing degree to be guided by purely mathematical, formal considerations in his search for a theory, because the physical experience of the experimenter cannot lead him up to the regions of highest abstraction. The predominantly inductive methods appropriate to the youth of science are giving place to tentative deduction. Such a theoretical structure needs to be very thoroughly elaborated before it can lead to conclusions which can be compared with experience. Here, too, the observed fact is undoubtedly the supreme arbiter; but it cannot pronounce sentence until the wide chasm separating the axioms from their verifiable consequences has been bridged by much intense, hard thinking.

The theorist has to set about this Herculean task fully aware that his efforts may only be destined to prepare the death blow to his theory. The theorist who undertakes such a labor should not be carped at as fanciful; on the contrary, he should be granted the right to give free reign to his fancy, for there is no other way to the goal. His is no idle daydreaming, but a search for the logically simplest possibilities and their consequences. This plea was needed in order to make the listener or reader more inclined to follow the ensuing train of ideas with attention; it is the line of thought which has led from the special to the general theory of relativity and thence to its latest offshoot, the unified field theory. In this exposition the use of mathematical symbols cannot be completely avoided.

We start with the special theory of relativity. This theory is still based directly on an empirical law, that of the constancy of the velocity of light. Let P be a point in empty space, P' an infinitely close point at a distance. Let a flash of light be emitted from P at a time t and reach P' at a time t + dt.

If dx, dx2, dx3 are the orthogonal projections of do, and the imaginary time coordinate -lct = x4 is introduced, then the above-mentioned law of the constancy of the velocity of light propagation takes the form

Since this formula expresses a real situation, we may attribute a real meaning to the quantity ds, even if the neighboring points of the four-dimensional continuum are so chosen that the corresponding ds does not vanish. This may be expressed by saying that the four-dimensional space (with an imaginary time-coordinate) of the special theory of relativity possesses a Euclidean metric.

The fact that such a metric is called Euclidean is connected with the following. The postulation of such a metric in a three-dimensional continuum is fully equivalent to the postulation of the axioms of Euclidean geometry. The defining equation of the metric is then nothing but the Pythagorean theorem applied to the differentials of the coordinates.

In the special theory of relativity those coordinate changes (by transformation) are permitted for which also in the new coordinate system the quantity ds2 (fundamental invariant) equals the sum of the squares of the coordinate differentials. Such transformations are called Lorentz transformations.

The heuristic method of the special theory of relativity is characterized by the following principle: only those equations are admissible as an expression of natural laws which do not change their form when the coordinates are changed by means of a Lorentz transformation (covariance of equations with respect to Lorentz transformations).

This method led to the discovery of the necessary connection between momentum and energy, between electric and magnetic field strength, electrostatic and electrodynamic forces, inert mass and energy; and the number of independent concepts and fundamental equations in physics was thereby reduced. This method pointed beyond itself. Is it true that the equations which express natural laws are covariant with respect to Lorentz transformations only and not with respect to other transformations) Well, formulated in that way the question really has no meaning, since every system of equations can be expressed in general coordinates. We must ask: Are not the laws of nature so constituted that they are not materially simplified through the choice of any one articular set of coordinates.

We will only mention in passing that our empirical law of the equality of inertial and gravitational masses prompts us to answer this question in the affirmative. If we elevate the equivalence of all coordinate systems for the formulation of natural laws into a principle, we arrive at the general theory of relativity, provided we retain the law of the constancy of the velocity of light or, in other words, the hypothesis of the objective significance of the Euclidean metric at least for infinitely small portions of four-dimensional space.

This means that finite regions of space the (physically meaningful) existence of a general Riemannian metric is postulated according to the formula s' = gµv xµ xv,

where the summation is to be extended to a1l index combinations from 1.1 to 4.4.

The structure a such a space differs quite basically in one respect from that of a Euclidean space. The coefficients guv are for the time being any functions whatever of the coordinates x1 to x4, and the structure of the space is not really determined until these function guv, are really known. One can also say: the structure of such a space is as such completely undetermined. It is only determined more closely by specifying laws which the metrical field of the guv satisfy. On physical grounds it was assumed that the metrical field was at the same time the gravitational field.

Since the gravitational field is determined by the configuration of masses and changes with it, the geometric structure of this space is also dependent on physical factors. Thus, according to this theory space is exactly as Riemann guessed no longer absolute; its structure depends on physical influences. (Physical) geometry is no longer an isolated self-contained science like the geometry of Euclid.

The problem of gravitation was thus reduced to a mathematical problem: it was required to find the simplest fundamental equations which are covariant with respect to arbitrary coordinate transformation. This was a well-defined problem that could at least be solved.

I will not speak here of the experimental confirmation of this theory, but explain at once why the theory could not rest permanently satisfied with this success. Gravitation had indeed been deduced from the structure of space, but besides the gravitational field there is also the electromagnetic field. This had, to begin with, to be introduced into the theory as an entity independent of gravitation. Terms which took account of the existence of the electromagnetic field had to be added to the fundamental field equations. But the idea that there exist two structures of space independent of each other, the metric-gravitational and the electromagnetic, was intolerable to the theoretical spirit. We are prompted to the believe that both sorts of field must correspond to a unified structure of space.