Physics as a Geometrical Necessity1

Arthur Haas.

An ancient saying, attributed to Plato, states that geometry is God’s favorite occupation, while the Pythagoreans regarded number as the basis of all things. In both of these assertions there finds expression an aspiration as old as scientific physics itself. This aspiration is to regard as a mathematical necessity those regularities which the attentive observer of nature readily discovers amid the variety of phenomena of the sensory world.

Often this aspiration led to naïve mystical errors, when attempts were made to establish a relation between physical phenomena and special properties of certain numbers or geometrical figures.2 On the other hand, it led to fruitful and important works, thanks to which it proved possible, to a considerable degree, to rework, systematize, and supplement physics by introducing into it the mathematical method. But the culmination of this aspiration was the recognition that not only can physical phenomena be described with the aid of mathematics, and mathematics find application in physics, but that mathematics and physics are in their very essence one; that they differ no more than two different languages, both capable of serving for the description of the same objects; in other words—that by virtue of geometrical necessity the world given to us as reality reveals the phenomena of gravitation and electricity, as well as of magnetism, light, motion, and heat.

Theoretical physics has arrived at these grand conceptions, of the highest importance for our general understanding of the world, in recent years. The new ideas arose on the ground of one of the greatest creations of the human spirit, the theory of relativity, founded by Einstein in 1905.

The occasion for the establishment of the theory of relativity was a famous dilemma. From the formulas of so-called “classical” physics it followed that optical phenomena make it possible to establish an abso-

relative motion, whereas on mechanical grounds the very concept of absolute motion proved to be devoid of meaning. Indeed, if we assume that light propagates uniformly with respect to some coordinate system, i.e., with one and the same velocity in all directions, and imagine a second coordinate system in motion relative to the first, then, on the basis of the formulae of classical physics, the propagation of light with respect to this second system cannot be uniform, i.e., it cannot take place with the same velocity in all directions. It follows from this that one can conceive of some ingenious optical experiment which ought to reveal, to a sufficient degree, the influence of the earth’s own motion; however, when Michelson carried out such a difficult experiment (1881), the expected influence was not detected in the slightest degree. This discovery greatly perplexed physicists, although a priori it should have seemed highly improbable that in some branch of physics—namely in optics—absolute motion could be discovered, while in another—mechanics—it has no meaning1.

Theoretical physics was confronted with a difficult task—to eliminate the contradiction that had thus arisen between experiment and classical theory.

Its complete solution was first indicated by the genius of Einstein, who uncovered the existence of a prejudice so deeply rooted in physics that it had remained entirely unnoticed—namely, the prejudice concerning the existence of absolute time.

In physical formulae there appear, generally speaking, four independent variables: three spatial coordinates and time. It seemed perfectly self-evident that only a relative meaning could be ascribed to spatial coordinates; but as for time—no such thought had arisen.

Einstein was the first to point out that the data of time, just like the spatial coordinates by means of which an observer describes physical processes, are something relative. The contradiction between experiment and theory spoken of above is resolved without any difficulty if one accepts that the time to which an observer on the earth refers events must by no means be identified with the time to which another observer, situated on the sun, assigns the same events. Michelson’s experiment shows that between these two different times there must necessarily exist a certain relation, expressed by the postulate of relativity. Namely, both observers, although moving relative to one another, must have the possibility, with equal right, to assert—

each from his own point of view,—that light propagates in all directions with one and the same velocity, provided only that this velocity is expressed in their relative spatial coordinates and their relative time.

The recognition of the relativity of time represents a theoretical discovery of the first importance, but its still deeper meaning was first indicated by the Göttingen mathematician Minkowski (Minkowski, 1908). It was Minkowski who found that the relation, expressed by the postulate of relativity, between relative spatial coordinates and relative time can be interpreted geometrically in the sense that time, multiplied by the velocity of light and by the imaginary unit, and the spatial coordinates are related to one another just like the four coordinates of four-dimensional geometry1.

In order to make the meaning of Minkowski’s idea more comprehensible, let us recall the basic notions of analytic geometry, the creation of the great philosopher Descartes (Descartes). Anyone who has ever studied analytic geometry, even only in the plane, will easily remember that if some curve is given in the plane—for example, an ellipse—then its analytic expression is a certain equation. It indicates the “functional dependence” connecting with one another the two coordinates of every point of the curve. But anyone who has studied analytic geometry in the plane also knows that analytic geometry rests on the fact that the general form of such an equation does not change if we replace an arbitrarily chosen rectangular coordinate system in the plane of the drawing by some other rectangular coordinate system. Mathematicians say (and this expression, in view of what has been set forth above, is understandable without explanation) that the equation of the curve is invariant (i.e., preserves its form) with respect to transformations of the coordinate system.

Exactly the same thing as in analytic planimetry, of course, also takes place in the analytic geometry of space.

A surface (for example, an ellipsoid, and the like) is determined by one equation among three coordinates, and in exactly the same way this equation is invariant with respect to any transformation of the coordinate system. From the point of view of abstract geometry, it presents no difficulty to imagine a four-dimensional geometry, as it were a super-stereometry (Überstereometrie), in which each point will, of course, be determined by four coordinates.

We can now formulate, in this way, the aforementioned principle of Minkowski: three spatial coordinates and a quantity obtained by multiplying time by the speed of light and by the imaginary unit constitute four-dimensional coordinates; with respect to any transformation of a coordinate system constructed in this way, the equations of physics are invariant.

On the basis of this representation, which is only a simple geometrical description of the principle of relativity, an old problem of the theory of knowledge receives an entirely new illumination. For physical knowledge there come true the poetic words that Richard Wagner has Gurnemanz say to Parsifal: “Du siehst, mein Sohn, zum Raum wird hier die Zeit”—“You see, my son, here time becomes space.” Indeed, in the four-dimensional manifold that we shall call the Minkowski world (or simply “the world”), the position of the coordinate axes remains entirely arbitrary, one of which we interpret as the time axis. The distinction between time and space is not absolute, but only relative.

Thanks to Minkowski’s astonishing discovery, yet another concept, one of primary importance for theoretical physics, underwent a complete transformation: this is the concept of the physical field. By a physical field one understands a region over which some physical state is distributed in such a way that to each place in the region there corresponds a certain quantity characterizing that state.

In the investigation of a physical field, classical physics was always concerned with two questions: first, the determination of the properties of the field at a given instant; second, the change of the field in time. Alongside the spatial differences of a state, it was also necessary to take into account the changes that the state undergoes, in the course of time, at some definite place. The new physics, built on Minkowski’s ideas, in general knows no concept of a changing field. The new physics knows only one question: how are the quantities characterizing the physical state distributed in the Minkowski world? Indeed, if, for example, we know the four-dimensional temperature field of the Minkowski world, then by that very fact we know what temperature prevails at every place for every moment. For of the four axes of any coordinate system constructed in the temperature field, one is interpreted by us as the time axis, so that differences in the direction of this axis represent the change of temperature in time. Thus the concept of the variability of a field loses all meaning in the four-dimensional world. The Minkowski world, considered as the arena of physical events, is the realization of the definition of eternity given by Thomas Aquinas: he called eternity “nunc stans” (the standing now).

Theoretical physics may, therefore, be understood as a set of regularities according to which, in the world of Minkowski, the quantities characterizing the physical state are distributed. Physical states may, of course, be very diverse. In the sense of experimental physics we distinguish the mechanical, electromagnetic, optical, and thermal state1. The enormous significance of the kinetic theory of heat consisted, as is well known, in the fact that it reduced thermal states to mechanical ones; and in exactly the same way Maxwell succeeded in substantiating the electromagnetic theory of optical processes. Finally, precisely on the basis of the theory of relativity it became clear that processes of motion must, in essence, be electromagnetic processes, so that mechanical states are reduced to electromagnetic ones.

Thus, in the final analysis, all of physics has been reduced to the manifestation of the electromagnetic field. But on the basis of the equations which, since the time of Maxwell, have been established for the electromagnetic field, it can be shown that in the four-dimensional representation all quantities characterizing the electromagnetic state can be reduced to a single one, from which they are easily obtained by a simple computational operation. This quantity can be represented by means of a segment of definite direction; it is the so-called vector quantity. From the principle underlying the elementary law of the parallelogram of forces it follows further that, on a plane, any directed segment is completely determined if its two components along the axes of some definite coordinate system constructed on that plane are known; in space, a directed segment is completely determined by three components; in the four-dimensional world of Minkowski, four components are necessary for its determination. The four-component vector by means of which the electromagnetic field is completely determined is called the electromagnetic vector potential. The circumstance that this vector in the world of Minkowski is not constant, but changes from place to place (both in magnitude and in direction), determines the existence of the electromagnetic field and at the same time provides the possibility for man, by means of his sense organs, to perceive mechanical, thermal, and optical phenomena. Thus physics, strictly speaking, is caused by the existence of a four-dimensional vector field. Only one phenomenon occupies a special place: gravitation.

Thus, physics is compelled to reduce all physical phenomena to the existence of two fields, which, of course, coincide in time and in space: the electromagnetic field and the field of gravitation. The role which electricity plays in the electromagnetic field, in the field of gravitation pri-

falls to the share of mass. It was already known to classical physics that the quantities characterizing the field of gravitation can be derived, by a purely mathematical operation, from one quantity, the so-called gravitational potential. In classical physics, mass, just like the gravitational potential, is a quantity that is determined by specifying only a single numerical value, and for which, consequently, a decomposition into components is impossible1.

However, the introduction of Minkowski’s ideas makes it possible, in four-dimensional physics, to replace mass by a single quantity which, of course, does not depend on the coordinate system: the so-called tensor with ten components. This quantity, inherent in four-dimensional geometry, has four components corresponding to the four coordinate axes and, in addition, six components corresponding to the six coordinate surfaces2. From this one must conclude (a more detailed justification lies beyond the limits of this article) that in relativistic physics (of which classical physics is a special case) the gravitational potential must likewise be a quantity with ten components.

Thus, the electromagnetic field is determined by a four-component vector, and the field of gravitation by a ten-component tensor. At the same time, the two fields coincide, and the existence of both is conditioned by the totality of phenomena which we regard as physical.

In the light of these conceptions, the question whether physics proceeds from geometrical necessity may be formulated in the following clear way. We ask: are there geometrical grounds for assigning, in a general geometry, one vector quantity and one tensor quantity to each point? In the present state of geometry we must answer this question in the affirmative. Already in the middle of the nineteenth century the great mathematician Riemann3, in developing Gauss’s ideas, pointed out that a general geometry acquires meaning only in the case where a certain tensor is assigned to each point.

Einstein merely applied Riemann’s conception, which is valid for any geometrical manifold (with any number of dimensions), to Minkowski’s world, and in this way was able to establish the theory of gravitation on purely geometrical grounds (1915)4.

But only in 1918 did Weyl show that in a general geometry, to each point, in addition to the tensor already mentioned, it is necessary to assign

also a certain vector; by transferring these considerations to the world of Minkowski, Weyl obtained the possibility, alongside Einstein’s purely geometrical theory of gravitation, of constructing a purely geometrical theory of electricity1.

In order that, in the manifold of any measurement, geometry in general should be possible, it is necessary, according to Weyl’s conception, that this manifold represent first of all a vector and at the same time a tensor field. To non-mathematicians the foregoing will hardly seem clear. This is because we are accustomed to dealing with so-called Euclidean geometry, which, according to modern views, should be regarded as an exceptional special case of a far more general geometry. Indeed, Euclidean geometry rests on two arbitrary assumptions, the first of which was revealed by Riemann, and the second by Weyl.

To understand more distinctly what both these assumptions consist in, we shall first confine ourselves to the simple case of two-dimensional geometry. Every educated person knows the elementary laws of so-called planimetry, for example, the law according to which the sum of the angles in a triangle is equal to 180°. But anyone who has heard of spherical trigonometry also knows that two-dimensional geometry is possible not only on a plane, but also on curved surfaces, for example on the surface of a sphere; he also knows that in such geometries different laws2 hold than in planimetry, that, for example, the sum of the angles of a spherical triangle is always greater than 180°.

Even a non-mathematician likewise feels—I am ready to say instinctively—why plane geometry stands out as a special case among all other geometries on curved surfaces. Only in plane geometry does it make sense to speak simply of direction; in all other curvilinear geometries one can speak only of direction at a given definite point. If through some point on the plane of a board we draw a straight line, then without any difficulty we can draw through any other point of the plane a straight line having the very same direction as the first. But everyone knows that on an arbitrary curved surface such a problem, generally speaking, has no meaning. Indeed, let us imagine that on the surface of a very large globe representing the earth, at the place of Vienna, we place a small needle in the direction of the meridian; because of the smallness of the needle one may ignore the fact that it does not quite coincide with the surface of the globe. We shall move the needle along the meridian until it reaches the polar circle. Since we can look from outside the surface of the sphere (from space), it is perfectly

it is clear that the needle now has a different direction than it had before in Vienna. And yet we moved the needle from Vienna to the polar circle over the surface of the globe in its own direction, so that from the point of view of two-dimensional geometry we could with full right assert that the direction of the needle did not change.

The indeterminacy of direction will stand out before us still more clearly if we draw on the globe two curves which lead from Vienna to Paris. We shall construct these curves so that one passes through Amsterdam and the other through Rome. In both cases we shall move the needle along the curves under consideration in such a way that the initial point of the needle moves along the curve, while the needle is displaced parallel to itself. We shall find that in both cases, upon its arrival in Paris, the needle has a different direction. ^1)

Already this simple consideration shows us that on a curved surface the mathematical theory of directed quantities must in any case be more complicated than on a plane. It is obvious that on a curved surface we cannot, generally speaking, construct a “geometry of the distant” (Ferngeometrie), as is done on the plane, but only the so-called geometry of the near (Nahegeometrie). But even if we confine ourselves to the geometry of the near, then, evidently, we must first of all be able to compare with one another directions at two neighboring points. Indeed, let \(A\) be some point, and let \(B\) be a point, as we say, infinitely close, and suppose that some direction is given to us at the point \(A\); then, in order to be at all able to construct even the geometry of the near, we must know which direction at the point \(B\) corresponds to the given direction at the point \(A\) as “the same direction.” It is clear that we can know this only in the case when we are given at the point \(A\) a certain quantity which determines, in a known manner, the way the direction changes in passing from the point \(A\) to a neighboring one. This quantity turns out to be tensorial in the above (narrow) sense of the word.

Gauss made yet another very important discovery: with the aid of these tensors the length of curves, the magnitude of angles, and the area of figures are determined; consequently, conversely, the tensor can be found by measurements in the part of the surface under consideration, and we need know nothing about how the surface is “situated” in space. Thus, by means of the tensor, measurable relations are established for the first time; therefore it is called the fundamental metric tensor, for by it the so-called metric is first determined; it goes without saying that in the general geometry of surfaces we regard the fundamental metric tensor as a quantity varying continuously from place to place, so that the given

^1) In mathematical language this circumstance is expressed by the law according to which, in non-Euclidean geometry, the problem of the transport of a direction is not integrable.

the surface at the same time represents the field of its own fundamental metric tensor. If this field is given to us, then we can construct the geometry on the surface without knowing anything about the form of this surface. So-called Euclidean geometry is a special case of the general geometry of surfaces, characterized by the fact that in it the components of the fundamental metric tensor in all coordinate directions become either zero or one1.

Until now we have intentionally spoken only of the geometry of two dimensions; for only with respect to a two-dimensional manifold is it possible for us to have a clear representation of how it is situated in a manifold whose number of dimensions is greater by one. For a three-dimensional manifold the possibility of an analogous clear representation is already absent. However, Riemann already knew that our ordinary geometry should be regarded only as a special case of a more general three-dimensional geometry, in which the fundamental metric tensor changes from place to place; and what is true for three-dimensional geometry is true—as Riemann also knew—for geometries of any number of dimensions. The number of components of the fundamental metric tensor is then always equal to the sum of the numbers of coordinate axes and coordinate surfaces; in two-dimensional geometry the tensor therefore has three components, in three-dimensional geometry—six, in four-dimensional—ten.

The first arbitrary assumption of Euclidean geometry consists, as we have seen, in the fact that in it one speaks simply of direction, whereas in a geometry free from arbitrariness one can speak only of direction at a definite point. If we formulate in this way the first arbitrary assumption of Euclid’s geometry, then we naturally also arrive at the elucidation of the second assumption, which was first discovered by Weyl in 1918. This second assumption consists in the fact that in classical, as well as in Riemannian, geometry we speak simply of length, whereas one should always speak only of length in a definite small region. As we have seen, from a direction at one point one cannot, generally speaking, conclude that it is the same as a direction at some distant point; in exactly the same way, the conclusion that two small segments drawn in two remote regions have one and the same length turns out to be entirely devoid of meaning. In a consistently carried-out geometry the nearest comparison between two lengths is, generally speaking, just as difficult as the comparison of two directions2.

Let us be given two points \(A\) and \(B\) close to one another, and let us imagine that around each of them there is in some way bounded a certain very small region; first of all we must be able to establish what length a segment in the second region must have in order that its length may be regarded as equal to a given segment of the first region. In other words, if for the first region a unit of length is given as a scale, then we must first establish what is to serve as the scale in the second region. To the point \(A\) it is necessary to attach a quantity which would determine the transport of the unit of scale in passing from the region lying near the point \(A\) to some neighboring region. This quantity, on closer examination, proves to be a vector quantity. It would be appropriate to call it the scale vector (Masschtabvektor). Thus we are only then in a position actually to construct a general geometry if at every point of space, in addition to the fundamental metric tensor, we also assign a scale vector.

But everything that is true for any geometry, with any number of dimensions, is, of course, also true for the four-dimensional Minkowski world. In it, too, a general geometry free of hypotheses is possible only when, for each place in it, it is determined how the transport of the unit of scale takes place there; in other words, when for each place we know the value of the ten-component fundamental metric tensor and of the four-component scale vector. On the other hand, we have seen above that all physics rests upon the ability to encompass the Minkowski world as a field of a ten-component tensor, the gravitational potential, and of a four-component vector, the electromagnetic vector-potential. The thought naturally suggests itself of identifying the physical concept of gravitational potential with the mathematical concept of the fundamental metric tensor, and likewise of identifying the physical concept of electromagnetic potential with the mathematical concept of the scale vector; and if we carry out both identifications, then we do indeed obtain the possibility of regarding physics as a geometrical necessity. For on the basis of these ideas we find ourselves able to connect general geometry with the existence of the gravitational field and the electromagnetic field.

Geometry and physics thus appear in an inseparably close connection. Electric charges and gravitating masses determine the geometry of the Minkowski world. Yet the reverse construction is also conceivable: namely, one could first of all give the geometry of the Minkowski world, for which it would be necessary to know how the four-dimensional Minkowski world is situated in a five-dimensional manifold, just as a surface is situated in space. Then it would be possible, for example, to reduce the apparent manifestation of gravitating masses to the manner of this situation.

However, not only the existence of the electromagnetic and gravitational fields is a geometrical necessity. The laws of both of these fields, which find their expression in definite equations, also receive an acceptable mathematical foundation.

In Riemann–Weyl geometry an important role is played by a certain quantity¹), for its vanishing constitutes the necessary and sufficient condition for the general Riemann–Weyl geometry to pass over into the special case of classical Euclidean geometry²). By means of simple computational operations one can form from this quantity the so-called square of its value and integrate it over any region of Minkowski space. The integral thus obtained could be called the geometrical quantum of the region of space under consideration.

If we make the assumption that the geometrical quantum does not change under all possible variations of each of the ten components of the fundamental metric tensor and of each of the four components of the scale vector within our region³), then we obtain fourteen equations, which, on the basis of the identifications made above, should be regarded as the fundamental equations of the gravitational and electromagnetic fields⁴). And these are precisely the equations upon which the whole system of modern theoretical physics is built.

From the purely mathematical postulate expressing the property of the geometrical quantum described above there follows a definite relation, established for each point, between the fundamental metric tensor and the scale vector. From the physical point of view this relation is the manifestation of the relation between gravitation and electricity.

In this sense Weyl’s theory leads to a new and very interesting conclusion. Namely, it can be shown (we cannot dwell on this here in greater detail) that the law, known from experimental physics, of the uncreatability and indestructibility of electricity is a necessary consequence of the law of gravitation governing the motion of the planets⁵).

The new conceptions open up, to the highest degree, important prospects in another direction as well. The quantity that we have called the geometrical quantum is an abstract number. But if we interpret the fundamental metric tensor as the gravitational potential, and the scale vector as the electromagnetic potential, then from the physical—

¹) The so-called tensor of the fourth rank.

²) In this transition directions and lengths (see the note on p. 10) become integrable; the former occurs in a world devoid of gravitation, the latter in a world free of electric charges.

³) “Within” our region means that at the boundaries of the region no variation is to take place.

⁴) The basic idea of this principle belongs to Mie and was further developed by Hilbert and Weyl.

⁵) In exactly the same way, the identity of inertial and gravitational mass is, in a certain sense, a consequence of the indicated relation between the fundamental tensor and the scale vector.

from the mathematical point of view, the geometrical quantum turns out to be the so-called action multiplied by the constant velocity of light; here by “action,” as is known, one understands the product of energy and time.

At the beginning of the twentieth century there arose a new theory that led to most important consequences in many areas of physics, namely in the theory of the atom. This is the so-called quantum theory, which is based on the assumption that the quantity of action associated with some physical phenomenon is an integral multiple of the so-called elementary quantum of action. But since, from the mathematical point of view, action multiplied by the constant velocity of light is, according to what was said above, an abstract number, the question of the foundation of quantum theory proves to be most intimately connected with the fundamental problem of arithmetic, with the question of the essence of number.

It becomes ever clearer that the foundations of physics are the same as the foundations of mathematics. Thanks to physics, general geometry for the first time acquires meaning and content; and conversely, perhaps physics is nothing other than the geometry of a four-dimensional manifold translated into another language, which, by virtue of our interpretation, decomposes into space and time.

Translated by G. S. Landsberg.

1. The components whose indices are the same digit repeated twice become one; the rest become zero (cf. the note on p. 8). ↩↩↩↩↩↩↩

2. Expressing ourselves in the language of mathematics, we shall say: the transfer of direction, as well as the transfer of length, is, generally speaking, a nonintegrable problem. ↩↩↩↩

3. Riemann. Mathematische Werke (2 Aufl. Leipzig, 1892), p. XIII. ↩

4. Cf. the booklet by Einstein cited above. ↩