presented to the general public, in terms that the average person could understand (or so he thought).

(³Olds and News of Field Theory²)

By Albert Einstein

Newton saw this with perfect clarity and consequently he called space "absolute". In his theoretical system, there was a third constituent of independent reality; the motive force acting between material particles, such forces being considered to depend only on the position of the particles. These forces between particles were regarded as unconditionally associated with the particles themselves and as distributed spatially according to an unchanging law.

The physicists of the nineteenth century considered that there existed two kinds of such matter, namely, ponderable matter and electricity. The particles of ponderable matter were supposed to act on each other by gravitational forces under Newton's law, the particles of electrical matter by Coulomb forces also inversely proportional to the square of the distance. No definite views prevailed regarding the nature of the forces acting between ponderable and electrical particles.

It need not, therefore, surprise us that this theoretical system, built up by Newton with his powerful and logical intellect, should have been overthrown precisely by a theory of light. This was brought about by the Huygens-Young-Fresnel wave theory of light which the facts of interference and diffraction forced on stubbornly resisting physicists. The great range of phenomena, which could be calculated and predicted to the finest detail by using this theory, delighted physicists and filled many fat and learned books. No wonder then that the learned men failed to notice the crack which this theory made in the statue of their eternal goddess. For, in fact, this theory upset the view that everything real can be conceived as the motion of particles in space. Light waves, were, after all, nothing more than undulatory states of empty space, and space thus gave up its passive role as a mere stage for physical events. The other hypothesis patched up the crack and made it invisible.

The ether was invented, penetrating everything, filling the whole of space, and was admitted as a new kind of matter. Thus it was overlooked that by this procedure space itself had been brought to life. It is clear that this had really happened, since the ether was considered to be a sort of matter which could nowhere be removed. It was thus to some degree identical with space itself; that is, something necessarily given with space. Light was thus viewed as a dynamical process undergone, as it were by space itself. In this way the field theory was born as an illegitimate child of Newtonian physics, though it was cleverly passed off a first as legitimate.

To become fully conscious of this change in outlook was a task for a highly original mind whose insight could go straight to essentials, a mind that never got stuck in formulas. Faraday was this favored spirit. His instinct revolted at the idea of forces acting directly at a distance which seemed contrary to every elementary observation. If one electrified body attracts or repels a second body, this was for him brought about not by a direct action from the first body on the second, but through an intermediary action. The first body brings the space immediately around it into a certain condition which spreads itself into more distant parts of space, according to a certain spatio-temporal law of propagation. This condition of space was called "the electric field." The second body experiences a force because it lies in the field of the first, and vice versa. The "field" thus provided a conceptual apparatus which rendered unnecessary the idea of action at a distance. Faraday also had the bold idea that under appropriate circumstances fields might detach themselves from the bodies producing them and speed away through space as free fields: this was his interpretation of light.

Maxwell then discovered the wonderful group of formulae which seems so simple to us nowadays and which finally build the bridge between the theory of electro-magnetism and the theory of light. It appered that light consists of rapidly oscillating electro magnetic fields.

After Hertz, in the '80s of the last century, had confirmed the existence of the electro-magnetic waves and displayed their identity with light by means of his wonderful experiments, the great intellectual revolution in physics gradually became complete. People slowly accustomed themselves to the idea that the physical states of space itself were the final physical reality, especially after Lorentz had shown in his penetrating theoretical researches that even inside ponderable bodies the electro-magnetic fields are not to be regarded as states of the matter, but essentially as states of the empty space in which the material atoms are to be considered as loosely distributed.

We shall not here discuss questions concerning molecular theory, but shall describe the improvements made in the field theory during this century.

On the other hand, the services tendered by the special theory of relativity to its parent, Maxwell theory of the electro-magnetic field, are less adequately recognized. Up to that time the electric field and the magnetic field were regarded as existing separately even if a close causal correlation between the two types of field was provided by Maxwell's field equations. But the special theory of relativity showed that this causal correlation corresponds to an essential identity of the two types of field. In fact, the same condition of space, which in one coordinate system appears as a pure magnetic field, appears simultaneously in another coordinate system in relative motion as an electric field, and vice versa. Relationship of this kind displaying an identity between different conceptions, which therefore reduce the number of independent hypotheses and concepts of field theory and heighten its logical selfcontainedness are a characteristic feature of the theory of relativity. For instance, the special theory also indicated the essential identity of the conceptions' inertial mass and energy. This is all generally known and is only mentioned here in order to emphasize the unitary tendency which dominates the whole development of the theory.

It furnished an exact field theory of gravitation and brought the latter into a fully determinate relationship to the metrical properties of the continuum. The theory of gravitation, which until then had not advanced beyond Newton, was thus brought within Faraday's conception of the field in a necessary manner; that is, without any essential arbitrariness in the selection of the field laws. At the same time gravitation and inertia were fused into an essential identity. The confirmation which this theory has received in recent years through the measurement of the deflection of light rays in a gravitational field and the spectroscoptic examination of binary stars is well known.

Yet in the meantime one thing may be said in defense of the theory. Advance in scientific knowledge must bring about the result that an increase in formal simplicity can only be won at the cost of an increased distance or gap between the fundamental hypothesis of the theory on the one hand and the directly observed facts on the other hand. Theory is compelled to pass more and more from the inductive to the deductive method, even though the most important demand to be made of every scientific theory will always remain: that it must fit the facts.

We now reach the difficult task of giving to the reader an idea of the methods used in the mathematical construction which led to the general theory of relativity and to the new unitary field theory.

The conceptions which are used in this connection can be explained just as well in a two-dimensional continuum (a surface) as in the four-dimensional continuum of space and time. Imagine a piece of paper ruled in millimeter squares. What does it mean if I say that the printed surface is two-dimensional? If any point P is marked on the paper, one can define its position by using two numbers. Thus, starting from the bottom left-hand corner, move a pointer toward the right until the lower end of the vertical through the point P is reached. Suppose that in doing this one has passed the the lower ends of X vertical (millimeter) lines. Then move the pointer up to the point P passing Y horizontal lines. The point P is then described without ambiguity by the numbers X Y (coordinates). If one had used, instead of ruled millimeter paper, a piece which had been stretched or deformed the same determination could still be carried out: but in this case the lines passed would no longer be horizontals or verticals or even straight lines. The same point would then, of course, yield different numbers, but the possibility of determining a point by means of two numbers (Gaussian coordinates) still remains. Moreover, if P and Q are two points which lie very close to one another, then their coordinates differ only very slightly. When a point can be described by two numbers in this way, we speak of a two-dimensional continuum (surface).

ds2 = g11dx2 + 2 g11 g22 dx dy + g22 dy2

Then it is called a Riemannian metric. If it is possible to choose the coordinates so that this expression takes the form:

ds2 = dx2 + dy2 (Pythagoras's theorem), then the continuum is Euclidean (a plane).

Thus it is clear that the Euclidean continuum is a special case of the Riemannian. Inversely, the Riemannian continuum is a metric continuum which is Euclidean in infinitely small regions, but not in finite regions. The quantities g11, g12, and g22 describe the metrical properties of the surface; that is, the metrical field.

By making use of empirically known properties of space, especially the law of the propagation of light; it is possible to show that the space - time continuum has a Riemannian metric. The quantities g11, g12, and g22 , appertaining to it, determine not only the metric of the continuum, but also the gravitational field. The law governing the gravitational field is found in answer to the question: Which are the simplest mathematical laws to which the metric (that is the g11, g12, and g22) can be subjected? The answer was given by the discovery of the field laws of gravitation, which have proved themselves more accurate than the Newtonian law. This rough outline is intended only to give a general idea of the sense in which I have spoken of the "speculative" methods of the general theory of relativity.

The new unitary field theory removes this fault by displaying both types of field as manifestations of one comprehensive type of spatial structure in the space-time continuum. The stimulus to the new theory arose from the discovery that there exists a structure between the Riemannian space structure and the Euclidean, which is richer in formal relationships than the former but poorer than the latter. Consider a two-dimension Riemannian space in the form of the surface of a hen's egg. Since this surface is embedded in our (accurately enough) Euclidean space, it possesses a Riemannian metric. In fact, it has a perfectly definite meaning to speak of the distance of two neighboring points P, Q on the surface. Similarly it has, of course, a meaning to say of two such pairs of points (PQ) (P'Q'), at separate parts of the surface of the egg, that the distance PQ is equal to the distance P'Q'. On the other hand, it is impossible now to compare the direction PQ with the direction P'Q'. In particular it is meaningless to demand that P'Q' shall be chosen parallel to PQ. In the corresponding Euclidean geometry of two dimensions, the Euclidean geometry of the plan, directions can be compared and the relationship of parallelism can exist between lines in regions of the plane at any distance from one another (distant parallelism). To this extend the Euclidean continuum is richer in relationships than the Riemannian.

First of all, in such a continuum there are lines whose elements are parallel to one another. We shall call those "straight lines". It also has a definite meaning to speak of two parallel straight lines as in the Euclidean case. Now choose two such parallels E1L1 and E2L2 and mark on each a point P1, P2.

On E1L1 choose in addition a point Q1. If we now draw through Q1 a straight line Q1-R parallel to the straight line P1, P2, then in Euclidean geometry this will cut the straight line E2L2 ; in the geometry now used the line Q1-R and the line do not in general cut one another. To this extent the geometry now used is not only a specialization of the Riemannian but also a generalization of the Euclidean geometry. My opinion is that our space-time continuum has a structure of the kind here outlined.

The mathematical problem whose solution, in my view, leads to the correct field laws is to be formulated thus: which are the simplest and most natural conditions to which a continuum of this kind can be subjected? The answer to this question which I have attempted to give in a new paper yields unitary field laws for gravitation and electro-magnetism.

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