5 7 4 D O C . 3 8 7 T H E N E W F I E L D T H E O R Y SEARCH FOR SIMPLICITY. 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 general problem is: Which are the simplest formal structures that can be attributed to a four-dimensional continuum, and which are the simplest laws that may be conceived to govern these structures? We then look for the mathematical expression of the physical fields in these formal structures and for the field laws of physics – already known to a certain approximation from earlier researches – in the simplest laws governing these structures. 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 millimetre 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 lower ends of X vertical (millimetre) lines. The point P is then described without ambiguity by the numbers X Y (co- ordinates). If one had used, instead of a ruled millimetre 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 ver- ticals or even straight lines. The same point would then, of course, yield different numbers, but the possibility of de- termining 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 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 millimetre 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 lower ends of X vertical (millimetre) lines. The point P is then described without ambiguity by the numbers X Y (co- ordinates). If one had used, instead of a ruled millimetre 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 ver- ticals or even straight lines. The same point would then, of course, yield different numbers, but the possibility of de- termining 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. Now consider two neighboring points P, Q on the surface and a little way off another pair of points P1, Q1. What does it mean to say that the distance PQ is equal to the distance P1Q1? This statement only has a clear meaning when we have a small measuring rod which we can take from one pair of points to the other and if the result of the comparison is independent of the particular measuring rod selected. If this is so, the magnitudes of the tracts PQ, P1Q1 can be compared. If a continuum is of this kind we say it has a metric. Of course, the distance of the two points PQ must depend on the coordinate differences (dx, dy). But the form if this dependence is not known a priori. If it is of the form: da2 = g11dx2+2g12dxdy+g22dy2 then it is called a Riemannian metric. If it is possible to choose the coordinates so that this expression takes the form: da2 = 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 re- gions, but not in finite regions. The g11, g12, g22 describe 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, dx. 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, dx.) 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. TWO FIELDS AS ONE. This theory, having brought together the metric and gravitation, would have been completely satisfactory if the world had only gravitational field and no electro-magnetic fields. Now it is true that the latter can be included within the general theory of relativity by taking over and appropriately modi- fying Maxwell’s equations of the electro-magnetic field, but they do not then appear like the gravitational fields as structural properties of the space-time continuum, but as logically independent constructions. The two types of field are causally linked in this theory, but still not fused to an identity. It can, however, scarcely be imagine that empty space has conditions or states of two essentially different kinds, and it is natural to suspect that this only appears to be so because the structures of the physical continuum is not completely described by the Riemannian Metric. The new Unitary Field Theory removes this fault by displaying both types of field as manifestations of one compre- hensive type of spatial structure in the space time continuum.The stimulus arose from the discovery that there exists a structure between the Riemannian space structure and the [4]