Einstein, the searcher

by Alexander Moszkowski

Now, it might occur to a particularly intelligent micro-being to undertake a voyage for the purpose of making measurements. He carefully marks his point of departure, walks Straight ahead in a certain direction, describing a circle on his sphere—a circle which he will necessarily regard as a straight line. He continues ever onwards in the firm conviction that he is getting farther and farther away from his starting-point. Suddenly, he discovers that he has reached it again. He discovers, by the mark he made, that he has not been describing a straight line, but a line that merges into itself.

The micro-professor would be compelled to declare: Our world, the only one known to me, is not infinite, although in a certain sense boundless. Moreover, it is not immeasurable, since it can be measured in at least one direction by the number of steps I have walked. From this we may infer that our former geometrical view was either wrong or incomplete, and that, in order to understand our world properly, we must build up a new geometry.

We may assume that the majority of the remaining micro-inhabitants would at first protest strongly against this decision. The idea that a line, which appears to them to be pointing always in the same direction, is curved, seems to them inconceivable and absurd. They would only gradually overcome their scruples of thought by getting an insight into a newly developed geometry that makes clear to them for the first time the conception of a sphere.

In our world of space, which includes all stars, we are the micro-inhabitants. We have been born with, or have inherited, the idea of a straight and ever-advancing path in space, and we become filled with the utmost astonishment if some one asks us to believe that if we undertake a voyage in one direction out into the universe, beyond Sirius and a million times farther, we should finally arrive at our starting-point again, although we had not changed our direction. But the macro-being, who belongs to a universe of higher dimensions and who looks on our world as we looked on the above spherical world one foot in diameter, sees the narrowness of our view. We, too, are in a position to rise above this narrow view by means of a theory founded on our experience, which will lead us to an extended world-geometry, just as the micro-professor used his experience to extend his theory of the circle to include the conception of a sphere.

After these preliminary remarks we shall endeavour to get an insight into Einstein's reasoning, not in the form in which it was originally presented (in the Report of the Proceedings of the Berlin Academy of Science of 8th February 1917), but in a very easy description which was given to me during a conversation. Here, too, I shall try to preserve the sense of Einstein's remarks without binding myself strictly to his words. For although I am indebted to him for his efforts to avoid difficult points, yet the aim of this book is, if possible, to make the explanation still easier. Any lack of accuracy arising from this last simplification is to be debited to me. The new form of representing the argument, which is as important as it is fascinating, is, of course, due to Einstein.

The final result stated by Einstein was: The universe, both as regards extent and mass, has finite limits and can be measured. If anyone asks whether this can be pictured, I shall not deprive him of the hope. All that is required is a power of imagination that is great enough to follow a pictorial description and that can take up the right attitude towards a sort of figurative representation.

Let us again imagine a sphere of modest dimensions with its two-dimensional surface. We are concerned only with the latter, and not with the cubical content. The sphere is to be considered as resting on an absolutely plane white table of unlimited extent in all directions. The sphere touches the table at a single point which we shall call its South Pole; on the top side directly opposite, we have the North Pole. To simplify matters we may make a sketch on paper of a vertical section through the centre of the sphere. This profile-picture will show us the sphere as a circle, and the white table as a straight line; the line joining the two poles is the axis of the globe, and the sectional circle is a meridian.

Let us further suppose a creature (resembling, say, a ladybird in shape) having length and breadth, but no thickness, to crawl along this meridian. Although it has no thickness, we shall imagine it to have one property of a solid body, that of being opaque, so that it can throw a shadow if properly illuminated. We assume the globe itself to be transparent. At the North Pole we suppose a very strong point-source of light, a little electric lamp, that sends out rays freely in all directions.

The insect begins its journey at the South Pole and sets out along the meridian to reach the North Pole. It is illuminated by the lamp all the way, so that it continually throws a shadow on the white table. The shadow moves along the table farther and farther from the South Pole, in proportion as the insect moves up the meridian, with the difference that while the insect is describing an arc of a circle, its shadow moves along a straight line. The position of the shadow can be determined at any moment by drawing the straight line connecting the lamp to the insect, and producing it to meet the white surface of the table; the point of intersection is the projection of the insect on the plane.

At the beginning of the excursion the shadow is exactly as large as the flat insect itself, if we assume that its dimensions are negligible compared with the surface of the sphere, for it will then coincide with its own shadow. But when the insect crawls upwards, its shadow will increase, because of the shortened distance between the insect and the lamp, and because the points of projection on the table separate more and more as their distances from their corresponding points on the sphere become greater. There is thus a twofold increase. The shadows move away more and more rapidly, and at the same time increase in size.