Arthur Eddington

“ It will be seen how welcome to such a philosophy is the theory that space and the inertial frame come into being with matter, and grow as it grows. Since the laws of inertia are part of the law of gravitation, Mach's philosophy was summed up—perhaps unconsciously—in the profound saying "If there were no matter in the universe, the law of gravitation would fall to the ground."
No doubt a world without matter, in which nothing could ever happen, would be very uninteresting; and some might deny its claim to be regarded as a world at all. ”

“ If real time is used the world is spherical in its space dimensions, but open towards plus and minus infinity in its time dimension, like an hyperboloid. This happily relieves us of the necessity of supposing that as we progress in time we shall ultimately come back to the instant we started from! History never repeats itself. But in the space dimensions we should, if we went on, ultimately come back to the starting point. This would have interesting physical results, and we shall see presently that Einstein has a theory of the world in which the return can actually happen ”

“ In that case there is no gravitational or centrifugal force at infinity; but there is still inertia, which is of the same nature. The distinction between force as requiring a cause and inertia as requiring no cause cannot be sustained. We shall not become any more solvent by commuting our debt into pure inertia. The debt is inevitable whatever mesh-system is used; we are only allowed to choose the form it shall take.
The debt after all is a very harmless one. At infinity we have the absolute geodesics in space-time, and we have our own arbitrarily drawn mesh-system. ”

“ We have hitherto been greatly indebted to the suggestions of philosophy in evolving this theory, because the suggestions related to the things we know about; and, as it turned out, they were confirmed by experiment. But as physicists we cannot take the same interest in the new demand; we do not necessarily challenge it, but it is outside our concern. Physics demands of its scheme of nature something else besides truth, namely a certain quality that we may call convergence. The law of conservation of energy is only strictly true when the whole universe is taken into account ”

“ Our new law of gravitation admits that a rapid motion of the attracting body will affect the field of force. If the earth is non-rotating, the stars must be going round it with terrific speed. May they not in virtue of their high velocities produce gravitationally a sensible field of force on the earth, which we recognise as the centrifugal field? This would be a genuine elimination of absolute rotation, attributing all effects indifferently to the rotation of the earth the stars being at rest, or to the revolution of the stars the earth being at rest ”

“ Einstein himself prefers a different theory of curved space-time. His world is cylindrical—curved in the three space dimensions and straight in the time dimension. Since time is no longer curved, the slowing of phenomena at great distances from the observer disappears, and with it the slight experimental support given to the theory by the observations of spiral nebulae. There is no longer a barrier of eternal rest, and a ray of light is able to go round the world. ”

“ An alternative way is to inflate the world from inside, as a balloon is blown out. In this case the stretching force is not relegated to infinity, and ruled outside the scope of experiment; it is acting at every point of space and time, curving the world to a sphere. We thus get the idea that space-time may have an essential curvature on a great scale independent of the small hummocks due to recognised matter. ”

“ For Einstein's cylindrical world it is necessary to postulate the existence of vast quantities of matter (not needed on de Sitter's theory) far in excess of what has been revealed by our telescopes. This additional material may either be in the form of distant stars and galaxies beyond our limits of vision, or it may be uniformly spread through space and escape notice by its low density. There is a definite relation between the average density of matter and the radius of the world; the greater the radius the smaller must be the average density. ”

“ This seems to be interpreted as placing matter on a plane above geodesic structure in regard to the formulation of physical laws, though it is not easy to see in what sense a distribution of matter can be regarded as more observable than the field of influence in surrounding space which makes us aware of its existence. The principle itself is debateable; that which is observable to us is determined by the accident of our own structure, and the law of causality seems to impose our own limitations on the free interplay of entities in the world outside us. ”

“ We would reply, "Your particle of matter is not fundamental; it has no meaning in itself; what you are really concerned with is its 'field'—the relation of the geodesics about it to the other geodesics in the world—and matter cannot be thought of apart from its field." It is all a tangle of relations; physical theory starts with the simplest constituents, philosophical theory with the most familiar constituents. They may reach the same goal ”

“ Spherical space is not very easy to imagine. We have to think of the properties of the surface of a sphere—the two-dimensional case—and try to conceive something similar applied to three-dimensional space. Stationing ourselves at a point let us draw a series of spheres of successively greater radii. The surface of a sphere of radius should be proportional to ; but in spherical space the areas of the more distant spheres begin to fall below the proper proportion. There is not so much room out there as we expected to find. ”

“ Physics, if it wishes to retain its advantage, must take its own course, formulating those laws which are approximately true for the limited data of sense, and extending them into the unknown. The relativity of rotation is not approximately true for the data of sense, although it may possibly be true when the unknown as well as the known are included. ”

“ Some may be inclined to challenge the right of the Einstein theory, at least as interpreted in this book, to be called a relativity theory. Perhaps it has not all the characteristics which have at one time or another been associated with that name; but the reader, who has followed us so far, will see how our search for an absolute world has been guided by a recognition of the relativity of the measurements of physics. It may be urged that our geodesics ought not to be regarded as fundamental; a geodesic has no meaning in itself ”

“ We should find nature there as active as ever. We thought time was standing still, but it was really proceeding there at the usual rate, as if in a fifth dimension of which we had no cognisance. Casting an eye back on our old home we should see that time apparently had stopped still there. Time in the two places is proceeding in directions at right angles, so that the progress of time at one point has no relation to the perception of time at the other point. ”

“ In any case, precise calculation shows that the centrifugal force could not be produced by the motion of the stars, so far as they are known.
We are therefore forced to give up the idea that the signs of the earth's—rotation the protuberance of its equator, the phenomena of the gyrocompass, etc. are due to a rotation relative to any matter we can recognise. The philosopher who persists that a rotation which is not relative to matter is unthinkable, will no doubt reply that the rotation must then be relative to some matter which we have not yet recognised. ”

“ The same considerations that apply to rotation apply to acceleration, although the difficulty is less striking. We can if we like attribute to the sun some arbitrary acceleration, balancing it by introducing a uniform gravitational field. Owing to this field the rest of the stars will move with the same acceleration and no phenomena will be altered. But then it seems necessary to find a cause for this field. It is not produced by the gravitation of the stars. Our only course is to pursue the cause further and further towards infinity ”

“ The reader will easily see that a being confined to the surface of a sphere and not cognisant of a third dimension, will, so to speak, lose one of his dimensions altogether when he watches things occurring at a point 90° away. He regains it if he visits the spot and so adapts himself to the two dimensions which prevail there.
It might seem that this kind of fantastic world-building can have little to do with practical problems. But that is not quite certain. May we not be able actually to observe the slowing down of natural phenomena at great distances from us? ”

“ Owing to curvature in the time dimension, as we examine the condition of things further and further from our starting point, our time begins to run faster and faster, or to put it another way natural phenomena and natural clocks slow down. ”

“ Physics is an exact science because the chief essentials of a problem are limited to a few conditions; and it draws near to the truth with ever-increasing approximation as it widens its purview. The approximations of physics form a convergent series. ”

“ In this book the tradition of Faraday and Maxwell still rules our outlook; and for us matter and electricity are but incidental points of complexity, the activity of nature being primarily in the so-called empty spaces between.
The vague universal frame to which rotation is referred is called the inertial frame. It is definite in the flat space-time far away from all matter. In the undulating country corresponding to the stellar universe it is not a precise conception ”

“ The most remote objects known are the spiral nebulae, whose distances may perhaps be of the order a million light years. If natural phenomena are slowed down there, the vibrations of an atom are slower, and its characteristic spectral lines will appear displaced to the red. ”

“ It is probable that here we part company from many of the continental relativists, who give prominent place to a principle known as the law of causality—that only those things are to be regarded as being in causal connection which are capable of being actually observed. ”