The problem of gravitation was thus reduced to a mathematical problem: it was required to find the simplest fundamental equations which are co-variant in relation to any transformation of co-ordinates whatever.
I will not speak here of the way this theory has been confirmed by experience, but explain at once why Theory could not rest permanently satisfied with this success. Gravitation had indeed been traced to the structure of space, but besides the gravitational field there is also the electro-magnetic field. This had, to begin with, to be introduced into the theory as an entity independent of gravitation. Additional terms which took account of the existence of the electromagnetic field had to be included in the fundamental equations for the field. But the idea that there were two structures of space independent of each other, the metric-gravitational and the electro-magnetic, was intolerable to the theoretical spirit. We are forced to the belief that both sorts of field must correspond to verified structure of space.
The “unitary field-theory,” which represents itself as a mathematically independent extension of the general theory of relativity, attempts to fulfil this last postulate of the field theory. The formal problem should be put as follows:—Is there a theory of the continuum in which a new structural element appears side by side with the metric such that it forms a single whole together with the metric? If so, what are the simplest field laws to which such a continuum can be made subject? And finally, are these field-laws well fitted to represent the properties of the gravitational field and the electromagnetic field? Then there is the further question whether the corpuscles (electrons and protons) can be regarded as positions of particularly dense fields, whose movements are determined by the field equations. At present there is only one way of answering the first three questions. The space structure on which it is based may be described as follows, and the description applies equally to a space of any number of dimensions.
Space has a Riemannian metric. This means that the Euclidean geometry holds good in the infinitesimal neighborhood of every point P. Thus for the neighborhood of every point P there is a local Cartesian system of co-ordinates, in reference to which the metric is calculated according to the Pythagorean theorem. If we now imagine the length I cut off from the positive axes of these local systems, we get the orthogonal “local n-leg.” Such a local n-leg is to be found in every other point P’ of space also. Thus, if a linear element (PG or P’G’) starting from the points P or P’, is given, then the magnitude of this linear element can be calculated by the aid of the relevant local n-leg from its local co-ordinates by means of Pythagoras’s theorem. There is therefore a definite meaning in speaking of the numerical equality of the linear elements PG and P’G’.
It is essential to observe now that the local orthogonal n-legs are not completely determined by the metric. For we can still select the orientation of the n-legs perfectly freely without causing any alteration in the result of calculating the size of the linear elements according to Pythagoras’s theorem. A corollary of this is that in a space whose structure consists exclusively of a Riemannian metric, two linear elements PG and P’G’, can be compared with regard to their magnitude but not their direction; in particular, there is no sort of point in saying that the two linear elements are parallel to one another. In this respect, therefore, the purely metrical (Riemannian) space is less rich in structure than the Euclidean.
Since we are looking for a space which exceeds Riemannian space in wealth of structure, the obvious thing is to enrich Riemannian space by adding the relation of direction or parallelism. Therefore for every direction through P let there be a definite direction through P’, and let this mutual relation be a determinate one. We call the directions thus related to each other “parallel.” Let this parallel relation further fulfil the condition of angular uniformity: If PG and PK are two directions in P, P’G’ and P’K’ the corresponding parallel directions through P’, then the angles KPG and K’P’G’ (measurable on Euclidean lines in the local system) should be equal.
The basic space-structure is thereby completely defined. It is most easily described mathematically as follows:—In the definite point P we suppose an orthogonal n-leg with definite, freely chosen orientation. In every other point P’ of space we so orient its local n-leg that its axes are parallel to the corresponding axes at the point P. Given the above structure of space and free choice in the orientation of the n-leg at one point P, all n-legs are thereby completely defined. In the space P let us now imagine any Gaussian system of co-ordinates and that in every point the axes of the n-leg there are projected on to it. This system of n2 components completely describes the structure of space.
This spatial structure stands, in a sense, midway between the Riemannian and the Euclidean. In contrast to the former, it has room for the straight-line, that is to say a line all of whose elements are parallel to each other in pairs. The geometry here described differs from the Euclidean in the non-existence of the parallelogram. If at the ends P and G of a length PG two equal and parallel lengths PP’ and GG’ are marked off, P’G’ is in general neither equal nor parallel to PG.
The mathematical problem now solved so far is this:—What are the simplest conditions to which a space-structure of the kind described can be subjected? The chief question which still remains to be investigated is this:—To what extent can physical fields and primary entities be represented by solutions, free from singularities, of the equations which answer the former question?
Notes on the Origin of the General Theory of Relativity
I GLADLY ACCEDE TO the request that I should say something about the history of my own scientific work. Not that I have an exaggerated notion of the importance of my own efforts, but to write the history of other men’s work demands a degree of absorption in other people’s ideas which is much more in the line of the trained historian; to throw light on one’s own earlier thinking appears incomparably easier. Here one has an immense advantage over everybody else, and one ought not to leave the opportunity unused out of modesty.
When, by the special theory of relativity I had arrived at the equivalence of all so-called inertial systems for the formulation of natural laws (1905), the question whether there was not a further equivalence of co-ordinate systems followed naturally, to say the least of it. To put it in another way, if only a relative meaning can be attached to the concept of velocity, ought we nevertheless to persevere in treating acceleration as an absolute concept?
From the purely kinematic point of view there was no doubt about the relativity of all motions whatever; but physically speaking, the inertial system seemed to occupy a privileged position, which made the use of co-ordinate systems moving in other ways appear artificial.
I was of course acquainted with Mach’s view, according to which it appeared conceivable that what inertial resistance counteracts is not acceleration as such but acceleration with respect to the masses of the other bodies existing in the world. There was something fascinating about this idea to me, but it provided no workable basis for a new theory.
I first came a step nearer to the solution of the problem when I attempted to deal with the law of gravity within the framework of the special theory of relativity. Like most writers at the time, I tried to frame a field-law for gravitation, since it was no longer possible, at least in any natural way, to introduce direct action at a distance owing to the abolition of the notion of absolute simultaneity.
The simplest thing was, of course, to retain the Laplacian scalar potential of gravity, and to complete the equation of Poisson in an obvious way by a term differentiated as to time in such a way that the special theory of relativity was satisfied. The law of motion of the mass point in a gravitational field had also to be adapted to the special theory of relativity. The path was not so unmistakably marked out here, since the inert mass of a body might depend on the gravitational potential. In fact this was to be expected on account of the principle of the inertia of energy.
These investigations, however, led to
a result which raised my strong suspicions. According to classical mechanics the vertical acceleration of a body in the vertical gravitational field is independent of the horizontal component of velocity. Hence in such a gravitational field the vertical acceleration of a mechanical system or of its center of gravity works out independently of its internal kinetic energy. But in the theory I advanced the acceleration of a falling body was not independent of the horizontal velocity or the internal energy of a system.
This did not fit in with the old experimental fact that all bodies have the same acceleration in a gravitational field. This law, which may also be formulated as the law of the equality of inertial and gravitational mass, was now brought home to me in all its significance. I was in the highest degree amazed at its persistence and guessed that in it must lie the key to a deeper understanding of inertia and gravitation. I had no serious doubts about its strict validity even without knowing the results of the admirable experiments of Eötvös, which—if my memory is right—I only came to know later. I now abandoned as inadequate the attempt to treat the problem of gravitation, in the manner outlined above, within the framework of the special theory of relativity. It clearly failed to do justice to the most fundamental property of gravitation. The principle of the equality of inertial and gravitational mass could now be formulated quite clearly as follows:—In a homogeneous gravitational field all motions take place in the same way as in the absence of a gravitational field in relation to a uniformly accelerated co-ordinate system. If this principle held good for any events whatever (the “principle of equivalence”), this was an indication that the principle of relativity needed to be extended to co-ordinate systems in non-uniform motion with respect to each other, if we were to reach an easy and natural theory of the gravitational fields. Such reflections kept me busy from 1908 to 1911, and I attempted to draw special conclusions from them, of which I do not propose to speak here. For the moment the one important thing was the discovery that a reasonable theory of gravitation could only be hoped for from an extension of the principle of relativity.
What was needed, therefore, was to frame a theory whose equations kept their form in the case of nonlinear transformations of the co-ordinates. Whether this was to apply to absolutely any (constant) transformations of co-ordinates or only to certain ones, I could not for the moment say.
I soon saw that bringing in non-linear transformations, as the principle of equivalence demanded, was inevitably fatal to the simple physical interpretation of the co-ordinates—i.e., that it could no longer be required that differentials of co-ordinates should signify direct results of measurement with ideal scales or clocks. I was much bothered by this piece of knowledge, for it took me a long time to see what co-ordinates in general really meant in physics. I did not find the way out of this dilemma till 1912, and then it came to me as a result of the following consideration :—
A new formulation of the law of inertia had to be found which in case of the absence of a real “gravitational field with application of an inertial system” as a co-ordinate system passed over into Galileo’s formula for the principle of inertia. The latter amounts to this:—A material point, which is acted on by no force, will be represented in four-dimensional space by a straight line, that is to say by a line that is as short as possible or more correctly, an extreme line. This concept presupposes that of the length of a linear element, that is to say, a metric. In the special theory of relativity, as Minkowski had shown, this metric was a quasi-Euclidean one, i.e., the square of the “length” ds of the linear element was a definite quadratic function of the differentials of the coordinates.
If other co-ordinates are introduced by means of a non-linear transformation, ds2 remains a homogeneous function of the differentials of the co-ordinates, but the co-efficients of this function (gμν) cease to be constant and become certain functions of the coordinates. In mathematical terms this means that physical (four-dimensional) space has a Riemannian metric. The time-like extremal lines of this metric furnish the law of motion of a material point which is acted on by no force apart from the forces of gravity. The co-efficients (gμν) of this metric at the same time describe the gravitational field with reference to the co-ordinate system selected. A natural formulation of the principle of equivalence had thus been found, the extension of which to any gravitational field whatever formed a perfectly natural hypothesis.
The solution of the above-mentioned dilemma was therefore as follows:—A physical significance attaches not to the differentials of the co-ordinates but only to the Riemannian metric co-ordinated with them. A workable basis had now been found for the general theory of relativity. Two further problems remained to be solved, however.
(1) If a field-law is given in the terminology of the special theory of relativity, how can it be transferred to the case of a Riemannian metric?
(2) What are the differential laws which determine the Riemannian metric (i.e., gμν) itself?
I worked on these problems from 1912 to 1914 together with my friend Grossmann. We found that the mathematical methods for solving problem (1) lay ready to our hands in the infinitesimal differential calculus of Ricci and Levi-Civita.
As for problem (2), its solution obviously needed invariant differential systems of the second order taken from gμν. We soon saw that these had already been established by Riemann (the tensor of curvature). We had already considered the right field-equation for gravitation for two years before the publication of the general theory of relativity, but we were unable to see how they could be used in physics. On the contrary I felt sure that they could not do justice to experience. Moreover I believed that I could show on general considerations that a law of gravitation invariant in relation to any transformation of co-ordinates whatever was inconsistent with the principle of causation. These were errors of thought which cost me two years of excessively hard work, until I finally recognized them as such at the end of 1915 and succeeded in linking the question up with the facts of astronomical experience, after which I ruefully returned to the Riemannian curvature.
In the light of knowledge attained, the happy achievement seems almost a matter of course, and any intelligent student can grasp it without too much trouble. But the years of anxious searching in the dark, with their intense longing, their alternations of confidence and exhaustion, and the final emergence into the light;—only those who have experienced it can understand that.
The Cause of the Formation of Meanders in the Courses of Rivers and of the So-Called Beer’s Law
IT IS COMMON KNOWLEDGE that streams tend to curve in serpentine shapes instead of following the line of the maximum declivity of the ground. It is also well known to geographers that the rivers of the northern hemisphere tend to erode chiefly on the right side. The rivers of the southern hemisphere behave in the opposite manner (Beer’s law). Many attempts have been made to explain this phenomenon, and I am not sure whether anything I say in the following pages will be new to the expert; some of the relevant considerations are in any case known. Nevertheless, having found nobody who thoroughly understood the elementary principles involved, I think it is proper for me to give the following short qualitative exposition of them.
First of all, it is clear that the erosion must be stronger the greater the velocity of the current where it touches the bank in question, or the more steeply it falls to zero at any particular point of the confining wall. This is equally true under all circumstances, whether the erosion depends on mechanical or on physico-chemical factors (decomposition of the ground). We must concentrate our attention on the circumstances which affect the steepness with which the velocity falls at the wall.
In both cases the asymmetry in relation to the fall in velocity in question is indirectly due to the occurrence of a circular motion to which we will next direct our attention. I begin with a little experiment which anybody can easily repeat.
Imagine a flat-bottomed cup full of tea. At the bottom there are some tea leaves, which stay there because they are r
ather heavier than the liquid they have displaced. If the liquid is made to rotate by a spoon, the leaves will soon collect in the center of the bottom of the cup. The explanation of this phenomenon is as follows:—The rotation of the liquid causes a centrifugal force to act on it. This in itself would give rise to no change in the flow of the liquid if the latter rotated like a solid body. But in the neighborhood of the walls of the cup the liquid is restrained by friction, so that the angular velocity with which it circulates is less there than in other places near the center. In particular, the angular velocity of circulation, and therefore the centrifugal force, will be smaller near the bottom than higher up. The result of this will be a circular movement of the liquid of the type illustrated in fig. 1. which goes on increasing until, under the influence of ground friction, it becomes stationary. The tea leaves are swept into the center by the circular movement and act as proof of its existence.
The same sort of thing happens with a curving stream (fig. 2). At every section of its course, where it is bent, a centrifugal force operates in the direction of the outside of the curve (from A to B). This force is less near the bottom, where the speed of the current is reduced by friction, than higher above the bottom. This causes a circular movement of the kind illustrated in the diagram. Even where there is no bend in the river, a circular movement of the kind shown in fig. 2 will still take place, if only on a small scale and as a result of the earth’s rotation. The latter produces a Coriolis-force, acting transversely to the direction of the current, whose right-hand horizontal component amounts to 2 ν Ω sin Φ per unit of mass of the liquid, where ν is the velocity of the current, Ω the speed of the earth’s rotation, and Φ the geographical latitude. As ground friction causes a diminution of this force towards the bottom, this force also gives rise to a circular movement of the type indicated in fig. 2.
Essays in Science Page 7