A Stubbornly Persistent Illusion Page 12

  § 4. CONCLUDING REMARKS

  The above reflexions show the possibility of a theoretical construction of matter out of gravitational field and electromagnetic field alone, without the introduction of hypothetical supplementary terms on the lines of Mie’s theory. This possibility appears particularly promising in that it frees us from the necessity of introducing a special constant λ for the solution of the cosmological problem. On the other hand, there is a peculiar difficulty. For, if we specialize (1) for the spherically symmetrical static case we obtain one equation too few for defining the gμv and ϕμv, with the result that any spherically symmetrical distribution of electricity appears capable of remaining in equilibrium. Thus the problem of the constitution of the elementary quanta cannot yet be solved on the immediate basis of the given field equations.

  *Cf. e.g. A. Einstein, Sitzungsber. d. Preuss. Akad. d. Wiss., 1916, pp. 187, 188.

  †Cf. D. Hilbert, Göttinger Nachr., 20 Nov., 1915.

  *Cf. H. Weyl, “Raum, Zoit, Matorie,” § 33.

  Relativity–The

  Special and General

  Theory

  The earth is a slightly squashed sphere, and yet from the ground, it appears flat, and was thought to be flat for several thousand years. Likewise, our universe appears to us to be “flat” in the sense that Euclid’s axioms seem obviously true; chief among these that two straight lines or beams of light could intersect at most just once. This “flat” picture of space is the simplest one, and the picture accepted by all physicists prior to Einstein.

  Einstein did not immediately overturn the flat model of the universe, but simply added another dimension to height, width, and breadth: time. In “Relativity—The Special and General Theory,” Einstein described physics in flat space, the domain of special relativity. His postulates were quite simple: first, the laws of physics are the same for all observers moving at constant velocity, and second, all such observers will measure the speed of light. Sir Isaac Newton would certainly have conceded the first point, but the second he would have deemed impossible. Einstein achieved this effect by noting that the laws of physics were unchanged not only under rotations between directions in space, but also under “rotations” between space and time.

  Einstein recognized that the theory did not include gravity, and thus was necessarily incomplete. To remedy this, as discussed in Part II, he argued that the universe may be curved as well. The curvature of space and time has a number of profound implications: Light does not travel in straight lines, but rather is curved around massive bodies. Clocks sitting near massive bodies run slower than clocks far away. In other words, Einstein noted that not only is space curved, but time is as well. With a simple set of “field equations,” Einstein derived not only both the laws of motion and of gravity put forth by Newton, but also paved the way for explanations of a number of hitherto inexplicable phenomena.

  Almost immediately after Einstein published his theory of general relativity in 1915, Karl Schwarzschild showed that Einstein’s field equations could be solved in the case of a single massive body. Although it was not realized at the time, and never admitted by Einstein, this solution describes compact objects from which not even light can escape: what we now call “black holes.” We now believe that some stars end their lives as black holes, and at the center of most, if not all, galaxies there lie supermassive black holes. In our own Milky Way Galaxy, recent evidence suggests that there is a black hole approximately 3 million times the mass of the sun.

  Since light is bent around massive objects, the images of distant galaxies can be distorted or even multiplied on their way to observers here on earth. This effect, termed “gravitational lensing,” is not unlike that of a curved piece of glass. One of the first observational confirmations of general relativity was a lensing effect seen by Sir Arthur Eddington during a solar eclipse in 1919. Eddington noted that the position of a star seemed to shift in the sky relative to its normal position. The shift was consistent with the result predicted by Einstein given the mass of the sun. The bending of space is not necessarily local. Much of modern astrophysics is concerned with the question of what the overall “shape” of the universe is, and whether it is “flat,” “closed” like a sphere (and thus finite), or “open” like a saddle (and thus infinite). Recent measurements from the Wilkinson Microwave Anisotropy Probe (WMAP) satellite suggest that the universe is flat, or so large that it cannot yet be distinguished from perfect flatness.

  When Einstein first proposed general relativity, he recognized that his theory predicted that the universe as a whole could not be static as had always been assumed: the attraction of gravity meant that the universe had to be either expanding or contracting. Therefore, he added a “cosmological constant” to balance the attraction of gravity and keep the universe static. In 1922, the astronomer Edwin Hubble measured the expansion of the universe through observation, an expansion wholly consistent with Einstein’s original theory, but not with his value of the cosmological constant. In Appendix 4 of this work, Einstein responds to the recent findings, and elsewhere notes that his ad hoc introduction of a cosmological constant was his “greatest blunder.” As an interesting epilogue, however measurements of distant supernova explosions during the mid-1990s indicated that, though not the value proposed by Einstein, there may be a cosmological constant after all.

  PREFACE

  The present book is intended, as far as possible, to give an exact insight into the theory of Relativity to those readers who, from a general scientific and philosophical point of view, are interested in the theory, but who are not conversant with the mathematical apparatus of theoretical physics. The work presumes a standard of education corresponding to that of a university matriculation examination, and, despite the shortness of the book, a fair amount of patience and force of will on the part of the reader. The author has spared himself no pains in his endeavour to present the main ideas in the simplest and most intelligible form, and on the whole, in the sequence and connection in which they actually originated. In the interest of clearness, it appeared to me inevitable that I should repeat myself frequently, without paying the slightest attention to the elegance of the presentation. I adhered scrupulously to the precept of that brilliant theoretical physicist L. Boltzmann, according to whom matters of elegance ought to be left to the tailor and to the cobbler. I make no pretence of having withheld from the reader difficulties which are inherent to the subject. On the other hand, I have purposely treated the empirical physical foundations of the theory in a “step-motherly” fashion, so that readers unfamiliar with physics may not feel like the wanderer who was unable to see the forest for trees. May the book bring some one a few happy hours of suggestive thought!

  A. EINSTEIN

  December 1916

  PART I: THE SPECIAL

  THEORY OF RELATIVITY

  ONE

  PHYSICAL MEANING OF GEOMETRICAL PROPOSITIONS

  In your schooldays most of you who read this book made acquaintance with the noble building of Euclid’s geometry, and you remember— perhaps with more respect than love—the magnificent structure, on the lofty staircase of which you were chased about for uncounted hours by conscientious teachers. By reason of your past experience, you would certainly regard everyone with disdain who should pronounce even the most out-of-the-way proposition of this science to be untrue. But perhaps this feeling of proud certainty would leave you immediately if some one were to ask you: “What, then, do you mean by the assertion that these propositions are true?” Let us proceed to give this question a little consideration.

  Geometry sets out from certain conceptions such as “plane,” “point,” and “straight line,” with which we are able to associate more or less definite ideas, and from certain simple propositions (axioms) which, in virtue of these ideas, we are inclined to accept as “true.” Then, on the basis of a logical process, the justification of which we feel ourselves compelled to admit, all remaining propositions are shown to follow from those axioms,
i.e. they are proven. A proposition is then correct (“true”) when it has been derived in the recognised manner from the axioms. The question of the “truth” of the individual geometrical propositions is thus reduced to one of the “truth” of the axioms. Now it has long been known that the last question is not only unanswerable by the methods of geometry, but that it is in itself entirely without meaning. We cannot ask whether it is true that only one straight line goes through two points. We can only say that Euclidean geometry deals with things called “straight lines,” to each of which is ascribed the property of being uniquely determined by two points situated on it. The concept “true” does not tally with the assertions of pure geometry, because by the word “true” we are eventually in the habit of designating always the correspondence with a “real” object; geometry, however, is not concerned with the relation of the ideas involved in it to objects of experience, but only with the logical connection of these ideas among themselves.

  It is not difficult to understand why, in spite of this, we feel constrained to call the propositions of geometry “true.” Geometrical ideas correspond to more or less exact objects in nature, and these last are undoubtedly the exclusive cause of the genesis of those ideas. Geometry ought to refrain from such a course, in order to give to its structure the largest possible logical unity. The practice, for example, of seeing in a “distance” two marked positions on a practically rigid body is something which is lodged deeply in our habit of thought. We are accustomed further to regard three points as being situated on a straight line, if their apparent positions can be made to coincide for observation with one eye, under suitable choice of our place of observation.

  If, in pursuance of our habit of thought, we now supplement the propositions of Euclidean geometry by the single proposition that two points on a practically rigid body always correspond to the same distance (line-interval), independently of any changes in position to which we may subject the body, the propositions of Euclidean geometry then resolve themselves into propositions on the possible relative position of practically rigid bodies.1 Geometry which has been supplemented in this way is then to be treated as a branch of physics. We can now legitimately ask as to the “truth” of geometrical propositions interpreted in this way, since we are justified in asking whether these propositions are satisfied for those real things we have associated with the geometrical ideas. In less exact terms we can express this by saying that by the “truth” of a geometrical proposition in this sense we understand its validity for a construction with ruler and compasses.

  Of course the conviction of the “truth” of geometrical propositions in this sense is founded exclusively on rather incomplete experience. For the present we shall assume the “truth” of the geometrical propositions, then at a later stage (in the general theory of relativity) we shall see that this “truth” is limited, and we shall consider the extent of its limitation.

  1It follows that a natural object is associated also with a straight line. Three points A, B and C on a rigid body thus lie in a straight line when, the points A and C being given, B is chosen such that the sum of the distances AB and B C is as short as possible. This incomplete suggestion will suffice for our present purpose.

  TWO

  THE SYSTEM OF CO-ORDINATES

  On the basis of the physical interpretation of distance which has been indicated, we are also in a position to establish the distance between two points on a rigid body by means of measurements. For this purpose we require a “distance” (rod S) which is to be used once and for all, and which we employ as a standard measure. If, now, A and B are two points on a rigid body, we can construct the line joining them according to the rules of geometry; then, starting from A, we can mark off the distance S time after time until we reach B. The number of these operations required is the numerical measure of the distance A B. This is the basis of all measurement of length.1

  Every description of the scene of an event or of the position of an object in space is based on the specification of the point on a rigid body (body of reference) with which that event or object coincides. This applies not only to scientific description, but also to everyday life. If I analyse the place specification “Trafalgar Square, London,”2 I arrive at the following result. The earth is the rigid body to which the specification of place refers; “Trafalgar Square, London,” is a well-defined point, to which a name has been assigned, and with which the event coincides in space.3

  This primitive method of place specification deals only with places on the surface of rigid bodies, and is dependent on the existence of points on this surface which are distinguishable from each other. But we can free ourselves from both of these limitations without altering the nature of our specification of position. If, for instance, a cloud is hovering over Trafalgar Square, then we can determine its position relative to the surface of the earth by erecting a pole perpendicularly on the Square, so that it reaches the cloud. The length of the pole measured with the standard measuring-rod, combined with the specification of the position of the foot of the pole, supplies us with a complete place specification. On the basis of this illustration, we are able to see the manner in which a refinement of the conception of position has been developed.

  (a) We imagine the rigid body, to which the place specification is referred, supplemented in such a manner that the object whose position we require is reached by the completed rigid body.

  (b) In locating the position of the object, we make use of a number (here the length of the pole measured with the measuring-rod) instead of designated points of reference.

  (c) We speak of the height of the cloud even when the pole which reaches the cloud has not been erected. By means of optical observations of the cloud from different positions on the ground, and taking into account the properties of the propagation of light, we determine the length of the pole we should have required in order to reach the cloud.

  From this consideration we see that it will be advantageous if, in the description of position, it should be possible by means of numerical measures to make ourselves independent of the existence of marked positions (possessing names) on the rigid body of reference. In the physics of measurement this is attained by the application of the Cartesian system of co-ordinates.

  This consists of three plane surfaces perpendicular to each other and rigidly attached to a rigid body. Referred to a system of co-ordinates, the scene of any event will be determined (for the main part) by the specification of the lengths of the three perpendiculars or co-ordinates (x, y, z) which can be dropped from the scene of the event to those three plane surfaces. The lengths of these three perpendiculars can be determined by a series of manipulations with rigid measuring-rods performed according to the rules and methods laid down by Euclidean geometry.

  In practice, the rigid surfaces which constitute the system of coordinates are generally not available; furthermore, the magnitudes of the co-ordinates are not actually determined by constructions with rigid rods, but by indirect means. If the results of physics and astronomy are to maintain their clearness, the physical meaning of specifications of position must always be sought in accordance with the above considerations.1

  We thus obtain the following result: Every description of events in space involves the use of a rigid body to which such events have to be referred. The resulting relationship takes for granted that the laws of Euclidean geometry hold for “distances,” the “distance” being represented physically by means of the convention of two marks on a rigid body.

  1Here we have assumed that there is nothing left over, i.e. that the measurement gives a whole number. This difficulty is got over by the use of divided measuring-rods, the introduction of which does not demand any fundamentally new method.

  2I have chosen this as being more familiar to the English reader than the “Potsdamer Platz, Berlin,” which is referred to in the original. (R. W. L.)

  3It is not necessary here to investigate further the significance of the expression “coincidence
in space.” This conception is sufficiently obvious to ensure that differences of opinion are scarcely likely to arise as to its applicability in practice.

  1 A refinement and modification of these views does not become necessary until we come to deal with the general theory of relativity, treated in the second part of this book.

  THREE

  SPACE AND TIME IN CLASSICAL MECHANICS

  The purpose of mechanics is to describe how bodies change their position in space with “time.” I should load my conscience with grave sins against the sacred spirit of lucidity were I to formulate the aims of mechanics in this way, without serious reflection and detailed explanations. Let us proceed to disclose these sins.

  It is not clear what is to be understood here by “position” and “space.” I stand at the window of a railway carriage which is travelling uniformly, and drop a stone on the embankment, without throwing it. Then, disregarding the influence of the air resistance, I see the stone descend in a straight line. A pedestrian who observes the misdeed from the footpath notices that the stone falls to earth in a parabolic curve. I now ask: Do the “positions” traversed by the stone lie “in reality” on a straight line or on a parabola? Moreover, what is meant here by motion “in space”? From the considerations of the previous section the answer is self-evident. In the first place we entirely shun the vague word “space,” of which, we must honestly acknowledge, we cannot form the slightest conception, and we replace it by “motion relative to a practically rigid body of reference.” The positions relative to the body of reference (railway carriage or embankment) have already been defined in detail in the preceding section. If instead of “body of reference” we insert “system of co-ordinates,” which is a useful idea for mathematical description, we are in a position to say: The stone traverses a straight line relative to a system of co-ordinates rigidly attached to the carriage, but relative to a system of co-ordinates rigidly attached to the ground (embankment) it describes a parabola. With the aid of this example it is clearly seen that there is no such thing as an independently existing trajectory (lit. “path-curve”1), but only a trajectory relative to a particular body of reference.