... Squashed down to read in about 25 minutes
"...according to this theory, time is robbed of its independence"
Wikipedia - Full Text - Print Edition: ISBN 0415253845
© This page does not contain Relativity, but a short abridgement for private study and research only. Copyright may exist on the original work.
Albert Einstein was born on March 14, 1879 at Ulm, Württemberg in Germany. His early years have become the stuff of legend. Either he shone by theorising about magnetism at five, playing the violin at six and excelling at school, or else he showed little scholastic ability and left without a diploma. Or possibly both. Either way, he soon began to take an interest in mathematics under the guidance of his uncle Cäsar Koch, a partner in the family electronics business. He pursued his interests in Switzerland at the Federal Polytechnic Academy in Zürich, and, after graduating, became a Swiss citizen and took up the position of an examiner at the Swiss Patent Office- a job which required him to assess inventions to see if the ideas behind them were genuinely new.
Einstein's special theory of relativity grew from an essay he wrote aged just 16. It is remarkable, even stunning, in that it shows that, for all frames of reference, the speed of light is constant but the progress of time is, not fixed, as Newton (and everyone else) had assumed, but that both time and motion are relative to the observer.
In 1915 he caused a veritable sensation in the scientific world by his explanation of gravitational attraction based on the wider or 'general' form of his theory of relativity, and the verification by the British solar eclipse expeditions in 1919 of a prediction of his as to the bending of light rays from the stars which passed close to the sun, made him world-famous.
Unusually in the Squashed Philosophers series, this is not an abridgement of Einstein's own work, the mathematics in which make it too complex to be effectively squashed, but a condensed version of the explanation by JWN Sullivan first published in 1920.
Squashed version edited by Glyn Hughes © 2011
An explanation by British mathematician JWN Sullivan
THE famous Einstein theory was published in two parts. The first part, the so-called 'Special' theory, was published in 1905, when Einstein was only twenty-six years of age. The 'General' theory which, besides greatly extending the special theory, gave also a solution of the problem of gravitation, was published ten years later. It is this theory that attracted the attention of the whole world, as well as the strictly scientific portion of it, by the dramatic verification, at the total solar eclipse of May 29, 1919, of one of the most startling predictions of the theory.
The book under consideration is Einstein's own exposition, for the general public, of both theories. It may be said at once that, judging from this book, Einstein had a rather exalted opinion of the intelligence of the general public. His exposition is superb, but it demands very close attention. He says what he has to say so compactly that the reader is in danger of missing the full significance of his statements.
He begins with a question which is fundamental for his whole theory, and that is the status of the axioms of geometry. His own words are:
We cannot ask whether it is true that only one straight line goes through two points. We can only say that Euclidian 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 connexion of these ideas among themselves.
This estimate of the status of Euclidean geometry is justified by the fact that any number of non-Euclidean geometries exist. For two thousand years Euclid's axioms were regarded as necessities of thought. But early in the nineteenth century it was discovered that certain of Euclid's axioms could be denied and others substituted for them, and yet that perfectly self-consistent systems of geometry could be constructed. It follows that Euclid's axioms are not necessary truths. They are 'conventions.' We may adopt them or not, as we please. Consequently, in applying geometry to the real world, we are at liberty to apply that system of geometry we find most convenient. All systems of geometry are equally logical and no one is more 'true' than another, just as it is no more true that there are three feet in a yard than it is that there are one hundred centimetres in a metre. Which system we employ is a matter of convenience. In his general theory of relativity Einstein finds it convenient to use a non-Euclidean geometry.
After some preliminary remarks dealing with our methods of measuring the positions of bodies Einstein enunciates his 'special' theory of relativity which is to the effect that two observers in uniform translatory motion with respect to one another find the same laws for natural phenomena. By uniform translatory motion is meant motion at a constant speed in a straight line, i.e. without rotation or acceleration of any kind. Now, Newton had said, long ago, that two such observers will find the same laws for mechanics. But will they find the same laws for optics and for electricity? Einstein says that they will, but this statement, when we come to think about it, is a very puzzling one. Consider, for instance, the fact that light travels at 186,000 miles per second for a given observer. Could it have the same velocity for a second observer moving relatively to the first? It seems obvious that it could not.
An aeroplane does not pass a moving train at the same pace that it passes one at rest. Nevertheless, Einstein asserts that light will have the same velocity for two observers, whatever their relative motion. He says:
As a result of an analysis of the physical conceptions of time and space, it became evident that in reality there is not the least incompatibility between the principle of relativity and the law of propagation of light, and that by systematically holding fast to both these laws a logically rigid theory could be arrived at.
He then proceeds to show that the notion of simultaneity is a relative one. Events which are simultaneous for one observer are not simultaneous for an observer moving relatively to the first. Two such observers will not agree in their estimates of the time-lapse between two events. Neither will their distance measurements agree. What are the relations between the space and time measurements of such observers, supposing them to get the same velocity for light? This is a purely mathematical problem, and Einstein gives the solution, which we need not quote. But we must realize clearly what he has done here, for this is the basis of the whole theory.
He has shown that observers in uniform relative motion will obtain the same laws of nature for phenomena provided they use different space and time measurements, and he has shown just what these differences would be. Now it is a fact of experiment that such observers do obtain the same laws of nature. Not only the famous Michelson-Morley experiment on light, described by Einstein, but many other experiments bear out this statement. It follows that observers in relative motion naturally adopt space and time measurements which differ in the way described by Einstein. In other words, each observer has his own space-time framework. There is no absolute space and time, the same for all observers.
If an observer B, carrying a clock and also carrying a yard measure pointing in the direction of his motion moves past an observer A then, from A's point of view, B's yard measure is short of a yard and his clock is going slow. And the discrepancy is greater the greater B's velocity relative to A. If B passed A with the velocity of light then we reach the highly astonishing result that from A's point of view B's yard measure would be of zero length and his clock would not be going at all! This means that the velocity of light is a limiting velocity. No object in the universe can possibly move at a speed greater than the speed of light.
Einstein proceeds to work out some of the consequences of this theory: The most important result of a general character to which the special theory of relativity has led is concerned with the conception of mass. Before the advent of relativity, physics recognized two conservation laws of fundamental importance, namely, the law of the conservation of energy and the law of the conservation of mass; these two fundamental laws appeared to be quite independent of each other. By means of the theory of relativity they have been united into one law.
Mass and energy have become, in fact, interchangeable terms. A body radiating energy thereby loses mass; a body receiving energy thereby gains mass. As a body moves faster its energy, and therefore its mass, increases. At the velocity of light its mass would be infinite. We may mention that these deductions from Einstein's theory have been verified by experiment.
Thus the swiftest electrons we can produce artificially have speeds within a few per cent. of that of light, and their mass is found to increase to just the extent calculated by Einstein. The reader should remember, in reading this book, that he is not dealing with speculations 'in the air.' Countless experiments have confirmed Einstein's conclusions.
Einstein concludes this part of his exposition with an account of Minkowski's 'four-dimensional space.' The central idea of this must be understood before the general theory can be tackled. It is thus described by Einstein:
Space is a three-dimensional continuum. By this we mean that it is possible to describe the position of a point (at rest) by means of three numbers (co-ordinates) x, y, z, and that there is an indefinite number of points in the neighbourhood of this one, the position of which can be described by coordinates such as Xi, yi, Zi, which may be as near as we choose to the respective values of the co-ordinates x, y, z, of the first point. In virtue of the latter property we speak of a 'continuum,' and owing to the fact that there are three co-ordinates we speak of it as being 'three-dimensional.' Similarly, the world of the physical phenomena which was briefly called 'world' by Minkowski is naturally four-dimensional in the space-time sense. For it is composed of individual events, each of which is described by four numbers, namely, three space coordinates x, y, z, and a time co-ordinate, the time-value t. The 'world' is in this sense also a continuum; for to every event there are as many 'neighbouring' events (realized or at least thinkable) as we care to choose, the co-ordinates Xi yi Zi ti of which differ by an indefinitely small amount from those of the event x y z t originally considered.
That we have not been accustomed to regard the world in this sense as a four-dimensional continuum is due to the fact that in physics, before the advent of the theory of relativity, time played a different and more independent role, as compared with the space co-ordinates. It is for this reason that we have been in the habit of treating time as an independent continuum.
As a matter of fact, according to classical mechanics, time is absolute, i.e. it is independent of the position and the condition of motion of the system of co-ordinates
The four-dimensional mode of consideration of the 'world' is natural on the theory of relativity, since according to this theory time is robbed of its independence.
We say that space has three dimensions because we require three measurements to specify the position of a point in space. For instance, to specify the position in a room of the tip of an electric light bulb we would have to give its distances from two walls and its distance from the floor or the ceiling. Whatever method we adopted we should have to give at least three measurements. That is why we call space three-dimensional. And space is continuous because we can have points in space as close together as we like. The three distances of a point from our frame of reference (such a frame, for instance, as the two walls and the ceiling) are called the 'co-ordinates' of the point.
But if we are specifying an event we want to say when as well as where it happened. We must give, therefore, the moment of time of its occurrence. This is called its time co-ordinate. In calculations the space co-ordinates are usually denoted by x, y, z, and the time co-ordinate by t.
Now Minkowski showed that the space and time co-ordinates of an event are not independent of one another. Two flashes of light may be separated by ten yards for one observer and occur at an interval of ten seconds. But for a second observer, moving relatively to the first, they may be more than ten yards apart and occur at an interval of more than ten seconds. A certain combination of distance and time will be the same for both observers, but the distances and times taken separately will not be the same. That particular combination of distance and time that all observers will find to be the same is called the 'interval'. Minkowski showed that the interval could be regarded as a 'distance in a four-dimensional space.' This four-dimensional space we split up into a three-dimensional space and a one-dimensional time-and each observer splits it up differently. The actual four-dimensional quantity involved- the 'interval'- is the same for all of them. but they split it up differently into so much of space and so much of time. This result is very interesting, but there is one particular aspect of it which is of the greatest importance for the relativity theory. This aspect is described by Einstein as follows:
But the discovery of Minkowski, which was of importance for the formal development of the theory of relativity, does not lie here. It is to be found rather in the fact of his recognition that the four-dimensional space-time continuum of the theory of relativity, in its most essential formal properties, shows a pronounced relationship to the three-dimensional continuum of Euclidean geometrical space.
That is to say the geometry of this four-dimensional space of Minkowski's is a Euclidean geometry. The whole 'special' theory of relativity can be explained as the geometry of a four-dimensional 'Euclidean' space. This fact gave Einstein a very important clue for developing his theory. For Einstein was dissatisfied with his special theory. We have pointed out that that theory says that the laws of nature are the same for observers in uniform translatory motion with respect to one another, and only for such observers. Einstein comments:
But no person whose mode of thought
is logical can rest satisfied with this condition of things. He asks: 'How does it come that certain reference-bodies (or their states of motion) are given priority over other reference-bodies (or their states of motion)? What is the reason for this preference?
Einstein wants to know why the laws of nature should not be the same for all observers, whatever their state of motion. We have found that uniform motion makes no difference. What difference does non-uniform motion make? His solution of this question is Einstein's most dazzling achievement. We begin with his famous account of the man in the box.
As reference-body let us imagine a spacious chest resembling a room with an observer inside who is equipped with apparatus. Gravitation naturally does not exist for this observer. (Einstein imagines the man in the chest to be right away in empty space.) He must fasten himself with strings to the floor, otherwise the slightest impact against the floor will cause him to rise slowly towards the ceiling of the room. To the middle of the lid of the chest is fixed externally a hook with rope attached, and now a 'being' (what kind of a being is immaterial to us) begins pulling at this with a constant force. The chest, together with the observer, begin to move 'upwards' with a uniformly accelerated motion. In course of time their velocity will reach unheard-of values-provided that we are viewing all this from another reference-body which is not being pulled with a rope. But how does the man in the chest regard the process. The acceleration of the chest will be transmitted to him by the reaction of the floor of the chest. He must therefore take up this pressure by means of his legs if he does not wish to be laid out full length on the floor. He is then standing in the chest in exactly the same way as anyone stands in a room of a house on our earth. If he release a body which he previously had in his hand, the acceleration of the chest will no longer be transmitted to this body, and for this reason the body will approach the floor of the chest with an accelerated relative motion. The observer will further convince himself that -the acceleration of the body towards the floor of the chest is always of the same magnitude, whatever kind of body he may happen to use for the experiment.
The point Einstein is leading up to is clear from the above passage in italics. For it is characteristic of the gravitational force, and of the gravitational force alone, that it is entirely independent of the physical or chemical constitution of the bodies on which it operates. The man in the chest will naturally conclude that he and his chest are in a gravitational field.
Einstein goes on:
Of course he will be puzzled for a moment as to why the chest does not fall in this gravitational field. Just then, however, he discovers the hook in the middle of the lid of the chest and the rope which is attached to it, and he consequently comes to the conclusion that the chest is suspended at rest in the gravitational field.
From this illustration we can grasp Einstein's conception of the essential difference between uniform and non-uniform motion. An observer in non-uniform motion may be regarded as existing in a gravitational field. The laws of phenomena for observers in non-uniform motion, therefore, are the laws of phenomena in gravitational fields. But we must be careful in reading this example not to suppose that Einstein means to say that a gravitational field is always merely apparent. There is a gravitational field for the man in the chest although there is no gravitational field from the point of view of an observer outside the chest. But there is no possible observer for whom the gravitational field of the earth does not exist.
Let us imagine the man looking through a window in his moving chest and observing the passage of a ray of light outside. Since the man's motion is an accelerated motion the ray of light would appear to him curved. But since, according to Einstein, there is no essential difference between accelerated motion and a gravitational field, it follows that light passing through a gravitational field should follow a curved path. Einstein prophesied, therefore, that this would be so and, as all the world knows, the prophecy was verified at the eclipse expedition of May 29, 1919
This case exemplified a procedure which is pretty general in the theory. We imagine an artificial gravitational field and find what would happen to phenomena in that field. We then use this result to say what would happen to phenomena in a real gravitational field. Also, by finding the laws obeyed by these artificial fields themselves we can deduce the laws obeyed by real gravitational fields. This is what Einstein has done, and it is perhaps the most celebrated part of his achievement. He has found the true laws for gravitation and shown that Newton's law is only approximate.
But, to do this, he had greatly to extend Minkowski's idea, mentioned above. He found that if Minkowski's four-dimensional space was permeated by a gravitational field, then its geometry was not Euclidean. He had to apply non-Euclidean geometry. Einstein, in his desire to omit no steps from the argument, gives the reader an outline of the method by which he did this. The reader will probably find this the most obscure part of his book.
There would be no point in quoting it and still less in summarising it. It is best for the reader who is not a mathematician to take this part of the argument for granted. Suffice it to say that the laws of motion in a non-Euclidean space can be worked out, and that they are found to give just the motions we observe in the case of the planets. Thus the actual motion of Mercury had never been satisfactorily explained on Newton's theory. The difference between observation and calculation was slight, but it was there, and the greatest mathematicians had exerted themselves in vain to explain it. Einstein's theory, in a perfectly natural and unforced way, clears up the whole mystery. Einstein's own words are:
Since the time of Leverrier, it has been known that the ellipse corresponding to the orbit of Mercury ... is not stationary with respect to the fixed stars, but that it rotates exceedingly slowly in the plane of the orbit and in the sense of the orbital motion. The value obtained for this rotary movement of the orbital ellipses was 43 seconds of arc per century, an amount ensured to be correct to within a few seconds of arc. This effect can be explained by means of classical mechanics only on the assumption of hypotheses which have little probability, and which were devised solely for this purpose. On the basis of the general theory of relativity, it is found that the ellipse of every planet round the sun must necessarily rotate in the manner indicated above; that for all the planets, with the exception of Mercury, this rotation is too small to be detected with the delicacy of observation possible at the present time; but that in the case of Mercury it must amount to 43 seconds of arc per century, a result which is strictly in agreement with observation.
This result, the deflection of light mentioned above, and the shift of the lines of the sun's spectrum towards the red, are the three great experimental confirmations of Einstein's general theory.
It may help the reader to grasp the central idea of Einstein's general theory if it be put in the following way. The natural motion of a body, left to itself, is, in Euclidean space, motion in a straight line and with a constant velocity. Now the planets do not move in straight lines with uniform velocity. Why? Newton said because there is a force of gravitation emanating from the sun and pulling them out of the straight line. Einstein says No, there is no force of gravitation at all. The planets do not move in straight lines because they are not moving in a Euclidean space at all, but in a non-Euclidean space, where their actual motions are their natural motions. And his general conditions for this non Euclidean space are what he calls the law of gravitation. His law of gravitation, therefore, is not the law of a force; it is a set of geometrical conditions. As a matter of fact, Einstein does not need the notion of 'force' at all.
In the last part of his book Einstein deals with his notion of a finite universe. We are accustomed to think of the universe as consisting of matter (in the form of stars) distributed pretty uniformly throughout infinite space. It can be shown that, on Newton's law of gravitation, such a universe is impossible. The gravitational force would be infinite. And there are grave objections to such a universe also on relativity theory. Einstein therefore supposes that our universe is finite. But, although finite, it is unbounded. He illustrates the idea by asking us to consider perfectly flat creatures living on the surface of a sphere. Such creatures could wander forward on this sphere for ever without meeting any boundary. Nevertheless their space, that is the area of their sphere, is of finite size. Analogously, the mathematics of a sort of three-dimensional spherical surface can be worked out. It will be of finite size but it will have no boundaries.
Einstein supposes that the space we live in is of this kind. We can get a rough idea of the size of this universe. A ray of light would go all round this spherical universe in about one thousand million years. Light sent out a thousand million years ago would come back to the place it started from and, if it had not become too scattered by passing through gravitational fields en route, would unite again in a focus at that point. It may be, therefore, that some of the stars we see are really ghosts-the images of stars that were there a thousand million years ago, but which have since moved on to other parts of the universe, or which have even become extinct or perished in a collision.
But this part of Einstein's theory, although it solves certain difficulties, must not be taken as on the same level of evidence as the rest of his theory. The rest of the theory may fairly be regarded as proved. The theory of the finite universe is still only a speculation. Even as it stands, however, the theory has been justly described by a great mathematician as 'the greatest synthetic achievement of the human mind.'
Einstein instructed that there be no funeral, grave or monument. His brain was donated to science and his ashes scattered in an unknown river in New Jersey.
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