Category Archives: Physics

Einstein 1938: General Relativity and Its Verification

Reference: Evolution of Physics

This paper presents Chapter III, section 13 from the book THE EVOLUTION OF PHYSICS by A. EINSTEIN and L. INFELD. The contents are from the original publication of this book by Simon and Schuster, New York (1942).

The paragraphs of the original material (in black) are accompanied by brief comments (in color) based on the present understanding.  Feedback on these comments is appreciated.

The heading below is linked to the original materials.

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General Relativity and Its Verification

The general theory of relativity attempts to formulate physical laws for all c.s. The fundamental problem of the theory is that of gravitation. The theory makes the first serious effort, since Newton’s time, to reformulate the law of gravitation. Is this really necessary? We have already learned about the achievements of Newton’s theory, about the great development of astronomy based upon his gravitational law. Newton’s law still remains the basis of all astronomical calculations. But we also learned about some objections to the old theory. Newton’s law is valid only in the inertia! c.s. of classical physics, in c.s. defined, we remember, by the condition that the laws of mechanics must be valid in them. The force between two masses depends upon their distance from each other. The connection between force and distance is, as we know, invariant with respect to the classical transformation. But this law does not fit the frame of special relativity. The distance is not invariant with respect to the Lorentz transformation. We could try, as we did so successfully with the laws of motion, to generalize the gravitational law, to make it fit the special relativity theory, or, in other words, to formulate it so that it would be invariant with respect to the Lorentz and not to the classical transformation. But Newton’s gravitational law opposed obstinately all our efforts to simplify and fit it into the scheme of the special relativity theory. Even if we succeeded in this, a further step would still be necessary: the step from the inertial c.s. of the special relativity theory to the arbitrary c.s. of the general relativity theory. On the other hand, the idealized experiments about the falling lift show clearly that there is no chance of formulating the general relativity theory without solving the problem of gravitation. From our argument we see why the solution of the gravitational problem will differ in classical physics and general relativity.

General relativity attempts to integrate gravitation into the laws of physics. Newton’s law of gravitation is valid only in the inertial c.s. in which inertia and characteristics of space remain constant, but that is not the case on a broader scale. We need to update the Gravitational law not only for the basis of motion used in special relativity, but also for the non-inertial c.s. In other words, we need to update the gravitational law for changing absolute value of inertia.

We have tried to indicate the way leading to the general relativity theory and the reasons forcing us to change our old views once more. Without going into the formal structure of the theory, we shall characterize some features of the new gravitational theory as compared with the old. It should not be too difficult to grasp the nature of these differences in view of all that has previously been said.

(1) The gravitational equations of the general relativity theory can be applied to any c.s. It is merely a matter of convenience to choose any particular c.s. in a special case. Theoretically all c.s. are permissible. By ignoring the gravitation, we automatically come back to the inertial c.s. of the special relativity theory.

A coordinate system is defined by its level of inertia and the rate at which that inertia is changing.

(2) Newton’s gravitational law connects the motion of a body here and now with the action of a body at the same time in the far distance. This is the law which formed a pattern for our whole mechanical view. But the mechanical view broke down. In Maxwell’s equations we realized a new pattern for the laws of nature. Maxwell’s equations are structure laws. They connect events which happen now and here with events which will happen a little later in the immediate vicinity. They are the laws describing the changes of the electromagnetic field. Our new gravitational equations are also structure laws describing the changes of the gravitational field. Schematically speaking, we could say: the transition from Newton’s gravitational law to general relativity resembles somewhat the transition from the theory of electric fluids with Coulomb’s law to Maxwell’s theory.

The gravitational field is a field of substance that is varying in inertia. Inertia is a measure of substantial-ness of the substance.  Inertia affects the motion of the substance in the field. There are laws describing the changes in the field.

(3) Our world is not Euclidean. The geometrical nature of our world is shaped by masses and their velocities. The gravitational equations of the general relativity theory try to disclose the geometrical properties of our world.

The geometrical properties in a gravitation field are really the relationships among inertia, space, time and motion. The space and time characteristics are changing according to the laws of inertia and motion.

Let us suppose, for the moment, that we have succeeded in carrying out consistently the programme of the general relativity theory. But are we not in danger of carrying speculation too far from reality? We know how well the old theory explains astronomical observations. Is there a possibility of constructing a bridge between the new theory and observation? Every speculation must be tested by experiment, and any results, no matter how attractive, must be rejected if they do not fit the facts. How did the new theory of gravitation stand the test of experiment? This question can be answered in one sentence: The old theory is a special limiting case of the new one. If the gravitational forces are comparatively weak, the old Newtonian law turns out to be a good approximation to the new laws of gravitation. Thus all observations which support the classical theory also support the general relativity theory. We regain the old theory from the higher level of the new one.

Even if no additional observation could be quoted in favour of the new theory, if its explanation were only just as good as the old one, given a free choice between the two theories, we should have to decide in favour of the new one. The equations of the new theory are, from the formal point of view, more complicated, but their assumptions are, from the point of view of fundamental principles, much simpler. The two frightening ghosts, absolute time and an inertial system, have disappeared. The clue of the equivalence of gravitational and inertial mass is not overlooked. No assumption about the gravitational forces and their dependence on distance is needed. The gravitational equations have the form of structure laws, the form required of all physical laws since the great achievements of the field theory.

Some new deductions, not contained in Newton’s gravitational law, can be drawn from the new gravitational laws. One, the bending of light rays in a gravitational field, has already been quoted. Two further consequences will now be mentioned.

Light rays bend in a gravitation field because light has a very small amount of inertia. General relativity predicts the amount of bend more accurately.

If the old laws follow from the new one when the gravitational forces are weak, the deviations from the Newtonian law of gravitation can be expected only for comparatively strong gravitational forces. Take our solar system. The planets, our earth among them, move along elliptical paths around the sun. Mercury is the planet nearest the sun. The attraction between the sun and Mercury is stronger than that between the sun and any other planet, since the distance is smaller. If there is any hope of finding a deviation from Newton’s law, the greatest chance is in the case of Mercury. It follows, from classical theory, that the path described by Mercury is of the same kind as that of any other planet except that it is nearer the sun. According to the general relativity theory, the motion should be slightly different. Not only should Mercury travel around the sun, but the ellipse which it describes should rotate very slowly, relative to the c.s. connected with the sun. This rotation of the ellipse expresses the new effect of the general relativity theory. The new theory predicts the magnitude of this effect. Mercury’s ellipse would perform a complete rotation in three million years! We see how small the effect is, and how hopeless it would be to seek it in the case of planets farther removed from the sun.

The deviation of the motion of the planet Mercury from the ellipse was known before the general relativity theory was formulated, and no explanation could be found. On the other hand, general relativity developed without any attention to this special problem. Only later was the conclusion about the rotation of the ellipse in the motion of a planet around the sun drawn from the new gravitational equations. In the case of Mercury, theory explained successfully the deviation of the motion from the Newtonian law.

General relativity successfully explains the path of Mercury around the sun, whereas Newtonian law gives an error. The error occurs due to stronger than usual gravitational forces between Mercury and Sun, which are accounted for in the general relativity equation but not by the Newtonian law.

But there is still another conclusion which was drawn from the general relativity theory and compared with experiment. We have already seen that a clock placed on the large circle of a rotating disc has a different rhythm from one placed on the smaller circle. Similarly, it follows from the theory of relativity that a clock placed on the sun would have a different rhythm from one placed on the earth, since the influence of the gravitational field is much stronger on the sun than on the earth.

We remarked on p. 103 that sodium, when incandescent, emits homogeneous yellow light of a definite wave-length. In this radiation the atom reveals one of its rhythms; the atom represents, so to speak, a clock and the emitted wave-length one of its rhythms. According to the general theory of relativity, the wave-length of light emitted by a sodium atom, say, placed on the sun should be very slightly greater than that of light emitted by a sodium atom on our earth.

Gravitational redshift is also predicted accurately by general relativity.

The problem of testing the consequences of the general relativity theory by observation is an intricate one and by no means definitely settled. As we are concerned with principal ideas, we do not intend to go deeper into this matter, and only state that the verdict of experiment seems, so far, to confirm the conclusions drawn from the general relativity theory.

General relativity has been tested only for higher gravity than earth’s gravity.

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FINAL COMMENTS

Newton’s law of gravitation is valid only in the inertial coordinate system in which inertia and characteristics of space remain constant. This law needs to be updated for coordinate systems that cover the whole range of inertia. Transformation among such systems must account for change in inertia.  

Inertia is a measure of substantial-ness of the substance.  The natural uniform motion of a body is reciprocal to its inertia. Gravity being equivalent to acceleration is a field of varying inertia. General relativity seems to provide the law describing the changes in inertia in the gravitational field.

The geometrical properties in a gravitation field are really the relationships among inertia and motion. The space and time characteristics are changing with changes in inertia and motion. General relativity attempts to integrate the laws of physics over the whole range of inertia.

Like special relativity, general relativity works only in the material domain because the basis of light does not exactly represent zero inertia. That approximation, however, is adequate for the material domain.

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Einstein 1938: Geometry and Experiment

Reference: Evolution of Physics

This paper presents Chapter III, section 12 from the book THE EVOLUTION OF PHYSICS by A. EINSTEIN and L. INFELD. The contents are from the original publication of this book by Simon and Schuster, New York (1942).

The paragraphs of the original material (in black) are accompanied by brief comments (in color) based on the present understanding.  Feedback on these comments is appreciated.

The heading below is linked to the original materials.

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Geometry and Experiment

Our next example will be even more fantastic than the one with the falling lift. We have to approach a new problem; that of a connection between the general relativity theory and geometry. Let us begin with the description of a world in which only two-dimensional and not, as in ours, three-dimensional creatures live. The cinema has accustomed us to two-dimensional creatures acting on a two-dimensional screen. Now let us imagine that these shadow figures, that is, the actors on the screen, really do exist, that they have the power of thought, that they can create their own science, that for them a two-dimensional screen stands for geometrical space. These creatures are unable to imagine, in a concrete way, a three-dimensional space just as we are unable to imagine a world of four dimensions. They can deflect a straight line; they know what a circle is, but they are unable to construct a sphere, because this would mean forsaking their two-dimensional screen. We are in a similar position. We are able to deflect and curve lines and surfaces, but we can scarcely picture a deflected and curved three-dimensional space.

By living, thinking, and experimenting, our shadow figures could eventually master the knowledge of the two-dimensional Euclidean geometry. Thus, they could prove, for example, that the sum of the angles in a triangle is 180 degrees. They could construct two circles with a common centre, one very small, the other large. They would find that the ratio of the circumferences of two such circles is equal to the ratio of their radii, a result again characteristic of Euclidean geometry. If the screen were infinitely great, these shadow beings would find that once having started a journey straight ahead, they would never return to their point of departure.

Let us now imagine these two-dimensional creatures living in changed conditions. Let us imagine that someone from the outside, the “third dimension”, transfers them from the screen to the surface of a sphere with a very great radius. If these shadows are very small in relation to the whole surface, if they have no means of distant communication and cannot move very far, then they will not be aware of any change. The sum of angles in small triangles still amounts to 180 degrees. Two small circles with a common centre still show that the ratio of their radii and circumferences are equal. A journey along a straight line never leads them back to the starting-point.

But let these shadow beings, in the course of time, develop their theoretical and technical knowledge. Let them find means of communication which will enable them to cover large distances swiftly. They will then find that starting on a journey straight ahead, they ultimately return to their point of departure. “Straight ahead” means along the great circle of the sphere. They will also find that the ratio of two circles with a common centre is not equal to the ratio of the radii, if one of the radii is small and the other great.

If our two-dimensional creatures are conservative, if they have learned the Euclidean geometry for generations past when they could not travel far and when this geometry fitted the facts observed, they will certainly make every possible effort to hold on to it, despite the evidence of their measurements. They could try to make physics bear the burden of these discrepancies. They could seek some physical reasons, say temperature differences, deforming the lines and causing deviation from Euclidean geometry. But, sooner or later, they must find out that there is a much more logical and convincing way of describing these occurrences. They will eventually understand that their world is a finite one, with different geometrical principles from those they learned. They will understand that, in spite of their inability to imagine it, their world is the two-dimensional surface of a sphere. They will soon learn new principles of geometry, which though differing from the Euclidean can, nevertheless, be formulated in an equally consistent and logical way for their two-dimensional world. For the new generation brought up with a knowledge of the geometry of the sphere, the old Euclidean geometry will seem more complicated and artificial since it does not fit the facts observed.

Moving from a two-dimensional to a three-dimensional world shall produce a great shift in perception and thinking.

Let us return to the three-dimensional creatures of our world.

What is meant by the statement that our three-dimensional space has a Euclidean character? The meaning is that all logically proved statements of the Euclidean geometry can also be confirmed by actual experiment. We can, with the help of rigid bodies or light rays, construct objects corresponding to the idealized objects of Euclidean geometry. The edge of a ruler or a light ray corresponds to the line; the sum of the angles of a triangle built of thin rigid rods is 180 degrees; the ratio of the radii of two circles with a common centre constructed from thin unbendable wire is equal to that of their circumference. Interpreted in this way, the Euclidean geometry becomes a chapter of physics, though a very simple one.

But we can imagine that discrepancies have been discovered: for instance, that the sum of the angles of a large triangle constructed from rods, which for many reasons had to be regarded as rigid, is not 180 degrees. Since we are already used to the idea of the concrete representation of the objects of Euclidean geometry by rigid bodies, we should probably seek some physical force as the cause of such unexpected misbehaviour of our rods. We should try to find the physical nature of this force and its influence on other phenomena. To save the Euclidean geometry, we should accuse the objects of not being rigid, of not exactly corresponding to those of Euclidean geometry. We should try to find a better representation of bodies behaving in the way expected by Euclidean geometry. If, however, we should not succeed in combining Euclidean geometry and physics into a simple and consistent picture, we should have to give up the idea of our space being Euclidean and seek a more convincing picture of reality under more general assumptions about the geometrical character of our space.

Is the geometric character of our space strictly Euclidean?

The necessity for this can be illustrated by an idealized experiment showing that a really relativistic physics cannot be based upon Euclidean geometry. Our argument will imply results already learned about inertial c.s. and the special relativity theory.

Imagine a large disc with two circles with a common centre drawn on it, one very small, the other very large. The disc rotates quickly. The disc is rotating relative to an outside observer, and there is an inside observer on the disc. We further assume that the c.s. of the outside observer is an inertial one. The outside observer may draw, in his inertial c.s., the same two circles, small and large, resting in his c.s. but coinciding with the circles on the rotating disc. Euclidean geometry is valid in his c.s. since it is inertial, so that he will find the ratio of the circumferences equal to that of the radii. But how about the observer on the disc? From the point of view of classical physics and also the special relativity theory, his c.s. is a forbidden one. But if we intend to find new forms for physical laws, valid in any c.s., then we must treat the observer on the disc and the observer outside with equal seriousness. We, from the outside, are now watching the inside observer in his attempt to find, by measurement, the circumferences and radii on the rotating disc. He uses the same small measuring stick used by the outside observer. “The same” means either really the same, that is, handed by the outside observer to the inside, or, one of two sticks having the same length when at rest in a c.s.

We compare a disc at rest with a rotating disc.

The inside observer on the disc begins measuring the radius and circumference of the small circle. His result must be the same as that of the outside observer. The axis on which the disc rotates passes through the centre. Those parts of the disc near the centre have very small velocities. If the circle is small enough, we can safely apply classical mechanics and ignore the special relativity theory. This means that the stick has the same length for the outside and inside observers, and the result of these two measurements will be the same for them both. Now the observer on the disc measures the radius of the large circle. Placed on the radius, the stick moves, for the outside observer. Such a stick, however, does not contract and will have the same length for both observers, since the direction of the motion is perpendicular to the stick. Thus three measurements are the same for both observers: two radii and the small circumference. But it is not so with the fourth measurement! The length of the large circumference will be different for the two observers. The stick placed on the circumference in the direction of the motion will now appear contracted to the outside observer, compared to his resting stick. The velocity is much greater than that of the inner circle, and this contraction should be taken into account. If, therefore, we apply the results of the special relativity theory, our conclusion here is: the length of the great circumference must be different if measured by the two observers. Since only one of the four lengths measured by the two observers is not the same for them both, the ratio of the two radii cannot be equal to the ratio of the two circumferences for the inside observer as it is for the outside one. This means that the observer on the disc cannot confirm the validity of Euclidean geometry in his c.s.

As the velocity goes higher the corresponding inertia decreases. This affects the characteristics of space and time. It is not dependent on the observer. It is the law of nature.

After obtaining this result, the observer on the disc could say that he does not wish to consider c.s. in which Euclidean geometry is not valid. The breakdown of the Euclidean geometry is due to absolute rotation, to the fact that his c.s. is a bad and forbidden one. But, in arguing in this way, he rejects the principal idea of the general theory of relativity. On the other hand, if we wish to reject absolute motion and to keep up the idea of the general theory of relativity, then physics must all be built on the basis of a geometry more general than the Euclidean. There is no way of escape from this consequence if all c.s. are permissible.

The changes brought about by the general relativity theory cannot be confined to space alone. In the special relativity theory we had clocks resting in every c.s., having the same rhythm and synchronized, that is, showing the same time simultaneously. What happens to a clock in a non-inertial c.s.? The idealized experiment with the disc will again be of use. The outside observer has in his inertial c.s. perfect clocks all having the same rhythm, all synchronized. The inside observer takes two clocks of the same kind and places one on the small inner circle and the other on the large outer circle. The clock on the inner circle has a very small velocity relative to the outside observer. We can, therefore, safely conclude that its rhythm will be the same as that of the outside clock. But the clock on the large circle has a considerable velocity, changing its rhythm compared to the clocks of the outside observer and, therefore, also compared to the clock placed on the small circle. Thus, the two rotating clocks will have different rhythms and, applying the results of the special relativity theory, we again see that in our rotating c.s. we can make no arrangements similar to those in an inertial c.s.

Space and time, and therefore, geometry is affected by change in inertia. Euclidean geometry is limited to the average inertia of the material universe.

To make clear what conclusions can be drawn from this and previously described idealized experiments, let us once more quote a dialogue between the old physicist O, who believes in classical physics, and the modern physicist M, who knows the general relativity theory. O is the outside observer, in the inertial c.s., whereas M is on the rotating disc.

O. In your c.s., Euclidean geometry is not valid. I watched your measurements and I agree that the ratio of the two circumferences is not, in your c.s., equal to the ratio of the two radii. But this shows that your c.s. is a forbidden one. My c.s., however, is of an inertial character, and I can safely apply Euclidean geometry. Your disc is in absolute motion and, from the point of view of classical physics, forms a forbidden c.s., in which the laws of mechanics are not valid.

M. I do not want to hear anything about absolute motion. My c.s. is just as good as yours. What I noticed was your rotation relative to my disc. No one can forbid me to relate all motions to my disc.

O. But did you not feel a strange force trying to keep you away from the centre of the disc? If your disc were not a rapidly rotating merry-go-round, the two things which you observed would certainly not have happened. You would not have noticed the force pushing you toward the outside nor would you have noticed that Euclidean geometry is not applicable in your c.s. Are not these facts sufficient to convince you that your c.s. is in absolute motion?

M. Not at all! I certainly noticed the two facts you mention, but I hold a strange gravitational field acting on my disc responsible for them both. The gravitational field, being directed toward the outside of the disc, deforms my rigid rods and changes the rhythm of my clocks. The gravitational field, non-Euclidean geometry, clocks with different rhythms are, for me, all closely connected. Accepting any c.s., I must at the same time assume the existence of an appropriate gravitational field with its influence upon rigid rods and clocks.

Gravitational field is related to change in inertia. It, therefore, has effect on space-time geometry.

O. But are you aware of the difficulties caused by your general relativity theory? I should like to make my point clear by taking a simple non-physical example. Imagine an idealized American town consisting of parallel streets with parallel avenues running perpendicular to them. The distance between the streets and also between the avenues is always the same. With these assumptions fulfilled, the blocks are of exactly the same size. In this way I can easily characterize the position of any block. But such a construction would be impossible without Euclidean geometry. Thus, for instance, we cannot cover our whole earth with one great ideal American town. One look at the globe will convince you. But neither could we cover your disc with such an “American town construction”. You claim that your rods are deformed by the gravitational field. The fact that you could not confirm Euclid’s theorem about the equality of the ratio of radii and circumferences shows clearly that if you carry such a construction of streets and avenues far enough you will, sooner or later, get into difficulties and find that it is impossible on your disc. Your geometry on your rotating disc resembles that on a curved surface, where, of course, the streets-and-avenues construction is impossible on a great enough part of the surface. For a more physical example take a plane irregularly heated with different temperatures on different parts of the surface. Can you, with small iron sticks expanding in length with temperature, carry out the “parallel-perpendicular” construction which I have drawn below? Of course not! Your “gravitational field” plays the same tricks on your rods as the change of temperature on the small iron sticks.

M. All this does not frighten me. The street-avenue construction is needed to determine positions of points, with the clock to order events. The town need not be American, it could just as well be ancient European. Imagine your idealized town made of plasticine and then deformed. I can still number the blocks and recognize the streets and avenues, though these are no longer straight and equidistant. Similarly, on our earth, longitude and latitude denote the positions of points, although there is no “American town” construction.

O.  But I still see a difficulty. You are forced to use your “European town structure”. I agree that you can order points, or events, but this construction will muddle all measurement of distances. It will not give you the metric properties of space as does my construction. Take an example. I know, in my American town, that to walk ten blocks I have to cover twice the distance of five blocks. Since I know that all blocks are equal, I can immediately determine distances.

M.  That is true. In my “European town” structure, I cannot measure distances immediately by the number of deformed blocks. I must know something more; I must know the geometrical properties of my surface. Just as everyone knows that from 0° to 10° longitude on the Equator is not the same distance as from 0° to 10° longitude near the North Pole. But every navigator knows how to judge the distance between two such points on our earth because he knows its geometrical properties. He can either do it by calculations based on the knowledge of spherical trigonometry, or he can do it experimentally, sailing his ship through the two distances at the same speed. In your case the whole problem is trivial, because all the streets and avenues are the same distance apart. In the case of our earth it is more complicated; the two meridians 0° and 10° meet at the earth’s poles and are farthest apart on the Equator. Similarly, in my “European town structure”, I must know something more than you in your “American town structure”, in order to determine distances. I can gain this additional knowledge by studying the geometrical properties of my continuum in every particular case.

Geometry based on constant inertia may be compared to the geometry of a flat surface. Geometry based on changing inertia may be compared to the geometry of a surface that is changing in a new dimension of inertia. It is no longer flat.

O. But all this only goes to show how inconvenient and complicated it is to give up the simple structure of the Euclidean geometry for the intricate scaffolding which you are bound to use. Is this really necessary?

M. I am afraid it is, if we want to apply our physics to any c.s., without the mysterious inertial c.s. I agree that my mathematical tool is more complicated than yours, but my physical assumptions are simpler and more natural.

We are simply adding a new dimension of inertia to Physics.

The discussion has been restricted to two-dimensional continua. The point at issue in the general relativity theory is still more complicated, since it is not the two-dimensional, but the four-dimensional time-space continuum. But the ideas are the same as those sketched in the two-dimensional case. We cannot use in the general relativity theory the mechanical scaffolding of parallel, perpendicular rods and synchronized clocks, as in the special relativity theory. In an arbitrary c.s. we cannot determine the point and the instant at which an event happens by the use of rigid rods, rhythmical and synchronized clocks, as in the inertial c.s. of the special relativity theory. We can still order the events with our non-Euclidean rods and our clocks out of rhythm. But actual measurements requiring rigid rods and perfect rhythmical and synchronized clocks can be performed only in the local inertial c.s. For this the whole special relativity theory is valid; but our “good” c.s. is only local, its inertial character being limited in space and time. Even in our arbitrary c.s. we can foresee the results of measurements made in the local inertial c.s. But for this we must know the geometrical character of our time-space continuum.

Our idealized experiments indicate only the general character of the new relativistic physics. They show us that our fundamental problem is that of gravitation. They also show us that the general relativity theory leads to further generalization of time and space concepts.

The mathematics of general relativity may be simplified and explained with the knowledge above.

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FINAL COMMENTS

In special relativity, the new idea added is that velocity of light is the basis of all motion. In general relativity the new idea added is that space-time has a new dimension of gravity.

Gravity is equivalent to acceleration. Acceleration is change in velocity. Velocity is equivalent to inertia. Therefore, a gravitation field exists because inertia in that space is changing.

Euclidean geometry corresponds to inertia of Earth. Geometry will be very different for a black hole whose inertia is much greater. We may say that in a gravitational field, the space-time characteristics of geometry are also changing along with inertia.
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Einstein 1938: Outside and Inside the Lift

Reference: Evolution of Physics

This paper presents Chapter III, section 11 from the book THE EVOLUTION OF PHYSICS by A. EINSTEIN and L. INFELD. The contents are from the original publication of this book by Simon and Schuster, New York (1942).

The paragraphs of the original material (in black) are accompanied by brief comments (in color) based on the present understanding.  Feedback on these comments is appreciated.

The heading below is linked to the original materials.

.

Outside and Inside the Lift

The law of inertia marks the first great advance in physics; in fact, its real beginning. It was gained by the contemplation of an idealized experiment, a body moving forever with no friction nor any other external forces acting. From this example, and later from many others, we recognized the importance of the idealized experiment created by thought. Here again, idealized experiments will be discussed. Although these may sound very fantastic, they will, nevertheless, help us to understand as much about relativity as is possible by our simple methods.

Fundamentally substance varies in terms of inertia and uniform velocity. At the lower end inertia appears as a spread out wave with a frequency and high velocity. At the upper end inertia appears as concentrated particle with mass and small velocity. Inertia and velocity are perceived when we compare the waves and particles. There is equivalence between inertia and velocity.

We had previously the idealized experiments with a uniformly moving room. Here, for a change, we shall have a falling lift.

Imagine a great lift at the top of a skyscraper much higher than any real one. Suddenly the cable supporting the lift breaks, and the lift falls freely toward the ground. Observers in the lift are performing experiments during the fall. In describing them, we need not bother about air resistance or friction, for we may disregard their existence under our idealized conditions. One of the observers takes a handkerchief and a watch from his pocket and drops them. What happens to these two bodies? For the outside observer, who is looking through the window of the lift, both handkerchief and watch fall toward the ground in exactly the same way, with the same acceleration. We remember that the acceleration of a falling body is quite independent of its mass and that it was this fact which revealed the equality of gravitational and inertial mass (p. 37). We also remember that the equality of the two masses, gravitational and inertial, was quite accidental from the point of view of classical mechanics and played no role in its structure. Here, however, this equality reflected in the equal acceleration of all falling bodies is essential and forms the basis of our whole argument.

In a gravitational field the force is proportional to the mass, and the proportionality constant is the acceleration. The gravitational field is defined by its acceleration, which is the same for all masses.

Let us return to our falling handkerchief and watch; for the outside observer they are both falling with the same acceleration. But so is the lift, with its walls, ceiling, and floor. Therefore: the distance between the two bodies and the floor will not change. For the inside observer the two bodies remain exactly where they were when he let them go. The inside observer may ignore the gravitational field, since its source lies outside his c.s. He finds that no forces inside the lift act upon the two bodies, and so they are at rest, just as if they were in an inertial c.s. Strange things happen in the lift! If the observer pushes a body in any direction, up or down for instance, it always moves uniformly so long as it does not collide with the ceiling or the floor of the lift. Briefly speaking, the laws of classical mechanics are valid for the observer inside the lift. All bodies behave in the way expected by the law of inertia. Our new c.s. rigidly connected with the freely falling lift differs from the inertial c.s. in only one respect. In an inertial c.s., a moving body on which no forces are acting will move uniformly for ever. The inertial c.s. as represented in classical physics is neither limited in space nor time. The case of the observer in our lift is, however, different. The inertial character of his c.s. is limited in space and time. Sooner or later the uniformly moving body will collide with the wall of the lift, destroying the uniform motion. Sooner or later the whole lift will collide with the earth, destroying the observers and their experiments. The c.s. is only a “pocket edition” of a real inertial c.s.

Within a freely falling lift the acceleration cancels out the gravitational field, leaving an inertial c.s. This shows the equivalence between acceleration and gravity.

This local character of the c.s. is quite essential. If our imaginary lift were to reach from the North Pole to the Equator, with the handkerchief placed over the North Pole and the watch over the Equator, then, for the outside observer, the two bodies would not have the same acceleration; they would not be at rest relative to each other. Our whole argument would fail! The dimensions of the lift must be limited so that the equality of acceleration of all bodies relative to the outside observer may be assumed.

With this restriction, the c.s. takes on an inertial character for the inside observer. We can at least indicate a c.s. in which all the physical laws are valid, even though it is limited in time and space. If we imagine another c.s., another lift moving uniformly, relative to the one falling freely, then both these c.s. will be locally inertial. All laws are exactly the same in both. The transition from one to the other is given by the Lorentz transformation.

Let us see in what way both the observers, outside and inside, describe what takes place in the lift.

The outside observer notices the motion of the lift and of all bodies in the lift, and finds them in agreement with Newton’s gravitational law. For him, the motion is not uniform, but accelerated, because of the action of the gravitational field of the earth.

However, a generation of physicists born and brought up in the lift would reason quite differently. They would believe themselves in possession of an inertial system and would refer all laws of nature to their lift, stating with justification that the laws take on a specially simple form in their c.s. It would be natural for them to assume their lift at rest and their c.s. the inertial one.

It is impossible to settle the differences between the outside and the inside observers. Each of them could claim the right to refer all events to his c.s. Both descriptions of events could be made equally consistent.

We see from this example that a consistent description of physical phenomena in two different c.s. is possible, even if they are not moving uniformly, relative to each other. But for such a description we must take into account gravitation, building, so to speak, the “bridge” which effects a transition from one c.s. to the other. The gravitational field exists for the outside observer; it does not for the inside observer. Accelerated motion of the lift in the gravitational field exists for the outside observer, rest and absence of the gravitational field for the inside observer. But the “bridge”, the gravitational field, making the description in both c.s. possible, rests on one very important pillar: the equivalence of gravitational and inertial mass. Without this clue, unnoticed in classical mechanics, our present argument would fail completely.

Newton considered a momentary push and a momentary change in uniform motion. Einstein is considering a continuous push and a continuous change in uniform motion.

Now for a somewhat different idealized experiment. There is, let us assume, an inertial c.s., in which the law of inertia is valid. We have already described what happens in a lift resting in such an inertial c.s. But we now change our picture. Someone outside has fastened a rope to the lift and is pulling, with a constant force, in the direction indicated in our drawing. It is immaterial how this is done. Since the laws of mechanics are valid in this c.s., the whole lift moves with a constant acceleration in the direction of the motion. Again we shall listen to the explanation of phenomena going on in the lift and given by both the outside and inside observers.

The outside observer: My c.s. is an inertial one. The lift moves with constant acceleration, because a constant force is acting. The observers inside are in absolute motion, for them the laws of mechanics are invalid. They do not find that bodies, on which no forces are acting, are at rest. If a body is left free, it soon collides with the floor of the lift, since the floor moves upward toward the body. This happens exactly in the same way for a watch and for a handkerchief. It seems very strange to me that the observer inside the lift must always be on the “floor”, because as soon as he jumps the floor will reach him again.

The inside observer: I do not see any reason for believing that my lift is in absolute motion. I agree that my c.s., rigidly connected with my lift, is not really inertial, but I do not believe that it has anything to do with absolute motion. My watch, my handkerchief, and all bodies are falling because the whole lift is in a gravitational field. I notice exactly the same kinds of motion as the man on the earth. He explains them very simply by the action of a gravitational field. The same holds good for me.

Acceleration is zero in an inertial system. The moment acceleration is added as a continuous push we see the manifestation of gravity within the moving system. NOTE: Accelerated motion is in reference to the body itself and not to another body.

These two descriptions, one by the outside, the other by the inside, observer, are quite consistent, and there is no possibility of deciding which of them is right. We may assume either one of them for the description of phenomena in the lift: either non-uniform motion and absence of a gravitational field with the outside observer, or rest and the presence of a gravitational field with the inside observer.

The outside observer may assume that the lift is in “absolute” non-uniform motion. But a motion which is wiped out by the assumption of an acting gravitational field cannot be regarded as absolute motion.

There is, possibly, a way out of the ambiguity of two such different descriptions, and a decision in favour of one against the other could perhaps be made. Imagine that a light ray enters the lift horizontally through a side window and reaches the opposite wall after a very short time. Again let us see how the path of the light would be predicted by the two observers.

The outside observer, believing in accelerated motion of the lift, would argue: The light ray enters the window and moves horizontally, along a straight line and with a constant velocity, toward the opposite wall. But the lift moves upward, and during the time in which the light travels toward the wall the lift changes its position. Therefore, the ray will meet at a point not exactly opposite its point of entrance, but a little below. The difference will be very slight, but it exists nevertheless, and the light ray travels, relative to the lift, not along a straight, but along a slightly curved line. The difference is due to the distance covered by the lift during the time the ray is crossing the interior.

The inside observer, who believes in the gravitational field acting on all objects in his lift, would say: there is no accelerated motion of the lift, but only the action of the gravitational field. A beam of light is weightless and, therefore, will not be affected by the gravitational field. If sent in a horizontal direction, it will meet the wall at a point exactly opposite to that at which it entered.

It seems from this discussion that there is a possibility of deciding between these two opposite points of view as the phenomenon would be different for the two observers. If there is nothing illogical in either of the explanations just quoted, then our whole previous argument is destroyed, and we cannot describe all phenomena in two consistent ways, with and without a gravitational field.

But there is, fortunately, a grave fault in the reasoning of the inside observer, which saves our previous conclusion. He said: “A beam of light is weightless and, therefore, will not be affected by the gravitational field.” This cannot be right! A beam of light carries energy and energy has mass. But every inertial mass is attracted by the gravitational field, as inertial and gravitational masses are equivalent. A beam of light will bend in a gravitational field exactly as a body would if thrown horizontally with a velocity equal to that of light. If the inside observer had reasoned correctly and had taken into account the bending of light rays in a gravitational field, then his results would have been exactly the same as those of an outside observer.

Near infinite velocity of light indicates the presence of infinitesimal inertia. This inertia is restraining some acceleration because the velocity is constant. Because of the equivalency of acceleration with gravity, we may say that the infinitesimal inertia of light has infinitesimal gravity.

The gravitational field of the earth is, of course, too weak for the bending of light rays in it to be proved directly, by experiment. But the famous experiments performed during the solar eclipses show, conclusively though indirectly, the influence of a gravitational field on the path of a light ray.

We may also say that high inertia of stars shall have high gravity. Gravitational fields interact with each other. Therefore, there shall be interaction between the gravity of light and the gravity of stars.

It follows from these examples that there is a well-founded hope of formulating a relativistic physics. But for this we must first tackle the problem of gravitation.

We saw from the example of the lift the consistency of the two descriptions. Non-uniform motion may, or may not, be assumed. We can eliminate “absolute” motion from our examples by a gravitational field. But then there is nothing absolute in the non-uniform motion. The gravitational field is able to wipe it out completely.

In uniform motion there is no acceleration, since all acceleration has reduced to the gravity of inertia Gravity, then, forms the gradient of uniform velocity or inertia in the spectrum of inertia. The highest gravity shall exist at the interface between particle and void, where the gradient is the highest.

The ghosts of absolute motion and inertial c.s. can be expelled from physics and a new relativistic physics built. Our idealized experiments show how the problem of the general relativity theory is closely connected with that of gravitation and why the equivalence of gravitational and inertial mass is so essential for this connection. It is clear that the solution of the gravitational problem in the general theory of relativity must differ from the Newtonian one. The laws of gravitation must, just as all laws of nature, be formulated for all possible c.s., whereas the laws of classical mechanics, as formulated by Newton, are valid only in inertial c.s.

Newton’s inertial c.s. is an idealized system that deals with uniform motion. But such systems do not really exist in nature. There exist non-inertial systems only where motion manifests itself as gravity.

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FINAL COMMENTS

Newton considered a momentary push and a momentary change in uniform motion. This gives us the sense of inertia, which keeps the body moving. Einstein, on the other hand, considered a continuous push and a continuous change in motion. This gives us the sense of gravitational field, which keeps the body accelerating.

There is equivalence between inertia and velocity on one hand; and between gravity and acceleration on the other. As velocity changes, inertia changes too and thus acceleration and gravity manifest themselves. External force produces change not only in velocity, but also in inertia.

Increasing inertia is balancing increasing acceleration, which appears as gravity. So, heavier is the mass, the stronger is its gravitational field. With infinite mass there is infinite inertia and infinite gravity. The body acts as a point of fixedness in space relative to which other locations around it are in motion. As inertia decreases, the velocity increases.

In this material region, inertia is so large that any changes in it are imperceptible. But the velocity is small and changes in it are visible. The velocity fluctuates in a narrow range. This material region is wide. It is followed by a very narrow atomic region, which is then followed by a wide field region.

In the narrow atomic region, there is a rapid decline of inertia accompanied by a rapid increase in velocity by many degrees of magnitude. Here we have electromagnetic wave-particles of rapidly declining mass.

Beyond the atomic region there is a wide field region, where, once again, inertia decreases and velocity increases gradually. The velocity in this region is approximated by the velocity of light. This velocity is so high that any changes in it are imperceptible. But the inertia is small and changes in it are visible. The inertia fluctuates in a narrow range.

The inertia appears as mass in the material region, electromagnetic wave-particles in the narrow atomic region, and as frequency in the field region. Inertia is very weak in the field region. Gravity is proportional to inertia.

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Einstein 1938: General Relativity

Reference: Evolution of Physics

This paper presents Chapter III, section 10 from the book THE EVOLUTION OF PHYSICS by A. EINSTEIN and L. INFELD. The contents are from the original publication of this book by Simon and Schuster, New York (1942).

The paragraphs of the original material (in black) are accompanied by brief comments (in color) based on the present understanding.  Feedback on these comments is appreciated.

The heading below is linked to the original materials.

.

General Relativity

There still remains one point to be cleared up. One of the most fundamental questions has not been settled as yet: does an inertial system exist? We have learned something about the laws of nature, their invariance with respect to the Lorentz transformation, and their validity for all inertial systems moving uniformly, relative to each other. We have the laws but do not know the frame to which to refer them.

We are aware of the laws of nature but we do not have their frame of reference.

In order to be more aware of this difficulty, let us interview the classical physicist and ask him some simple questions:

“What is an inertial system?”

“It is a c.s. in which the laws of mechanics are valid. A body on which no external forces are acting moves uniformly in such a c.s. This property thus enables us to distinguish an inertial c.s. from any other.”

“But what does it mean to say that no forces are acting on a body?”

“It simply means that the body moves uniformly in an inertial c.s.”

Here we could once more put the question: “What is an inertial c.s.?” But since there is little hope of obtaining an answer differing from the above, let us try to gain some concrete information by changing the question:

“Is a c.s. rigidly connected with the earth an inertial one?”

“No, because the laws of mechanics are not rigorously valid on the earth, due to its rotation. A c.s. rigidly connected with the sun can be regarded for many problems as an inertial c.s. ; but when we speak of the rotating sun, we again understand that a c.s. connected with it cannot be regarded as strictly inertial.”

“Then what, concretely, is your inertial c.s., and how is its state of motion to be chosen?”

“It is merely a useful fiction and I have no idea how to realize it. If I could only get far away from all material bodies and free myself from all external influences, my c.s. would then be inertial.”

“But what do you mean by a c.s. free from all external influences?”

“I mean that the c.s. is inertial.”

Once more we are back at our initial question!

Our interview reveals a grave difficulty in classical physics. We have laws, but do not know what frame to refer them to, and our whole physical structure seems to be built on sand.

The classical frame of reference cannot account for rotation.

We can approach this same difficulty from a different point of view. Try to imagine that there is only one body, forming our c.s., in the entire universe. This body begins to rotate. According to classical mechanics, the physical laws for a rotating body are different from those for a non-rotating body. If the inertial principle is valid in one case, it is not valid in the other. But all this sounds very suspicious. Is it permissible to consider the motion of only one body in the entire universe? By the motion of a body we always mean its change of position in relation to a second body. It is, therefore, contrary to common sense to speak about the motion of only one body. Classical mechanics and common sense disagree violently on this point. Newton’s recipe is: if the inertial principle is valid, then the c.s. is either at rest or in uniform motion. If the inertial principle is invalid, then the body is in non-uniform motion. Thus, our verdict of motion or rest depends upon whether or not all the physical laws are applicable to a given c.s.

Take two bodies, the sun and the earth, for instance. The motion we observe is again relative. It can be described by connecting the c.s. with either the earth or the sun. From this point of view, Copernicus’ great achievement lies in transferring the c.s. from the earth to the sun. But as motion is relative and any frame of reference can be used, there seems to be no reason for favouring one c.s. rather than the other.

The inertial principle supports uniform motion and no acceleration. Visual sense of motion requires two bodies—one moving relative to another. Sense of motion with only one body requires the sense of acceleration. Motion as rotation provides that sense of acceleration in terms of being pulled by a centripetal force.

Physics again intervenes and changes our commonsense point of view. The c.s. connected with the sun resembles an inertial system more than that connected with the earth. The physical laws should be applied to Copernicus’ c.s. rather than to Ptolemy’s. The greatness of Copernicus’ discovery can be appreciated only from the physical point of view. It illustrates the great advantage of using a c.s. connected rigidly with the sun for describing the motion of planets.

No absolute uniform motion exists in classical physics. If two c.s. are moving uniformly, relative to each other, then there is no sense in saying, “This c.s. is at rest and the other is moving”. But if two c.s. are moving non-uniformly, relative to each other, then there is very good reason for saying, “This body moves and the other is at rest (or moves uniformly) “. Absolute motion has here a very definite meaning. There is, at this point, a wide gulf between common sense and classical physics. The difficulties mentioned, that of an inertial system and that of absolute motion, are strictly connected with each other. Absolute motion is made possible only by the idea of an inertial system, for which the laws of nature are valid.

Velocity refers to another body, but acceleration uses the single body itself as its reference. Therefore, acceleration is closer to being absolute motion. Here the body is its own reference.

It may seem as though there is no way out of these difficulties, as though no physical theory can avoid them. Their root lies in the validity of the laws of nature for a special class of c.s. only, the inertial. The possibility of solving these difficulties depends on the answer to the following question. Can we formulate physical laws so that they are valid for all c.s., not only those moving uniformly, but also those moving quite arbitrarily, relative to each other? If this can be done, our difficulties will be over. We shall then be able to apply the laws of nature to any c.s. The struggle, so violent in the early days of science, between the views of Ptolemy and Copernicus would then be quite meaningless. Either c.s. could be used with equal justification. The two sentences, “the sun is at rest and the earth moves”, or “the sun moves and the earth is at rest”, would simply mean two different conventions concerning two different c.s.

Could we build a real relativistic physics valid in all c.s.; a physics in which there would be no place for absolute, but only for relative, motion? This is indeed possible!

But the theory of special relativity makes the motion absolute by using velocity of light as its reference point. What is missing is the relationship between motion (as acceleration) and inertia.

We have at least one indication, though a very weak one, of how to build the new physics. Really relativistic physics must apply to all c.s. and, therefore, also to the special case of the inertial c.s. We already know the laws for this inertial c.s. The new general laws valid for all c.s. must, in the special case of the inertial system, reduce to the old, known laws.

The problem of formulating physical laws for every c.s. was solved by the so-called general relativity theory; the previous theory, applying only to inertial systems, is called the special relativity theory. The two theories cannot, of course, contradict each other, since we must always include the old laws of the special relativity theory in the general laws for an inertial system. But just as the inertial c.s. was previously the only one for which physical laws were formulated, so now it will form the special limiting case, as all c.s. moving arbitrarily, relative to each other, are permissible.

The physical laws under general relativity reduce to those under special relativity for inertial systems.

This is the programme for the general theory of relativity. But in sketching the way in which it was accomplished we must be even vaguer than we have been so far. New difficulties arising in the development of science force our theory to become more and more abstract. Unexpected adventures still await us. But our final aim is always a better understanding of reality. Links are added to the chain of logic connecting theory and observation. To clear the way leading from theory to experiment of unnecessary and artificial assumptions, to embrace an ever-wider region of facts, we must make the chain longer and longer. The simpler and more fundamental our assumptions become, the more intricate is our mathematical tool of reasoning; the way from theory to observation becomes longer, more subtle, and more complicated. Although it sounds paradoxical, we could say: Modern physics is simpler than the old physics and seems, therefore, more difficult and intricate. The simpler our picture of the external world and the more facts it embraces, the more strongly it reflects in our minds the harmony of the universe.

The general relativity has, however, become mathematical and abstract and has moved farther away from reality.

Our new idea is simple: to build a physics valid for all c.s. Its fulfilment brings formal complications and forces us to use mathematical tools different from those so far employed in physics. We shall show here only the connection between the fulfilment of this programme and two principal problems: gravitation and geometry.

Formal complications arise in general relativity as it is not explained in terms of the reality we are familiar with.

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FINAL COMMENTS

Classical mechanics applies to inertial systems. Inertial systems comply with Galileo’s law of inertia. This law treats inertia qualitatively and not quantitatively. These inertial systems are limited to the familiar material domain. Motion in different inertial systems is related according to the Galilean transformation. Such motion is perceived in a relative sense only. It has no absolute basis.

Special relativity replaces the Galilean transformation of classical mechanics by Lorentz transformation. The Lorentz transformation uses the velocity of light as the reference point for all motion. This makes measurement of motion closer to being absolute and more consistent. Hence special relativity gives better results; but it is still limited to inertial frames that deal only with non-varying inertia and uniform velocity. Inertial frames do not account for rotation and acceleration.

A universal theory of “relativity” is expected to work not only for inertial frames but also for the frames containing rotation, acceleration and variations of inertia. It will work not only in the material domain but also in the electromagnetic and the field domains. It will measure motion on an absolute basis. It shall not require external reference and will work with internal reference like inertia.

The last point about internal reference requires that we measure the velocity of a body based on its internal characteristics and not relative to other bodies. The acceleration of a body is based on its internal characteristics and so does its inertia. The condition of uniform velocity comes about when the acceleration of a body is balanced by its inertia. Considering the broad range of a universal theory from field to matter, it is obvious that motion is reciprocal of inertia. Therefore, the natural uniform velocity of a body may be described in terms of its inertia. This will truly be a universal theory because it does not require an external reference point.

The universe has no external reference. It acts as the reference to everything within the universe. It can provide an absolute basis to all motions including rotations within the universe. It provides an absolute scale from infinite velocity of zero inertia to zero velocity of infinite inertia.

We are aware of the laws of nature. We are now aware that their ultimate frame of reference is the universe itself. Here we may have the universal observer of general relativity. We hope that this universal observer can help relate the abstract mathematics of general relativity to a reality that we can grasp.

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Einstein 1938: The Time-Space Continuum

Reference: Evolution of Physics

This paper presents Chapter III, section 9 from the book THE EVOLUTION OF PHYSICS by A. EINSTEIN and L. INFELD. The contents are from the original publication of this book by Simon and Schuster, New York (1942).

The paragraphs of the original material (in black) are accompanied by brief comments (in color) based on the present understanding.  Feedback on these comments is appreciated.

The heading below is linked to the original materials.

.

The Time-Space Continuum

“The French revolution began in Paris on the 14th of July 1789.” In this sentence the place and time of an event are stated. Hearing this statement for the first time, one who does not know what “Paris” means could be taught: it is a city on our earth situated in long. 2° East and lat. 49° North. The two numbers would then characterize the place, and “14th of July 1789” the time, at which the event took place. In physics, much more than in history, the exact characterization of when and where an event takes place is very important, because these data form the basis for a quantitative description.

For the sake of simplicity, we considered previously only motion along a straight line. A rigid rod with an origin but no end-point was our c.s. Let us keep this restriction. Take different points on the rod; their positions can be characterized by one number only, by the co-ordinate of the point. To say the co-ordinate of a point is 7.586 feet means that its distance is 7.586 feet from the origin of the rod. If, on the contrary, someone gives me any number and a unit, I can always find a point on the rod corresponding to this number. We can state: a definite point on the rod corresponds to every number, and a definite number corresponds to every point. This fact is expressed by mathematicians in the following sentence: all points on the rod form a one-dimensional continuum. There exists a point arbitrarily near every point on the rod. We can connect two distant points on the rod by steps as small as we wish. Thus the arbitrary smallness of the steps connecting distant points is characteristic of the continuum.

Arbitrarily small units of distance are possible. Thus we have a one-dimensional continuum.

Now another example. We have a plane, or, if you prefer something more concrete, the surface of a rectangular table. The position of a point on this table can be characterized by two numbers and not, as before, by one. The two numbers are the distances from two perpendicular edges of the table. Not one number, but a pair of numbers corresponds to every point on the plane; a definite point corresponds to every pair of numbers. In other words: the plane is a two-dimensional continuum. There exist points arbitrarily near every point on the plane. Two distant points can be connected by a curve divided into steps as small as we wish. Thus the arbitrary smallness of the steps connecting two distant points, each of which can be represented by two numbers, is again characteristic of a two-dimensional continuum.

We also have a two-dimensional continuum in terms of the same arbitrarily small unit.

One more example. Imagine that you wish to regard your room as your c.s. This means that you want to describe all positions with respect to the rigid walls of the room. The position of the end-point of the lamp, if the lamp is at rest, can be described by three numbers: two of them determine the distance from two perpendicular walls, and the third that from the floor or ceiling. Three definite numbers correspond to every point of the space; a definite point in space corresponds to every three numbers. This is expressed by the sentence: Our space is a three-dimensional continuum. There exist points very near every point of the space. Again the arbitrary smallness of the steps connecting the distant points, each of them represented by three numbers, is characteristic of a three-dimensional continuum.

Our space is a three-dimensional continuum described by arbitrarily small units.

But all this is scarcely physics. To return to physics, the motion of material particles must be considered. To observe and predict events in nature we must consider not only the place but also the time of physical happenings. Let us again take a very simple example.

A small stone, which can be regarded as a particle, is dropped from a tower. Imagine the tower 256 feet high. Since Galileo’s time we have been able to predict the co-ordinate of the stone at any arbitrary instant after it was dropped. Here is a “timetable” describing the positions of the stone after 0, 1,2, 3, and 4 seconds. Five events are registered in our “timetable”, each represented by two numbers, the time and space coordinates of each event. The first event is the dropping of the stone from 256 feet above the ground at the zero second. The second event is the coincidence of the stone with our rigid rod (the tower) at 240 feet above the ground. This happens after the first second. The last event is the coincidence of the stone with the earth.

We could represent the knowledge gained from our “timetable” in a different way. We could represent the five pairs of numbers in the “timetable” as five points on a surface. Let us first establish a scale. One segment will correspond to a foot and another to a second. For example:

We then draw two perpendicular lines, calling the horizontal one, say, the time axis and the vertical one the space axis. We see immediately that our “timetable” can be represented by five points in our time-space plane.

The distances of the points from the space axis represent the time co-ordinates as registered in the first column of our “timetable”, and the distances from the time axis their space co-ordinates.

Exactly the same thing is expressed in two different ways: by the “timetable” and by the points on the plane. Each can be constructed from the other. The choice between these two representations is merely a matter of taste, for they are, in fact, equivalent.

Let us now go one step farther. Imagine a better “timetable” giving the positions not for every second, but for, say, every hundredth or thousandth of a second. We shall then have very many points on our time-space plane. Finally, if the position is given for every instant or, as the mathematicians say, if the space co-ordinate is given as a function of time, then our set of points becomes a continuous line. Our next drawing therefore represents not just a fragment as before, but a complete knowledge of the motion.

We may associate space with time in arbitrarily small units of both space and time.

The motion along the rigid rod (the tower), the motion in a one-dimensional space, is here represented as a curve in a two-dimensional time-space continuum. To every point in our time-space continuum there corresponds a pair of numbers, one of which denotes the time, and the other the space, co-ordinate. Conversely: a definite point in our time-space plane corresponds to every pair of numbers characterizing an event. Two adjacent points represent two events, two happenings, at slightly different places and at slightly different instants.

The time-space coordinate describes an event.

You could argue against our representation thus: there is little sense in representing a unit of time by a segment, in combining it mechanically with the space, forming the two-dimensional continuum from the two one-dimensional continua. But you would then have to protest just as strongly against all the graphs representing, for example, the change of temperature in New York City during last summer, or against those representing the changes in the cost of living during the last few years, since the very same method is used in each of these cases. In the temperature graphs the one-dimensional temperature continuum is combined with the one-dimensional time continuum into the two-dimensional temperature-time continuum.

Just like there is a relationship among the units of the three dimensions of space, there may be a relationship between natural units of space and time. There could be a conversion factor between space and time.

Let us return to the particle dropped from a 256-foot tower. Our graphic picture of motion is a useful convention since it characterizes the position of the particle at an arbitrary instant. Knowing how the particle moves, we should like to picture its motion once more. We can do this in two different ways.

We remember the picture of the particle changing its position with time in the one-dimensional space. We picture the motion as a sequence of events in the one-dimensional space continuum. We do not mix time and space, using a dynamic picture in which positions change with time.

But we can picture the same motion in a different way. We can form a static picture, considering the curve in the two-dimensional time-space continuum. Now the motion is represented as something which is, which exists in the two-dimensional time-space continuum, and not as something which changes in the one-dimensional space continuum.

Both these pictures are exactly equivalent, and preferring one to the other is merely a matter of convention and taste.

Nothing that has been said here about the two pictures of the motion has anything whatever to do with the relativity theory. Both representations can be used with equal right, though classical physics favoured rather the dynamic picture describing motion as happenings in space and not as existing in time-space. But the relativity theory changes this view. It was distinctly in favour of the static picture and found in this representation of motion as something existing in time-space a more convenient and more objective picture of reality. We still have to answer the question: why are these two pictures, equivalent from the point of view of classical physics, not equivalent from the point of view of the relativity theory?

A static time-space picture is more useful.

The answer will be understood if two c.s. moving uniformly, relative to each other, are again taken into account.

According to classical physics, observers in two c.s. moving uniformly, relative to each other, will assign different space co-ordinates, but the same time coordinate, to a certain event. Thus in our example, the coincidence of the particle with the earth is characterized in our chosen c.s. by the time co-ordinate “4” and by the space co-ordinate “0”. According to classical mechanics, the stone will still reach the earth after four seconds for an observer moving uniformly, relative to the chosen c.s. But this observer will refer the distance to his c.s. and will, in general, connect different space co-ordinates with the event of collision, although the time co-ordinate will be the same for him and for all other observers moving uniformly, relative to each other. Classical physics knows only an “absolute” time flow for all observers. For every c.s. the two-dimensional continuum can be split into two one-dimensional continua: time and space. Because of the “absolute” character of time, the transition from the “static” to the “dynamic” picture of motion has an objective meaning in classical physics.

With change in inertia, both space and time units change.

But we have already allowed ourselves to be convinced that the classical transformation must not be used in physics generally. From a practical point of view it is still good for small velocities, but not for settling fundamental physical questions.

According to the relativity theory the time of the collision of the stone with the earth will not be the same for all observers. The time co-ordinate and the space co-ordinate will be different in two c.s., and the change in the time co-ordinate will be quite distinct if the relative velocity is close to that of light. The two-dimensional continuum cannot be split into two one-dimensional continua as in classical physics. We must not consider space and time separately in determining the time-space co-ordinates in another c.s. The splitting of the two-dimensional continuum into two one-dimensional ones seems, from the point of view of the relativity theory, to be an arbitrary procedure without objective meaning.

We must not consider space and time separately in determining the time-space co-ordinates in another c.s. because time and space are closely related as duration and extents of the substance through its inertia.

It will be simple to generalize all that we have just said for the case of motion not restricted to a straight line. Indeed, not two, but four, numbers must be used to describe events in nature. Our physical space as conceived through objects and their motion has three dimensions, and positions are characterized by three numbers. The instant of an event is the fourth number. Four definite numbers correspond to every event; a definite event corresponds to any four numbers. Therefore: The world of events forms a four-dimensional continuum. There is nothing mysterious about this, and the last sentence is equally true for classical physics and the relativity theory. Again, a difference is revealed when two c.s. moving relatively to each other are considered. The room is moving, and the observers inside and outside determine the time-space co-ordinates of the same events. Again, the classical physicist splits the four-dimensional continua into the three-dimensional spaces and the one-dimensional time-continuum. The old physicist bothers only about space transformation, as time is absolute for him. He finds the splitting of the four-dimensional world-continua into space and time natural and convenient. But from the point of view of the relativity theory, time as well as space is changed by passing from one c.s. to another, and the Lorentz transformation considers the transformation properties of the four-dimensional time-space continuum of our four-dimensional world of events.

Inertia is considered constant in classical mechanics, but in long range as covered by the theory of relativity it is a variable. Change in inertia means change in the substance, its extent (space) and its duration (time). Duration (time) cannot be taken as constant.

The world of events can be described dynamically by a picture changing in time and thrown on to the background of the three-dimensional space. But it can also be described by a static picture thrown on to the background of a four-dimensional time-space continuum. From the point of view of classical physics the two pictures, the dynamic and the static, are equivalent. But from the point of view of the relativity theory the static picture is the more convenient and the more objective.

Even in the relativity theory we can still use the dynamic picture if we prefer it. But we must remember that this division into time and space has no objective meaning since time is no longer “absolute”. We shall still use the “dynamic” and not the “static” language in the following pages, bearing in mind its limitations.

Time makes the picture dynamic, but it can be portrayed in static terms in relationship to space.

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FINAL COMMENTS

Inertia is considered constant in classical mechanics, but in the range covered by the theory of relativity inertia becomes a variable. Change in inertia means change in the characteristics of substance. The key characteristics of substance are its extents and duration. Extent (space) and duration (time) cannot be assumed to have fixed characteristics.

Arbitrarily small units of both distance and time are possible. We, therefore, have a continuum of three space dimensions and one time dimension. These dimensions represent the extents and duration of a substance respectively. Since it is the same substance, its extents and duration are not independent of each other. There seems to be an exact relationship among space, time and inertia. There could even be a conversion factor between space and time.

Time makes the picture dynamic, but it can be portrayed in static terms in relationship to space. A static time-space picture is more useful as used in the theory of relativity. The time-space coordinate describes an event.

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