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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