LECTURE I GENERAL A. RELATIVITY OF MOTION, LOCATION AND TIME The theory of relativity as developed by Einstein and his collaborators has revohdionized science by sweeping aside many of the limitations which hitherto fettered the human intellect. But, being essentially mathematical, a general conception of it can be given to the non-mathematician only by the use of analogies and illustrations, and this inevitably involves a certain looseness of argumentation. The following pages therefore may serve to give a general idea of the theory of relativity and its consequences, but not to revieiv it critically. The theory of relativity starts from two premises : 1. All phenomena of space, time and motion are relative; that is, there is no absolute motion, etc., but motion, location and time have a meaning only relative to some other location, time, etc. 2. The laws of nature are universal ; that is, they apply in the same form everywhere, whether in a speeding railway train on earth or in the empty space between the fixed stars. So far, these two premises appear simple and rather obvious, but startling and revolutionary ideas appear when carrying the reasoning from these premises to their ultimate conclusions, as Einstein has done. Suppose, for instance, you happen to run your car at 30 miles per hour against a stone wall. There seems nothing relative about this. The wreck is very real; the stone wall does not budge, and when a rapidly moving mass 1 2 RELATIVITY AND SPACE meets an immovable body mechanical energy is set free destructively. But is the stone wall really immovable? Is it not a part of the earth, which spins around its axis at 800 miles per hour so that both the stone wall and your car were moving. And perhaps if you happened to drive the car in a westward direction — that is, against the rotation of the earth — your car really was moving more slowly than the stone wall — was going only 770 miles per hour, and the stone wall 800. But think further: Is not the stone wall, as a part of the earth, revolving around the sun at 70,000 miles per hour, and is not the sun, and with it the earth, and the stone wall, and your car, also moving on an unknown path among the fixed stars? So that really you know nothing and can know nothing about the actual or absolute speed of the car. All you know is that the relative speed of the car^ — ^that is, the speed relative to the earth and thus to the stone wall — was 30 miles per hour. But that is sufficient to let you understand the effects of the car meeting the stone wall. So with location. The room in which you are sitting while reading appears fixed and definite. But the only way you can describe its location is by referring it to some other body or location as reference point, by saying, for instance, that your room is located x feet north and y feet west and z feet above the surveyor's markstone on City Hall Square. Or you can give its latitude, longitude and altitude, stating that from the starting point of latitude, longitude and altitude — -that is, where the equator meets the zero meridian at ocean level — you go so many degrees north (or south), then so many degrees west (or east) , and then so many feet up (or down), and thereby reach your room. Three distances thus are required, measured in three chosen directions, from a chosen starting point, to locate a point or an object in space, and therefore we say that space has three dimensions. But do these three distances really locate you in your room? Suppose somebody, reading the directions, should try to locate your room 1000 years RELATIVITY OF MOTION 3 hence. He would not find it. Thus one more thing must be given — the time, measured from some arbitrary starting point, for instance, anno domini. Thus, you see, to locate anything in this world of ours requires four measurements, three distances and one time; and we thus can say that the world and its events have four dimensions, three dimensions of space and one of time. But all such location of events in the four-dimensional world can only be relative to the arbitrarily chosen reference points in space and time. In bygone ages, when people thought the earth flat and immovable as the center of the universe, they could dream of referring location to an absolute stationary reference point, say the Capitol of Rome. But when we learned that the earth is a sphere, spinning around its axis and revolving around the sun, the earth ceased to offer any fixed and permanent reference point in space. The sun then was chosen. But the astronomers found that the sun also is moving among the fixed stars. And the ''fixed" stars do not stand still, but are moving "every which way," so that all the attempts to find some- thing immovable and fixed in the universe have failed, and thus all motion, all location, can be relative only to other objects, which are also moving. B. EFFECT OF RELATIVE MOTION ON LENGTH AND TIME Suppose you toss a stone across your room. Observing the point at which the stone leaves your hand, the direction in which it leaves, and the speed, the physicists can calcu- late the path of the stone as it curves downward and finally comes to rest on the floor of the room. Suppose now you are on a railway train, moving at constant speed on a straight, level track, and toss a stone across the car in which you are riding. From the same three observations — the point in the moving railway car where the stone leaves your hand, the direction, and the speed relative to the carat which 4 RELATIVITY AND SPACE the stone leaves — the physicist by the same laws calculates the path which the tossed stone traverses in the car. Whether the car is moving at constant speed on a straight, level track or standing still makes no difference; the path of the stone is the same, as the same laws of nature apply everywhere. If the laws of nature are the same in the railway train moving at constant speed on straight, level track as they are on the "rigid" platform of the earth or in the empty space among the fixed stars, then the speed of light must also be the same, 186,000 miles per second, and so must be the speed with which the electric current travels in its circuit, which is the speed of light. This is important because all observations depend on it. Any event is either observed by seeing it or recorded by some electrical arrange- ment, and in either case we do not get the exact time when the event occurs but a time later by the time it takes the light to reach our eye or the electric current to flow from the event to the recording device, and to get the exact time of the event, we therefore have to allow for the time taken by the light or the electric current. Owing to the enormous speed of the light, this time difference between the moment when the event occurs and the moment when we observe or record it usually is so extremely small as to be negligible. But not always. For instance, when on ship- board out on the ocean the chronometer has stopped and the mariner tries to find the location of his ship from the stars, he might use the eclipses of the moons of Jupiter for this purpose. But when he sees the eclipse it has already passed by from 30 to 50 minutes — ^depending on the relative position of the earth and Jupiter — owing to the time which it takes the light to go from Jupiter to the earth over the hundreds of millions of miles of distance. But if the speed of hght in the moving train must be the same as on the stationary track, we get some rather strange conclusions. Suppose we place a lamp on the track, back of the receding train, so that the light shines RELATIVITY OF LOCATION AND TIME 5 along the track (for instance, a signal light). The beam of light travels along the track at 186,000 miles per second. The train moves along the track, in the same direction, at 100 feet per second. Therefore, relative to the train, we should expect the beam of light to travel at 186,000 miles less 100 feet per second. It would be thus with a rifle bullet. If I shoot a rifle along the track at the receding train, and the rifle bullet travels along the track at 2000 feet per second, while the train travels in the same direction at 100 feet per second, then the rifle bullet will catch up with the train and pass through the train at the relative speed of 2000 less 100, or 1900, feet per second. But the constancy of the laws of nature teaches us that if the beam of light travels along the track at 186,000 miles per second, and the train in the same direction at 100 feet per second, the speed of the beam of light measured in the train (that is, its relative speed to the moving train) cannot be 186,000 miles less 100 feet, as we would expect, but must be 186,000 miles per second, the same as its relative speed to the track. Now, the only way we can explain this con- tradiction is to say that when we measured the speed of light on the train our measuring rods were shorter, or, using the length of the train as measure, the train was shorter, or the time was slower, or both. These three possibilities really are one. It can be shown that if the length of the train is shorter, the time must be slower in the same proportion. Thus this leads to the strange conclusions that, when the train is moving, to the beam of light coming from the outside, and to an outside observer, the length of the train has shortened and the time in the train has slowed down. But if we now stop the train and remeasure it, we find the same length and the same time as before. This conclusion from the two premises of the theory of relativity is so against our accustomed ideas that we would be inclined to reject it if it could not be verified by experi- ment, and the experiment has been made repeatedly. It is 6 RELATIVITY AND SPACE true a difference of 100 feet per second out of 186,000 miles per second is so extremely small that it could not be measured. But we can speed up the train and, instead of 100 feet per second, run it at 100,000 feet, or 20 miles, per second. We have such a train. The earth on its path around the sun moves about 20 miles per second, and the speed of light going with the motion of the earth then should be 20 miles less, going against the motion of the earth 20 miles greater. But the experiment shows that it is the same, and experiment has proven this with an accuracy many times greater than the difference in the speed of light which we should expect but do not find, so that the fact of the constancy of the speed of light is beyond question. Beyond question, then, also, is it that for an outside observer motion shortens the length and slows down the time on the moving body; but not for an observer moving with the train — for him length and time are the same. C. RELATIVITY OF LENGTH AND DURATION What does this mean? The train stands on the track. I measure it from the outside, you measure it from the inside, and we find the same length. We compare our watches and find them to go alike. Now the train starts and runs at high speed. While it is passing me I measure its length again and find it shorter than before, while at the same time, you, traveling with the train, measure it again from the inside and find the same length we both found when the train was standing still. But while passing over it you measure a piece of the track and find it shorter than I find it when measuring it from the outside. While you pass me on the train I compare your watch with mine and find your watch slower than mine. But, at the same time, you, comparing your watch with mine, while passing me, find my watch slower. Then the train stops, and both our measure- ments agree again. What then is the "true" length of the train and the *'true" time — that which I get when measur- RELATIVITY OF LOCATION AND TIME 7 ing the train while it passes me at high speed or that which you get while moving with the train? Both, and neither. It means that length is not a fixed and invariable property of a body, but depends on the condition under which it is observed. The train has one length to the observer stand- ing still with regard to it, that is, the observer in the train; a different and a shorter length to the observer whom it passes at 100 feet per second; and if I could go outside of the earth and measure the length of the train, while the train and earth rush by me at 20 miles per second, I would find a third still shorter length. Length and time, therefore, are relative properties of things, depending on the conditions under which they are observed, particularly the relative speed of the body to the observer. This really is so startling only because it is novel, since at all speeds which we find around us, even the highest speeds of rifle bullets, the change of length and time is so extremely small as to be inappreciable, and we therefore are used to finding length and time constant. Appreciable changes occur only at the speed of 10,000 to 100,000 miles per second, while the most accurate methods of measure- ment would fail to show an appreciable shortening of the railway train going at 60 miles per hour, because the short- ening is so small. But it is there just the same. However, the relativity of the length of a body- — ^that is, the dependence of the length on the conditions of observa- tion—is no more strange than the relativity of the color of a body. Off-hand we will say that a body has a fixed and definite color; the grass is green, the snow is white. Never- theless, when we think of it we know it is not so. The lady buying material for a dress in the dry-goods store during the daytime may select a nice heliotrope. But when the dress is finished, in the ballroom, she finds its color a clear soft pink. And when, to have a photograph taken, she goes to a photographer using mercury lamps in his studio, she finds the dress a clear blue. Which is its "true" color? Helio- trope, or pink, or blue? Any of the three is the true color 8 RELATIVITY AND SPACE in the condition under which it is observed. So, Einstein's theory of relativity proves to us, it is with length and with time. There is no single length of a body, nor time on the body, but length and time are relative and vary with the conditions under which they are observed, with the relative speed of the observer, just as the color of a body varies with the kind of light under which it is seen. D. RELATIVITY OF MASS If, then, in a body moving rapidly past us, the distance appears shortened and the time slowed down, the speed, which is distance divided by time, must also appear slower. Now, the energy of the moving body depends on its mass and its speed, and with the same energy put into the body, if the speed appears slower, the mass must appear larger. We thus draw the conclusion from Einstein's theory of relativity that the mass of a moving body is not constant, but increases with the speed, and the oldest of the great fundamental laws of nature, the law of conservation of matter, thus goes into the discard. For nearly two cen- turies we have accepted the law of conservation of matter and believed that matter — that is, mass — can neither be created nor destroyed, and now we find that mass varies with the speed, so that speed- — that is, energy- — ^can create mass, and mass or matter probably is merely a manifesta- tion of energy. And this can be and has been verified experimentally. The decrease of length, the slowing down of time, the increase of mass, becomes appreciable only at velocities approaching that of the light. Thus at ordinary every- day velocities length, time and mass are constant; but in the vacuum tubes used in our big wireless stations to pro- duce electrical vibrations which carry the message through space across oceans and continents, or to receive the faint signal arriving from far-distant stations, the current is carried through the empty space of the tube by minute RELATIVITY OF LOCATION AND TIME 9 particles, so-called electrons, and measuring the speed and the mass of these electrons, the physicists find that they move at speeds of 10,000 and 100,000 miles per second and that their mass is not constant, but increases with the speed, in the manner required by Einstein's theory. This was the first experimental proof of the change of mass, and it was found before Einstein gave the explanation in his relativity theory. E. ACCELERATION AND THE LAW OF GRAVITATION Suppose you have a billiard table in your house. You put a ball in the middle of the table. It stays there until something pushes it, and this something we call " force." Or you shoot a ball across the billiard table. It moves in a straight line until it strikes the boundary, rebounds and again moves in a straight line at constant speed. Suppose now we have a billiard table in a train, and the train is running at constant speed on a straight level track. You again put a ball in the middle of the table and it stays there, just as was the case in your house, at rest with regard to the table, though I, standing outside near the track, see that train and table and ball all three move together at constant speed. You shoot the ball across the table, and it moves in a straight line at constant speed, thus in the moving railway train obeying the same law of nature as in your stationary house, the law that any body keeps the same state, whether rest or motion, until something changes its state. But suppose the train is speeding up, its speed increasing while you put the ball in the middle of the billiard table in the train. Now you find that this ball does not remain at rest, but it begins to move toward the back of the train, first slowly and then more and more rapidly until it comes to rest against the back boundary of the table, just as a stone which I drop does not remain at rest, suspended in the air, but begins to move downward with increasing speed — -''falls." So the billiard ball in the speeding train 10 RELATIVITY AND SPACE "falls" toward the back of the train. You shoot a ball across the billiard table while the train is speeding up. It does not move in a straight line, but curves toward the back of the train, just as a thrown stone, on earth, does not move in a straight line at constant speed, but curves down- ward. You say then that in the speeding railway train some force acts on the billiard ball, pulling it toward the back of the train, just as the attraction of the earth pulls downward. You may speculate on this force which attracts things toward the back of the speeding railway train and find its laws just as Newton found the laws governing the force of gravitation. But I, standing on the embankment, near the track, while the speeding railway train passes, see that there is no real force acting on the billiard ball, but when you put it in the middle of the table, left to itself, it continues to move in a straight line at the speed which it and the train had when you put it there. What happens is that the billiard table and train, speeding up, slide forward under the ball, and the ball thus seems to fall backward toward the end of the train. So, when you shoot a ball across the billiard table in the speeding railway train, I from the outside see the ball move in a straight line at constant speed, but see billiard table and train slide forward under it, so giving you, who are moving with the speeding railway train, the impression of an attracting force pulling the ball toward the back of the train. You try to find the laws of this force; that is, the laws obeyed by the relative motion which you see. But to me these motions are those of a body left to itself, in a straight line at constant speed, and, knowing the motion of the speeding railway train, the mathematician can calculate the motion which you observe, without any physical assumption, merely as a mathematical transformation from the straight- line motion which I see from the outside to the complicated motion relative to the speeding train which you observe, and so derive the law of the latter motion — ^that is, the law of the fictitious attracting force — to which you ascribe RELATIVITY OF LOCATION AND TIME 11 these motions. This Einstein has done, and so has derived a new and more general expression for the law of gravitation, in a way which does not depend on any hypothesis con- cerning the nature of the force. This law is more general than Newton's law of gravitation, and the latter appears as the first approximation of Einstein's law of gravitation. The more general law of gravitation given by Einstein does not mean that Newton's law of gravitation is wrong. Both laws give so nearly the same results in almost all cases, even in the calculation of cosmic motions, that usually the difference cannot be discovered even by the most accurate measurements. This is to say that Newton's law is a very close approximation of Einstein's. There are a few cases only in the universe as we know it today where the difference becomes noticeable. Such, for instance, is the motion of the planet Mercury. This planet has been observed for thousands of years, but all attempts to calcu- late its motion accurately by Newton's law have failed, while the application of Einstein's law has succeeded, thus once again corroborating his theory of relativity. To summarize the conclusions at which we have arrived, the theory of relativity means: All phenomena of motion, space and time are relative. The laws of nature, including the speed of light, are the same everywhere. From these principles it follows that length, time and mass are relative also, are not fixed properties of things, but vary with the relative speed of the observer. A more general law of gravitation is derived as a mathe- matical transformation of straight-line inertial motion to the apparent motion relative to a speeding system (the railway train in above illustration) and shows that gravi- tation is not a real force, but a manifestation of inertia, just as centrifugal force is.