by Isaac Asimov
Newton visualized the earth’s gravitational force as spreading out in all directions like a vast, expanding sphere. The surface area A of a sphere is proportional to the square of its radius r:
A = 4πr2.
He therefore reasoned that the gravitational force, spreading out over the spherical area, must weaken as the square of the radius. The intensity of light and of sound weakens as the square of the distance from the source. Why not the force of gravity as well?
The distance from the earth’s center to an apple on its surface is roughly 4,000 miles. The distance from the earth’s center to the moon is roughly 240,000 miles. Since the distance to the moon was 60 times greater than to the apple, the force of the earth’s gravity at the moon must be 602, or 3,600, times weaker than at the apple. Divide 16 feet by 3,600, and you come out with roughly 1/20 of an inch. It seemed clear to Newton that the moon does indeed move in the grip of the earth’s gravity.
Newton was persuaded further to consider mass in relation to gravity. Ordinarily, we measure mass as weight. But weight is only the result of the attraction of the earth’s gravitational force. If there were no gravity, an object would be weightless; nevertheless, it would still contain the same amount of matter. Mass, therefore, is independent of weight and should be capable of measurement by a means not involving weight.
Suppose you tried to pull an object on a perfectly frictionless surface in a direction horizontal to the earth’s surface, so that there was no resistance from gravity. It would take effort to set the body in motion and to accelerate its motion, because of the body’s inertia.
If you measured the applied force accurately—say, by pulling on a spring balance attached to the object—you would see that the force f required to bring about a given acceleration a would be directly proportional to the mass m. If you doubled the mass, it would take double the force. For a given mass, the force required would be directly proportional to the acceleration desired.
Mathematically, this is expressed in the equation:
f = ma.
The equation is known as Newton’s Second Law of Motion.
Now, as Galileo had found, the pull of the earth’s gravity accelerates all bodies, heavy or light, at precisely the same rate. (Air resistance may slow the fall of very light bodies; but in a vacuum, a feather will fall as rapidly as a lump of lead, as can easily be demonstrated.) If the Second Law of Motion is to hold, one must conclude that the earth’s gravitational pull on a heavy body must be greater than on a light body, in order to produce the same acceleration. To accelerate a mass that is eight times as great as another, for instance, takes eight times as much force. It follows that the earth’s gravitational pull on any body must be exactly proportional to the mass of that body. (That, in fact, is why mass on the earth’s surface can be measured quite accurately as weight.)
Newton evolved a Third Law of Motion, too: “For every action there is an equal and opposite reaction.” This law applies to force. In other words, if the earth pulls at the moon with a certain force, then the moon pulls on the earth with an equal force. If the moon were suddenly doubled in mass, the earth’s gravitational force upon it would also be doubled, in accordance with the Second Law; of course, the moon’s gravitational force on the earth would then have to be doubled in accordance with the Third Law.
Similarly, if it were the earth rather than the moon that doubled in mass, it would be the moon’s gravitational force on the earth that would double, according to the Second Law, and the earth’s gravitational force on the moon that would double, in accordance with the Third.
If both the earth and the moon were to double in mass, there would be a doubled doubling, each body doubling its gravitational force twice, for a fourfold increase all told.
Newton could only conclude, by this sort of reasoning, that the gravitational force between any two bodies in the universe was directly proportional to the product of the masses of the bodies. And, of course, as he had decided earlier, it is inversely proportional to the square of the distance (center to center) between the bodies. This is Newton’s Law of Universal Gravitation.
If we let f represent the gravitational force, m1 and m2 the masses of the two bodies concerned, and d the distance between them, then the law can be stated:
f = Gm1m2
d2
G is the gravitational constant; the determination of which made it possible to “weigh the earth” (see chapter 4). It was Newton’s surmise that G has a fixed value throughout the universe. As time went on, it was found that new planets, undiscovered in Newton’s time, temper their motions to the requirements of Newton’s law; even double stars incredibly far away dance in time to Newton’s analysis of the universe.
All this came from the new quantitative view of the universe pioneered by Galileo. As you see, much of the mathematics involved was really very simple. Those parts of it I have quoted here are high-school algebra.
In fact, all that was needed to introduce one of the greatest intellectual revolutions of all time was:
1. A simple set of observations any high-school student of physics might make with a little guidance.
2. A simple set of mathematical generalizations at high school level.
3. The transcendent genius of Galileo and Newton, who had the insight and originality to make these observations and generalizations for the first time.
Relativity
The laws of motion as worked out by Galileo and Newton depended on the assumption that such a thing as absolute motion exists—that is, motion with reference to something at rest. But everything that we know of in the universe is in motion: the earth, the sun, our galaxy, the systems of galaxies. Where in the universe, then, can we find absolute rest against which to measure absolute motion?
THE MICHELSON-MORLEY EXPERIMENT
It was this line of thought that led to the Michelson-Morley experiment, which in turn led to a scientific revolution as great, in some respects, as that initiated by Galileo (see chapter 8). Here, too, the basic mathematics is rather simple.
The experiment was an attempt to detect the absolute motion of the earth against an ether that was supposed to fill all space and to be at rest. The reasoning behind the experiment was as follows.
Suppose that a beam of light is sent out in the direction in which the earth is traveling through the ether; and that at a certain distance in that direction, there is a fixed mirror which reflects the light back to the source. Let us symbolize the velocity of light as c, the velocity of the earth through the ether as v, and the distance of the mirror as d. The light starts with the velocity c + v: its own velocity plus the earth’s velocity. (It is traveling with a tail wind, so to speak.) The time it takes to reach the mirror is d divided by (c + v).
On the return trip, however, the situation is reversed. The reflected light now is bucking the head wind of the earth’s velocity, and its net velocity is c − v. The time it takes to return to the source is d divided by (c − v).
The total time for the round trip is:
d + d
c + v c − v
Combining the terms algebraically, we get:
d(c − v) + d(c + v)
(c + v) (c − v)
= dc − dv + dc + dv = 2dc
c2 − v2 c2 − v2
Now suppose that the light-beam is sent out to a mirror at the same distance in a direction at right angles to the earth’s motion through the ether.
The beam of light is aimed from S (the source) to M (the mirror) over the distance d. However, during the time it takes the light to reach the mirror, the earth’s motion has carried the mirror from M to M', so that the actual path traveled by the light beam is from S to M'. This distance we call x, and the distance from M to M' we call y (see diagram above).
While the light is moving the distance x at its velocity c, the mirror is moving the distance y at the velocity of the earth’s motion Y. Since both the light and the mirror arrive at M' simultaneously, the distances traveled must be exac
tly proportional to the respective velocities. Therefore:
y = v
x c
or:
y = vx
c
Now we can solve for the value of x by use of the Pythagorean theorem, which states that the sum of the squares of the sides of a right triangle is equal to the square of the hypotenuse. In the right triangle SMM', then, substituting vx / c for y:
x2 = d2 + ( vx ) 2
c
x2 − ( vx ) 2 = d2
c
x2 − v2x2 = d2
c2
c2x2 − v2x2 = d2
c2
(c2x2 − v2)x2 = d2c2
x2 = d2c2
c2 − v2
x = d2c2
√c2 − v2
The light is reflected from the mirror at M to the source, which meanwhile has traveled on to S'. Since the distance S'S" is equal to SS', the distance MS" is equal to x. The total path traveled by the light beam is therefore 2x, or 2dc/√c2 − v2.
The time taken by the light beam to cover this distance at its velocity c is:
2dc ÷ c = 2d
√c2 − v2 √c2 − v2
How does this compare with the time that light takes for the round trip in the direction of the earth's motion? Let us divide the time in the parallel case (2dc/(c2 − v2) ) by the time in the perpendicular case (2d/√c2 − v2):
2dc ÷ 2d
c2 − v2 √c2 − v2
= 2dc × √c2 − v2 = c√c2 − v2
c2 − v2 2d c2 − v2
Now any number divided by its square root gives the same square root as a quotient, that is, x / √x = √x. Conversely, √x / x = 1 / √x. So the last equation simplifies to:
c
√c2 − v2
This expression can be further simplified if we multiply both the numerator and the denominator by √1 /c2 (which is equal to 1/c):
c√1 /c2
√c2 − v2 √1 /c2
= c/c = 1
√c2/c2 − v2/c2 √1 − v2/c2
And there you are. That is the ratio of the time that light should take to travel in the direction of the earth’s motion as compared with the time it should take in the direction perpendicular to the earth’s motion. For any value of y greater than zero, the expression 1/√1 − v2/c2 is greater than 1. Therefore, if the earth is moving through a motionless ether, it should take longer for light to travel in the direction of the earth’s motion than in the perpendicular direction. (In fact, the parallel motion should take the maximum time and the perpendicular motion the minimum time.)
Michelson and Morley set up their experiment to try to detect the directional difference in the travel time of light. By trying their beam of light in all directions, and measuring the time of return by their incredibly delicate interferometer, they felt they ought to get differences in apparent velocity. The direction in which they found the velocity of light to be at a minimum should be parallel to the earth’s absolute motion, and the direction in which the velocity would be at a maximum should be perpendicular to the earth’s motion. From the difference in velocity, the amount (as well as the direction) of the earth’s absolute motion could be calculated.
They found no differences at all in the velocity of light with changing direction! To put it another way, the velocity of light was always equal to c, regardless of the motion of the source—a clear contradiction of the Newtonian laws of motion. In attempting to measure the absolute motion of the earth, Michelson and Morley had thus managed to cast doubt not only on the existence of the ether, but on the whole concept of absolute rest and absolute motion, and upon the very basis of the Newtonian system of the universe.
THE FITZGERALD EQUATION
The Irish physicist G. F. FitzGerald conceived a way to save the situation. He suggested that all objects decrease in length in the direction in which they are moving by an amount equal to √1 − v2/c2. Thus:
L' = L √1 − v2/c2
where L' is the length of a moving body in the direction of its motion and L is what the length would be if it were at rest.
The foreshortening fraction √1 − v2/c2, FitzGerald showed, would just cancel the ratio 1/√1 − v2/c2, which related the maximum and minimum velocities of light in the Michelson-Morley experiment. The ratio would become unity, and the velocity of light would seem to our foreshortened instruments and sense organs to be equal in all directions, regardless of the movement of the source of light through the ether.
Under ordinary conditions, the amount of foreshortening is very small. Even if a body were moving at one-tenth the velocity of light, or 18,628 miles per second, its length would be foreshortened only slightly, according to the FitzGerald equation. Taking the velocity of light as 1, the equation says:
L' = L √(1 − 0.1 ) 2
1
L' = L √1 − 0.01
L' = L √0.99
Thus L' turns out to be approximately equal to 0.995L, a foreshortening of about half of 1 percent.
For moving bodies, velocities such as this occur only in the realm of the subatomic particles. The foreshortening of an airplane traveling at 2,000 miles per hour is infinitesimal, as you can calculate for yourself.
At what velocity will an object be foreshortened to half its rest-length? With L' equal to one-half L, the FitzGerald equation is:
L/2 = L √1 − v2/c2
or, dividing by L:
½ = √1 − v2/c2
Squaring both sides of the equation:
¼ = 1 − v2/c2
v2/c2 = ¾
v = √3c / 4 = 0.866c
Since the velocity of light in a vacuum is 186,282 miles per second, the velocity at which an object is foreshortened to half its length is 0.866 times 186,282, or roughly 161,300 miles per second.
If a body moves at the speed of light, so that v equals c, the FitzGerald equation becomes:
L' = L √1 − c2/c2 = L √0 = 0
At the speed of light, then, length in the direction of motion becomes zero.
It would seem, therefore, that no velocity faster than that of light is possible.
THE LORENTZ EQUATION
In the decade after FitzGerald had advanced his equation, the electron was discovered, and scientists began to examine the properties of tiny charged particles. Lorentz worked out a theory that the mass of a particle with a given charge is inversely proportional to its radius. In other words, the smaller the volume into which a particle crowds its charge, the greater its mass.
Now if a particle is foreshortened because of its motion, its radius in the direction of motion is reduced in accordance with the FitzGerald equation.
Substituting the symbols R and R' for L and L', we write the equation:
R' = R √1 − v2/c2
R'/R = √1 − v2/c2
The mass of a particle is inversely proportional to its radius. Therefore:
R' = M
R M'
where M is the mass of the particle at rest and M' is its mass when in motion.
Substituting M/M' for R'/R in the preceding equation, we have:
M/M' = √1 − v2/c2
M' = M
√1 − v2/c2
The Lorentz equation can be handled just as the FitzGerald equation was.
It shows, for instance, that for a particle moving at a velocity of 18,628 miles per second (one-tenth the speed of light), the mass M would appear to be 0.5 percent higher than the rest-mass M. At a velocity of 161,300 miles per second, the apparent mass of the particle would be twice the rest-mass.
Finally, for a particle moving at a velocity equal to that of light, so that v is equal to c, the Lorentz equation becomes:
M' = M = M
√1 − c2/c2 0
Now as the denominator of any fraction with a fixed numerator becomes smaller and smaller (approaches zero), the value of the fraction itself becomes larger and larger without limit. In other words, from the equation preceding, it would seem that the mass of any object traveling at a velocity approaching
that of light becomes infinitely large. Again, the velocity of light would seem to be the maximum possible.
All this led Einstein to recast the laws of motion and of gravitation. He considered a universe, in other words, in which the results of the Michelson-Morley experiments were to be expected.
Yet even so we are not quite through. Please note that the Lorentz equation assumes some value for M that is greater than zero. This is true for most of the particles with which we are familiar and for all bodies, from atoms to stars, that are made up of such particles. There are, however, neutrinos and antineutrinos for which M, the mass at rest, or rest-mass, is equal to zero. This is also true of photons.
Such particles travel at the speed of light in a vacuum, provided they are indeed in a vacuum. The moment they are formed they begin to move at such a velocity without any measurable period of acceleration.
We might wonder how it is possible to speak of the rest-mass of a photon or a neutrino, if they are never at rest but can only exist while traveling (in the absence of interfering matter) at a constant speed of 186,280 miles per second. The physicists Olexa-Myron Bilaniuk and Ennackal Chandy George Sudarshan have therefore suggested that M be spoken of as proper mass. For a particle with mass greater than zero, the proper mass is equal to the mass measured when the particle is at rest relative to the instruments and observer making the measurement. For a particle with mass equal to zero, the proper mass is obtained by indirect reasoning. Bilaniuk and Sudarshan also suggest that all particles with a proper mass of zero be called luxons (from the Latin word for “light”) because they travel at light-speed, while particles with a proper mass greater than zero be called tardyons because they travel at less than light-speed, or at subluminal velocities.
In 1962, Bilaniuk and Sudarshan began to speculate on the consequences of faster-than-light velocities (superluminal velocities). Any particle traveling with faster-than-light velocities would have an imaginary mass. That is, the mass would be some ordinary value multiplied by the square root of −1.
Suppose, for instance, a particle were going at twice the speed of light, so that in the Lorentz equation v = 2c. In that case:
