The pursuit of quantitative precision is as arduous as it is important.
Physical measurements are made with extraordinary exactitude; if
they were made less carefully, such minute discrepancies as form
the experimental data for the theory of relativity could never be
revealed. Mathematical physics, before the coming of relativity, used
a set of conceptions which were supposed to be as precise as physical
measurements, but it has turned out that they were logically defective,
and that this defectiveness showed itself in very small deviations from
expectations based upon calculation. In this chapter I want to show how
the fundamental ideas of pre-relativity physics are affected, and what
modifications they have had to undergo.
We have already had occasion to speak of mass. For purposes of
daily life, mass is much the same as weight; the usual measures of
weight—ounces, grams, etc.—are really measures of mass. But as
soon as we begin to make accurate measurements, we are compelled to
distinguish between mass and weight. Two different methods of weighing
are in common use, one, that of scales, the other that of the spring
balance. When you go a journey and your luggage is weighed, it is not
put on scales, but on a spring; the weight depresses the spring a
certain amount, and the result is indicated by a needle on a dial. The
same principle is used in automatic machines for finding your weight.
The spring balance shows weight, but scales show _mass_. So long as
you stay in one part of the world, the difference does not matter;
but if you test two weighing machines of different kinds in a number
of different places, you will find, if they are accurate, that their
results do not always agree. Scales will give the same result anywhere,
but a spring balance will not. That is to say, if you have a lump of
lead weighing ten pounds by the scales, it will also weigh ten pounds
by scales in any other part of the world. But if it weighs ten pounds
by a spring balance in London, it will weigh more at the North Pole,
less at the equator, less high up in an aeroplane, and less at the
bottom of a coal mine, if it is weighed in all those places on the same
spring balance. The fact is that the two instruments measure quite
different quantities. The scales measure what may be called (apart from
refinements which will concern us presently) “quantity of matter.”
There is the same “quantity of matter” in a pound of feathers as in a
pound of lead. Standard “weights,” which are really standard “masses,”
will measure the amount of mass in any substance put into the opposite
scales. But “weight” is a properly due to the earth’s gravitation: It
is the amount of the force by which the earth attracts a body. This
force varies from place to place. In the first place, anywhere outside
the earth the attraction varies inversely as the square of the distance
from the center of the earth; it is therefore less at great heights.
In the second place, when you go down a coal mine, part of the earth
is above you, and attracts matter upwards instead of downwards, so
that the net attraction downwards is less than on the surface of the
earth. In the third place, owing to the rotation of the earth, there is
what is called a “centrifugal force,” which acts against gravitation.
This is greatest at the equator, because there the rotation of the
earth involves the fastest motion; at the poles it does not exist,
because they are on the axis of rotation. For all these reasons, the
force with which a given body is attracted to the earth is measureably
different at different places. It is this force that is measured by a
spring balance; that is why a spring balance gives different results
in different places. In the case of scales, the standard “weights” are
altered just as much as the body to be weighed, so that the result is
the same everywhere; but the result is the “mass,” not the “weight.”
A standard “weight” has the same mass everywhere, but not the same
“weight”; it is in fact a unit of mass, not of weight. For theoretical
purposes, mass, which is almost invariable for a given body, is much
more important than weight, which varies according to circumstances.
Mass may be regarded, to begin with, as “quantity of matter”; we shall
see that this view is not strictly correct, but it will serve as a
starting point for subsequent refinements.
For theoretical purposes, a mass is defined as being determined by the
amount of force required to produce a given acceleration: The more
massive a body is, the greater will be the force required to alter its
velocity by a given amount in a given time. It takes a more powerful
engine to make a long train attain a speed of ten miles an hour at the
end of the first half-minute, than it does to make a short train do so.
Or we may have circumstances where the force is the same for a number
of different bodies; in that case, if we can measure the accelerations
produced in them, we can tell the ratios of their masses: the greater
the mass, the smaller the acceleration. We may take, in illustration
of this method, an example which is important in connection with
relativity. Radio-active bodies emit beta-particles (electrons) with
enormous velocities. We can observe their path by making them travel
through water vapor and form a cloud as they go. We can at the same
time subject them to known electric and magnetic forces, and observe
how much they are bent out of a straight line by these forces. This
makes it possible to compare their masses. It is found that the faster
they travel, the greater is their mass, as measured by the stationary
observer; the increase is greatest as applied to their mass as measured
by the effect of a force in the line of motion. In regard to forces at
right angles to the line of motion, there is a change of mass with
velocity in the same proportion as the changes of length and time. It
is known otherwise that, apart from the effect of motion, all electrons
have the same mass.
All this was known before the theory of relativity was invented, but
it showed that the traditional conception of mass had not quite the
definiteness that had been ascribed to it. Mass used to be regarded as
“quantity of matter,” and supposed to be quite invariable. Now mass was
found to be relative to the observer, like length and time, and to be
altered by motion in exactly the same proportion. However, this could
be remedied. We could take the “proper mass,” the mass as measured by
an observer who shares the motion of the body. This was easily inferred
from the measured mass, by taking the same proportion as in the case of
lengths and times.
But there is a more curious fact, and that is, that after we have
made this correction we still have not obtained a quantity which is
at all times exactly the same for the same body. When a body absorbs
energy—for example, by growing hotter—its “proper mass” increases
slightly. The increase is very slight, since it is measured by
dividing the increase of energy by the square of the velocity of
light. On the other hand, when a body parts with energy it loses mass.
The most notable case of this is that four hydrogen atoms can come
together to make one helium atom, but a helium atom has rather less
than four times the mass of one hydrogen atom.
We have thus two kinds of mass, neither of which quite fulfils the old
ideal. The mass as measured by an observer who is in motion relative
to the body in question is a relative quantity, and has no physical
significance as a property of the body. The “proper mass” is a genuine
property of the body, not dependent upon the observer; but it, also,
is not strictly constant. As we shall see shortly, the notion of mass
becomes absorbed into the notion of energy; it represents, so to speak,
the energy which the body expends internally, as opposed to that which
it displays to the outer world.
Conservation of mass, conservation of momentum, and conservation of
energy were the great principles of classical mechanics. Let us next
consider conservation of momentum.
The momentum of a body in a given direction is its velocity in that
direction multiplied by its mass. Thus a heavy body moving slowly may
have the same momentum as a light body moving fast. When a number of
bodies interact in any way, for instance by collisions, or by mutual
gravitation, so long as no outside influences come in, the total
momentum of all the bodies in any direction remains unchanged. This law
remains true in the theory of relativity. For different observers, the
mass will be different, but so will the velocity; these two differences
neutralize each other, and it turns out that the principle still
remains true.
The momentum of a body is different in different directions. The
ordinary way of measuring it is to take the velocity in a given
direction (as measured by the observer) and multiply it by the mass (as
measured by the observer). Now the velocity in a given direction is
the distance traveled in that direction in unit time. Suppose we take
instead the distance traveled in that direction while the body moves
through unit “interval.” (In ordinary cases, this is only a very slight
change, because, for velocities considerably less than that of light,
interval is very nearly equal to lapse of time.) And suppose that
instead of the mass as measured by the observer we take the proper
mass. These two changes increase the velocity and diminish the mass,
both in, the same proportion. Thus the momentum remains the same, but
the quantities that vary according to the observer have been replaced
by quantities which are fixed independently of the observer—with the
exception of the distance traveled by the body in the given direction.
When we substitute space-time for time, we find that the measured
mass (as opposed to the proper mass) is a quantity of the same kind
as the momentum in a given direction; it might be called the momentum
in the time direction. The measured mass is obtained by multiplying
the invariant mass by the _time_ traversed in traveling through unit
interval; the momentum is obtained by multiplying the same invariant
mass by the _distance_ traversed (in the given direction) in traveling
through unit interval. From a space-time point of view, these naturally
belong together.
Although the measured mass of a body depends upon the way the observer
is moving relatively to the body, it is none the less a very important
quantity. For any given observer, the measured mass of the whole
physical universe is constant.[8] The proper mass of all the bodies
in the world is not necessarily the same at one time as at another,
so that in this respect the measured mass has an advantage. The
conservation of measured mass is the same thing as the conservation of
energy. This may seem surprising, since at first sight mass and energy
are very different things. But it has turned out that energy is the
same thing as measured mass. To explain how this comes about is not
easy; nevertheless we will make the attempt.
[8] This is subject to the explanations given below as regards
conservation of energy.
In popular talk, “mass” and “energy” do not mean at all the same thing.
We associate “mass” with the idea of a fat man in a chair, very slow to
move, while “energy” suggests a thin person full of hustle and “pep.”
Popular talk associates “mass” and “inertia,” but its view of inertia
is one-sided: it includes slowness in beginning to move, but not
slowness in stopping, which is equally involved. All these terms have
technical meanings in physics, which are only more or less analogous
to the meanings of the terms in popular talk. For the present, we are
concerned with the technical meaning of “energy.”
Throughout the latter half of the nineteenth century, a great deal was
made of the “conservation of energy,” or the “persistence of force,”
as Herbert Spencer preferred to call it. This principle was not easy
to state in a simple way, because of the different forms of energy;
but the essential point was that energy is never created or destroyed,
though it can be transformed from one kind into another. The principle
acquired its position through Joule’s discovery of “the mechanical
equivalent of heat,” which showed that there was a constant proportion
between the work required to produce a given amount of heat and the
work required to raise a given weight through a given height: in fact,
the same sort of work could be utilized for either purpose according to
the mechanism. When heat was found to consist in motion of molecules,
it was seen to be natural that it should be analogous to other forms of
energy. Broadly speaking, by the help of a certain amount of theory,
all forms of energy were reduced to two, which were called respectively
“kinetic” and “potential.” These were defined as follows:
The kinetic energy of a particle is half the mass multiplied by the
square of the velocity. The kinetic energy of a number of particles is
the sum of the kinetic energies of the separate particles.
The potential energy is more difficult to define. It represents any
state of strain, which can only be preserved by the application of
force. To take the easiest case: If a weight is lifted to a height and
kept suspended, it has potential energy, because, if left to itself, it
will fall. Its potential energy is equal to the kinetic energy which it
would acquire in falling through the same distance through which it was
lifted. Similarly when a comet goes round the sun in a very eccentric
orbit, it moves much faster when it is near the sun than when it is far
from it, so that its kinetic energy is much greater when it is near the
sun. On the other hand, its potential energy is greatest when it is
farthest from the sun, because it is then like the stone which has been
lifted to a height. The sum of the kinetic and potential energies of
the comet is constant, unless it suffers collisions or loses matter by
forming a tail. We can determine accurately the _change_ of potential
energy in passing from one position to another, but the total amount of
it is to a certain extent arbitrary, since we can fix the zero level
where we like. For example, the potential energy of our stone may be
taken to be the kinetic energy it would acquire in falling to the
surface of the earth, or what it would acquire in falling down a well
to the center of the earth, or any assigned lesser distance. It does
not matter which we take, so long as we stick to our decision. We are
concerned with a profit-and-loss account, which is unaffected by the
amount of the assets with which we start.
Both the kinetic and the potential energies of a given set of bodies
will be different for different observers. In classical dynamics,
the kinetic energy differed according to the state of motion of the
observer, but only by a constant amount; the potential energy did not
differ at all. Consequently, for each observer, the total energy was
constant—assuming always that the observers concerned were moving
in straight lines with uniform velocities, or, if not, were able to
refer their motions to bodies which were so moving. But in relativity
dynamics the matter becomes more complicated. We cannot profitably
adapt the idea of potential energy to the theory of relativity, and
therefore the conservation of energy, in a strict sense, cannot
be maintained. But we obtain a property, closely analogous to
conservation, which applies to kinetic energy alone. As Eddington
puts it: the kinetic energy is not always strictly conserved, and the
classical theory therefore introduces a supplementary quantity, the
potential energy, so that the sum of the two is strictly conserved. The
relativity treatment, on the other hand, discovers another formula,
analogous to the one expressing conservation, which holds always for
the kinetic energy. “The relativity treatment adheres to the physical
quantity and modifies the law; the classical treatment adheres to
the law and modifies the physical quantity.” The new formula, he
continues, may be spoken of “as the law of conservation of energy and
momentum, because, though it is not formally a law of conservation, it
expresses exactly the phenomena which classical mechanics attributes to
conservation.”[9] It is only in this modified and less rigorous sense
that the conservation of energy remains true.
[9] _Mathematical Theory of Relativity_, p. 135.
What is meant by “conservation” in practice is not exactly what it
means in theory. In theory we say that a quantity is conserved when the
amount of it in the world is the same at any one time as at any other.
But in practice we cannot survey the whole world, so we have to mean
something more manageable. We mean that, taking any given region, if
the amount of the quantity in the region has changed, it is because
some of the quantity has passed across the boundary of the region. If
there were no births and deaths, population would be conserved; in that
case the population of a country could only change by emigration or
immigration, that is to say, by passing across the boundaries. We might
be unable to take an accurate census of China or Central Africa, and,
therefore, we might not be able to ascertain the total population of
the world. But we should be justified in assuming it to be constant if,
wherever statistics were possible, the population never changed except
through people crossing the frontiers. In fact, of course, population
is not conserved. A physiologist of my acquaintance once put four mice
into a thermos. Some hours later, when he went to take them out, there
were eleven of them. But mass is not subject to these fluctuations:
the mass of the eleven mice at the end of the time was no greater than
the mass of the four at the beginning.
This brings us back to the problem for the sake of which we have been
discussing energy. We stated that, in relativity theory, measured mass
and energy are regarded as the same thing, and we undertook to explain
why. It is now time to embark upon this explanation. But here, as at
the end of Chapter VI, the totally unmathematical reader will do well
to skip, and begin again at the following paragraph.
Let us take the velocity of light as the unit of velocity; this is
always convenient in relativity theory. Let _m_ be the proper mass of a
particle, _v_ its velocity relative to the observer. Then its measured
mass will be
_m_
——————————
√(1 - _v²_)
while its kinetic energy, according to the usual formula, will be
½ _mv²_
As we saw before, energy only occurs in a profit-and-loss account,
so that we can add any constant quantity to it that we like. We may
therefore take the energy to be
_m_ + ½(_mv²_).
Now if _v_ is a small fraction of the velocity of light,
_m_ + ½ _mv²_
is almost exactly equal to
_m_
—————————
√(1 - _v²_).
Consequently, for velocities such as large bodies have, the energy and
the measured mass turn out to be indistinguishable within the limits of
accuracy attainable. In fact, it is better to alter our definition of
energy, and take it to be
_m_
——————————
√(1 - _v²_),
because this is the quantity for which the law analogous to
conservation holds. And when the velocity is very great, it gives a
better measure of energy than the traditional formula. The traditional
formula must therefore be regarded as an approximation, of which the
new formula gives the exact version. In this way, energy and measured
mass become identified.
I come now to the notion of “action,” which is less familiar to
the general public than energy, but has become more important in
relativity physics, as well as in the theory of quanta.[10] (The
quantum is a small amount of action.) The word “action” is used to
denote energy multiplied by time. That is to say, if there is one unit
of energy in a system, it will exert one unit of action in a second,
100 units of action in 100 seconds, and so on; a system which has
100 units of energy will exert 100 units of action in a second, and
10,000 in 100 seconds, and so on. “Action” is thus, in a loose sense,
a measure of how much has been accomplished: it is increased both by
displaying more energy and by working for a longer time. Since energy
is the same thing as measured mass, we may also take action to be
measured mass multiplied by time. In classical mechanics, the “density”
of matter in any region is the mass divided by the volume; that is
to say, if you know the density in a small region, you discover the
total amount of matter by multiplying the density by the volume of the
small region. In relativity mechanics, we always want to substitute
space-time for space; therefore a “region” must no longer be taken to
be merely a volume, but a volume lasting for a time; a small region
will be a small volume lasting for a small time. It follows that, given
the density, a small region in the new sense contains, not a small mass
merely, but a small mass multiplied by a small time, that is to say, a
small amount of “action.” This explains why it is to be expected that
“action” will prove of fundamental importance in relativity mechanics.
And so in fact it is.
[10] On this subject, see the present author’s _A.B.C. of Atoms_,
chaps. VI and XIII.
All the laws of dynamics have been put together into one principle,
called “The Principle of Least Action.” This states that, in passing
from one state to another, a body chooses a route involving less action
than any slightly different route—again a law of cosmic laziness. The
principle is subject to certain limitations, which have been pointed
out by Eddington,[11] but it remains one of the most comprehensive
ways of stating the purely formal part of mechanics. The fact that
the quantum is a unit of action seems to show that action is also
fundamental in the empirical structure of the world. But at present
there is no bridge connecting the quantum with the theory of relativity.
[11] _Op. cit._ § 60.