CHAPTER 6
THE IMPOSSIBILITY OF ISOLATION
When we are studying a particular
phenomenon in nature, such as the way two objects interact with each other, it
is customary to focus attention on the objects, to the exclusion of everything
else in the universe. We say that we examine their behavior "in
isolation". What I will argue is that isolation is never in practice
achievable. This means that the effect of the rest of the universe must be
considered. If it appears to be neglected it shows up, in disguise, some other
way.
In this chapter I will be looking at the
laws of motion of objects as proposed in the seventeenth century by Sir Isaac
Newton in his famous Principia. My intention is to determine their
information content. Newton's laws
mention the term body by which he means an object, or a material thing. Laws in physics should really be
restricted to dealing only with fundamental objects if we want to label them as
fundamental laws. This is because we believe that, if we know the laws that
describe the behavior of the fundamental objects, we can derive all the other
laws. In Newton's time the idea of a fundamental object had not been formed so
that, in a sense, Newton's laws are not fundamental laws. From our present point of view we can
think of the entire universe as being made up of three fundamental objects, or
particles as we usually call them.
These are electrons, protons, and neutrons. There are other particles but we can neglect them in this
model. The protons and neutrons group together to form the nucleus of an atom
and the electrons move about this nucleus. Niels Bohr devised a model of the atom which was like a
solar system in which the electrons moved in orbits around the nucleus, like
planets around the sun. You often
see drawings of atoms showing several electrons moving in orbits. The number of electrons moving about
the nucleus in an ordinary atom is always the same as the number of protons in
the nucleus.
The simplest atom is hydrogen. It has one proton in the nucleus and
one electron moving about it. I do
not say "moving in orbit" about it because we now know that the Bohr
model of the atom is not true. But we still believe that the electron and
proton are there and are interacting with one another. The electron and the proton are
electric charges. We say
arbitrarily that the electron is a negative charge and the proton is a positive
charge. The magnitude, or size, of
their two charges is equal but opposite so that when they are together in an
atom their total electrical effect nearly cancels out unless something is close
to the atom.
Atoms that are heavier than hydrogen have
more protons in the nucleus and, of course, more electrons moving about. There is always the same number of
electrons as protons in an atom.
In heavier atoms there are always neutrons in the nucleus as well as the
protons. The number of neutrons varies.
The atom with eight protons in the nucleus and eight electrons moving
about this nucleus is called an oxygen atom. Most oxygen atoms that we find have eight neutrons in the
nucleus as well. But a small
percentage (0.2%) have ten neutrons and a smaller percentage (0.04%) have
nine. These different kinds of
oxygen atoms are called isotopes of oxygen.
In heavy atoms there are more neutrons
than protons in the nuclei of isotopes found in nature. The istopes we find around now Ð
the naturally occurring ones Ð are the stable isotopes. The fittest
survive. Whenever we create the
isotope of an atom whose number of neutrons is far from those which occur
naturally, the isotope is unstable and breaks up. We say it is radioactive. We can judge the relative stability of different isotopes of
an atom by their relative abundances in nature.
This has been a bit of a diversion, but
it is necessary to point out that Newton's laws do not deal with electrons,
protons, and neutrons but bodies made up of complex structures of these
fundamental particles. The particles form atoms; the atoms form molecules; and
the molecules form solids. The
bodies Newton was talking about are solids of various shapes. To keep things
somewhat simple we will think of bodies that have a simple shape Ð spherical
like a ball Ð and bodies that are small enough so that we can think of them as
being located all in one place, at a point. We will call these ideal bodies point particles, or just
particles. Then our study of the
laws of motion will be restricted to particles and the way they interact with
one another. We will be looking at what is called particle dynamics. Everything about more complex shapes
can be explained in terms of the interaction of the particles that make up the
complex shapes.
Now I am ready to bring on Newton's laws
of motion. Here they are:
Law I:
Every body perseveres in its state of rest or uniform motion in a straight line unless change in that
state is compelled by impressed forces.
Law II:
Change of motion is proportional to the force impressed and takes place in the
direction of the straight line in which such force is impressed.
Law III:
Reaction is always equal and opposite to action; that is, the mutual actions of
two bodies upon each other are always equal and directly opposite.
In these laws various terms are
introduced: "impressed forces", "change of motion",
"reaction" and "action". Really there are only two new concepts; one is force. "Impressed force" is just a
force; "action" is a force one body exerts on another;
"reaction" is the force the second body exerts on the first. "Change of motion" involves
the other new concept, namely mass.
By motion, Newton meant the product of the mass times the velocity. This product is sometimes called
momentum. The "change in
motion" is the time rate of change of the mass multiplied by the velocity. Since, to Newton, mass was assumed to
be a constant independent of how fast the body went, this "change in
motion" is just the mass multiplied by the time rate of change of velocity
(which is what we call acceleration).
The second law is often written as a formula which is
F = ma
where F is the force acting on a body, m is the mass of the body, and a is the
acceleration of the body produced by the "impressed force" F. This
formula, as it stands, does not tell the fact that the direction of the
acceleration is the same as the direction of the force. This can be indicated by drawing an
arrow over both F and a to indicate that the directions are the same.
F = ma
The third law is sometimes written as
F12 = ÐF21
where what is meant by F12 (with the arrow) is the force exerted by particle
number l on particle number 2 (there are two particles interacting and we label
them l and 2). Then F21 (with the arrow) is the force exerted by particle
number 2 on particle number l.
These two forces are along the same line but point in opposite
directions. The minus sign says
they are in opposite directions.
I am going to look now at the information
content of these laws. Newton tried to define the concepts of mass and force
independently of the laws, but failed.
For example, he said that the mass of a body was its volume multiplied
by its density. He then defined
density as the mass per unit volume, which is a complete circle. His definition of force basically said
it was the cause of acceleration which is said in the laws. So Newton's three laws must provide a
definition of mass and force, concepts that Newton introduced, as well as have
some other content.
I have often argued in an elementary
class of university physics students that the physical content of Newton's
three laws of motion is zero. Here
is the argument: Law I is a special case of Law II since it says that if there
is no force there is no acceleration.
(Remember Law II says that the acceleration is proportional to the
force.) So we can eliminate Law I
since it adds nothing not given in Law II. Laws II and III are required to
define mass and force. If we use
the two-body interaction case, Law II says that:
F12 = m2a2
and
F21 = m1 a1
Then using Law III we get:
m2a2 = Ðm1 a1
I can write this last equation as:
m2/m1 = Ða1/a2
This then defines the ratio of the masses of the two
particles as the inverse of the ratio of their accelerations when they
interact. Force can be defined
by taking this mass and returning
to Law II
F = ma
Since now the mass is known (relative to some
standard of mass), acceleration can be measured and force calculated. And so we have absolutely no physical
content to the laws! Or do we?
Mach, in his Science of Mechanics, analyzes Newton's laws at length and comes to this conclusion:
... its main result will be found to be the
perception, that bodies mutually determine in each other accelerations dependent on definite spatial and material
circumstances and that there are masses. In reality only one great fact was established... Different pairs of bodies determine,
independently of each other, and mutually in themselves pairs of accelerations,
whose terms exhibit a constant ratio, the criterion and characteristic of each
pair. [1]
Mach puts his finger on one piece of physical content
when he says "pairs of accelerations whose terms exhibit a constant ratio,
the criterion and characteristic of each pair". This gives mass, as defined this way, a physical meaning
because the ratio of acceleration is "constant". Constant over what range of
circumstances? One circumstance is
that no matter how far apart the interacting bodies are, the ratio is the
same. For all interactions the
actual size of each acceleration decreases as the bodies get farther apart; but
the ratio stays the same. The
accelerations get smaller in proportion to each other. Another circumstamce is that it does
not matter how the bodies are moving when they interact. The ratio of accelerations does not
depend on the velocities of the bodies.
So the concept of mass is one that is velocity independent since its
value, as measured by comparing it with another body, does not depend on the
velocities of the bodies.
Because the ratio of accelerations is the
same no matter how far apart the
bodies are, we can get more information: that the interaction seems
instantaneous. It does not take
any time at all for the effect of a change in position of body 2 to be felt at
body l, and vice versa. The acceleration of each body changes instantaneously
so as to keep the ratio constant.
So let me summarize the information
content we have so far. When two
bodies interact, each is accelerated and the ratio of the two accelerations is
a constant no matter how far apart they are and what their velocities are at
the time of the interaction (interaction being instantaneous). We use this constant ratio to assign to
each body a mass, a property of the body which is independent of its motion. If
the bodies are particles, the accelerations are along the line joining the
particles and point in opposite directions. This means that particles either
attract each other or repel each other.
(All this last information comes from the fact that the ratio a2/a1 is a constant.) We seem to
be getting more information content now.
Is this all? Newton had a
second concept in addition to mass, namely force, and he had arranged his
definition of force so that the forces of two bodies on each other were equal
and opposite. This appealed to Newton.
Here is a passage from his Rules of Reasoning section of the Principia:
We are certainly not to relinquish the evidence of
experiments for the sake of dreams and vain fictions of our own devising; nor
are we to recede from the analogy of Nature, which is wont to be simple and
always consonant to itself. Nature does nothing in vain, and more is in vain
when less will serve; for Nature
is pleased with simplicity, and affects not the pomp of superfluous causes. [2]
As far as I am concerned the concept of force is
superfluous but Newton's passion for simplicity in Nature overcame his feeling
that "Nature does nothing in vain". If we do allow concepts in addition to what we really need
we must label them clearly as artifacts Ð man-made, not natural. Let us not admire Nature for its
simplicity when it is our own (Newton's) creation.
There is a simplicity here though in that
the accelerations are oppositely directed along a line joining the particles
and have a constant ratio. It is
here "explained" by the fact that the particles have a property
called mass. But is that an
adequate explanation of this simplicity?
As I mentioned earlier in the chapter, the fundamental particles that
interact are the electron, proton, and neutron. The proton and neutron interact only if they are at close
range as they are in the nucleus and the interactions which Newton was talking
about do not in any way involve this close-range nuclear interaction. The electron and the proton interact
with each other at any distance so it is this fundamental interaction that
accounts for many Newtonian interactions.
(I leave gravitational interaction aside here.) The interaction between the electron
and the proton, or between two electrons, or two protons is not correctly
described by Newton's laws. It is
not instantaneous; the accelerations do not have a constant ratio, are not
always oppositely directed along a line and are not independent of the velocity
of the interacting particles. I will
be looking at this interaction in the chapter dealing with electromagnetic interaction but, for
the moment, what about Newton? We know his laws do not apply to the fundamental
interaction; what, beside gravitational interaction, do they apply to? They apply to large-scale objects
pushing and pulling each other in contact with one another. What is more, they are adequate in
describing this behavior so that engineers can build cars, airplanes, and
bridges. But they are not
information about the behavior of the fundamental particles and thus can be
ignored completely as far as additional information content is concerned. This is because we will be able to
explain why Newton's laws hold for large scale objects once we have the proper
laws for fundamental interactions.
We need only worry about accounting for the laws of the fundamental
interactions.
But wait! Not so fast!
There is more content lurking around. Mach includes it when he says that "different bodies determine
independently of each other and mutually in themselves pairs of
accelerations". This fact,
that the interaction between any two bodies is independent of the presence of
other bodies that might be interacting with them, is called the principle of
superposition. The principle of
superposition is a law which says that the effects of a number of different
bodies are just superimposed one on top of the other. They do not interfere with the interactions of one
another. The final acceleration
that any one body experiences is the resultant of the individual accelerations
produced by all other bodies present, each acceleration being computed just as
if there were no other bodies there.
The principle of superposition does
contain information. It is a basic
part of the theory of the interaction between fundamental particles, whereas
Newton's laws are not. The principle of superposition was not mentioned
directly by Newton; what he did speak about is how the forces are added if more
than one force acts on a body.
Force has magnitude and direction (as does acceleration) and thus two
forces pointing in different directions are not just added like 2 and 2 to make 4. In fact, if equal forces are in
opposite directions, they add to zero.
If they are at right angles, we can represent them as two sides of a
rectangle and the resultant is the diagonal of the rectangle starting from the
point of application of the forces.
In general any two forces define a parallelogram of forces. Many of you will know all about how to
compute the resultant of quantities like forces or acceleration that have
magnitude and direction. Such
quantities are called vector quantities.
We talk about adding vectors.
All this brings me to the most exciting
part, the law we threw away so casually Ð Law I. The first law has the most fundamental piece of information of
all. Here is the law again:
Law I:
Every body perseveres in its state of rest or uniform motion in a
straight line unless change in that state is compelled by impressed forces.
We know that impressed forces come from other bodies
so that we can say that a body will continue to be at rest or move in a
straight line at a uniform speed if it is isolated from other bodies which
might exert forces on it (cause
accelerations). The body must be
alone in space. Mach calls our
attention to a little problem here:
When we say that a body K alters its direction and
velocity solely through the influence of another body K', we have asserted a conception that it is
impossible to come at unless other bodies A,B,C... are present with reference
to which the motion of the body K has been estimated... If now we suddenly
neglect A,B,C... and attempt to speak of the deportment of body K in absolute
space, we implicate ourselves in a twofold error. In the first place, we cannot know how K would act in the
absence of A,B,C...; and in the second place every means would be wanting of
forming a judgment of the behavior of K... [3]
If the body were alone in space we could not tell
whether or not its motion were uniform, or accelerated, or if it were standing
still. We need other bodies to provide a background against which motion can be
measured. We sometimes say that
these other background bodies are a frame of reference for making measurements
of position. What Mach says is
that our error is twofold in neglecting reference bodies. Beside making
measurement impossible we are presuming that the body K under observation would
behave the same way with and without the reference bodies. When we make measurements of the motion
of a system of bodies that we claim are isolated, they are not isolated at all;
isolation is an impossibility. We
have the rest of the universe present and we can never know how a body would
behave if the rest of the universe were not present.
When there are no other bodies near a
body that we intend to observe, Newton says that the body will be in either of
two states: resting or moving at constant speed in a straight line. Either of these two states is a natural
state for a body. We can
reasonably assume that these two states are equivalent as far as the body is
concerned. The information content
of this is that the effect of the rest of the universe on a body is identical
in these two states. Suppose that we have a frame of reference and in that
frame a body is moving at a constant speed. Now imagine that we are on the body
and have another frame of reference moving with the body. We would say that the
universe was equivalent to us at rest in the moving frame of reference to what
it is to a body at rest in the original frame.
A frame of reference in which a body,
uninfluenced by any nearby bodies, is at rest (or moving uniformly) is called
an inertial frame of reference.
Any frame of reference moving at constant velocity relative to an
inertial frame is also an inertial frame.
In an inertial frame a body
can maintain a resting position.
All this is well accepted material. Newton knew about it. He wrote down what has become called
his principle of relativity. Here
it is:
The motions of bodies included in a given space are
the same among themselves whether that space is at rest or moves uniformly forward
in a straight line. [4]
The space referred to here is an inertial space. I will call it an inertial
environment. I believe, like Mach,
that an inertial environment is more than a background against which
measurements can be made. Bridgman
says this:
We do not have a simple event A causally connected
with a simple event B, but the whole background of the system in which the
events occur is included in the concept, and is a vital part of it. [5]
The theory of special relativity is based on the
Einstein principle of relativity: that the laws of physics are invariant (the
same) from one inertial frame of reference to another. This is usually taken to indicate that
inertial frames are equivalent but what does that mean? There must be some effect on the
particle in an inertial environment that is very definite and different from a
particle "isolated" in an accelerated frame. When a particle is all by itself in an
accelerated (non-inertial) frame it will accelerate just as if there were some
other particle causing it to accelerate.
But there is no other particle.
Its environment at rest in this accelerated frame must be different in
the same way as it is different when another body acts on it in an inertial
frame, because in both cases it accelerates.
If we presume that the particle
accelerating in the non-inertial frame is in its natural state we would assume
that it had achieved an inertial environment. Newton's First Law says that the inertial environment is the
natural state of the particle. Is
it not reasonable to assume that a particle will always move in such a way as
to achieve an inertial
environment? This would
mean that in an inertial frame the combination of the environment produced by a
body that is influencing our observed body and the environment experienced by
it accelerating in the inertial frame superimpose to make an inertial
environment. The particle
accelerating in this way would be in its natural state.
Newton's laws could then be summed up by
saying that bodies behave in such a way as to maintain an inertial environment
for themselves. In this point of
view the rest of the universe is not inert or benign. How could it be if a
particle in a frame accelerating with respect to the universe cannot remain at
rest without holding it at rest?
But normally we just ignore the universe and pretend the particle is
isolated. One of the reasons for
neglecting the effect of the universe is that most people believe in the idea
of laws as part of a grand design; and the law of the equivalence of inertial frames
seems eminently suitable.
Sir Fred Hoyle says this in an article on
"The Future of Physics and Astronomy":
There is also a second reason for the astronomer not
to remain idle: the universe, in the large, may be relevant to physics. The current and conventional point of
view is that, while the universe may set boundary conditions for the operation
of the physical laws, the laws themselves are independent of large-scale
structure and could, in principle, be determined by entirely local
experiments. The opposite,
unorthodox point of view argues that the physical laws as we discover them in
the laboratory already involve the influence of the universe as a whole...
There are two clues indicating that the unorthodox, nonlocal point of view may
be correct. The first is that by
taking account of an influence of the universe it is possible to avoid the
assumption that the local laws of physics are lopsided with respect to time...
The second clue comes in a somewhat roundabout way, from considering particle
masses to arise from interactions with other particles. [6]
The point that Hoyle makes regarding "the local
laws of physics being lopsided with respect to time" makes reference to
the fact that when we have large numbers of particles (atoms) interacting with
each other as we have in gas contained in a box, there is an irreversible
nature to their behavior. They always move from a more orderly to a more disorderly
state, never the reverse. Bridgman
speaks about this too:
What prevents the following out through all future
time of a definite sequence is the walls [of the box], the atoms of which are
supposed to be in such a complex state of motion because they are in connection
with the entire outer universe and to a certain extent reflect its
complexities, that no resultant regularities are to be expected in the motion
which the atoms of the wall impress on the atoms of the gas. [7]
As his second clue Hoyle believes that the masses of
the fundamental particles may also be connected with their interactions with
the other particles in the universe.
I am trying to convince you that an inertial environment is not empty. It is the natural environment of
particles. If a particle moves in
a straight line at a constant speed relative to an inertial environment what it
perceives is an inertial environment.
This is a property of whatever produces the inertial environment, which
I say is the universe. It
indicates that the effect of universe is the same when we move in different
directions. There is no special
frame that we can say is really "the" frame or "absolute
space" as Newton called it.
Many frames are equivalent.
This is a fact about the universe, not a law governing the
universe. Heisenberg hints at this
active rather than passive nature of the inertial environment (space) here:
From our modern point of view we would say that the
empty space between the atoms in the philosophy of Democritus [the void] was
not nothing; it was the carrier for geometry and kinematics, making possible
the various arrangements and movements of atoms. [8]
By examining Newton's laws of motion I
have come to the conclusion that the behavior of particles of matter is
determined by their environment. Their natural state is to be in an inertial
environment. In an inertial frame, that is, one in which a particle at rest
experiences an inertial environment, a particle moving at constant speed in a
straight line is also in a natural state. This means that a frame of reference
moving at constant velocity relative to an inertial frame is also an inertial
frame. In it the moving particle of the original frame could be at rest. This
leads to the principle of relativity which Newton stated and which was the
basis of Einstein's theory of special relativity: that inertial frames were all
equivalent to each other as environments for the interactions between any two
bodies. As well, there is no one inertial frame that has superior status to the
others. This is stated by saying that there is no such thing as absolute space.
There is an infinite number of equivalent inertial spaces. The environment in
these spaces is produced, I believe, by all the matter in the universe. As
Heisenberg says it is "not nothing", as space is often perceived to
be, but it is "the carrier for geometry [measurements of distance and
direction] and kinematics [measurements of motion which include distance,
direction, and time]". Heisenberg goes on to say that "this [space] makes
possible the various arrangements and movements of atoms" meaning that,
without the environment, the atoms would not be what they are. So the
examination of the information content in Newton's laws leads me to focus
attention on "empty space". The main point of this chapter is to get
you interested in things that might be overlooked. I have maintained that the
idea of an isolated body is never achievable so can be of no interest to
us. But the environment produced
by the universe should be thought about very carefully.
Mario Bunge in his book on Causality
and Modern Science has quite a bit
to say about the assumption of isolation:
The isolation of a system from its surroundings, of a
thing or process from its context ... are indispensible not only for the
applicability of causal ideas but for any research, whether empirical or
theoretical... Analysis is the sole known method of attaining a rational
understanding of the whole: first it is decomposed into artificially isolated
elements, then an attempt is made to synthesize the components. The best grasp
of reality is not obtained by respecting fiction but by vexing fact and
controlling fiction ... perfect isolation is a theoretical fiction. [9]
Bunge points out that the uncertainties in quantum
mechanics have been considered by some thinkers to be "a result of
external perturbations, that is, as a consequence of imperfect isolation".
But Bunge adds "However, inertial motion goes of itself in complete
isolation and in the absence of causes". I, following Mach, maintain that
an inertial environment is not one
of "completeisolation".
SUMMARY
1. The
environment of any body is produced by the other bodies in the universe. (Mach)
2. The
natural state of a body is for it to be in an inertial environment.
3. A
body will move relative to a frame of reference in such a way as to achieve an
inertial environment.
4. In
an accelerating (non-inertial) frame of reference a body, which does not have
other bodies nearby, will accelerate. It thereby achieves an inertial
environment. Acceleration in any frame changes the environment experienced by
the body.
5. If
a body is in an otherwise inertial environment (in an inertial frame of
reference) but is influenced by another nearby body, it will accelerate either
towards or away from that body. Its total environment will thereby become an
inertial environment.
6. Moving
uniformly in an inertial frame does not change the environment of a body from
what it is when at rest. This means that the effect of the remainder of the
universe is equivalent in the two states.
7. Because there are many equivalent inertial frames, learning about the behavior of objects in one inertial frame gives us information that is useful in many such frames. The information has generality.
Copyright © 1983 J.N.P. Hume All rights in this book reserved