21.1: Conceptual Distinctions
It has been pointed out by others that the word 'mass' has three
distinct meanings. Arthur Zinzen, for instance, discussing these in his
Praktische Naturphilosophie (1953, Westkulturverlag), quotes G. Hamel,
who had previously distinguished between these meanings in 1927 (Hand-
buch der Physik, Vol. 5, Julius Springer). The three meanings are named
by him as follows:
(1) Inert mass
(2) Attracted (weight) mass
(3) Attracting mass
The first two of these meanings have also been discussed by Einstein
in his The Meaning of Relativity. He used the respective words 'inert' and
'gravitational' mass. I shall use the same ones here, but Einstein had no
need to discuss the third meaning and so he gave no name to it. I shall
therefore use Hamel's name 'attracting mass'.
Before its significance is discussed it has to be made clear why mass in
its first two meanings is important in general relativity. For this purpose it
is best to use Einstein's own, very precise, words. On page 55 of The Meaning of Relativity, Sixth Edition, he said:
'The ratio of the masses of two bodies is defined in mechanics in two
ways which differ from each other fundamentally; in the first place, as the
reciprocal ratio of the accelerations which the same motive force imparts to
them (inert mass), and in the second place, as the ratio of the forces which
act upon them in the same gravitational field (gravitational mass).'
In this passage the inert mass of a body is said to be the property the
value of which determines the acceleration of the body under the influence
of a given applied force. The gravitational mass, on the other hand, is
said to be the property of the same body the value of which determines its
weight in a gravitational field. The inert mass of a billiard ball, Einstein
would have said, causes the ball to receive a finite acceleration when
propelled by a billiard cue over a smooth, level surface. In the absence of
its inert mass the acceleration would be infinite. The gravitational mass
causes the same ball to deflect a spring balance. In the absence of
gravitational mass the ball would not be attracted by the earth. The
distinction between these two kinds of mass is made even more precise
in a later passage:
'Newton's equation of motion in a gravitational field, written out in
(Inert mass). (Accelerafion) = (lntensity of the gravitational field).
The left hand side of the above equation defines the performance of a
billiard ball when a force, F, is applied to it by a billiard cue. If the inert
mass is m1 and the acceleration f1, one can write:
F = m1f ...... (21a)
The right hand side defines the weight of the ball on a spring balance. Let
this weight be W and the gravitational mass mg The intensity of the
gravitational field near the earth is g, so the algebraic expression for the
right hand side is
W = gmg ...... (21b)
Equations (21a) and (21b) are found to give numerically identical results
for the same ball when f = g. From this Einstein inferred that mg = mi. He
spoke of 'the law of the equality between the inert and the gravitational
mass' and made it clear that 'the numerical equality is reduced to an
equality of the real nature of the two concepts'.
General relativity is based on this identity, so that Einstein could say:
'The possibility of explaining the numerical equality of inertia and
gravitation by the unity of their nature gives to the general theory of
relativity, according to my conviction, such superiority over the conceptions of classical mechanics, that all the difficulties encountered in development must be considered as small in comparison with this progress.'
This is very clear and, I think, irrefutable. We observe that the ratio
mg / mi is always a constant and ask why. Einstein's answer is that, if general relativity is accepted, they are one and the same thing.
Zinzen, in the book referred to above, speaks, nevertheless, of 'a really
great confusion of thought' about the nature of mass, and I have been led
to the reluctant conclusion that he is right. For I have found very little
appreciation of the distinction between attracting mass and the other two
kinds. If the distinction is hardly ever mentioned, it might be because it is
too obvious to need mention, and I thought at one time that it was so. In
an earlier draft of this book I therefore took the distinction for granted,
But comments from sundry authorities who saw the draft convinced me
that the true reason why the distinction is not mentioned is that it is not
even thought about. I was told, for instance, that the word 'mass' can never;
have any other but the two meanings that figure in the above quotations
from Einstein; that attracting mass and gravitational mass are known
by everyone to be identical both numerically and in their nature; that this
identity is proved by Newton's third law of motion, according to which
action and reaction are equal and opposite; that general relativity does not
rest on the identity of the first two kinds of mass only, but on that of all
three kinds; that conceptual distinctions are hair-splitting and fruitless.
All these statements are, I think, erroneous.
If the distinction between the three kinds of mass and its importance
are clear to many, they are evidently not as universally known as they need
to be. I have therefore no choice but to draw attention to some rather
elementary facts about mass and gravitation before I can proceed further
with the present inquiry. One of those unfortunate situations arises here
that do occasionally arise in science when an obstacle to research and
discussion is not raised by the difficulty of the subject but by an all-too-prevalent notion that the subject does not present any problem at all. Here
the problem to be discussed is why the ratio of attracting mass to the other
two kinds is observed to have a constant value. One cannot begin to answer
this question until one has achieved an understanding of the nature of
attracting mass, and in view of the widespread confusion that I have
encountered about it I make no apology for presenting some elementary
Equation (21 a) represents the performance of a billiard ball in one kind
of circumstance, namely, when it is being propelled by a cue. Equation
(21b) represents its performance in another kind of circumstance, namely,
when it is being weighed on a spring balance. In the first instance the
performance is accelerated motion, in the second, deflection of the spring.
Einstein did not need to ask questions about the circumstances. He did
not ask, for instance, how a push came to be given to the billiard cue. The
question would be as irrelevant to his theme as the names of the players.
Nor did he ask what was the source of the gravitational field in which the
ball is weighed. It sufficed for his purpose to note that there was a field. He
needed to speak of its intensity only, not of its source or cause. But it is
now necessary for us to turn our attention to the source.
The intensity of the gravitational field has the letter symbol g. If M is
the mass of the earth, one can write
g = GM / x2
where G is the gravitational constant and x the distance between the centres
of gravity of the earth and the ball. The question now is what suffix to give
to M. The mass is the property of the earth by virtue of which it attracts
the billiard ball. It is what Hamel called the 'attracting mass'. So I shall
use suffix a. For the case when f = g Newton's law can then be expressed
algebraically in such a way that all the three kinds of mass are distinguished
by their appropriate suffixes as follows:
F = mi g = W = ( G Ma / x2 ) mg ...... (21c)
21.2: The Behaviour of Masses in Newtonian Mechanics
Equation (21c) expresses the behaviour of masses in Newtonian
mechanics when appropriate suffixes are used to distinguish between the
three meanings of mass. Translated into words the equation yields three
important statements in each of which the word mass has a different one of
its three meanings. They are:
(1) The velocity of a body having inert mass is changed only when a
force is applied to it. The relation between the force and the acceleration
is as given by equation (21a), which shows that, for a given acceleration, the
force is proportional to the inert mass, mi of the body.
(2) A body having gravitational mass experiences a force when it is in a
gravitational field. The force is proportional to the gravitational mass of
the body, mg , and the intensity of the field, as shown in equation (21b).
(3) A body having attracting mass is the origin of a gravitational field,
the intensity of the field being proportional to the attracting mass, ma.
Be it remarked in passing that Newton did not use the word 'field', but
this word is today a suitable one for making statements in Newtonian
Here is a definition of inertia: The property of a body by virtue of which
it resists a change in its velocity. Hence statement (1) is true by definition.
The measure used to express the magnitude of the inertia is, in Newtonian
mechanics, the force needed to procure unit acceleration. The formula
(21a) can thus be used as a means of defining this force. It expresses the
definition in quantitative terms. Statement (1) is a means of recognizing
inertia when one meets it and does not do more than define inert mass.
To make the first statement is but a way of saying that inert mass is inert
mass. Like many useful statements in physics it is a tautology.
If one defines gravitational mass as the property by virtue of which a
body experiences a force in a gravitational field, the second statement is
also a tautology. It is true by definition of gravitational mass. But it is not
a tautology to say that inert and gravitational mass are always equal. In
Newtonian mechanics this statement appears as a necessary clause in the
Cosmic Statute Book. It was Einstein who showed that the equality need
not appear there as it can be inferred from the more basic principle of
general relativity. This was a great step towards the unification of physics.
The third statement is also a tautology in so far as it serves to define
attracting mass. Again, what is not a tautology is the statement that there
is a constant ratio between the attracting mass of a body and its other two
masses. This is the fact that can at present only be gleaned from observation.
It is not implicit in the definitions of any of the three kinds of mass. The
definition of inert mass tells us that a body possessing inert mass will be
accelerated if it is pushed or pulled by something. The definition does not
tell us that the mass will itself do any pushing or pulling. The definition
of gravitational mass tells us how a body possessing this kind of mass is
influenced by its environment. The definition does not tell us how a body
having gravitational mass influences its environment.
The distinction can be expressed in a different way. Inert and gravitational mass can be measured only by observing the body that possesses
them and in order to make the measurement something has to be done
to this body. It has to be pushed or pulled in order to measure the
inert mass and it has to be placed in a gravitational field in order to
measure the gravitational mass. Attracting mass on the other hand can
be measured only by observing its effect on a different body and in order
to make the measurement something has to be done to this other
body. Thus the inert mass of a train could be measured by doing
something to the train itself, namely by accelerating it and measuring the
tension in the draw-bar. The gravitational mass of the train could be
measured by placing it on a weighing device. But one could not measure
the attracting mass of the train by any observation on the train itself. One
could, however, measure this kind of mass, in theory though not in
practice, by bringing a different body, a pendulum for instance, near the
train and observing the effect of the train's attracting mass on the deflection
of the pendulum. The gravitational mass is measured by the effect of the
earth on the train, the attracting mass by the effect of the train on some
other gravitational mass. (This other gravitational mass might be the
earth, as will be shown later).
One could express the difference, if perhaps a little loosely, by saying
that inert and gravitational mass are properties of the body while attracting
mass is a capacity for affecting other bodies. Or one could say alternatively
that inert and gravitational mass are observed while attracting mass can
only be inferred. But how best to express the difference is not as important
as to remember that there is a difference. Use of the one word 'mass' for
things that are conceptually so distinct has had some unfortunate results.
21.3: Mutuality in Gravitation
The notion that the equality of action and reaction proves the identity
of gravitational and attracting mass is based on a misapprehension about
elementary mechanics. But as it seems to be rather prevalent it ought not
to be ignored.
Let us return to equation (21c). It gives a numerically correct result; so
it contains everything needed for weighing the billiard ball. But it fails to
embody a statement that is to be found in elementary textbooks. This is
that the billiard ball attracts the earth at the same time as the earth attracts
the billiard ball. This statement attributes an attracting mass, ma to the ball,
which does not appear in the equation. We must find out where and how to
Mutuality implies that the ball attracts other objects with a faint
gravitational field. So we can represent the field intensity of the ball as gm
We then have to write gM for the field intensity due to the earth. The
intensity, gm is of course very small indeed and can be neglected for
practical purposes. But when the ball is replaced by the moon it is no
longer so. The moon does not have an orbit around the centre of the earth
but around the common centre of gravity of both bodies, and this is
because the moon has a significant field of its own. In computing their
relative movement one gives an acceleration, gM to the moon toward
the earth and another acceleration, gm of the earth towards the moon
The case of the billiard ball is similar. When the billiard ball is free to
move, it does so in the earth's field, gM, while the earth moves towards the
ball in the field of the latter, gm. If one wants to be pedantically accurate
one must therefore define the relative motion of the billiard ball and the
earth as determined by
g = gM + gm ...... (21d)
In this equation
gM = k Ma / x2
gm = k ma / x2
The pull of the earth on the ball is gM,mg , while the pull of the ball on the earth is gm,Mg. The very small field intensity due to the ball is compensated for by the large attracted mass. One can say that the heavy earth is being weighed in the faint field due to the ball at the same time when the
light ball is being weighed in the strong field due to the earth. In these
circumstances both show the same weight and contribute equally to the
deflection of the spring. The ball presses downwards on the top end of the
spring and the earth presses upwards on the bottom end. In producing a
compression the forces are additive; hence the plus sign in equation (21d).
Let the pull of the earth on the billiard ball be Fe and the pull of the
billiard ball on the earth Fb. One can then write
Fe = ( k Ma / x2 ) mg
Fb = ( k ma2 ) Mg
These two forces are equal and opposite. Action and reaction are also
equal and opposite. Therefore, it has been argued to my surprise, these
forces are action and reaction. But are we, according to this argument, to
regard Fe as action and Fb as reaction, or vice versa? The difficulty of
saying which is which ought to serve as a warning against so careless a conclusion.
The true situation can be understood with the help of the diagram in
Fig. 4. Fe is shown as acting on the top of the spring and has its own equal
and opposite reaction. The same holds for Fb, which is shown as acting on
the bottom of the spring. Each force could equallv well be shown in anv
other place between the ball and the earth. It is only for convenience of
presentation that they are shown separated. The important fact is that if
either Fe, or Fb disappeared, the other force would still be there with its own reaction.
The reader may have difficulty in believing that the very weak field of
the billiard ball has any effect at all on the spring. It will help him if he
imagines that mass is being continually transferred from below the spring
balance to the tray where the billiard ball rests. If this goes on until half
the mass of the earth is on the tray and half left below the spring balance
Fig. 4. Action and Reaction produced by gravitation
the situation will be reached when half the earth is weighed in the field of
the other half. A gramme mass will then have half a gramme weight. If
the process continues until no more than a billiard ball is left beneath the
spring balance practically the whole of the earth will be weighed in the
field of this billiard ball. In this field a gramme mass will weigh very little.
The compression of the spring will be the same as before the transfer of
substance was begun.
From the above considerations it follows that the pedantically precise
algebraic expression for the weight of the billiard ball is
W = ( 1/2 G Ma / x2 ) mg + ( 1/2 Gma / x2 ) Mg ...... (21e)
from which it appears that k , above, equals 1/2G.
Equations (21e) and (21c) are numerically equal, but this is only
because Ma mg = ma Mg. One cannot infer from this equality that attracting
and gravitational mass are of the same nature.
Their numerical equality, it must also be noted, has been obtained by a
suitable choice of G. If one had arbitrarily chosen G as unity, one would
say that one unit of gravitational mass is always equal to G units of attracting mass.
The value of G has been determined by experiments with more or less
massive spheres by Cavendish, Poynting, Boys and others. These experiments show that the spheres have attracting mass as well as inert and gravitational mass. But they do not prove that less massive bodies also have
attracting mass. It is here that equation (21e) is valuable. If one were
weigh a body that had gravitational mass only and no attracting mass,
would have half the observed weight. But the smallest objects that give
observable deflection on a spring balance have the weight that one would
predict on the assumption that they have attracting mass. One may therefore safely conclude from observation and experiment that the law of
constant proportionality between inert, gravitational and attracting mass
holds for very small accumulations of matter.
It is, nevertheless, important to bear in mind that this mutuality in
gravitation is based on observation and experiment and not derived from
any more fundamental principle, at least in Newtonian mechanics. If
one were to find a body that had no attracting mass, it would not be weightless. Action and reaction would apply to its pressure on a spring
balance. Every other principle of which I am aware would also be preserved. I can think of no way of proving the impossibility of a body that.
has no attracting mass having gravitational mass except by the inductive
kind of reasoning that takes the line: This has never been observed; therefore it cannot happen.
Before I leave the discussion of the three kinds of mass in terms of
Newtonian mechanics I should like to point out that one cannot discover
whether a body does or does not possess attracting mass by observing
it while it falls freely. One can do so only by weighing it. For the equation
of the free fall is
gm1 = ( GMa / X2 ) mg ...... . (21f)
The attracting mass, mag = gM + gm to a minute extent. A ball that had no attracting
mass would fall with practically the same acceleration as an actual one
One could only observe the lack of attracting mass in a falling body if the
ratio gm / gM were large enough to be significant.
It should be remembered that it was from equation (2If) that Einstein
developed general relativity. This equation established for him the identity
in nature as well as numerically, of the inert and gravitational mass of
mass m, namely mi and mg. And this equation gives no information about
the attracting mass Ma. It is therefore impossible to reach any conclusion about the attracting mass of m so long as one uses equation (21f).
The question arises what part, if any, attracting mass plays in general
relativity. Only experts can give the answer, but I should like to say that I
have been able to find but little discussion of this question. This may,
however, be due to my meagre reading. General relativity does explain
quite clearly why a gravitational mass is accelerated when it finds itself in a
gravitational field, but, so far as I can make out, it does not account for
(and does not need to account for) the field. What is basic in general
relativity seems to me to be equally true whatever the object may be
that is the source and cause of the field, by whatever process the field is
Some relativists to whom I have spoken about this have taken it for
granted that in all relativity equations the symbol that represents gravitational mass automatically represents attracting mass as well. It is an
assumption that calls for great caution. I am sure that the basic conclusion
of general relativity does not depend on it. I do not know whether any
subsidiary conclusions do. If so, I venture to suggest that they ought to be
That general relativity does not depend basically on the identity of
attracting and gravitational mass seems to me not only to follow from the
equations but also to be apparent when one translates into the language of
general relativity the three statements about mass that have been expressed
above in the language of Newtonian mechanics. This will be done in a
moment, but it is necessary to lead up to the translation by first explaining
the nature of the gravitational field as it appears in general relativity.
21.4: Gravitation as Interpreted in Relativity Theory
In Newtonian mechanics a gravitational field appears as a region in
which force acts at a distance from its source. Action at a distance had
always been regarded as an unsatisfactory concept. One thinks of a force as
applied by physical contact between something and ponderable matter,
such as when a thing is pushed by a stick or pulled by a string. The notion
could never be easily accepted that a force could be exerted at a distance
from any physical object. Yet this is what seemed to happen to a stone
that was attracted by the earth. To make this notion a little less unsatisfactory an hypothetical ether was postulated. It was assumed that a
gravitational mass produced some kind of effect, called a strain, in the
ether and that the force was applied by physical contact between the stone
and this strained ether. To give a more concrete meaning to this assumption the strained ether was later said to contain tubes of force that extended
between the attracting earth and the attracted stone. Thus every massive
body was believed to have a twofold environment, to be surrounded
at the same time by a featureless, undifferentiated space and by a featured
It has been said that Einstein abolished the notion of a featured ether.
One might equally well say that he abolished the notion of a featureless
space. He retained the concept of a featured environment with physical
properties but showed that there was no need to assume an additional
featureless environment. For this single environment with physical properties he wisely retained the name space. But what he did abolish was the
concept of space as the container of the material universe. Einstein's
featured space is a constituent of it.
One of Einstein's great achievements was to define precisely the physical
condition of a gravitational field. It had been previously vaguely thought of
as a strain. He showed that this condition can be described as a departure
from Euclidean geometry. Space that shows such a departure is technically
called curved. Einstein further proved that the kind of motion that would
appear to be at constant velocity in flat space would appear to be accelerated motion in curved space. If a stone moved out of nearly flat space into a
gravitational field, as would happen if it fell from a very great height, its
velocity would change, but that would not mean that it was subjected to a
force. In other words, one must not think of the acceleration of a falling
stone as the consequence of what is done to the stone but as the consequence of where it is, i.e. the acceleration is a function of the local
The new outlook made it necessary to define inertia more precisely
than had been necessary with Newtonian mechanics. In relativity language
inertia is the property by virtue of which the acceleration of a body that is
free of all restraint depends on the curvature of the space in which it finds
itself. If the space has zero curvature, the acceleration is zero. In this respect
relativity is not a denial, but an extension, of Newtonian mechanics.
One can conceive a line in four-dimensional space-time that a freely
moving body would trace when not subjected to a force. According to
Newtonian mechanics it would always be a straight line and represent a
constant velocity. According to relativity theory it would only be a straight
line when the space was flat. When the space was curved, though the body
was not subjected to a force, the line would be curved and represent an
acceleration. Thus to the question: How can a falling stone be subjected
to a force without physical contact with ponderable matter? Einstein's
answer was as beautifully simple as it was unexpected. He said that the
falling stone is not subjected to a force. A force is no more needed to maintain an acceleration in curved space than to maintain constant velocity in
flat space. On the contrary a force would be needed to prevent acceleration
in curved space. A stone is subjected to such a restraining force when it is
at rest on a shelf, the force being exerted by physical contact with ponderable matter, namely the shelf. In flat space an observer who is being
accelerated experiences a force. In curved space it would be an observer
who was not being accelerated in the direction and at the rate defined by the
curvature who would experience a force.
21.5: The Behaviour of Masses in Relativity Theory
Now let the three statements about mass that were formulated above in
the language of Newtonian mechanics be translated into the language
(1) The acceleration of a body having inert mass and not subjected
to any force depends on the geometry of the space in which the body is.
The acceleration is zero when the space is flat and has a finite value when
the space is curved.' It would be idle to discuss whether this is a tautology.
Whether it is or not it is a definition of inert mass in the language of
relativity just as the equivalent statement given earlier here is a definition
formulated in the language of Newtonian mechanics.
(2) 'A body that is free of restraint is accelerated when it is in the curved
space that constitutes a gravitational field.' This is but to mention a special
case of (1).
(3) 'A body having attracting mass is itself the cause of a curvature in
space, the value of the curvature being proportional to the value of the
mass.' This is no tautology. It is not implicit in the definition of inertia.
There does not appear to be any reason why a body that experiences an
acceleration when the space around it is curved should itself have any
influence on the curvature of the space. No one has been able to show that
the one fact is a logical consequence of the other. Though relativity has
explained much, it has not explained the apparent invariable association of inert and attracting mass. One is still left with the question:
Why has one never observed an inert mass that leaves the space around
It has become clear that both Newtonian mechanics and general
relativity leave us with the same puzzle about attracting mass. Observationally, this and the other two kinds of mass, inert and gravitational, are always
coupled. But no reason has been given why it should be so. Why, one is led
to ask, does one never observe a body that has inert and gravitational
mass but has no attracting mass? Why, in relativity language, does
one never observe a body with inert mass that leaves the space around it
I shall suggest a possible answer later on, but much ground remains
to be covered before this can be done.
21.6: Two Observers or One?
After this book had been sent to the printer I came by accident upon
two papers published in the Physikalische Zeitschrift of 1921. They are by
one E. Reichenbacher. In these papers much is said that has been said in
the present chapter. It is said very clearly and cogently, if in a different
way. Reichenbacher urges many sound arguments against identifying the
gravitational with what he calls, rather unfortunately, the inert field. At
the same time he shows complete understanding of the reasoning that led
Einstein to identify inert with gravitational mass. Reichenbacher distinguishes between passive and active gravitational mass. This is the distinction that has been made in the present chapter between gravitational
and attracting mass.
It is strange indeed that this distinction has not received greater
prominence in spite of the fact that attention was drawn to it with some
insistence in the very early days of general relativity. In these two articles
Reichenbacher is revealed as a clear and powerful thinker. To understand
why his contribution was so ineffective one must recall the atmosphere in
the early nineteen-twenties.
Concerning general relativity theory scientists found themselves in
two opposing camps. Those in one camp rejected the whole of the theory,
not without emotion. Those who did not express themselves violently on
the subject did so peevishly. Scientists in the other camp supported
relativity theory equally vehemently. Aware of the immense clarification
and unification that it had achieved they saw themselves as crusaders for
a new revelation. Loyalty to the cause, loyalty to Einstein impelled them.
Those who did not understand relativity theory in the least rejected the
whole of it. Those who did understand it, more or less, accepted it in its
entirety, often uncritically. As defenders of the Faith they could tolerate
no doubts. They reacted as we all do at times against the nagging suspicion
that something needed further clarification.
In such an atmosphere Reichenbacher could hardly escape being
unpopular with both camps. He understood general relativity so well that
he could venture to explore its limitations. For him doubts about the
identity of the inertial and the gravitational field were not nagging. He
faced them squarely. Doing so he came to realize that the gravitational
field can be described as a distortion of space such as to give it a non-Euclidean geometry and that this distortion is independent of an observer
in a way that causes the gravitational field to differ from the 'field' that he
saw as the result of accelerated motion and that he called one of acceleration. That Reichenbacher's analysis led him to a clearer explanation of
centrifugal force in relativistic terms than has been provided by other
relativists makes it all the more regrettable that his work should have
received so little notice. Relativists of his time would seem to have disliked
the awkward questions that he propounded all too inescapably. Unable
to refute him thev ignored him.
I am bold enough to suggest that this reveals one of the less healthy
aspects of the science of relativity. Today opportunities for sanitation are
rarer than they were in the nineteen twenties. In those years many of the
best intellects trained in physics devoted a great deal of their time and
thought to relativity theory. Today experts in this field are but few. Special
relativity occupies no more than a minute fraction of the physics syllabus
in our universities and general relativity is hardly taught there at all. This
is unfortunate. We are in danger of losing the great benefit, the considerable
insight, that relativity theory brought. It has more to tell us, much more,
than the formulae expressing the Lorentz transformation. I am hoping
that the following chapters will illustrate how a revival of interest in general
relativity can prove rewarding.
If different interpretations of general relativity were possible in 1921
they are certainly still possible today, for there is little, if anything, to
show that the difficulties to which Reichenbacher referred have been
resolved, or even much appreciated. There can be no change unless
interest in general relativity becomes more widespread and it may encourage some to turn their attention to this subject if I give a very brief account
of its basis. For this purpose the mathematical treatment is not relevant
and I shall not attempt it. I shall follow as closely as possible the lines taken
by Einstein and other brilliant expositors of the past, while departing
from these lines just enough to show why those who ignored, and have
continued to ignore, the difficulties have not been guided by understanding
but by careless thinking.
In the early days of relativity we were introduced to the notion of a
box in space, far distant from any ponderable matter. Einstein was the
first to invent this illustration. A man was supposed to be inside the box
and to make all the observations that could enable him to learn something about its motion through space. The man could secure himself to
the floor of the box. He could climb to the ceiling and catch hold of a
strap attached to it. He could let go of the strap and observe what happened. He could measure any forces that were exercised on his body. When
he had observed and measured he could apply logic and mathematical
reasoning to the results and arrive therefrom at some conclusions about the
nature of space, time and gravitation.
It will be convenient to refer to the observing, reasoning man in the box
by a name. I should call him Mr. Einstein were it not that I might then
mislead the reader and also divert attention from the subject of relativity
to the question what, in fact, Einstein did or did not say, what he did or
did not think, what he ought or ought not to have said or thought. For
what I have read and what I have learnt in conversation has shown that
different people have different ideas about these matters. So the man inside
the box shall be called Mr. Smith.
Let us begin by considering an occasion when the box is in uniform
motion. Mr Smith tries to measure its velocity. He would like to select
a point in space outside the box and to measure its velocity relative to that
Joint. But he finds, to his disappointment, that he cannot even define a
point in space in any helpful way, let alone measure a velocity relative
One point in empty space is exactly like any other point. It can only
be defined as its distance from a reference object and the only available
reference object is the box. Mr Smith could choose a point defined as
being at a particular distance from this. But then the point remains, by
definition, always at the same distance from the box, whatever the speed
of the box may be. If the box is considered to be moving, so is the point.
If the box is at rest, so is the point.
Mr Smith cannot avoid the conclusion that he has no means of
knowing whether his box is at rest or in uniform motion. It is, indeed,
meaningless to ask the question. In the technical language of relativity there
is no privileged frame of reference by which to define a uniform velocity.
Mr Smith is perfectly free to ascribe any velocity that he likes to the box.
The figure that he mentions cannot be disputed.
Relativists have another way of expressing this. Let us suppose that
someone has constructed a four-dimensional model of the space-time in
which the box is situated. One cannot, of course, make such a model out
of wood or metal; but one can represent it by algebraic symbols. For our
purpose such a representation is just as serviceable. Nothing would be
gained here by writing the symbols down. It suffices that it can be done and
that the space-time relations can be expressed mathematically as clearly
and precisely as they would be if a material model could replace the
algebra. How this is done is not relevant.
If successive positions of the box in uniform motion are represented
by points in the model the points lie along a straight line. This may be
vertical or sloping; it depends on how the model is tilted. One can, if one
likes, adopt the convention that zero velocity is represented by points that
lie vertically above each other. Points along a sloping line would then
represent a finite velocity, which would be greater the more the line
departed from the vertical.
If Mr Smith knew which was the right side up for his model he could
then read the velocity of the box off from it by measuring the slope of the
line. But he finds that there is no means of determining a right side up. He
is free to tilt the model in any way he likes. He can so tilt it that the line
representing successive points of the box is vertical and say that it shows
the box to be at rest. But he can also tilt it so as to show the box to be
moving at a uniform velocity. No inclination of the model could be
claimed to represent a wrong velocity, which is another way of saying
that there is no such thing as absolute velocity.
This is the basic tenet of special relativity, but not, of course, the whole
of it. What has been said above does not show the part played in special
relativity by the velocity of light. But this omission need not detain us. Our
immediate concern is with general relativity.
Let us next suppose that there is a hook in the ceiling of the box on the
outside and that a rope is attached to this hook. Einstein asked us to
imagine that the rope is being pulled and pointed out that it is
relevant to say by what agency. For descriptive purposes it is, however
convenient to name the pulling agent, so I shall say that an angel is
pulling the rope from time to time and with a constant force. As a
consequence the box, with Mr Smith inside it, receives at intervals a
Action and reaction being equal and opposite Mr Smith is subjected to
a force in the opposite direction from that in which the angel is pulling.
While the pull is upwards away from the ceiling the force observed
Mr Smith is downwards towards the floor. Being inside the box Mr Smith
cannot see the rope and does not know what causes the force on his box
He reasons as follows:
'A moment ago there was no force, now there is one. So there is a
physical change to the system in which I find myself. What is the cause of
this change? I can think of two possible ones. The first is that this box is
being accelerated in the direction towards the ceiling. The second is that
some undetectable device is attached to me and is pulling me downwards
towards the floor. Have I any means of discovering which of these two
theories is the correct one?
'Can my model of space-time help? If the first theory is the true one,
my box is being accelerated, successive positions when plotted in the model
will not lie on a straight line. They will lie on a parabola. Let me do the
plotting. Now let me tilt the model in such a way that it represents my
box at rest. I find that it cannot be done. I can tilt the model so that a
little bit of the line connecting the points is vertical. But then all the
remainder of the line is sloping. Velocity can be made to disappear on
the model, but not acceleration. The inference is that a body in uniform
motion can rightly be regarded as at rest all the time; but a body in
accelerated motion can be regarded, at most, as at rest for a moment
It now occurs to Mr Smith to construct a different model of space-time. He calls it model B and calls the first one model A. In model A
equal distances in space-time were represented by equal distances in the
model. In model B it is not so. Distances are represented to functional
scales, somewhat like log scales on graph paper. The selected scales are
unusual and represent relations in non-Euclidean geometry. They are
arranged in such a way that successive positions of the box lie along a
curved line when the box is in uniform motion and along a straight line
when the box is being accelerated at a uniform rate. As for model A there
is no reason why model B should be tilted in one way rather than in another. Like the first model the second one correctly represents the fact that there is no absolute velocity.
The second model can be so tilted that the points lie along a vertical
straight line. It then represents a condition when the box is at rest. Let us
learn how Mr Smith reasons about this:
'When I constructed my second model I thought of it only as a convenient plotting device that would enable me to place all the points along a
straight line. I regarded model A as the true scale model and model B as
one constructed deliberately to be out of scale, to be a distorted representation of reality. But can I have been wrong in this? Is model B more than a
mere plotting device? Can it be a true-to-scale model of the space-time in
which my box is situated? Can space-time have the strange shape that is
represented by the model? If so space-time itself has a physical property,
'This suggests a third way of accounting for the force that I am experiencing. The reason may be neither that the box is being accelerated upwards nor that I am being pulled downwards. That notion that I am being
pulled is unsatisfactory anyhow so long as I cannot detect any devices
attached to me by means of which the pull can be transmitted to my body.
So the third explanation, which has only just occurred to me, deserves
consideration. It is that the box is in a region where the space-time is
If this is the explanation the box is not being accelerated. It can be
regarded as either at rest or in uniform motion. Model B correctly represents
the situation. I can tilt this model so that the line is vertical if I like. So I
shall give up the old notion that something is pulling me downwards. It
ought to be discredited anyhow. I shall consider instead only the two
possibilities that remain, namely that the box is being accelerated in a
space-time correctly represented by model A or is at rest in a space-time
correctly represented by model B.
'Can I make any observation here and now by means of which I can
decide between these alternative possibilities?,
'No. The force exerted on my body must be just the same whether I
am being accelerated in flat space or am at rest in curved space. The only
difference that I can mention is that I attribute the force to the inert
mass of my body when its cause is an acceleration and I attribute it to a
different property (I shall call it the gravitational mass of my body) when
the cause is the curvature of space. But if the only difference between two
properties is the name that convention has given them there is no real
difference. So it is reasonable to regard inert and gravitational mass as
By gaining this piece of insight Einstein took a great step forward
towards the unification of science. What has been said here so far conforms
to the traditional view of general relativity and differs only slightly from
the conventional manner of its presentation. But it now becomes necessary
to consider some of the false conclusions at which one can arrive if one
thinks carelessly and superficially. If any of those who speak on relativity
theory with authority have in fact arrived at any of these false conclusions,
as I fear may have happened, it has become very necessary to do some
rethinking. If I am wrong and no errors have been made it will nevertheless
help towards clarity to appreciate what errors could be made.
Mr Smith might be tempted for a moment to think that it was meaningful to speak of a 'field of acceleration' in the sense in which one speaks of
a gravitational field. (After all, Reichenbacher did so.) Mr Smith would
suggest therewith that a field of acceleration occurs in flat space when a
force is the result of an acceleration just as a gravitational field is present
in curved space. But he would, I hope, soon realize that such a manner of
speaking was misleading and imprecise. The word 'field' should be retained
for the condition of a region and not for what happens in it. The acceleration of a body does not change anything in its surroundings. If the space
is flat it remains flat when something in it is being accelerated.
Should Mr Smith persist however in speaking of a field of acceleration
he might fall into the further error of thinking that the space around him
really did change its curvature when a body was accelerated in that space.
Suppose he were to climb to the ceiling of his box and hang there from the
strap for a short while. If he let go his body would be accelerated relative
to the box. It would be so whether the box was being accelerated in smooth
space or was at rest in curved space. If it was the former Mr Smith would
be at rest after he had let go while the box would continue to be accelerated.
If it were the latter Mr Smith would be accelerated after he had let go, while
the box would continue to be at rest. By falling Mr Smith does not smooth
out the space that surrounds him. He just falls.
The error of thinking that the act of falling smooths out space-time
can only arise if one believes that the choice between model A and model B
depends on the motion of the observer. But it will have become clear that
it does not. As I have said already, Mr Smith can find nothing inside his
box that will enable him to know which model to choose. His own motion
relative to the box does not tell him; nor do any other observations made
Having appreciated the fact that he cannot alter the curvature of space-time around him by falling any more than he can by dropping his watch
Mr Smith will replace his previous assertion by the following:
'When I feel a force on my body and still have hold of the strap my
successive positions lie along a curved line if plotted in model A and along
a straight line if plotted in model B. After I have let go I do not feel any
force and the shape of the lines in the model is the other way about. While I
am falling my successive positions are represented by a straight line in
model A and by a curved line in model B. I can represent this n
conveniently in tabular form as follows:
|Condition of space-time
Body subjected to a force when
Body not subjected to a force when
Uniform motion represented by
Accelerated motion represented by
Force attributed to
’I must refrain in future from saying, or implying, that my actions
change the curvature of space-time. As I do not even know from any
observation that I can make from where I am, whether the correct model
is A or B, I can certainly not claim to know that when I am falling I am
changing from the one model to the other.'
There is the possibility of yet another error. Mr Smith may reach the
erroneous conclusion that he possesses no means at all of deciding between
the two models. He may, in other words, believe that observations made
by an observer outside the box are as inconclusive as those made by
If he makes this mistake he will feel justified in asserting that the two
models, as any others, are equally valid in the same way as two different
tilts of either model are equally valid. He will then enunciate the theory of
full equivalence between a body at rest in curved space-time and a body in
accelerated motion in smooth space-time. But Mr Smith is fortunate in
having a friend outside the box from whom he can obtain some useful
The friend's name is Mr Jones. He is standing beside the angel. The
two gentlemen establish telephonic communication. The following
conversation takes place:
SMITH: 'What is happening where you are?'
JONES: 'The angel is pulling himself hand over hand along the rope
towards your box.'
SMITH : I believe you are wrong. It seems to me that the angel must be
pulling the box towards himself.'
JONES : 'Have it that way if you like. All that I can observe is that the
rope is taut and is getting shorter. It is doing so at a rate of 9.80665 metres
per second, per second. The explanation might equally well be that the
angel is being accelerated towards the box or that the box is being accelerated towards the angel. I have no means here of knowing which it is.
The two statements are fully equivalent. To ask which it is is a meaningless
SMITH: 'But I have means of knowing which is which. I am experiencing a force towards the floor of the box. As my body has inert mass I
conclude that it is the box and not the angel that is being accelerated.'
JONES : 'After what you have told me that seems to be the more probable
explanation. My observation alone left the question open, but our joint
observations make it reasonable to assume that the box is being accelerated
and not the angel.'
SMITH: 'Yes. And my observation alone left the possibility open that
there was no pull at all on the rope but that, instead, the box was in a non-Euclidean region of space-time. Our combined observations have provided
the answer to that question too.'
At this moment it occurs to the two observers that Mr Jones would have
known whether the angel was being accelerated or not if the angel had
possessed inert mass. They agree that acceleration of a body P relative to Q
is not equivalent to acceleration of Q relative to P. If the bodies are in
smooth space-time the force on each is the product of its acceleration and
its inert mass. If the acceleration is zero its product with inert mass is also
zero. Conversely, if the force is zero and the space-time is smooth the
acceleration is also zero.
After a while the angel gets tired of intermittently pulling at the rope.
He changes his position and places himself underneath the box. Mr Jones
accompanies him. Presently Mr Smith calls Mr Jones on their telephone.
SMITH: 'Jones. The angel is pulling at that rope again.'
JONES : 'No he is not. The rope is curled up neatly and lies on top of
the box. The angel is underneath the box. I am here with him.'
SMITH: 'What has the angel been doing?'
JONES : 'He has placed a big sphere underneath the box. Its circumference
is 40,000 kilometres.'
SMITH: Is that all?'
JONES: 'Yes. All I can see.'
SMITH : 'As the angel is no longer pulling at the rope I must assume that
the box is no longer being accelerated. And yet I feel that same force as I
did while the acceleration lasted. So I have to conclude either that some
undetected device is attached to my body and pulling downwards or that
the box is in a non-Euclidean region of space-time. I deprecate the mystical
hypothesis of undetectable devices and so I have to assume the latter.
Something must therefore have changed the physical nature of space-time
around here in such a way as to cause curvature. The only physical change
that you can report is the appearance of the big sphere. So I conclude that
this is influencing the geometry of space-time in its vicinity.'
Mr Smith and Mr Jones confer further together and reach agreement
on several points. They are satisfied that there is no difference between
inert and gravitational mass; that there is nevertheless a difference between
acceleration and uniform motion; that relative acceleration has real meaning; that the process that causes a stone to be subjected to a force during
its acceleration is not the same as the process that causes it to be subjected
to a force while it is at rest on a shelf; that an observer may be conceived
to be so situated that he cannot discover which of the two processes is operating; that an observer may also be so situated that he can discover just
this; that, for instance, Smith inside the box can only observe the effects
of the processes, which are identical for both, while Jones outside the box
can observe the sources of the processes, which are different for both; that
it is the big sphere underneath the box that causes the geometry of space-time to become non-Euclidean.
Having thus clarified the subject they are able to formulate the big
question that had previously been obscured by sundry misconceptions:
By what process is the distortion of space-time effected?
Top of Page