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
full, is:
*(Inert mass). (Accelerafion) = (lntensity of the gravitational field).
(Gravitational mass).' *

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 *m*_{1} and the acceleration *f*_{1}, one can write:

*
F = m*_{1}f ...... (21a)

The right hand side defines the weight of the ball on a spring balance. Let
this weight be *W* and the gravitational mass *m*_{g} The intensity of the
gravitational field near the earth is *g,* so the algebraic expression for the
right hand side is

*
W = gm*_{g} ...... (21b)

Equations (21a) and (21b) are found to give numerically identical results
for the same ball when *f = g*. From this Einstein inferred that *m*_{g} = m_{i}. 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 *
m*_{g} / m_{i} 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
facts here.

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 / x*^{2}

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 = m*_{i} g = W = ( G M_{a} / x^{2} ) m_{g} ...... (21c)

In this equation

*
g*_{M} = k M_{a} / x^{2}

g_{m} = k m_{a} / x^{2}

The pull of the earth on the ball is *g*_{M},m_{g} , while the pull of the ball on the earth is *g*_{m},M_{g}. 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 *F*_{e} and the pull of the
billiard ball on the earth *F*_{b}. One can then write

*
F*_{e} = ( k M_{a} / x^{2 ) mg
Fb = ( k ma2} ) M_{g}

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 *F*_{e} as action and *F*_{b} 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. *F*_{e} is shown as acting on the top of the spring and has its own equal
and opposite reaction. The same holds for *F*_{b}, 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 *F*_{e}, or *F*_{b} 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 M*_{a} / x^{2} ) m_{g} + ( 1/2 Gm_{a} / x^{2} ) M_{g} ...... (21e)

from which it appears that *k *, above, equals *1/2G*.

Equations (21e) and (21c) are numerically equal, but this is only
because *M*_{a} m_{g} = m_{a} M_{g}. 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

*
gm*_{1} = ( GM_{a} / X^{2} ) m_{g} ...... . (21f)

The attracting mass, *m*_{a}g = g_{M} + g_{m} 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 *g*_{m} / g_{M} 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 *m*_{i} and *m*_{g}. And this equation gives no information about
the attracting mass *M*_{a}. 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
produced.

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
carefully scrutinized.

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.

’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
himself.

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
information.

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
question.'

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?