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On the basis of the physical interpretation of distance which has been
indicated, we are also in a position to establish the distance between
two points on a rigid body by means of measurements. For this purpose
we require a " distance " (rod S) which is to be used once and for
all, and which we employ as a standard measure. If, now, A and B are
two points on a rigid body, we can construct the line joining them
according to the rules of geometry ; then, starting from A, we can
mark off the distance S time after time until we reach B. The number
of these operations required is the numerical measure of the distance
AB. This is the basis of all measurement of length. *

Every description of the scene of an event or of the position of an
object in space is based on the specification of the point on a rigid
body (body of reference) with which that event or object coincides.
This applies not only to scientific description, but also to everyday
life. If I analyse the place specification " Times Square, New York,"
**A I arrive at the following result. The earth is the rigid body
to which the specification of place refers; " Times Square, New York,"
is a well-defined point, to which a name has been assigned, and with
which the event coincides in space.**B

This primitive method of place specification deals only with places on
the surface of rigid bodies, and is dependent on the existence of
points on this surface which are distinguishable from each other. But
we can free ourselves from both of these limitations without altering
the nature of our specification of position. If, for instance, a cloud
is hovering over Times Square, then we can determine its position
relative to the surface of the earth by erecting a pole
perpendicularly on the Square, so that it reaches the cloud. The
length of the pole measured with the standard measuring-rod, combined
with the specification of the position of the foot of the pole,
supplies us with a complete place specification. On the basis of this
illustration, we are able to see the manner in which a refinement of
the conception of position has been developed.

(a) We imagine the rigid body, to which the place specification is
referred, supplemented in such a manner that the object whose position
we require is reached by. the completed rigid body.

(b) In locating the position of the object, we make use of a number
(here the length of the pole measured with the measuring-rod) instead
of designated points of reference.

(c) We speak of the height of the cloud even when the pole which
reaches the cloud has not been erected. By means of optical
observations of the cloud from different positions on the ground, and
taking into account the properties of the propagation of light, we
determine the length of the pole we should have required in order to
reach the cloud.

From this consideration we see that it will be advantageous if, in the
description of position, it should be possible by means of numerical
measures to make ourselves independent of the existence of marked
positions (possessing names) on the rigid body of reference. In the
physics of measurement this is attained by the application of the
Cartesian system of co-ordinates.

This consists of three plane surfaces perpendicular to each other and
rigidly attached to a rigid body. Referred to a system of
co-ordinates, the scene of any event will be determined (for the main
part) by the specification of the lengths of the three perpendiculars
or co-ordinates (x, y, z) which can be dropped from the scene of the
event to those three plane surfaces. The lengths of these three
perpendiculars can be determined by a series of manipulations with
rigid measuring-rods performed according to the rules and methods laid
down by Euclidean geometry.

In practice, the rigid surfaces which constitute the system of
co-ordinates are generally not available ; furthermore, the magnitudes
of the co-ordinates are not actually determined by constructions with
rigid rods, but by indirect means. If the results of physics and
astronomy are to maintain their clearness, the physical meaning of
specifications of position must always be sought in accordance with
the above considerations. ***

We thus obtain the following result: Every description of events in
space involves the use of a rigid body to which such events have to be
referred. The resulting relationship takes for granted that the laws
of Euclidean geometry hold for "distances;" the "distance" being
represented physically by means of the convention of two marks on a
rigid body.


  Notes

* Here we have assumed that there is nothing left over i.e. that
the measurement gives a whole number. This difficulty is got over by
the use of divided measuring-rods, the introduction of which does not
demand any fundamentally new method.

**A Einstein used "Potsdamer Platz, Berlin" in the original text.
In the authorised translation this was supplemented with "Tranfalgar
Square, London". We have changed this to "Times Square, New York", as
this is the most well known/identifiable location to English speakers
in the present day. [Note by the janitor.]

**B It is not necessary here to investigate further the significance
of the expression "coincidence in space." This conception is
sufficiently obvious to ensure that differences of opinion are
scarcely likely to arise as to its applicability in practice.

*** A refinement and modification of these views does not become
necessary until we come to deal with the general theory of relativity,
treated in the second part of this book.



SPACE AND TIME IN CLASSICAL MECHANICS


The purpose of mechanics is to describe how bodies change their
position in space with "time." I should load my conscience with grave
sins against the sacred spirit of lucidity were I to formulate the
aims of mechanics in this way, without serious reflection and detailed
explanations. Let us proceed to disclose these sins.

It is not clear what is to be understood here by "position" and
"space." I stand at the window of a railway carriage which is
travelling uniformly, and drop a stone on the embankment, without
throwing it. Then, disregarding the influence of the air resistance, I
see the stone descend in a straight line. A pedestrian who observes
the misdeed from the footpath notices that the stone falls to earth in
a parabolic curve. I now ask: Do the "positions" traversed by the
stone lie "in reality" on a straight line or on a parabola? Moreover,
what is meant here by motion "in space" ? From the considerations of
the previous section the answer is self-evident. In the first place we
entirely shun the vague word "space," of which, we must honestly
acknowledge, we cannot form the slightest conception, and we replace
it by "motion relative to a practically rigid body of reference." The
positions relative to the body of reference (railway carriage or
embankment) have already been defined in detail in the preceding
section. If instead of " body of reference " we insert " system of
co-ordinates," which is a useful idea for mathematical description, we
are in a position to say : The stone traverses a straight line
relative to a system of co-ordinates rigidly attached to the carriage,
but relative to a system of co-ordinates rigidly attached to the
ground (embankment) it describes a parabola. With the aid of this
example it is clearly seen that there is no such thing as an
independently existing trajectory (lit. "path-curve"*), but only
a trajectory relative to a particular body of reference.

In order to have a complete description of the motion, we must specify
how the body alters its position with time ; i.e. for every point on
the trajectory it must be stated at what time the body is situated
there. These data must be supplemented by such a definition of time
that, in virtue of this definition, these time-values can be regarded
essentially as magnitudes (results of measurements) capable of
observation. If we take our stand on the ground of classical
mechanics, we can satisfy this requirement for our illustration in the
following manner. We imagine two clocks of identical construction ;
the man at the railway-carriage window is holding one of them, and the
man on the footpath the other. Each of the observers determines the
position on his own reference-body occupied by the stone at each tick
of the clock he is holding in his hand. In this connection we have not
taken account of the inaccuracy involved by the finiteness of the
velocity of propagation of light. With this and with a second
difficulty prevailing here we shall have to deal in detail later.


  Notes

*) That is, a curve along which the body moves.



THE GALILEIAN SYSTEM OF CO-ORDINATES


As is well known, the fundamental law of the mechanics of
Galilei-Newton, which is known as the law of inertia, can be stated
thus: A body removed sufficiently far from other bodies continues in a
state of rest or of uniform motion in a straight line. This law not
only says something about the motion of the bodies, but it also
indicates the reference-bodies or systems of coordinates, permissible
in mechanics, which can be used in mechanical description. The visible
fixed stars are bodies for which the law of inertia certainly holds to
a high degree of approximation. Now if we use a system of co-ordinates
which is rigidly attached to the earth, then, relative to this system,
every fixed star describes a circle of immense radius in the course of
an astronomical day, a result which is opposed to the statement of the
law of inertia. So that if we adhere to this law we must refer these
motions only to systems of coordinates relative to which the fixed
stars do not move in a circle. A system of co-ordinates of which the
state of motion is such that the law of inertia holds relative to it
is called a " Galileian system of co-ordinates." The laws of the
mechanics of Galflei-Newton can be regarded as valid only for a
Galileian system of co-ordinates.



THE PRINCIPLE OF RELATIVITY
(IN THE RESTRICTED SENSE)


In order to attain the greatest possible clearness, let us return to
our example of the railway carriage supposed to be travelling
uniformly. We call its motion a uniform translation ("uniform" because
it is of constant velocity and direction, " translation " because
although the carriage changes its position relative to the embankment
yet it does not rotate in so doing). Let us imagine a raven flying
through the air in such a manner that its motion, as observed from the
embankment, is uniform and in a straight line. If we were to observe
the flying raven from the moving railway carriage. we should find that
the motion of the raven would be one of different velocity and
direction, but that it would still be uniform and in a straight line.
Expressed in an abstract manner we may say : If a mass m is moving
uniformly in a straight line with respect to a co-ordinate system K,
then it will also be moving uniformly and in a straight line relative
to a second co-ordinate system K1 provided that the latter is
executing a uniform translatory motion with respect to K. In
accordance with the discussion contained in the preceding section, it
follows that:

If K is a Galileian co-ordinate system. then every other co-ordinate
system K' is a Galileian one, when, in relation to K, it is in a
condition of uniform motion of translation. Relative to K1 the
mechanical laws of Galilei-Newton hold good exactly as they do with
respect to K.

We advance a step farther in our generalisation when we express the
tenet thus: If, relative to K, K1 is a uniformly moving co-ordinate
system devoid of rotation, then natural phenomena run their course
with respect to K1 according to exactly the same general laws as with
respect to K. This statement is called the principle of relativity (in
the restricted sense).

As long as one was convinced that all natural phenomena were capable
of representation with the help of classical mechanics, there was no
need to doubt the validity of this principle of relativity. But in
view of the more recent development of electrodynamics and optics it
became more and more evident that classical mechanics affords an
insufficient foundation for the physical description of all natural
phenomena. At this juncture the question of the validity of the
principle of relativity became ripe for discussion, and it did not
appear impossible that the answer to this question might be in the
negative.

Nevertheless, there are two general facts which at the outset speak

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should retain the principle of relativity, which appeals so
convincingly to the intellect because it is so natural and simple. The
law of the propagation of light in vacuo would then have to be
replaced by a more complicated law conformable to the principle of
relativity. The development of theoretical physics shows, however,
that we cannot pursue this course. The epoch-making theoretical
investigations of H. A. Lorentz on the electrodynamical and optical
phenomena connected with moving bodies show that experience in this
domain leads conclusively to a theory of electromagnetic phenomena, of
which the law of the constancy of the velocity of light in vacuo is a
necessary consequence. Prominent theoretical physicists were theref
ore more inclined to reject the principle of relativity, in spite of
the fact that no empirical data had been found which were
contradictory to this principle.

At this juncture the theory of relativity entered the arena. As a
result of an analysis of the physical conceptions of time and space,
it became evident that in realily there is not the least
incompatibilitiy between the principle of relativity and the law of
propagation of light, and that by systematically holding fast to both
these laws a logically rigid theory could be arrived at. This theory
has been called the special theory of relativity to distinguish it
from the extended theory, with which we shall deal later. In the
following pages we shall present the fundamental ideas of the special
theory of relativity.



ON THE IDEA OF TIME IN PHYSICS


Lightning has struck the rails on our railway embankment at two places
A and B far distant from each other. I make the additional assertion
that these two lightning flashes occurred simultaneously. If I ask you
whether there is sense in this statement, you will answer my question
with a decided "Yes." But if I now approach you with the request to
explain to me the sense of the statement more precisely, you find
after some consideration that the answer to this question is not so
easy as it appears at first sight.

After some time perhaps the following answer would occur to you: "The
significance of the statement is clear in itself and needs no further
explanation; of course it would require some consideration if I were
to be commissioned to determine by observations whether in the actual
case the two events took place simultaneously or not." I cannot be
satisfied with this answer for the following reason. Supposing that as
a result of ingenious considerations an able meteorologist were to
discover that the lightning must always strike the places A and B
simultaneously, then we should be faced with the task of testing
whether or not this theoretical result is in accordance with the
reality. We encounter the same difficulty with all physical statements
in which the conception " simultaneous " plays a part. The concept
does not exist for the physicist until he has the possibility of
discovering whether or not it is fulfilled in an actual case. We thus
require a definition of simultaneity such that this definition
supplies us with the method by means of which, in the present case, he
can decide by experiment whether or not both the lightning strokes
occurred simultaneously. As long as this requirement is not satisfied,
I allow myself to be deceived as a physicist (and of course the same
applies if I am not a physicist), when I imagine that I am able to
attach a meaning to the statement of simultaneity. (I would ask the
reader not to proceed farther until he is fully convinced on this
point.)

After thinking the matter over for some time you then offer the
following suggestion with which to test simultaneity. By measuring
along the rails, the connecting line AB should be measured up and an
observer placed at the mid-point M of the distance AB. This observer
should be supplied with an arrangement (e.g. two mirrors inclined at
90^0) which allows him visually to observe both places A and B at the
same time. If the observer perceives the two flashes of lightning at
the same time, then they are simultaneous.

I am very pleased with this suggestion, but for all that I cannot
regard the matter as quite settled, because I feel constrained to
raise the following objection:

"Your definition would certainly be right, if only I knew that the
light by means of which the observer at M perceives the lightning
flashes travels along the length A arrow M with the same velocity as
along the length B arrow M. But an examination of this supposition
would only be possible if we already had at our disposal the means of
measuring time. It would thus appear as though we were moving here in
a logical circle."

After further consideration you cast a somewhat disdainful glance at
me -- and rightly so -- and you declare:

"I maintain my previous definition nevertheless, because in reality it
assumes absolutely nothing about light. There is only one demand to be
made of the definition of simultaneity, namely, that in every real
case it must supply us with an empirical decision as to whether or not
the conception that has to be defined is fulfilled. That my definition
satisfies this demand is indisputable. That light requires the same
time to traverse the path A arrow M as for the path B arrow M is in
reality neither a supposition nor a hypothesis about the physical
nature of light, but a stipulation which I can make of my own freewill
in order to arrive at a definition of simultaneity."

It is clear that this definition can be used to give an exact meaning
not only to two events, but to as many events as we care to choose,
and independently of the positions of the scenes of the events with
respect to the body of reference * (here the railway embankment).
We are thus led also to a definition of " time " in physics. For this
purpose we suppose that clocks of identical construction are placed at
the points A, B and C of the railway line (co-ordinate system) and
that they are set in such a manner that the positions of their
pointers are simultaneously (in the above sense) the same. Under these
conditions we understand by the " time " of an event the reading
(position of the hands) of that one of these clocks which is in the
immediate vicinity (in space) of the event. In this manner a
time-value is associated with every event which is essentially capable
of observation.

This stipulation contains a further physical hypothesis, the validity
of which will hardly be doubted without empirical evidence to the
contrary. It has been assumed that all these clocks go at the same
rate if they are of identical construction. Stated more exactly: When
two clocks arranged at rest in different places of a reference-body
are set in such a manner that a particular position of the pointers of
the one clock is simultaneous (in the above sense) with the same

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GENERAL RESULTS OF THE THEORY


It is clear from our previous considerations that the (special) theory
of relativity has grown out of electrodynamics and optics. In these
fields it has not appreciably altered the predictions of theory, but
it has considerably simplified the theoretical structure, i.e. the
derivation of laws, and -- what is incomparably more important -- it
has considerably reduced the number of independent hypothese forming
the basis of theory. The special theory of relativity has rendered the
Maxwell-Lorentz theory so plausible, that the latter would have been
generally accepted by physicists even if experiment had decided less
unequivocally in its favour.

Classical mechanics required to be modified before it could come into
line with the demands of the special theory of relativity. For the
main part, however, this modification affects only the laws for rapid
motions, in which the velocities of matter v are not very small as
compared with the velocity of light. We have experience of such rapid
motions only in the case of electrons and ions; for other motions the
variations from the laws of classical mechanics are too small to make
themselves evident in practice. We shall not consider the motion of
stars until we come to speak of the general theory of relativity. In
accordance with the theory of relativity the kinetic energy of a
material point of mass m is no longer given by the well-known
expression

                        eq. 15: file eq15.gif

but by the expression

                        eq. 16: file eq16.gif


This expression approaches infinity as the velocity v approaches the
velocity of light c. The velocity must therefore always remain less
than c, however great may be the energies used to produce the
acceleration. If we develop the expression for the kinetic energy in
the form of a series, we obtain

                        eq. 17: file eq17.gif


When eq. 18 is small compared with unity, the third of these terms is
always small in comparison with the second,

which last is alone considered in classical mechanics. The first term
mc^2 does not contain the velocity, and requires no consideration if
we are only dealing with the question as to how the energy of a
point-mass; depends on the velocity. We shall speak of its essential
significance later.

The most important result of a general character to which the special
theory of relativity has led is concerned with the conception of mass.
Before the advent of relativity, physics recognised two conservation
laws of fundamental importance, namely, the law of the canservation of
energy and the law of the conservation of mass these two fundamental
laws appeared to be quite independent of each other. By means of the
theory of relativity they have been united into one law. We shall now
briefly consider how this unification came about, and what meaning is
to be attached to it.

The principle of relativity requires that the law of the concervation
of energy should hold not only with reference to a co-ordinate system
K, but also with respect to every co-ordinate system K1 which is in a
state of uniform motion of translation relative to K, or, briefly,
relative to every " Galileian " system of co-ordinates. In contrast to
classical mechanics; the Lorentz transformation is the deciding factor
in the transition from one such system to another.

By means of comparatively simple considerations we are led to draw the
following conclusion from these premises, in conjunction with the
fundamental equations of the electrodynamics of Maxwell: A body moving
with the velocity v, which absorbs * an amount of energy E[0] in
the form of radiation without suffering an alteration in velocity in
the process, has, as a consequence, its energy increased by an amount

                        eq. 19: file eq19.gif

In consideration of the expression given above for the kinetic energy
of the body, the required energy of the body comes out to be

                        eq. 20: file eq20.gif


Thus the body has the same energy as a body of mass

                         eq.21: file eq21.gif

moving with the velocity v. Hence we can say: If a body takes up an
amount of energy E[0], then its inertial mass increases by an amount

                        eq. 22: file eq22.gif


the inertial mass of a body is not a constant but varies according to
the change in the energy of the body. The inertial mass of a system of
bodies can even be regarded as a measure of its energy. The law of the
conservation of the mass of a system becomes identical with the law of
the conservation of energy, and is only valid provided that the system
neither takes up nor sends out energy. Writing the expression for the
energy in the form

                        eq. 23: file eq23.gif

we see that the term mc^2, which has hitherto attracted our attention,
is nothing else than the energy possessed by the body ** before it
absorbed the energy E[0].

A direct comparison of this relation with experiment is not possible
at the present time (1920; see *** Note, p. 48), owing to the fact that
the changes in energy E[0] to which we can Subject a system are not
large enough to make themselves perceptible as a change in the
inertial mass of the system.

                                eq. 22: file eq22.gif


is too small in comparison with the mass m, which was present before
the alteration of the energy. It is owing to this circumstance that
classical mechanics was able to establish successfully the

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of drawing on hypotheses as to the physical nature of the liquid.

The second class of facts to which we have alluded has reference to
the question whether or not the motion of the earth in space can be
made perceptible in terrestrial experiments. We have already remarked
in Section 5 that all attempts of this nature led to a negative
result. Before the theory of relativity was put forward, it was
difficult to become reconciled to this negative result, for reasons
now to be discussed. The inherited prejudices about time and space did
not allow any doubt to arise as to the prime importance of the
Galileian transformation for changing over from one body of reference
to another. Now assuming that the Maxwell-Lorentz equations hold for a
reference-body K, we then find that they do not hold for a
reference-body K1 moving uniformly with respect to K, if we assume
that the relations of the Galileian transformstion exist between the
co-ordinates of K and K1. It thus appears that, of all Galileian
co-ordinate systems, one (K) corresponding to a particular state of
motion is physically unique. This result was interpreted physically by
regarding K as at rest with respect to a hypothetical æther of space.
On the other hand, all coordinate systems K1 moving relatively to K
were to be regarded as in motion with respect to the æther. To this
motion of K1 against the æther ("æther-drift " relative to K1) were
attributed the more complicated laws which were supposed to hold
relative to K1. Strictly speaking, such an æther-drift ought also to
be assumed relative to the earth, and for a long time the efforts of
physicists were devoted to attempts to detect the existence of an
æther-drift at the earth's surface.

In one of the most notable of these attempts Michelson devised a
method which appears as though it must be decisive. Imagine two
mirrors so arranged on a rigid body that the reflecting surfaces face
each other. A ray of light requires a perfectly definite time T to
pass from one mirror to the other and back again, if the whole system
be at rest with respect to the æther. It is found by calculation,
however, that a slightly different time T1 is required for this
process, if the body, together with the mirrors, be moving relatively
to the æther. And yet another point: it is shown by calculation that
for a given velocity v with reference to the æther, this time T1 is
different when the body is moving perpendicularly to the planes of the
mirrors from that resulting when the motion is parallel to these
planes. Although the estimated difference between these two times is
exceedingly small, Michelson and Morley performed an experiment
involving interference in which this difference should have been
clearly detectable. But the experiment gave a negative result -- a
fact very perplexing to physicists. Lorentz and FitzGerald rescued the
theory from this difficulty by assuming that the motion of the body
relative to the æther produces a contraction of the body in the
direction of motion, the amount of contraction being just sufficient
to compensate for the differeace in time mentioned above. Comparison
with the discussion in Section 11 shows that also from the
standpoint of the theory of relativity this solution of the difficulty
was the right one. But on the basis of the theory of relativity the
method of interpretation is incomparably more satisfactory. According
to this theory there is no such thing as a " specially favoured "
(unique) co-ordinate system to occasion the introduction of the
æther-idea, and hence there can be no æther-drift, nor any experiment
with which to demonstrate it. Here the contraction of moving bodies
follows from the two fundamental principles of the theory, without the
introduction of particular hypotheses ; and as the prime factor
involved in this contraction we find, not the motion in itself, to
which we cannot attach any meaning, but the motion with respect to the
body of reference chosen in the particular case in point. Thus for a
co-ordinate system moving with the earth the mirror system of
Michelson and Morley is not shortened, but it is shortened for a
co-ordinate system which is at rest relatively to the sun.


  Notes

*) The general theory of relativity renders it likely that the
electrical masses of an electron are held together by gravitational
forces.



MINKOWSKI'S FOUR-DIMENSIONAL SPACE


The non-mathematician is seized by a mysterious shuddering when he
hears of "four-dimensional" things, by a feeling not unlike that
awakened by thoughts of the occult. And yet there is no more
common-place statement than that the world in which we live is a
four-dimensional space-time continuum.

Space is a three-dimensional continuum. By this we mean that it is
possible to describe the position of a point (at rest) by means of
three numbers (co-ordinales) x, y, z, and that there is an indefinite
number of points in the neighbourhood of this one, the position of
which can be described by co-ordinates such as x[1], y[1], z[1], which
may be as near as we choose to the respective values of the
co-ordinates x, y, z, of the first point. In virtue of the latter
property we speak of a " continuum," and owing to the fact that there
are three co-ordinates we speak of it as being " three-dimensional."

Similarly, the world of physical phenomena which was briefly called "
world " by Minkowski is naturally four dimensional in the space-time
sense. For it is composed of individual events, each of which is
described by four numbers, namely, three space co-ordinates x, y, z,
and a time co-ordinate, the time value t. The" world" is in this sense
also a continuum; for to every event there are as many "neighbouring"
events (realised or at least thinkable) as we care to choose, the
co-ordinates x[1], y[1], z[1], t[1] of which differ by an indefinitely
small amount from those of the event x, y, z, t originally considered.
That we have not been accustomed to regard the world in this sense as
a four-dimensional continuum is due to the fact that in physics,
before the advent of the theory of relativity, time played a different
and more independent role, as compared with the space coordinates. It
is for this reason that we have been in the habit of treating time as
an independent continuum. As a matter of fact, according to classical
mechanics, time is absolute, i.e. it is independent of the position
and the condition of motion of the system of co-ordinates. We see this
expressed in the last equation of the Galileian transformation (t1 =
t)

The four-dimensional mode of consideration of the "world" is natural
on the theory of relativity, since according to this theory time is
robbed of its independence. This is shown by the fourth equation of
the Lorentz transformation:

                        eq. 24: file eq24.gif

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state of the body. For instance, a piece of lead and a piece of wood
fall in exactly the same manner in a gravitational field (in vacuo),
when they start off from rest or with the same initial velocity. This
law, which holds most accurately, can be expressed in a different form
in the light of the following consideration.

According to Newton's law of motion, we have

(Force) = (inertial mass) x (acceleration),

where the "inertial mass" is a characteristic constant of the
accelerated body. If now gravitation is the cause of the acceleration,
we then have

(Force) = (gravitational mass) x (intensity of the gravitational
field),

where the "gravitational mass" is likewise a characteristic constant
for the body. From these two relations follows:

                                eq. 26: file eq26.gif


If now, as we find from experience, the acceleration is to be
independent of the nature and the condition of the body and always the
same for a given gravitational field, then the ratio of the
gravitational to the inertial mass must likewise be the same for all
bodies. By a suitable choice of units we can thus make this ratio
equal to unity. We then have the following law: The gravitational mass
of a body is equal to its inertial law.

It is true that this important law had hitherto been recorded in
mechanics, but it had not been interpreted. A satisfactory
interpretation can be obtained only if we recognise the following fact
: The same quality of a body manifests itself according to
circumstances as " inertia " or as " weight " (lit. " heaviness '). In
the following section we shall show to what extent this is actually
the case, and how this question is connected with the general
postulate of relativity.




THE EQUALITY OF INERTIAL AND GRAVITATIONAL MASS
AS AN ARGUMENT FOR THE GENERAL POSTULE OF RELATIVITY


We imagine a large portion of empty space, so far removed from stars
and other appreciable masses, that we have before us approximately the
conditions required by the fundamental law of Galilei. It is then
possible to choose a Galileian reference-body for this part of space
(world), relative to which points at rest remain at rest and points in
motion continue permanently in uniform rectilinear motion. As
reference-body let us imagine a spacious chest resembling a room with
an observer inside who is equipped with apparatus. Gravitation
naturally does not exist for this observer. He must fasten himself
with strings to the floor, otherwise the slightest impact against the
floor will cause him to rise slowly towards the ceiling of the room.

To the middle of the lid of the chest is fixed externally a hook with
rope attached, and now a " being " (what kind of a being is immaterial
to us) begins pulling at this with a constant force. The chest
together with the observer then begin to move "upwards" with a
uniformly accelerated motion. In course of time their velocity will
reach unheard-of values -- provided that we are viewing all this from
another reference-body which is not being pulled with a rope.

But how does the man in the chest regard the Process ? The
acceleration of the chest will be transmitted to him by the reaction
of the floor of the chest. He must therefore take up this pressure by
means of his legs if he does not wish to be laid out full length on
the floor. He is then standing in the chest in exactly the same way as
anyone stands in a room of a home on our earth. If he releases a body
which he previously had in his land, the accelertion of the chest will
no longer be transmitted to this body, and for this reason the body
will approach the floor of the chest with an accelerated relative
motion. The observer will further convince himself that the
acceleration of the body towards the floor of the chest is always of
the same magnitude, whatever kind of body he may happen to use for the
experiment.

Relying on his knowledge of the gravitational field (as it was
discussed in the preceding section), the man in the chest will thus
come to the conclusion that he and the chest are in a gravitational
field which is constant with regard to time. Of course he will be
puzzled for a moment as to why the chest does not fall in this
gravitational field. just then, however, he discovers the hook in the
middle of the lid of the chest and the rope which is attached to it,
and he consequently comes to the conclusion that the chest is
suspended at rest in the gravitational field.

Ought we to smile at the man and say that he errs in his conclusion ?
I do not believe we ought to if we wish to remain consistent ; we must
rather admit that his mode of grasping the situation violates neither
reason nor known mechanical laws. Even though it is being accelerated
with respect to the "Galileian space" first considered, we can
nevertheless regard the chest as being at rest. We have thus good
grounds for extending the principle of relativity to include bodies of
reference which are accelerated with respect to each other, and as a
result we have gained a powerful argument for a generalised postulate
of relativity.

We must note carefully that the possibility of this mode of
interpretation rests on the fundamental property of the gravitational
field of giving all bodies the same acceleration, or, what comes to
the same thing, on the law of the equality of inertial and
gravitational mass. If this natural law did not exist, the man in the
accelerated chest would not be able to interpret the behaviour of the
bodies around him on the supposition of a gravitational field, and he
would not be justified on the grounds of experience in supposing his
reference-body to be " at rest."

Suppose that the man in the chest fixes a rope to the inner side of
the lid, and that he attaches a body to the free end of the rope. The
result of this will be to strech the rope so that it will hang "
vertically " downwards. If we ask for an opinion of the cause of
tension in the rope, the man in the chest will say: "The suspended
body experiences a downward force in the gravitational field, and this
is neutralised by the tension of the rope ; what determines the
magnitude of the tension of the rope is the gravitational mass of the
suspended body." On the other hand, an observer who is poised freely
in space will interpret the condition of things thus : " The rope must
perforce take part in the accelerated motion of the chest, and it
transmits this motion to the body attached to it. The tension of the
rope is just large enough to effect the acceleration of the body. That
which determines the magnitude of the tension of the rope is the
inertial mass of the body." Guided by this example, we see that our
extension of the principle of relativity implies the necessity of the
law of the equality of inertial and gravitational mass. Thus we have
obtained a physical interpretation of this law.

From our consideration of the accelerated chest we see that a general
theory of relativity must yield important results on the laws of
gravitation. In point of fact, the systematic pursuit of the general
idea of relativity has supplied the laws satisfied by the
gravitational field. Before proceeding farther, however, I must warn
the reader against a misconception suggested by these considerations.
A gravitational field exists for the man in the chest, despite the
fact that there was no such field for the co-ordinate system first
chosen. Now we might easily suppose that the existence of a
gravitational field is always only an apparent one. We might also
think that, regardless of the kind of gravitational field which may be
present, we could always choose another reference-body such that no
gravitational field exists with reference to it. This is by no means
true for all gravitational fields, but only for those of quite special
form. It is, for instance, impossible to choose a body of reference
such that, as judged from it, the gravitational field of the earth (in
its entirety) vanishes.

We can now appreciate why that argument is not convincing, which we
brought forward against the general principle of relativity at theend
of Section 18. It is certainly true that the observer in the
railway carriage experiences a jerk forwards as a result of the
application of the brake, and that he recognises, in this the
non-uniformity of motion (retardation) of the carriage. But he is
compelled by nobody to refer this jerk to a " real " acceleration
(retardation) of the carriage. He might also interpret his experience
thus: " My body of reference (the carriage) remains permanently at
rest. With reference to it, however, there exists (during the period
of application of the brakes) a gravitational field which is directed
forwards and which is variable with respect to time. Under the
influence of this field, the embankment together with the earth moves
non-uniformly in such a manner that their original velocity in the
backwards direction is continuously reduced."



IN WHAT RESPECTS ARE THE FOUNDATIONS OF CLASSICAL MECHANICS AND OF THE
SPECIAL THEORY OF RELATIVITY UNSATISFACTORY?


We have already stated several times that classical mechanics starts
out from the following law: Material particles sufficiently far
removed from other material particles continue to move uniformly in a
straight line or continue in a state of rest. We have also repeatedly
emphasised that this fundamental law can only be valid for bodies of
reference K which possess certain unique states of motion, and which
are in uniform translational motion relative to each other. Relative
to other reference-bodies K the law is not valid. Both in classical
mechanics and in the special theory of relativity we therefore
differentiate between reference-bodies K relative to which the
recognised " laws of nature " can be said to hold, and
reference-bodies K relative to which these laws do not hold.

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Euclidean three-dimensional space, then there exists for this surface
a two-dimensional geometry, just as much as for a plane surface. Gauss
undertook the task of treating this two-dimensional geometry from
first principles, without making use of the fact that the surface
belongs to a Euclidean continuum of three dimensions. If we imagine
constructions to be made with rigid rods in the surface (similar to
that above with the marble slab), we should find that different laws
hold for these from those resulting on the basis of Euclidean plane
geometry. The surface is not a Euclidean continuum with respect to the
rods, and we cannot define Cartesian co-ordinates in the surface.
Gauss indicated the principles according to which we can treat the
geometrical relationships in the surface, and thus pointed out the way
to the method of Riemman of treating multi-dimensional, non-Euclidean
continuum. Thus it is that mathematicians long ago solved the formal
problems to which we are led by the general postulate of relativity.



GAUSSIAN CO-ORDINATES


According to Gauss, this combined analytical and geometrical mode of
handling the problem can be arrived at in the following way. We
imagine a system of arbitrary curves (see Fig. 4) drawn on the surface
of the table. These we designate as u-curves, and we indicate each of
them by means of a number. The Curves u= 1, u= 2 and u= 3 are drawn in
the diagram. Between the curves u= 1 and u= 2 we must imagine an
infinitely large number to be drawn, all of which correspond to real
numbers lying between 1 and 2. fig. 04 We have then a system of
u-curves, and this "infinitely dense" system covers the whole surface
of the table. These u-curves must not intersect each other, and
through each point of the surface one and only one curve must pass.
Thus a perfectly definite value of u belongs to every point on the
surface of the marble slab. In like manner we imagine a system of
v-curves drawn on the surface. These satisfy the same conditions as
the u-curves, they are provided with numbers in a corresponding
manner, and they may likewise be of arbitrary shape. It follows that a
value of u and a value of v belong to every point on the surface of
the table. We call these two numbers the co-ordinates of the surface
of the table (Gaussian co-ordinates). For example, the point P in the
diagram has the Gaussian co-ordinates u= 3, v= 1. Two neighbouring
points P and P1 on the surface then correspond to the co-ordinates

                       P:       u,v

                       P1:     u + du, v + dv,

where du and dv signify very small numbers. In a similar manner we may
indicate the distance (line-interval) between P and P1, as measured
with a little rod, by means of the very small number ds. Then
according to Gauss we have

                ds2 = g[11]du2 + 2g[12]dudv = g[22]dv2

where g[11], g[12], g[22], are magnitudes which depend in a perfectly
definite way on u and v. The magnitudes g[11], g[12] and g[22],
determine the behaviour of the rods relative to the u-curves and
v-curves, and thus also relative to the surface of the table. For the
case in which the points of the surface considered form a Euclidean
continuum with reference to the measuring-rods, but only in this case,
it is possible to draw the u-curves and v-curves and to attach numbers
to them, in such a manner, that we simply have :

                           ds2 = du2 + dv2

Under these conditions, the u-curves and v-curves are straight lines
in the sense of Euclidean geometry, and they are perpendicular to each
other. Here the Gaussian coordinates are samply Cartesian ones. It is
clear that Gauss co-ordinates are nothing more than an association of
two sets of numbers with the points of the surface considered, of such
a nature that numerical values differing very slightly from each other
are associated with neighbouring points " in space."

So far, these considerations hold for a continuum of two dimensions.
But the Gaussian method can be applied also to a continuum of three,
four or more dimensions. If, for instance, a continuum of four
dimensions be supposed available, we may represent it in the following
way. With every point of the continuum, we associate arbitrarily four
numbers, x[1], x[2], x[3], x[4], which are known as " co-ordinates."
Adjacent points correspond to adjacent values of the coordinates. If a
distance ds is associated with the adjacent points P and P1, this
distance being measurable and well defined from a physical point of
view, then the following formula holds:

ds2 = g[11]dx[1]^2 + 2g[12]dx[1]dx[2] . . . . g[44]dx[4]^2,

where the magnitudes g[11], etc., have values which vary with the
position in the continuum. Only when the continuum is a Euclidean one
is it possible to associate the co-ordinates x[1] . . x[4]. with the
points of the continuum so that we have simply

ds2 = dx[1]^2 + dx[2]^2 + dx[3]^2 + dx[4]^2.

In this case relations hold in the four-dimensional continuum which
are analogous to those holding in our three-dimensional measurements.

However, the Gauss treatment for ds2 which we have given above is not
always possible. It is only possible when sufficiently small regions
of the continuum under consideration may be regarded as Euclidean
continua. For example, this obviously holds in the case of the marble
slab of the table and local variation of temperature. The temperature
is practically constant for a small part of the slab, and thus the
geometrical behaviour of the rods is almost as it ought to be
according to the rules of Euclidean geometry. Hence the imperfections
of the construction of squares in the previous section do not show
themselves clearly until this construction is extended over a
considerable portion of the surface of the table.

We can sum this up as follows: Gauss invented a method for the
mathematical treatment of continua in general, in which "
size-relations " (" distances " between neighbouring points) are
defined. To every point of a continuum are assigned as many numbers
(Gaussian coordinates) as the continuum has dimensions. This is done
in such a way, that only one meaning can be attached to the
assignment, and that numbers (Gaussian coordinates) which differ by an
indefinitely small amount are assigned to adjacent points. The
Gaussian coordinate system is a logical generalisation of the
Cartesian co-ordinate system. It is also applicable to non-Euclidean
continua, but only when, with respect to the defined "size" or
"distance," small parts of the continuum under consideration behave
more nearly like a Euclidean system, the smaller the part of the
continuum under our notice.



THE SPACE-TIME CONTINUUM OF THE SPEICAL THEORY OF RELATIVITY CONSIDERED AS A
EUCLIDEAN CONTINUUM


We are now in a position to formulate more exactly the idea of
Minkowski, which was only vaguely indicated in Section 17. In
accordance with the special theory of relativity, certain co-ordinate
systems are given preference for the description of the
four-dimensional, space-time continuum. We called these " Galileian
co-ordinate systems." For these systems, the four co-ordinates x, y,
z, t, which determine an event or -- in other words, a point of the
four-dimensional continuum -- are defined physically in a simple
manner, as set forth in detail in the first part of this book. For the
transition from one Galileian system to another, which is moving
uniformly with reference to the first, the equations of the Lorentz
transformation are valid. These last form the basis for the derivation
of deductions from the special theory of relativity, and in themselves
they are nothing more than the expression of the universal validity of
the law of transmission of light for all Galileian systems of
reference.

Minkowski found that the Lorentz transformations satisfy the following
simple conditions. Let us consider two neighbouring events, the
relative position of which in the four-dimensional continuum is given
with respect to a Galileian reference-body K by the space co-ordinate
differences dx, dy, dz and the time-difference dt. With reference to a
second Galileian system we shall suppose that the corresponding
differences for these two events are dx1, dy1, dz1, dt1. Then these
magnitudes always fulfil the condition*

     dx2 + dy2 + dz2 - c^2dt2 = dx1 2 + dy1 2 + dz1 2 - c^2dt1 2.

The validity of the Lorentz transformation follows from this
condition. We can express this as follows: The magnitude

                   ds2 = dx2 + dy2 + dz2 - c^2dt2,

which belongs to two adjacent points of the four-dimensional
space-time continuum, has the same value for all selected (Galileian)
reference-bodies. If we replace x, y, z, sq. rt. -I . ct , by x[1],
x[2], x[3], x[4], we also obtaill the result that

             ds2 = dx[1]^2 + dx[2]^2 + dx[3]^2 + dx[4]^2.

is independent of the choice of the body of reference. We call the
magnitude ds the " distance " apart of the two events or
four-dimensional points.

Thus, if we choose as time-variable the imaginary variable sq. rt. -I

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If the reader has followed all our previous considerations, he will
have no further difficulty in understanding the methods leading to the
solution of the problem of gravitation.

We start off on a consideration of a Galileian domain, i.e. a domain
in which there is no gravitational field relative to the Galileian
reference-body K. The behaviour of measuring-rods and clocks with
reference to K is known from the special theory of relativity,
likewise the behaviour of "isolated" material points; the latter move
uniformly and in straight lines.

Now let us refer this domain to a random Gauss coordinate system or to
a "mollusc" as reference-body K1. Then with respect to K1 there is a
gravitational field G (of a particular kind). We learn the behaviour
of measuring-rods and clocks and also of freely-moving material points
with reference to K1 simply by mathematical transformation. We
interpret this behaviour as the behaviour of measuring-rods, docks and
material points tinder the influence of the gravitational field G.
Hereupon we introduce a hypothesis: that the influence of the
gravitational field on measuringrods, clocks and freely-moving
material points continues to take place according to the same laws,
even in the case where the prevailing gravitational field is not
derivable from the Galfleian special care, simply by means of a
transformation of co-ordinates.

The next step is to investigate the space-time behaviour of the
gravitational field G, which was derived from the Galileian special
case simply by transformation of the coordinates. This behaviour is
formulated in a law, which is always valid, no matter how the
reference-body (mollusc) used in the description may be chosen.

This law is not yet the general law of the gravitational field, since
the gravitational field under consideration is of a special kind. In
order to find out the general law-of-field of gravitation we still
require to obtain a generalisation of the law as found above. This can
be obtained without caprice, however, by taking into consideration the
following demands:

(a) The required generalisation must likewise satisfy the general
postulate of relativity.

(b) If there is any matter in the domain under consideration, only its
inertial mass, and thus according to Section 15 only its energy is
of importance for its etfect in exciting a field.

(c) Gravitational field and matter together must satisfy the law of
the conservation of energy (and of impulse).

Finally, the general principle of relativity permits us to determine
the influence of the gravitational field on the course of all those
processes which take place according to known laws when a
gravitational field is absent i.e. which have already been fitted into
the frame of the special theory of relativity. In this connection we
proceed in principle according to the method which has already been
explained for measuring-rods, clocks and freely moving material
points.

The theory of gravitation derived in this way from the general
postulate of relativity excels not only in its beauty ; nor in
removing the defect attaching to classical mechanics which was brought
to light in Section 21; nor in interpreting the empirical law of
the equality of inertial and gravitational mass ; but it has also
already explained a result of observation in astronomy, against which
classical mechanics is powerless.

If we confine the application of the theory to the case where the
gravitational fields can be regarded as being weak, and in which all
masses move with respect to the coordinate system with velocities
which are small compared with the velocity of light, we then obtain as
a first approximation the Newtonian theory. Thus the latter theory is
obtained here without any particular assumption, whereas Newton had to
introduce the hypothesis that the force of attraction between mutually
attracting material points is inversely proportional to the square of
the distance between them. If we increase the accuracy of the
calculation, deviations from the theory of Newton make their
appearance, practically all of which must nevertheless escape the test
of observation owing to their smallness.

We must draw attention here to one of these deviations. According to
Newton's theory, a planet moves round the sun in an ellipse, which
would permanently maintain its position with respect to the fixed
stars, if we could disregard the motion of the fixed stars themselves
and the action of the other planets under consideration. Thus, if we
correct the observed motion of the planets for these two influences,
and if Newton's theory be strictly correct, we ought to obtain for the
orbit of the planet an ellipse, which is fixed with reference to the
fixed stars. This deduction, which can be tested with great accuracy,
has been confirmed for all the planets save one, with the precision
that is capable of being obtained by the delicacy of observation
attainable at the present time. The sole exception is Mercury, the
planet which lies nearest the sun. Since the time of Leverrier, it has
been known that the ellipse corresponding to the orbit of Mercury,
after it has been corrected for the influences mentioned above, is not
stationary with respect to the fixed stars, but that it rotates
exceedingly slowly in the plane of the orbit and in the sense of the
orbital motion. The value obtained for this rotary movement of the
orbital ellipse was 43 seconds of arc per century, an amount ensured
to be correct to within a few seconds of arc. This effect can be
explained by means of classical mechanics only on the assumption of
hypotheses which have little probability, and which were devised
solely for this purponse.

On the basis of the general theory of relativity, it is found that the
ellipse of every planet round the sun must necessarily rotate in the
manner indicated above ; that for all the planets, with the exception
of Mercury, this rotation is too small to be detected with the
delicacy of observation possible at the present time ; but that in the
case of Mercury it must amount to 43 seconds of arc per century, a
result which is strictly in agreement with observation.

Apart from this one, it has hitherto been possible to make only two
deductions from the theory which admit of being tested by observation,
to wit, the curvature of light rays by the gravitational field of the
sun,*x and a displacement of the spectral lines of light reaching
us from large stars, as compared with the corresponding lines for
light produced in an analogous manner terrestrially (i.e. by the same
kind of atom).**  These two deductions from the theory have both
been confirmed.