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Relativity: The Special and General Theory

by Albert Einstein


Authorised Translation by Robert W. Lawson




ALBERT EINSTEIN REFERENCE ARCHIVE
RELATIVITY: THE SPECIAL AND GENERAL THEORY
BY ALBERT EINSTEIN


Written: 1916 (this revised edition: 1924)
Source: Relativity: The Special and General Theory (1920)
Publisher: Methuen & Co Ltd
First Published: December, 1916
Translated: Robert W. Lawson (Authorised translation)
Transcription/Markup: Brian Basgen
Transcription to text: Gregory B. Newby
Thanks to: Einstein Reference Archive (marxists.org)
The Einstein Reference Archive is online at:
http://www.marxists.org/reference/archive/einstein/index.htm




Contents

 Preface

 Part I: The Special Theory of Relativity
 I. Physical Meaning of Geometrical Propositions
 II. The System of Co-ordinates
 III. Space and Time in Classical Mechanics
 IV. The Galileian System of Co-ordinates
 V. The Principle of Relativity (in the Restricted Sense)
 VI. The Theorem of the Addition of Velocities employed in Classical Mechanics
 VII. The Apparent Incompatability of the Law of Propagation of Light with the Principle of Relativity
 VIII. On the Idea of Time in Physics
 IX. The Relativity of Simultaneity
 X. On the Relativity of the Conception of Distance
 XI. The Lorentz Transformation
 XII. The Behaviour of Measuring-Rods and Clocks in Motion
 XIII. Theorem of the Addition of Velocities. The Experiment of Fizeau
 XIV. The Heuristic Value of the Theory of Relativity
 XV. General Results of the Theory
 XVI. Experience and the Special Theory of Relativity
 XVII. Minkowski’s Four-dimensional Space

 Part II: The General Theory of Relativity
 XVIII. Special and General Principle of Relativity
 XIX. The Gravitational Field
 XX. The Equality of Inertial and Gravitational Mass as an Argument for the General Postulate of Relativity
 XXI. In What Respects are the Foundations of Classical Mechanics and of the Special Theory of Relativity Unsatisfactory?
 XXII. A Few Inferences from the General Principle of Relativity
 XXIII. Behaviour of Clocks and Measuring-Rods on a Rotating Body of Reference
 XXIV. Euclidean and non-Euclidean Continuum
 XXV. Gaussian Co-ordinates
 XXVI. The Space-Time Continuum of the Special Theory of Relativity Considered as a Euclidean Continuum
 XXVII. The Space-Time Continuum of the General Theory of Relativity is Not a Euclidean Continuum
 XXVIII. Exact Formulation of the General Principle of Relativity
 XXIX. The Solution of the Problem of Gravitation on the Basis of the General Principle of Relativity

 Part III: Considerations on the Universe as a Whole
 XXX. Cosmological Difficulties of Newton’s Theory
 XXXI. The Possibility of a “Finite” and yet “Unbounded” Universe
 XXXII. The Structure of Space According to the General Theory of Relativity

 Appendices:
 I. Simple Derivation of the Lorentz Transformation (supplementary to section XI)
 II. Minkowski’s Four-Dimensional Space (“World”) (supplementary to section XVII)
 III. The Experimental Confirmation of the General Theory of Relativity
 IV. The Structure of Space According to the General Theory of Relativity (supplementary to section XXXII)
 V. Relativity and the Problem of Space


Note: The fifth Appendix was added by Einstein at the time of the
fifteenth re-printing of this book; and as a result is still under
copyright restrictions so cannot be added without the permission of the
publisher.




PREFACE


The present book is intended, as far as possible, to give an exact
insight into the theory of Relativity to those readers who, from a
general scientific and philosophical point of view, are interested in
the theory, but who are not conversant with the mathematical apparatus
of theoretical physics. The work presumes a standard of education
corresponding to that of a university matriculation examination, and,
despite the shortness of the book, a fair amount of patience and force
of will on the part of the reader. The author has spared himself no
pains in his endeavour to present the main ideas in the simplest and
most intelligible form, and on the whole, in the sequence and
connection in which they actually originated. In the interest of
clearness, it appeared to me inevitable that I should repeat myself
frequently, without paying the slightest attention to the elegance of
the presentation. I adhered scrupulously to the precept of that
brilliant theoretical physicist L. Boltzmann, according to whom matters
of elegance ought to be left to the tailor and to the cobbler. I make
no pretence of having withheld from the reader difficulties which are
inherent to the subject. On the other hand, I have purposely treated
the empirical physical foundations of the theory in a “step-motherly”
fashion, so that readers unfamiliar with physics may not feel like the
wanderer who was unable to see the forest for the trees. May the book
bring some one a few happy hours of suggestive thought!

December, 1916

 A. EINSTEIN




PART I: THE SPECIAL THEORY OF RELATIVITY




I.
PHYSICAL MEANING OF GEOMETRICAL PROPOSITIONS


In your schooldays most of you who read this book made acquaintance
with the noble building of Euclid’s geometry, and you remember—perhaps
with more respect than love—the magnificent structure, on the lofty
staircase of which you were chased about for uncounted hours by
conscientious teachers. By reason of our past experience, you would
certainly regard everyone with disdain who should pronounce even the
most out-of-the-way proposition of this science to be untrue. But
perhaps this feeling of proud certainty would leave you immediately if
some one were to ask you: “What, then, do you mean by the assertion
that these propositions are true?” Let us proceed to give this question
a little consideration.

Geometry sets out from certain conceptions such as “plane,” “point,”
and “straight line,” with which we are able to associate more or less
definite ideas, and from certain simple propositions (axioms) which, in
virtue of these ideas, we are inclined to accept as “true.” Then, on
the basis of a logical process, the justification of which we feel
ourselves compelled to admit, all remaining propositions are shown to
follow from those axioms, _i.e._ they are proven. A proposition is then
correct (“true”) when it has been derived in the recognised manner from
the axioms. The question of “truth” of the individual geometrical
propositions is thus reduced to one of the “truth” of the axioms. Now
it has long been known that the last question is not only unanswerable
by the methods of geometry, but that it is in itself entirely without
meaning. We cannot ask whether it is true that only one straight line
goes through two points. We can only say that Euclidean geometry deals
with things called “straight lines,” to each of which is ascribed the
property of being uniquely determined by two points situated on it. The
concept “true” does not tally with the assertions of pure geometry,
because by the word “true” we are eventually in the habit of
designating always the correspondence with a “real” object; geometry,
however, is not concerned with the relation of the ideas involved in it
to objects of experience, but only with the logical connection of these
ideas among themselves.

It is not difficult to understand why, in spite of this, we feel
constrained to call the propositions of geometry “true.” Geometrical
ideas correspond to more or less exact objects in nature, and these
last are undoubtedly the exclusive cause of the genesis of those ideas.
Geometry ought to refrain from such a course, in order to give to its
structure the largest possible logical unity. The practice, for
example, of seeing in a “distance” two marked positions on a
practically rigid body is something which is lodged deeply in our habit
of thought. We are accustomed further to regard three points as being
situated on a straight line, if their apparent positions can be made to
coincide for observation with one eye, under suitable choice of our
place of observation.

If, in pursuance of our habit of thought, we now supplement the
propositions of Euclidean geometry by the single proposition that two
points on a practically rigid body always correspond to the same
distance (line-interval), independently of any changes in position to
which we may subject the body, the propositions of Euclidean geometry
then resolve themselves into propositions on the possible relative
position of practically rigid bodies.[1] Geometry which has been
supplemented in this way is then to be treated as a branch of physics.
We can now legitimately ask as to the “truth” of geometrical
propositions interpreted in this way, since we are justified in asking
whether these propositions are satisfied for those real things we have
associated with the geometrical ideas. In less exact terms we can
express this by saying that by the “truth” of a geometrical proposition
in this sense we understand its validity for a construction with rule
and compasses.


 [1] It follows that a natural object is associated also with a
 straight line. Three points _A, B_ and _C_ on a rigid body thus lie in
 a straight line when the points _A_ and _C_ being given, _B_ is chosen
 such that the sum of the distances _AB_ and _BC_ is as short as
 possible. This incomplete suggestion will suffice for the present
 purpose.


Of course the conviction of the “truth” of geometrical propositions in
this sense is founded exclusively on rather incomplete experience. For
the present we shall assume the “truth” of the geometrical
propositions, then at a later stage (in the general theory of
relativity) we shall see that this “truth” is limited, and we shall
consider the extent of its limitation.




II.
THE SYSTEM OF CO-ORDINATES


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.[2]


 [2] 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.


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 “Trafalgar Square,
London”[3] I arrive at the following result. The earth is the rigid
body to which the specification of place refers; “Trafalgar Square,
London” is a well-defined point, to which a name has been assigned, and
with which the event coincides in space.[4]


 [3]

I have chosen this as being more familiar to the English reader than
the “Potzdammer Platz, Berlin,” which is referred to in the original.
(R. W. L.)


 [4] 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.


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 Trafalgar 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.[5]


 [5] 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.


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.


III.

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”[6], but only a trajectory
relative to a particular body of reference.


 [6] That is, a curve along which the body moves.


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.


IV. 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 Galilei-Newton can be regarded as valid only for a Galileian system
of co-ordinates.


V.

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 _K′_ 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 _K′_ 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_, _K′_ is a uniformly moving co-ordinate
system devoid of rotation, then natural phenomena run their course with
respect to _K′_ 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
very much in favour of the validity of the principle of relativity.
Even though classical mechanics does not supply us with a sufficiently
broad basis for the theoretical presentation of all physical phenomena,
still we must grant it a considerable measure of “truth,” since it
supplies us with the actual motions of the heavenly bodies with a
delicacy of detail little short of wonderful. The principle of
relativity must therefore apply with great accuracy in the domain of
_mechanics_. But that a principle of such broad generality should hold
with such exactness in one domain of phenomena, and yet should be
invalid for another, is _a priori_ not very probable.

We now proceed to the second argument, to which, moreover, we shall
return later. If the principle of relativity (in the restricted sense)
does not hold, then the Galileian co-ordinate systems _K, K′, K″_,
etc., which are moving uniformly relative to each other, will not be
_equivalent_ for the description of natural phenomena. In this case we
should be constrained to believe that natural laws are capable of being
formulated in a particularly simple manner, and of course only on
condition that, from amongst all possible Galileian co-ordinate
systems, we should have chosen _one_ (_K0_) of a particular state of
motion as our body of reference. We should then be justified (because
of its merits for the description of natural phenomena) in calling this
system “absolutely at rest,” and all other Galileian systems _K_ “in
motion.” If, for instance, our embankment were the system _K0_ then our
railway carriage would be a system _K_, relative to which less simple
laws would hold than with respect to _K0_. This diminished simplicity
would be due to the fact that the carriage _K_ would be in motion
(_i.e._ “really”)with respect to _K0_. In the general laws of nature
which have been formulated with reference to _K_, the magnitude and
direction of the velocity of the carriage would necessarily play a
part. We should expect, for instance, that the note emitted by an
organpipe placed with its axis parallel to the direction of travel
would be different from that emitted if the axis of the pipe were
placed perpendicular to this direction.

Now in virtue of its motion in an orbit round the sun, our earth is
comparable with a railway carriage travelling with a velocity of about
30 kilometres per second. If the principle of relativity were not valid
we should therefore expect that the direction of motion of the earth at
any moment would enter into the laws of nature, and also that physical
systems in their behaviour would be dependent on the orientation in
space with respect to the earth. For owing to the alteration in
direction of the velocity of revolution of the earth in the course of a
year, the earth cannot be at rest relative to the hypothetical system
_K0_ throughout the whole year. However, the most careful observations
have never revealed such anisotropic properties in terrestrial physical
space, _i.e._ a physical non-equivalence of different directions. This
is very powerful argument in favour of the principle of relativity.


VI.

THE THEOREM OF THE ADDITION OF VELOCITIES EMPLOYED IN CLASSICAL
MECHANICS

Let us suppose our old friend the railway carriage to be travelling
along the rails with a constant velocity _v_, and that a man traverses
the length of the carriage in the direction of travel with a velocity
_w_. How quickly or, in other words, with what velocity _W_ does the
man advance relative to the embankment during the process? The only
possible answer seems to result from the following consideration: If
the man were to stand still for a second, he would advance relative to
the embankment through a distance _v_ equal numerically to the velocity
of the carriage. As a consequence of his walking, however, he traverses
an additional distance w relative to the carriage, and hence also
relative to the embankment, in this second, the distance w being
numerically equal to the velocity with which he is walking. Thus in
total he covers the distance _W = v + w_ relative to the embankment in
the second considered. We shall see later that this result, which
expresses the theorem of the addition of velocities employed in
classical mechanics, cannot be maintained; in other words, the law that
we have just written down does not hold in reality. For the time being,
however, we shall assume its correctness.


VII.

THE APPARENT INCOMPATIBILITY OF THE LAW OF PROPAGATION OF LIGHT WITH
THE PRINCIPLE OF RELATIVITY

There is hardly a simpler law in physics than that according to which
light is propagated in empty space. Every child at school knows, or
believes he knows, that this propagation takes place in straight lines
with a velocity _c_ = 300,000 km./sec. At all events we know with great
exactness that this velocity is the same for all colours, because if
this were not the case, the minimum of emission would not be observed
simultaneously for different colours during the eclipse of a fixed star
by its dark neighbour. By means of similar considerations based on
observations of double stars, the Dutch astronomer De Sitter was also
able to show that the velocity of propagation of light cannot depend on
the velocity of motion of the body emitting the light. The assumption
that this velocity of propagation is dependent on the direction “in
space” is in itself improbable.

In short, let us assume that the simple law of the constancy of the
velocity of light _c_ (in vacuum) is justifiably believed by the child
at school. Who would imagine that this simple law has plunged the
conscientiously thoughtful physicist into the greatest intellectual
difficulties? Let us consider how these difficulties arise.

Of course we must refer the process of the propagation of light (and
indeed every other process) to a rigid reference-body (co-ordinate
system). As such a system let us again choose our embankment. We shall
imagine the air above it to have been removed. If a ray of light be
sent along the embankment, we see from the above that the tip of the
ray will be transmitted with the velocity _c_ relative to the
embankment. Now let us suppose that our railway carriage is again
travelling along the railway lines with the velocity _v_, and that its
direction is the same as that of the ray of light, but its velocity of
course much less. Let us inquire about the velocity of propagation of
the ray of light relative to the carriage. It is obvious that we can
here apply the consideration of the previous section, since the ray of
light plays the part of the man walking along relatively to the
carriage. The velocity _W_ of the man relative to the embankment is
here replaced by the velocity of light relative to the embankment. _w_
is the required velocity of light with respect to the carriage, and we
have

_w = c – v._

The velocity of propagation ot a ray of light relative to the carriage
thus comes out smaller than _c_.

But this result comes into conflict with the principle of relativity
set forth in Section V. For, like every other general law of nature,
the law of the transmission of light _in vacuo_ [in vacuum] must,
according to the principle of relativity, be the same for the railway
carriage as reference-body as when the rails are the body of reference.
But, from our above consideration, this would appear to be impossible.
If every ray of light is propagated relative to the embankment with the
velocity _c_, then for this reason it would appear that another law of
propagation of light must necessarily hold with respect to the
carriage—a result contradictory to the principle of relativity.

In view of this dilemma there appears to be nothing else for it than to
abandon either the principle of relativity or the simple law of the
propagation of light _in vacuo_. Those of you who have carefully
followed the preceding discussion are almost sure to expect that we
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 therefore
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 reality 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.


VIII.

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°) 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_ → _M_ with
the same velocity as along the length _B_ → _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_ → _M_ as for the path _B_ → _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[7] (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.


 [7] We suppose further that, when three events _A, B_ and _C_ occur in
 different places in such a manner that, if _A_ is simultaneous with
 _B_, and _B_ is simultaneous with _C_ (simultaneous in the sense of
 the above definition), then the criterion for the simultaneity of the
 pair of events _A, C_ is also satisfied. This assumption is a physical
 hypothesis about the law of propagation of light; it must certainly be
 fulfilled if we are to maintain the law of the constancy of the
 velocity of light _in vacuo_.


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_
position, of the pointers of the other clock, then identical “settings”
are always simultaneous (in the sense of the above definition).


IX.

THE RELATIVITY OF SIMULTANEITY

Up to now our considerations have been referred to a particular body of
reference, which we have styled a “railway embankment.” We suppose a
very long train travelling along the rails with the constant velocity v
and in the direction indicated in Fig 1. People travelling in this
train will with a vantage view the train as a rigid reference-body
(co-ordinate system); they regard all events in reference to the train.
Then every event which takes place along the line also takes place at a
particular point of the train. Also the definition of simultaneity can
be given relative to the train in exactly the same way as with respect
to the embankment. As a natural consequence, however, the following
question arises:

image001


Are two events (_e.g._ the two strokes of lightning _A_ and _B_) which
are simultaneous _with reference to the railway embankment_ also
simultaneous _relatively to the train?_ We shall show directly that the
answer must be in the negative.

When we say that the lightning strokes _A_ and _B_ are simultaneous
with respect to be embankment, we mean: the rays of light emitted at
the places _A_ and _B_, where the lightning occurs, meet each other at
the mid-point _M_ of the length _A_ → _B_ of the embankment. But the
events _A_ and _B_ also correspond to positions _A_ and _B_ on the
train. Let _M′_ be the mid-point of the distance _A_ → _B_ on the
travelling train. Just when the flashes (as judged from the embankment)
of lightning occur, this point _M′_ naturally coincides with the point
_M_ but it moves towards the right in the diagram with the velocity v
of the train. If an observer sitting in the position _M′_ in the train
did not possess this velocity, then he would remain permanently at M,
and the light rays emitted by the flashes of lightning _A_ and _B_
would reach him simultaneously, _i.e._ they would meet just where he is
situated. Now in reality (considered with reference to the railway
embankment) he is hastening towards the beam of light coming from _B_,
whilst he is riding on ahead of the beam of light coming from _A_.
Hence the observer will see the beam of light emitted from _B_ earlier
than he will see that emitted from _A_. Observers who take the railway
train as their reference-body must therefore come to the conclusion
that the lightning flash _B_ took place earlier than the lightning
flash _A_. We thus arrive at the important result:

Events which are simultaneous with reference to the embankment are not
simultaneous with respect to the train, and _vice versa_ (relativity of
simultaneity). Every reference-body (co-ordinate system) has its own
particular time; unless we are told the reference-body to which the
statement of time refers, there is no meaning in a statement of the
time of an event.

Now before the advent of the theory of relativity it had always tacitly
been assumed in physics that the statement of time had an absolute
significance, _i.e._ that it is independent of the state of motion of
the body of reference. But we have just seen that this assumption is
incompatible with the most natural definition of simultaneity; if we
discard this assumption, then the conflict between the law of the
propagation of light _in vacuo_ and the principle of relativity
(developed in Section VII) disappears.

We were led to that conflict by the considerations of Section VI, which
are now no longer tenable. In that section we concluded that the man in
the carriage, who traverses the distance _w per second_ relative to the
carriage, traverses the same distance also with respect to the
embankment _in each second_ of time. But, according to the foregoing
considerations, the time required by a particular occurrence with
respect to the carriage must not be considered equal to the duration of
the same occurrence as judged from the embankment (as reference-body).
Hence it cannot be contended that the man in walking travels the
distance _w_ relative to the railway line in a time which is equal to
one second as judged from the embankment.

Moreover, the considerations of Section VI are based on yet a second
assumption, which, in the light of a strict consideration, appears to
be arbitrary, although it was always tacitly made even before the
introduction of the theory of relativity.


X.

ON THE RELATIVITY OF THE CONCEPTION OF DISTANCE

Let us consider two particular points on the train [8] travelling along
the embankment with the velocity _v_, and inquire as to their distance
apart. We already know that it is necessary to have a body of reference
for the measurement of a distance, with respect to which body the
distance can be measured up. It is the simplest plan to use the train
itself as reference-body (co-ordinate system). An observer in the train
measures the interval by marking off his measuring-rod in a straight
line (_e.g._ along the floor of the carriage) as many times as is
necessary to take him from the one marked point to the other. Then the
number which tells us how often the rod has to be laid down is the
required distance.


 [8] _e.g._ the middle of the first and of the hundredth carriage.


It is a different matter when the distance has to be judged from the
railway line. Here the following method suggests itself. If we call
_A′_ and _B′_ the two points on the train whose distance apart is
required, then both of these points are moving with the velocity v
along the embankment. In the first place we require to determine the
points _A_ and _B_ of the embankment which are just being passed by the
two points _A′_ and _B′_ at a particular time t—judged from the
embankment. These points _A_ and _B_ of the embankment can be
determined by applying the definition of time given in Section VIII.
The distance between these points A and B is then measured by repeated
application of the measuring-rod along the embankment.

_A priori_ it is by no means certain that this last measurement will
supply us with the same result as the first. Thus the length of the
train as measured from the embankment may be different from that
obtained by measuring in the train itself. This circumstance leads us
to a second objection which must be raised against the apparently
obvious consideration of Section VI. Namely, if the man in the carriage
covers the distance _w_ in a unit of time—_measured from the
train_,—then this distance—_as measured from the embankment_ is not
necessarily also equal to _w_.


XI.

THE LORENTZ TRANSFORMATION

The results of the last three sections show that the apparent
incompatibility of the law of propagation of light with the principle
of relativity (Section VII) has been derived by means of a
consideration which borrowed two unjustifiable hypotheses from
classical mechanics; these are as follows:

(1) The time-interval (time) between two events is independent of the
condition of motion of the body of reference.


(2) The space-interval (distance) between two points of a rigid body is
independent of the condition of motion of the body of reference.


If we drop these hypotheses, then the dilemma of Section VII
disappears, because the theorem of the addition of velocities derived
in Section VI becomes invalid. The possibility presents itself that the
law of the propagation of light _in vacuo_ may be compatible with the
principle of relativity, and the question arises: How have we to modify
the considerations of Section VI in order to remove the apparent
disagreement between these two fundamental results of experience? This
question leads to a general one. In the discussion of Section VI we
have to do with places and times relative both to the train and to the
embankment. How are we to find the place and time of an event in
relation to the train, when we know the place and time of the event
with respect to the railway embankment? Is there a thinkable answer to
this question of such a nature that the law of transmission of light
_in vacuo_ does not contradict the principle of relativity? In other
words: Can we conceive of a relation between place and time of the
individual events relative to both reference-bodies, such that every
ray of light possesses the velocity of transmission _c_ relative to the
embankment and relative to the train? This question leads to a quite
definite positive answer, and to a perfectly definite transformation
law for the space-time magnitudes of an event when changing over from
one body of reference to another.

Before we deal with this, we shall introduce the following incidental
consideration. Up to the present we have only considered events taking
place along the embankment, which had mathematically to assume the
function of a straight line. In the manner indicated in Section II we
can imagine this reference-body supplemented laterally and in a
vertical direction by means of a framework of rods, so that an event
which takes place anywhere can be localised with reference to this
framework. Similarly, we can imagine the train travelling with the
velocity _v_ to be continued across the whole of space, so that every
event, no matter how far off it may be, could also be localised with
respect to the second framework. Without committing any fundamental
error, we can disregard the fact that in reality these frameworks would
continually interfere with each other, owing to the impenetrability of
solid bodies. In every such framework we imagine three surfaces
perpendicular to each other marked out, and designated as “co-ordinate
planes” (“co-ordinate system”). A co-ordinate system _K_ then
corresponds to the embankment, and a co-ordinate system _K′_ to the
train. An event, wherever it may have taken place, would be fixed in
space with respect to _K_ by the three perpendiculars _x, y, z_ on the
co-ordinate planes, and with regard to time by a time value _t_.
Relative to _K′, the same event_ would be fixed in respect of space and
time by corresponding values _x′, y′, z′, t′_, which of course are not
identical with _x, y, z, t_. It has already been set forth in detail
how these magnitudes are to be regarded as results of physical
measurements.

image002


Obviously our problem can be exactly formulated in the following
manner. What are the values _x′, y′, z′, t′_, of an event with respect
to _K′_, when the magnitudes _x, y, z, t_, of the same event with
respect to _K_ are given? The relations must be so chosen that the law
of the transmission of light in vacuo is satisfied for one and the same
ray of light (and of course for every ray) with respect to _K_ and
_K′_. For the relative orientation in space of the co-ordinate systems
indicated in the diagram (Fig. 2), this problem is solved by means of
the equations:

image003


_y′_ = _y_

_z′_ = _z_


image004


This system of equations is known as the “Lorentz transformation.”[9]


 [9] A simple derivation of the Lorentz transformation is given in
 Appendix I.


If in place of the law of transmission of light we had taken as our
basis the tacit assumptions of the older mechanics as to the absolute
character of times and lengths, then instead of the above we should
have obtained the following equations:

_x′_ = _x_ – _vt_


_y′_ = _y_


_z′_ = _z_


_t′_ = _t_


This system of equations is often termed the “Galilei transformation.”
The Galilei transformation can be obtained from the Lorentz
transformation by substituting an infinitely large value for the
velocity of light _c_ in the latter transformation.

Aided by the following illustration, we can readily see that, in
accordance with the Lorentz transformation, the law of the transmission
of light _in vacuo_ is satisfied both for the reference-body _K_ and
for the reference-body _K′_. A light-signal is sent along the positive
_x_-axis, and this light-stimulus advances in accordance with the
equation

_x_ = _ct_,


_i.e._ with the velocity _c_. According to the equations of the Lorentz
transformation, this simple relation between _x_ and _t_ involves a
relation between _x′_ and _t′_. In point of fact, if we substitute for
_x_ the value _ct_ in the first and fourth equations of the Lorentz
transformation, we obtain:

image005


from which, by division, the expression

_x′_ = _ct′_


immediately follows. If referred to the system _K′_, the propagation of
light takes place according to this equation. We thus see that the
velocity of transmission relative to the reference-body _K′_ is also
equal to _c_. The same result is obtained for rays of light advancing
in any other direction whatsoever. Of cause this is not surprising,
since the equations of the Lorentz transformation were derived
conformably to this point of view.


XII.

THE BEHAVIOUR OF MEASURING-RODS AND CLOCKS IN MOTION

Place a metre-rod in the _x′_-axis of _K′_ in such a manner that one
end (the beginning) coincides with the point _x′_ = 0 whilst the other
end (the end of the rod) coincides with the point _x′_ = 1. What is the
length of the metre-rod relatively to the system _K_? In order to learn
this, we need only ask where the beginning of the rod and the end of
the rod lie with respect to _K_ at a particular time _t_ of the system
_K_. By means of the first equation of the Lorentz transformation the
values of these two points at the time _t_ = 0 can be shown to be

image006


the distance between the points being

image007


But the metre-rod is moving with the velocity _v_ relative to _K_. It
therefore follows that the length of a rigid metre-rod moving in the
direction of its length with a velocity _v_ is

image008


of a metre. The rigid rod is thus shorter when in motion than when at
rest, and the more quickly it is moving, the shorter is the rod. For
the velocity _v_ = _c_ we should have

image009


and for still greater velocities the square-root becomes imaginary.
From this we conclude that in the theory of relativity the velocity _c_
plays the part of a limiting velocity, which can neither be reached nor
exceeded by any real body.

Of course this feature of the velocity _c_ as a limiting velocity also
clearly follows from the equations of the Lorentz transformation, for
these became meaningless if we choose values of _v_ greater than _c_.

If, on the contrary, we had considered a metre-rod at rest in the
_x_-axis with respect to _K_, then we should have found that the length
of the rod as judged from _K′_ would have been

image010


this is quite in accordance with the principle of relativity which
forms the basis of our considerations.

_A priori_ it is quite clear that we must be able to learn something
about the physical behaviour of measuring-rods and clocks from the
equations of transformation, for the magnitudes _z, y, x, t_, are
nothing more nor less than the results of measurements obtainable by
means of measuring-rods and clocks. If we had based our considerations
on the Galileian transformation we should not have obtained a
contraction of the rod as a consequence of its motion.

Let us now consider a seconds-clock which is permanently situated at
the origin (_x′_ = 0) of _K′_. _t′_ = 0 and _t′_ = 1 are two successive
ticks of this clock. The first and fourth equations of the Lorentz
transformation give for these two ticks:

_t_ = 0

and

image011


As judged from _K_, the clock is moving with the velocity _v_; as
judged from this reference-body, the time which elapses between two
strokes of the clock is not one second, but

image012


seconds, _i.e._ a somewhat larger time. As a consequence of its motion
the clock goes more slowly than when at rest. Here also the velocity
_c_ plays the part of an unattainable limiting velocity.


XIII.

THEOREM OF THE ADDITION OF VELOCITIES. THE EXPERIMENT OF FIZEAU

Now in practice we can move clocks and measuring-rods only with
velocities that are small compared with the velocity of light; hence we
shall hardly be able to compare the results of the previous section
directly with the reality. But, on the other hand, these results must
strike you as being very singular, and for that reason I shall now draw
another conclusion from the theory, one which can easily be derived
from the foregoing considerations, and which has been most elegantly
confirmed by experiment.

In Section VI we derived the theorem of the addition of velocities in
one direction in the form which also results from the hypotheses of
classical mechanics. This theorem can also be deduced readily from the
Galilei transformation (Section XI). In place of the man walking inside
the carriage, we introduce a point moving relatively to the co-ordinate
system _K′_ in accordance with the equation

_x′_ = _wt′_

By means of the first and fourth equations of the Galilei
transformation we can express _x′_ and _t′_ in terms of _x_ and _t_,
and we then obtain

_x_ = (_v_ + _w_)_t_

This equation expresses nothing else than the law of motion of the
point with reference to the system _K_ (of the man with reference to
the embankment). We denote this velocity by the symbol _W_, and we then
obtain, as in Section VI,

_W_ = _v_ + _w_ . . . . . . . (A).

But we can carry out this consideration just as well on the basis of
the theory of relativity. In the equation

_x′_ = _wt′_

we must then express _x′_ and _t′_ in terms of _x_ and _t_, making use
of the first and fourth equations of the _Lorentz transformation_.
Instead of the equation (A) we then obtain the equation

image013


which corresponds to the theorem of addition for velocities in one
direction according to the theory of relativity. The question now
arises as to which of these two theorems is the better in accord with
experience. On this point we are enlightened by a most important
experiment which the brilliant physicist Fizeau performed more than
half a century ago, and which has been repeated since then by some of
the best experimental physicists, so that there can be no doubt about
its result. The experiment is concerned with the following question.
Light travels in a motionless liquid with a particular velocity _w_.
How quickly does it travel in the direction of the arrow in the tube
_T_ (see the accompanying diagram, Fig. 3) when the liquid above
mentioned is flowing through the tube with a velocity _v_?

image014


In accordance with the principle of relativity we shall certainly have
to take for granted that the propagation of light always takes place
with the same velocity _w with respect to the liquid_, whether the
latter is in motion with reference to other bodies or not. The velocity
of light relative to the liquid and the velocity of the latter relative
to the tube are thus known, and we require the velocity of light
relative to the tube.

It is clear that we have the problem of Section VI again before us. The
tube plays the part of the railway embankment or of the co-ordinate
system _K_, the liquid plays the part of the carriage or of the
co-ordinate system _K′_, and finally, the light plays the part of the
man walking along the carriage, or of the moving point in the present
section. If we denote the velocity of the light relative to the tube by
_W_, then this is given by the equation (A) or (B), according as the
Galilei transformation or the Lorentz transformation corresponds to the
facts. Experiment[10] decides in favour of equation (B) derived from
the theory of relativity, and the agreement is, indeed, very exact.
According to recent and most excellent measurements by Zeeman, the
influence of the velocity of flow _v_ on the propagation of light is
represented by formula (B) to within one per cent.


 [10] Fizeau found


image015


where


image016


is the index of refraction of the liquid. On the other hand, owing to
the smallness of


image017


as compared with 1, we can replace (B) in the first place by


image018


or to the same order of approximation by


image019


which agrees with Fizeau’s result.


Nevertheless we must now draw attention to the fact that a theory of
this phenomenon was given by H. A. Lorentz long before the statement of
the theory of relativity. This theory was of a purely electrodynamical
nature, and was obtained by the use of particular hypotheses as to the
electromagnetic structure of matter. This circumstance, however, does
not in the least diminish the conclusiveness of the experiment as a
crucial test in favour of the theory of relativity, for the
electrodynamics of Maxwell-Lorentz, on which the original theory was
based, in no way opposes the theory of relativity. Rather has the
latter been developed trom electrodynamics as an astoundingly simple
combination and generalisation of the hypotheses, formerly independent
of each other, on which electrodynamics was built.


XIV.

THE HEURISTIC VALUE OF THE THEORY OF RELATIVITY

Our train of thought in the foregoing pages can be epitomised in the
following manner. Experience has led to the conviction that, on the one
hand, the principle of relativity holds true and that on the other hand
the velocity of transmission of light _in vacuo_ has to be considered
equal to a constant _c_. By uniting these two postulates we obtained
the law of transformation for the rectangular co-ordinates _x, y, z_
and the time _t_ of the events which constitute the processes of
nature. In this connection we did not obtain the Galilei
transformation, but, differing from classical mechanics, the _Lorentz
transformation_.

The law of transmission of light, the acceptance of which is justified
by our actual knowledge, played an important part in this process of
thought. Once in possession of the Lorentz transformation, however, we
can combine this with the principle of relativity, and sum up the
theory thus:

Every general law of nature must be so constituted that it is
transformed into a law of exactly the same form when, instead of the
space-time variables _x, y, z, t_ of the original coordinate system
_K_, we introduce new space-time variables _x′, y′, z′, t′_ of a
co-ordinate system _K′_. In this connection the relation between the
ordinary and the accented magnitudes is given by the Lorentz
transformation. Or in brief: General laws of nature are co-variant with
respect to Lorentz transformations.

This is a definite mathematical condition that the theory of relativity
demands of a natural law, and in virtue of this, the theory becomes a
valuable heuristic aid in the search for general laws of nature. If a
general law of nature were to be found which did not satisfy this
condition, then at least one of the two fundamental assumptions of the
theory would have been disproved. Let us now examine what general
results the latter theory has hitherto evinced.


XV.

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 hypotheses 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

image020


but by the expression

image021


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

image022


When

image023


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 conservation 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 conservation
of energy should hold not only with reference to a co-ordinate system
_K_, but also with respect to every co-ordinate system _K′_ 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[11] 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

image024



 [11] _E_0 is the energy taken up, as judged from a co-ordinate system
 moving with the body.


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

image025


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

image026


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

image027


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

image028


we see that the term _mc_2, which has hitherto attracted our attention,
is nothing else than the energy possessed by the body[12] before it
absorbed the energy _E_0.


 [12] As judged from a co-ordinate system moving with the body.


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.

image027


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 conservation
of mass as a law of independent validity.


 [Note] The equation E = mc2 has been thoroughly proved time and again
 since this time.


Let me add a final remark of a fundamental nature. The success of the
Faraday-Maxwell interpretation of electromagnetic action at a distance
resulted in physicists becoming convinced that there are no such things
as instantaneous actions at a distance (not involving an intermediary
medium) of the type of Newton’s law of gravitation.

According to the theory of relativity, action at a distance with the
velocity of light always takes the place of instantaneous action at a
distance or of action at a distance with an infinite velocity of
transmission. This is connected with the fact that the velocity _c_
plays a fundamental role in this theory. In Part II we shall see in
what way this result becomes modified in the general theory of
relativity.


XVI.

EXPERIENCE AND THE SPECIAL THEORY OF RELATIVITY

To what extent is the special theory of relativity supported by
experience? This question is not easily answered for the reason already
mentioned in connection with the fundamental experiment of Fizeau. The
special theory of relativity has crystallised out from the
Maxwell-Lorentz theory of electromagnetic phenomena. Thus all facts of
experience which support the electromagnetic theory also support the
theory of relativity. As being of particular importance, I mention here
the fact that the theory of relativity enables us to predict the
effects produced on the light reaching us from the fixed stars. These
results are obtained in an exceedingly simple manner, and the effects
indicated, which are due to the relative motion of the earth with
reference to those fixed stars are found to be in accord with
experience. We refer to the yearly movement of the apparent position of
the fixed stars resulting from the motion of the earth round the sun
(aberration), and to the influence of the radial components of the
relative motions of the fixed stars with respect to the earth on the
colour of the light reaching us from them. The latter effect manifests
itself in a slight displacement of the spectral lines of the light
transmitted to us from a fixed star, as compared with the position of
the same spectral lines when they are produced by a terrestrial source
of light (Doppler principle). The experimental arguments in favour of
the Maxwell-Lorentz theory, which are at the same time arguments in
favour of the theory of relativity, are too numerous to be set forth
here. In reality they limit the theoretical possibilities to such an
extent, that no other theory than that of Maxwell and Lorentz has been
able to hold its own when tested by experience.

But there are two classes of experimental facts hitherto obtained which
can be represented in the Maxwell-Lorentz theory only by the
introduction of an auxiliary hypothesis, which in itself—_i.e._ without
making use of the theory of relativity—appears extraneous.

It is known that cathode rays and the so-called β-rays emitted by
radioactive substances consist of negatively electrified particles
(electrons) of very small inertia and large velocity. By examining the
deflection of these rays under the influence of electric and magnetic
fields, we can study the law of motion of these particles very exactly.

In the theoretical treatment of these electrons, we are faced with the
difficulty that electrodynamic theory of itself is unable to give an
account of their nature. For since electrical masses of one sign repel
each other, the negative electrical masses constituting the electron
would necessarily be scattered under the influence of their mutual
repulsions, unless there are forces of another kind operating between
them, the nature of which has hitherto remained obscure to us.[13] If
we now assume that the relative distances between the electrical masses
constituting the electron remain unchanged during the motion of the
electron (rigid connection in the sense of classical mechanics), we
arrive at a law of motion of the electron which does not agree with
experience. Guided by purely formal points of view, H. A. Lorentz was
the first to introduce the hypothesis that the form of the electron
experiences a contraction in the direction of motion in consequence of
that motion. the contracted length being proportional to the expression

image029


This, hypothesis, which is not justifiable by any electrodynamical
facts, supplies us then with that particular law of motion which has
been confirmed with great precision in recent years.


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


The theory of relativity leads to the same law of motion, without
requiring any special hypothesis whatsoever as to the structure and the
behaviour of the electron. We arrived at a similar conclusion in
Section XIII in connection with the experiment of Fizeau, the result of
which is foretold by the theory of relativity without the necessity 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 V 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 _K′_ moving uniformly with respect to _K_, if we assume
that the relations of the Galileian transformation exist between the
co-ordinates of _K_ and _K′_. 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 _K′_ moving relatively to _K_
were to be regarded as in motion with respect to the æther. To this
motion of _K′_ against the æther (“æther-drift” relative to _K′_) were
attributed the more complicated laws which were supposed to hold
relative to _K′_. 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 _T′_ 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 _T′_ 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 difference in
time mentioned above. Comparison with the discussion in Section XII
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.


XVII.

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-ordinates) _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 _x1, y1, z1_, 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 _x1, y1, z1, t1_ 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 rôle, 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 (_t′_ = _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:

image030


Moreover, according to this equation the time difference Δ_t′_ of two
events with respect to _K′_ does not in general vanish, even when the
time difference Δ_t_ of the same events with reference to _K_ vanishes.
Pure “space-distance” of two events with respect to _K_ results in
“time-distance ” of the same events with respect to _K_. But the
discovery of Minkowski, which was of importance for the formal
development of the theory of relativity, does not lie here. It is to be
found rather in the fact of his recognition that the four-dimensional
space-time continuum of the theory of relativity, in its most essential
formal properties, shows a pronounced relationship to the
three-dimensional continuum of Euclidean geometrical space.[14] In
order to give due prominence to this relationship, however, we must
replace the usual time co-ordinate t by an imaginary magnitude

image031


proportional to it. Under these conditions, the natural laws satisfying
the demands of the (special) theory of relativity assume mathematical
forms, in which the time co-ordinate plays exactly the same role as the
three space co-ordinates. Formally, these four co-ordinates correspond
exactly to the three space co-ordinates in Euclidean geometry. It must
be clear even to the non-mathematician that, as a consequence of this
purely formal addition to our knowledge, the theory perforce gained
clearness in no mean measure.


 [14] Cf. the somewhat more detailed discussion in Appendix II.


These inadequate remarks can give the reader only a vague notion of the
important idea contributed by Minkowski. Without it the general theory
of relativity, of which the fundamental ideas are developed in the
following pages, would perhaps have got no farther than its long
clothes. Minkowski’s work is doubtless difficult of access to anyone
inexperienced in mathematics, but since it is not necessary to have a
very exact grasp of this work in order to understand the fundamental
ideas of either the special or the general theory of relativity, I
shall leave it here at present, and revert to it only towards the end
of Part II.


PART II: THE GENERAL THEORY OF RELATIVITY


XVIII.

SPECIAL AND GENERAL PRINCIPLE OF RELATIVITY

The basal principle, which was the pivot of all our previous
considerations, was the _special_ principle of relativity, _i.e._ the
principle of the physical relativity of all _uniform_ motion. Let as
once more analyse its meaning carefully.

It was at all times clear that, from the point of view of the idea it
conveys to us, every motion must be considered only as a relative
motion. Returning to the illustration we have frequently used of the
embankment and the railway carriage, we can express the fact of the
motion here taking place in the following two forms, both of which are
equally justifiable:

(_a_) The carriage is in motion relative to the embankment,


(_b_) The embankment is in motion relative to the carriage.


In (_a_) the embankment, in (_b_) the carriage, serves as the body of
reference in our statement of the motion taking place. If it is simply
a question of detecting or of describing the motion involved, it is in
principle immaterial to what reference-body we refer the motion. As
already mentioned, this is self-evident, but it must not be confused
with the much more comprehensive statement called “the principle of
relativity,” which we have taken as the basis of our investigations.

The principle we have made use of not only maintains that we may
equally well choose the carriage or the embankment as our
reference-body for the description of any event (for this, too, is
self-evident). Our principle rather asserts what follows: If we
formulate the general laws of nature as they are obtained from
experience, by making use of

(_a_) the embankment as reference-body,


(_b_) the railway carriage as reference-body,


then these general laws of nature (_e.g._ the laws of mechanics or the
law of the propagation of light _in vacuo_) have exactly the same form
in both cases. This can also be expressed as follows: For the physical
description of natural processes, neither of the reference bodies _K,
K′_ is unique (lit. “specially marked out”) as compared with the other.
Unlike the first, this latter statement need not of necessity hold _a
priori;_ it is not contained in the conceptions of “motion” and
“reference-body” and derivable from them; only _experience_ can decide
as to its correctness or incorrectness.

Up to the present, however, we have by no means maintained the
equivalence of _all_ bodies of reference _K_ in connection with the
formulation of natural laws. Our course was more on the following
Iines. In the first place, we started out from the assumption that
there exists a reference-body _K_, whose condition of motion is such
that the Galileian law holds with respect to it: A particle left to
itself and sufficiently far removed from all other particles moves
uniformly in a straight line. With reference to K (Galileian
reference-body) the laws of nature were to be as simple as possible.
But in addition to K, all bodies of reference _K′_ should be given
preference in this sense, and they should be exactly equivalent to _K_
for the formulation of natural laws, provided that they are in a state
of _uniform rectilinear and non-rotary motion_ with respect to _K_; all
these bodies of reference are to be regarded as Galileian
reference-bodies. The validity of the principle of relativity was
assumed only for these reference-bodies, but not for others (_e.g._
those possessing motion of a different kind). In this sense we speak of
the _special_ principle of relativity, or special theory of relativity.

In contrast to this we wish to understand by the “general principle of
relativity” the following statement: All bodies of reference _K, K′_,
etc., are equivalent for the description of natural phenomena
(formulation of the general laws of nature), whatever may be their
state of motion. But before proceeding farther, it ought to be pointed
out that this formulation must be replaced later by a more abstract
one, for reasons which will become evident at a later stage.

Since the introduction of the special principle of relativity has been
justified, every intellect which strives after generalisation must feel
the temptation to venture the step towards the general principle of
relativity. But a simple and apparently quite reliable consideration
seems to suggest that, for the present at any rate, there is little
hope of success in such an attempt; Let us imagine ourselves
transferred to our old friend the railway carriage, which is travelling
at a uniform rate. As long as it is moving uniformly, the occupant of
the carriage is not sensible of its motion, and it is for this reason
that he can without reluctance interpret the facts of the case as
indicating that the carriage is at rest, but the embankment in motion.
Moreover, according to the special principle of relativity, this
interpretation is quite justified also from a physical point of view.
If the motion of the carriage is now changed into a non-uniform motion,
as for instance by a powerful application of the brakes, then the
occupant of the carriage experiences a correspondingly powerful jerk
forwards. The retarded motion is manifested in the mechanical behaviour
of bodies relative to the person in the railway carriage. The
mechanical behaviour is different from that of the case previously
considered, and for this reason it would appear to be impossible that
the same mechanical laws hold relatively to the non-uniformly moving
carriage, as hold with reference to the carriage when at rest or in
uniform motion. At all events it is clear that the Galileian law does
not hold with respect to the non-uniformly moving carriage. Because of
this, we feel compelled at the present juncture to grant a kind of
absolute physical reality to non-uniform motion, in opposition to the
general principle of relativity. But in what follows we shall soon see
that this conclusion cannot be maintained.


XIX.

THE GRAVITATIONAL FIELD

“If we pick up a stone and then let it go, why does it fall to the
ground?” The usual answer to this question is: “Because it is attracted
by the earth.” Modern physics formulates the answer rather differently
for the following reason. As a result of the more careful study of
electromagnetic phenomena, we have come to regard action at a distance
as a process impossible without the intervention of some intermediary
medium. If, for instance, a magnet attracts a piece of iron, we cannot
be content to regard this as meaning that the magnet acts directly on
the iron through the intermediate empty space, but we are constrained
to imagine—after the manner of Faraday—that the magnet always calls
into being something physically real in the space around it, that
something being what we call a “magnetic field.” In its turn this
magnetic field operates on the piece of iron, so that the latter
strives to move towards the magnet. We shall not discuss here the
justification for this incidental conception, which is indeed a
somewhat arbitrary one. We shall only mention that with its aid
electromagnetic phenomena can be theoretically represented much more
satisfactorily than without it, and this applies particularly to the
transmission of electromagnetic waves. The effects of gravitation also
are regarded in an analogous manner.

The action of the earth on the stone takes place indirectly. The earth
produces in its surrounding a gravitational field, which acts on the
stone and produces its motion of fall. As we know from experience, the
intensity of the action on a body dimishes according to a quite
definite law, as we proceed farther and farther away from the earth.
From our point of view this means: The law governing the properties of
the gravitational field in space must be a perfectly definite one, in
order correctly to represent the diminution of gravitational action
with the distance from operative bodies. It is something like this: The
body (_e.g._ the earth) produces a field in its immediate neighbourhood
directly; the intensity and direction of the field at points farther
removed from the body are thence determined by the law which governs
the properties in space of the gravitational fields themselves.

In contrast to electric and magnetic fields, the gravitational field
exhibits a most remarkable property, which is of fundamental importance
for what follows. Bodies which are moving under the sole influence of a
gravitational field receive an acceleration, _which does not in the
least depend either on the material or on the physical 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:

image032


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_ mass.

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.


XX.

THE EQUALITY OF INERTIAL AND GRAVITATIONAL MASS AS AN ARGUMENT FOR THE
GENERAL POSTULATE 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 stretch 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 the end
of Section XVIII. 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.”


XXI.

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.

But no person whose mode of thought is logical can rest satisfied with
this condition of things. He asks: “How does it come that certain
reference-bodies (or their states of motion) are given priority over
other reference-bodies (or their states of motion)? _What is the reason
for this preference?_” In order to show clearly what I mean by this
question, I shall make use of a comparison.

I am standing in front of a gas range. Standing alongside of each other
on the range are two pans so much alike that one may be mistaken for
the other. Both are half full of water. I notice that steam is being
emitted continuously from the one pan, but not from the other. I am
surprised at this, even if I have never seen either a gas range or a
pan before. But if I now notice a luminous something of bluish colour
under the first pan but not under the other, I cease to be astonished,
even if I have never before seen a gas flame. For I can only say that
this bluish something will cause the emission of the steam, or at least
_possibly_ it may do so. If, however, I notice the bluish something in
neither case, and if I observe that the one continuously emits steam
whilst the other does not, then I shall remain astonished and
dissatisfied until I have discovered some circumstance to which I can
attribute the different behaviour of the two pans.

Analogously, I seek in vain for a real something in classical mechanics
(or in the special theory of relativity) to which I can attribute the
different behaviour of bodies considered with respect to the reference
systems _K_ and _K′_.[15] Newton saw this objection and attempted to
invalidate it, but without success. But E. Mach recognised it most
clearly of all, and because of this objection he claimed that mechanics
must be placed on a new basis. It can only be got rid of by means of a
physics which is conformable to the general principle of relativity,
since the equations of such a theory hold for every body of reference,
whatever may be its state of motion.


 [15] The objection is of importance more especially when the state of
 motion of the reference-body is of such a nature that it does not
 require any external agency for its maintenance, _e.g._ in the case
 when the reference-body is rotating uniformly.


XXII.

A FEW INFERENCES FROM THE GENERAL PRINCIPLE OF RELATIVITY

The considerations of Section XX show that the general principle of
relativity puts us in a position to derive properties of the
gravitational field in a purely theoretical manner. Let us suppose, for
instance, that we know the space-time “course” for any natural process
whatsoever, as regards the manner in which it takes place in the
Galileian domain relative to a Galileian body of reference _K_. By
means of purely theoretical operations (_i.e._ simply by calculation)
we are then able to find how this known natural process appears, as
seen from a reference-body _K′_ which is accelerated relatively to _K_.
But since a gravitational field exists with respect to this new body of
reference _K′_, our consideration also teaches us how the gravitational
field influences the process studied.

For example, we learn that a body which is in a state of uniform
rectilinear motion with respect to _K_ (in accordance with the law of
Galilei) is executing an accelerated and in general curvilinear motion
with respect to the accelerated reference-body _K′_ (chest). This
acceleration or curvature corresponds to the influence on the moving
body of the gravitational field prevailing relatively to _K_. It is
known that a gravitational field influences the movement of bodies in
this way, so that our consideration supplies us with nothing
essentially new.

However, we obtain a new result of fundamental importance when we carry
out the analogous consideration for a ray of light. With respect to the
Galileian reference-body _K_, such a ray of light is transmitted
rectilinearly with the velocity _c_. It can easily be shown that the
path of the same ray of light is no longer a straight line when we
consider it with reference to the accelerated chest (reference-body
_K′_). From this we conclude, _that, in general, rays of light are
propagated curvilinearly in gravitational fields._ In two respects this
result is of great importance.

In the first place, it can be compared with the reality. Although a
detailed examination of the question shows that the curvature of light
rays required by the general theory of relativity is only exceedingly
small for the gravitational fields at our disposal in practice, its
estimated magnitude for light rays passing the sun at grazing incidence
is nevertheless 1.7 seconds of arc. This ought to manifest itself in
the following way. As seen from the earth, certain fixed stars appear
to be in the neighbourhood of the sun, and are thus capable of
observation during a total eclipse of the sun. At such times, these
stars ought to appear to be displaced outwards from the sun by an
amount indicated above, as compared with their apparent position in the
sky when the sun is situated at another part of the heavens. The
examination of the correctness or otherwise of this deduction is a
problem of the greatest importance, the early solution of which is to
be expected of astronomers.[16]


 [16] By means of the star photographs of two expeditions equipped by a
 Joint Committee of the Royal and Royal Astronomical Societies, the
 existence of the deflection of light demanded by theory was first
 confirmed during the solar eclipse of 29th May, 1919. (Cf. Appendix
 III.)


In the second place our result shows that, according to the general
theory of relativity, the law of the constancy of the velocity of light
in vacuo, which constitutes one of the two fundamental assumptions in
the special theory of relativity and to which we have already
frequently referred, cannot claim any unlimited validity. A curvature
of rays of light can only take place when the velocity of propagation
of light varies with position. Now we might think that as a consequence
of this, the special theory of relativity and with it the whole theory
of relativity would be laid in the dust. But in reality this is not the
case. We can only conclude that the special theory of relativity cannot
claim an unlimited domain of validity; its results hold only so long as
we are able to disregard the influences of gravitational fields on the
phenomena (_e.g._ of light).

Since it has often been contended by opponents of the theory of
relativity that the special theory of relativity is overthrown by the
general theory of relativity, it is perhaps advisable to make the facts
of the case clearer by means of an appropriate comparison. Before the
development of electrodynamics the laws of electrostatics were looked
upon as the laws of electricity. At the present time we know that
electric fields can be derived correctly from electrostatic
considerations only for the case, which is never strictly realised, in
which the electrical masses are quite at rest relatively to each other,
and to the co-ordinate system. Should we be justified in saying that
for this reason electrostatics is overthrown by the field-equations of
Maxwell in electrodynamics? Not in the least. Electrostatics is
contained in electrodynamics as a limiting case; the laws of the latter
lead directly to those of the former for the case in which the fields
are invariable with regard to time. No fairer destiny could be allotted
to any physical theory, than that it should of itself point out the way
to the introduction of a more comprehensive theory, in which it lives
on as a limiting case.

In the example of the transmission of light just dealt with, we have
seen that the general theory of relativity enables us to derive
theoretically the influence of a gravitational field on the course of
natural processes, the laws of which are already known when a
gravitational field is absent. But the most attractive problem, to the
solution of which the general theory of relativity supplies the key,
concerns the investigation of the laws satisfied by the gravitational
field itself. Let us consider this for a moment.

We are acquainted with space-time domains which behave (approximately)
in a “Galileian” fashion under suitable choice of reference-body,
_i.e._ domains in which gravitational fields are absent. If we now
refer such a domain to a reference-body _K′_ possessing any kind of
motion, then relative to _K′_ there exists a gravitational field which
is variable with respect to space and time.[17] The character of this
field will of course depend on the motion chosen for _K′._ According to
the general theory of relativity, the general law of the gravitational
field must be satisfied for all gravitational fields obtainable in this
way. Even though by no means all gravitationial fields can be produced
in this way, yet we may entertain the hope that the general law of
gravitation will be derivable from such gravitational fields of a
special kind. This hope has been realised in the most beautiful manner.
But between the clear vision of this goal and its actual realisation it
was necessary to surmount a serious difficulty, and as this lies deep
at the root of things, I dare not withhold it from the reader. We
require to extend our ideas of the space-time continuum still farther.


 [17] This follows from a generalisation of the discussion in Section
 XX.


XXIII.

BEHAVIOUR OF CLOCKS AND MEASURING-RODS ON A ROTATING BODY OF REFERENCE

Hitherto I have purposely refrained from speaking about the physical
interpretation of space- and time-data in the case of the general
theory of relativity. As a consequence, I am guilty of a certain
slovenliness of treatment, which, as we know from the special theory of
relativity, is far from being unimportant and pardonable. It is now
high time that we remedy this defect; but I would mention at the
outset, that this matter lays no small claims on the patience and on
the power of abstraction of the reader.

We start off again from quite special cases, which we have frequently
used before. Let us consider a space time domain in which no
gravitational field exists relative to a reference-body _K_ whose state
of motion has been suitably chosen. _K_ is then a Galileian
reference-body as regards the domain considered, and the results of the
special theory of relativity hold relative to _K_. Let us suppose the
same domain referred to a second body of reference _K′_, which is
rotating uniformly with respect to _K_. In order to fix our ideas, we
shall imagine _K′_ to be in the form of a plane circular disc, which
rotates uniformly in its own plane about its centre. An observer who is
sitting eccentrically on the disc _K′_ is sensible of a force which
acts outwards in a radial direction, and which would be interpreted as
an effect of inertia (centrifugal force) by an observer who was at rest
with respect to the original reference-body _K_. But the observer on
the disc may regard his disc as a reference-body which is “at rest”; on
the basis of the general principle of relativity he is justified in
doing this. The force acting on himself, and in fact on all other
bodies which are at rest relative to the disc, he regards as the effect
of a gravitational field. Nevertheless, the space-distribution of this
gravitational field is of a kind that would not be possible on Newton’s
theory of gravitation.[18] But since the observer believes in the
general theory of relativity, this does not disturb him; he is quite in
the right when he believes that a general law of gravitation can be
formulated—a law which not only explains the motion of the stars
correctly, but also the field of force experienced by himself.


 [18] The field disappears at the centre of the disc and increases
 proportionally to the distance from the centre as we proceed outwards.


The observer performs experiments on his circular disc with clocks and
measuring-rods. In doing so, it is his intention to arrive at exact
definitions for the signification of time- and space-data with
reference to the circular disc _K′_, these definitions being based on
his observations. What will be his experience in this enterprise?

To start with, he places one of two identically constructed clocks at
the centre of the circular disc, and the other on the edge of the disc,
so that they are at rest relative to it. We now ask ourselves whether
both clocks go at the same rate from the standpoint of the non-rotating
Galileian reference-body _K_. As judged from this body, the clock at
the centre of the disc has no velocity, whereas the clock at the edge
of the disc is in motion relative to _K_ in consequence of the
rotation. According to a result obtained in Section XII, it follows
that the latter clock goes at a rate permanently slower than that of
the clock at the centre of the circular disc, _i.e._ as observed from
_K_. It is obvious that the same effect would be noted by an observer
whom we will imagine sitting alongside his clock at the centre of the
circular disc. Thus on our circular disc, or, to make the case more
general, in every gravitational field, a clock will go more quickly or
less quickly, according to the position in which the clock is situated
(at rest). For this reason it is not possible to obtain a reasonable
definition of time with the aid of clocks which are arranged at rest
with respect to the body of reference. A similar difficulty presents
itself when we attempt to apply our earlier definition of simultaneity
in such a case, but I do not wish to go any farther into this question.

Moreover, at this stage the definition of the space co-ordinates also
presents insurmountable difficulties. If the observer applies his
standard measuring-rod (a rod which is short as compared with the
radius of the disc) tangentially to the edge of the disc, then, as
judged from the Galileian system, the length of this rod will be less
than 1, since, according to Section XII, moving bodies suffer a
shortening in the direction of the motion. On the other hand, the
measuring-rod will not experience a shortening in length, as judged
from _K_, if it is applied to the disc in the direction of the radius.
If, then, the observer first measures the circumference of the disc
with his measuring-rod and then the diameter of the disc, on dividing
the one by the other, he will not obtain as quotient the familiar
number π = 3.14 . . ., but a larger number,[19] whereas of course, for
a disc which is at rest with respect to _K_, this operation would yield
π exactly. This proves that the propositions of Euclidean geometry
cannot hold exactly on the rotating disc, nor in general in a
gravitational field, at least if we attribute the length 1 to the rod
in all positions and in every orientation. Hence the idea of a straight
line also loses its meaning. We are therefore not in a position to
define exactly the co-ordinates _x, y, z_ relative to the disc by means
of the method used in discussing the special theory, and as long as the
co-ordinates and times of events have not been defined, we cannot
assign an exact meaning to the natural laws in which these occur.


 [19] Throughout this consideration we have to use the Galileian
 (non-rotating) system _K_ as reference-body, since we may only assume
 the validity of the results of the special theory of relativity
 relative to _K_ (relative to _K′_ a gravitational field prevails).


Thus all our previous conclusions based on general relativity would
appear to be called in question. In reality we must make a subtle
detour in order to be able to apply the postulate of general relativity
exactly. I shall prepare the reader for this in the following
paragraphs.


XXIV.

EUCLIDEAN AND NON-EUCLIDEAN CONTINUUM

The surface of a marble table is spread out in front of me. I can get
from any one point on this table to any other point by passing
continuously from one point to a “neighbouring” one, and repeating this
process a (large) number of times, or, in other words, by going from
point to point without executing “jumps.” I am sure the reader will
appreciate with sufficient clearness what I mean here by “neighbouring”
and by “jumps” (if he is not too pedantic). We express this property of
the surface by describing the latter as a continuum.

Let us now imagine that a large number of little rods of equal length
have been made, their lengths being small compared with the dimensions
of the marble slab. When I say they are of equal length, I mean that
one can be laid on any other without the ends overlapping. We next lay
four of these little rods on the marble slab so that they constitute a
quadrilateral figure (a square), the diagonals of which are equally
long. To ensure the equality of the diagonals, we make use of a little
testing-rod. To this square we add similar ones, each of which has one
rod in common with the first. We proceed in like manner with each of
these squares until finally the whole marble slab is laid out with
squares. The arrangement is such, that each side of a square belongs to
two squares and each corner to four squares.

It is a veritable wonder that we can carry out this business without
getting into the greatest difficulties. We only need to think of the
following. If at any moment three squares meet at a corner, then two
sides of the fourth square are already laid, and, as a consequence, the
arrangement of the remaining two sides of the square is already
completely determined. But I am now no longer able to adjust the
quadrilateral so that its diagonals may be equal. If they are equal of
their own accord, then this is an especial favour of the marble slab
and of the little rods, about which I can only be thankfully surprised.
We must experience many such surprises if the construction is to be
successful.

If everything has really gone smoothly, then I say that the points of
the marble slab constitute a Euclidean continuum with respect to the
little rod, which has been used as a “distance” (line-interval). By
choosing one corner of a square as “origin” I can characterise every
other corner of a square with reference to this origin by means of two
numbers. I only need state how many rods I must pass over when,
starting from the origin, I proceed towards the “right” and then
“upwards,” in order to arrive at the corner of the square under
consideration. These two numbers are then the “Cartesian co-ordinates”
of this corner with reference to the “Cartesian co-ordinate system”
which is determined by the arrangement of little rods.

By making use of the following modification of this abstract
experiment, we recognise that there must also be cases in which the
experiment would be unsuccessful. We shall suppose that the rods
“expand” by in amount proportional to the increase of temperature. We
heat the central part of the marble slab, but not the periphery, in
which case two of our little rods can still be brought into coincidence
at every position on the table. But our construction of squares must
necessarily come into disorder during the heating, because the little
rods on the central region of the table expand, whereas those on the
outer part do not.

With reference to our little rods—defined as unit lengths—the marble
slab is no longer a Euclidean continuum, and we are also no longer in
the position of defining Cartesian co-ordinates directly with their
aid, since the above construction can no longer be carried out. But
since there are other things which are not influenced in a similar
manner to the little rods (or perhaps not at all) by the temperature of
the table, it is possible quite naturally to maintain the point of view
that the marble slab is a “Euclidean continuum.” This can be done in a
satisfactory manner by making a more subtle stipulation about the
measurement or the comparison of lengths.

But if rods of every kind (_i.e._ of every material) were to behave _in
the same way_ as regards the influence of temperature when they are on
the variably heated marble slab, and if we had no other means of
detecting the effect of temperature than the geometrical behaviour of
our rods in experiments analogous to the one described above, then our
best plan would be to assign the distance one to two points on the
slab, provided that the ends of one of our rods could be made to
coincide with these two points; for how else should we define the
distance without our proceeding being in the highest measure grossly
arbitrary? The method of Cartesian coordinates must then be discarded,
and replaced by another which does not assume the validity of Euclidean
geometry for rigid bodies.[20] The reader will notice that the
situation depicted here corresponds to the one brought about by the
general postulate of relativity (Section XXIII).


 [20] Mathematicians have been confronted with our problem in the
 following form. If we are given a surface (_e.g._ an ellipsoid) in
 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 Riemann 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.


XXV.

GAUSSIAN CO-ORDINATES

image033


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. 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 _P′_ on the surface then correspond to the co-ordinates

_P_: _u, v_

_P′_: _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 _P′_, as
measured with a little rod, by means of the very small number _ds_.
Then according to Gauss we have

_ds_2 = _g_11_du_2 + 2_g_12_du dv_ + _g_22_dv_2,

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:

_ds_2 = _du_2 + _dv_2

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 simply 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 _P′_, this
distance being measurable and well defined from a physical point of
view, then the following formula holds:

_ds_2 = _g_11_dx_12 + 2_g_12_dx_1_dx_2 . . . . + _g_44_dx_42,

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

_ds_2 = _dx_12 + _dx_22 + _dx_32 + _dx_42.

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 _ds_2 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.


XXVI.

THE SPACE-TIME CONTINUUM OF THE SPECIAL 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 XVII. 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 _dx′, dy′, dz′, dt′_. Then these
magnitudes always fulfill the condition.[21]


 [21] Cf. Appendixes I and II. The relations which are derived there
 for the co-ordinates themselves are valid also for co-ordinate
 _differences_, and thus also for co-ordinate differentials
 (indefinitely small differences).


_dx_2 + _dy_2 + _dz_2 – _c_2_dt_2 = _dx′_2 + _dy′_2 + _dz′_2 –
_c_2_dt′_2.


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

_ds_2 = _dx_2 + _dy_2 + _dz_2 – _c_2 _dt_2,


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_,

image034


by _x_1, _x_2, _x_3, _x_4, we also obtain the result that

_ds_2 = _dx_12 + _dx_22 + _dx_32 + _dx_42.


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

image035


instead of the real quantity _t_, we can regard the space-time
contintium—accordance with the special theory of relativity—as a
“Euclidean” four-dimensional continuum, a result which follows from the
considerations of the preceding section.


XXVII.

THE SPACE-TIME CONTINUUM OF THE GENERAL THEORY OF RELATIVITY IS NOT A
EUCLIDEAN CONTINUUM

In the first part of this book we were able to make use of space-time
co-ordinates which allowed of a simple and direct physical
interpretation, and which, according to Section XXVI, can be regarded
as four-dimensional Cartesian co-ordinates. This was possible on the
basis of the law of the constancy of the velocity of light. But
according to Section XXI the general theory of relativity cannot retain
this law. On the contrary, we arrived at the result that according to
this latter theory the velocity of light must always depend on the
co-ordinates when a gravitational field is present. In connection with
a specific illustration in Section XXIII, we found that the presence of
a gravitational field invalidates the definition of the coordinates and
the time, which led us to our objective in the special theory of
relativity.

In view of the resuIts of these considerations we are led to the
conviction that, according to the general principle of relativity, the
space-time continuum cannot be regarded as a Euclidean one, but that
here we have the general case, corresponding to the marble slab with
local variations of temperature, and with which we made acquaintance as
an example of a two-dimensional continuum. Just as it was there
impossible to construct a Cartesian co-ordinate system from equal rods,
so here it is impossible to build up a system (reference-body) from
rigid bodies and clocks, which shall be of such a nature that
measuring-rods and clocks, arranged rigidly with respect to one
another, shall indicate position and time directly. Such was the
essence of the difficulty with which we were confronted in Section
XXIII.

But the considerations of Sections XXV and XXVI show us the way to
surmount this difficulty. We refer the four-dimensional space-time
continuum in an arbitrary manner to Gauss co-ordinates. We assign to
every point of the continuum (event) four numbers, _x_1, _x_2, _x_3,
_x_4 (co-ordinates), which have not the least direct physical
significance, but only serve the purpose of numbering the points of the
continuum in a definite but arbitrary manner. This arrangement does not
even need to be of such a kind that we must regard _x_1, _x_2, _x_3, as
“space” co-ordinates and _x_4, as a “time” co-ordinate.

The reader may think that such a description of the world would be
quite inadequate. What does it mean to assign to an event the
particular co-ordinates _x_1, _x_2, _x_3, _x_4, if in themselves these
co-ordinates have no significance? More careful consideration shows,
however, that this anxiety is unfounded. Let us consider, for instance,
a material point with any kind of motion. If this point had only a
momentary existence without duration, then it would to described in
space-time by a single system of values _x_1, _x_2, _x_3, _x_4. Thus
its permanent existence must be characterised by an infinitely large
number of such systems of values, the co-ordinate values of which are
so close together as to give continuity; corresponding to the material
point, we thus have a (uni-dimensional) line in the four-dimensional
continuum. In the same way, any such lines in our continuum correspond
to many points in motion. The only statements having regard to these
points which can claim a physical existence are in reality the
statements about their encounters. In our mathematical treatment, such
an encounter is expressed in the fact that the two lines which
represent the motions of the points in question have a particular
system of co-ordinate values, _x_1, _x_2, _x_3, _x_4, in common. After
mature consideration the reader will doubtless admit that in reality
such encounters constitute the only actual evidence of a time-space
nature with which we meet in physical statements.

When we were describing the motion of a material point relative to a
body of reference, we stated nothing more than the encounters of this
point with particular points of the reference-body. We can also
determine the corresponding values of the time by the observation of
encounters of the body with clocks, in conjunction with the observation
of the encounter of the hands of clocks with particular points on the
dials. It is just the same in the case of space-measurements by means
of measuring-rods, as a little consideration will show.

The following statements hold generally: Every physical description
resolves itself into a number of statements, each of which refers to
the space-time coincidence of two events _A_ and _B_. In terms of
Gaussian co-ordinates, every such statement is expressed by the
agreement of their four co-ordinates _x_1, _x_2, _x_3, _x_4. Thus in
reality, the description of the time-space continuum by means of Gauss
co-ordinates completely replaces the description with the aid of a body
of reference, without suffering from the defects of the latter mode of
description; it is not tied down to the Euclidean character of the
continuum which has to be represented.


XXVIII.

EXACT FORMULATION OF THE GENERAL PRINCIPLE OF RELATIVITY

We are now in a position to replace the provisional formulation of the
general principle of relativity given in Section XVIII by an exact
formulation. The form there used, “All bodies of reference _K, K′_,
etc., are equivalent for the description of natural phenomena
(formulation of the general laws of nature), whatever may be their
state of motion,” cannot be maintained, because the use of rigid
reference-bodies, in the sense of the method followed in the special
theory of relativity, is in general not possible in space-time
description. The Gauss co-ordinate system has to take the place of the
body of reference. The following statement corresponds to the
fundamental idea of the general principle of relativity: “_All Gaussian
co-ordinate systems are essentially equivalent for the formulation of
the general laws of nature._”

We can state this general principle of relativity in still another
form, which renders it yet more clearly intelligible than it is when in
the form of the natural extension of the special principle of
relativity. According to the special theory of relativity, the
equations which express the general laws of nature pass over into
equations of the same form when, by making use of the Lorentz
transformation, we replace the space-time variables _x, y, z, t_, of a
(Galileian) reference-body _K_ by the space-time variables _x′, y′, z′,
t′_, of a new reference-body _K′_. According to the general theory of
relativity, on the other hand, by application of _arbitrary
substitutions_ of the Gauss variables _x_1, _x_2, _x_3, _x_4, the
equations must pass over into equations of the same form; for every
transformation (not only the Lorentz transformation) corresponds to the
transition of one Gauss co-ordinate system into another.

If we desire to adhere to our “old-time” three-dimensional view of
things, then we can characterise the development which is being
undergone by the fundamental idea of the general theory of relativity
as follows: The special theory of relativity has reference to Galileian
domains, _i.e._ to those in which no gravitational field exists. In
this connection a Galileian reference-body serves as body of reference,
_i.e._ a rigid body the state of motion of which is so chosen that the
Galileian law of the uniform rectilinear motion of “isolated” material
points holds relatively to it.

Certain considerations suggest that we should refer the same Galileian
domains to _non-Galileian_ reference-bodies also. A gravitational field
of a special kind is then present with respect to these bodies (cf.
Sections XX and XXIII).

In gravitational fields there are no such things as rigid bodies with
Euclidean properties; thus the fictitious rigid body of reference is of
no avail in the general theory of relativity. The motion of clocks is
also influenced by gravitational fields, and in such a way that a
physical definition of time which is made directly with the aid of
clocks has by no means the same degree of plausibility as in the
special theory of relativity.

For this reason non-rigid reference-bodies are used, which are as a
whole not only moving in any way whatsoever, but which also suffer
alterations in form _ad lib._ during their motion. Clocks, for which
the law of motion is of any kind, however irregular, serve for the
definition of time. We have to imagine each of these clocks fixed at a
point on the non-rigid reference-body. These clocks satisfy only the
one condition, that the “readings” which are observed simultaneously on
adjacent clocks (in space) differ from each other by an indefinitely
small amount. This non-rigid reference-body, which might appropriately
be termed a “reference-mollusc”, is in the main equivalent to a
Gaussian four-dimensional co-ordinate system chosen arbitrarily. That
which gives the “mollusc” a certain comprehensibility as compared with
the Gauss co-ordinate system is the (really unjustified) formal
retention of the separate existence of the

space co-ordinates as opposed to the time co-ordinate. Every point on
the mollusc is treated as a space-point, and every material point which
is at rest relatively to it as at rest, so long as the mollusc is
considered as reference-body. The general principle of relativity
requires that all these molluscs can be used as reference-bodies with
equal right and equal success in the formulation of the general laws of
nature; the laws themselves must be quite independent of the choice of
mollusc.

The great power possessed by the general principle of relativity lies
in the comprehensive limitation which is imposed on the laws of nature
in consequence of what we have seen above.


XXIX.

THE SOLUTION OF THE PROBLEM OF GRAVITATION ON THE BASIS OF THE GENERAL
PRINCIPLE OF RELATIVITY

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 _K′_. Then with respect to _K′_ 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 _K′_ simply by mathematical
transformation. We interpret this behaviour as the behaviour of
measuring-rods, clocks and material points under the influence of the
gravitational field _G_. Hereupon we introduce a hypothesis: that the
influence of the gravitational field on measuring-rods, 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 Galileian special case, 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 XV only its energy is
of importance for its effect 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 XXI; 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,[22] 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).[23] These two deductions from the theory have both been
confirmed.


 [22] First observed by Eddington and others in 1919. (Cf. Appendix
 III).


 [23] Established by Adams in 1924. (Cf. p. 132)


PART III: CONSIDERATIONS ON THE UNIVERSE AS A WHOLE


XXX.

COSMOLOGICAL DIFFICULTIES OF NEWTON’S THEORY

Part from the difficulty discussed in Section XXI, there is a second
fundamental difficulty attending classical celestial mechanics, which,
to the best of my knowledge, was first discussed in detail by the
astronomer Seeliger. If we ponder over the question as to how the
universe, considered as a whole, is to be regarded, the first answer
that suggests itself to us is surely this: As regards space (and time)
the universe is infinite. There are stars everywhere, so that the
density of matter, although very variable in detail, is nevertheless on
the average everywhere the same. In other words: However far we might
travel through space, we should find everywhere an attenuated swarm of
fixed stars of approrimately the same kind and density.

This view is not in harmony with the theory of Newton. The latter
theory rather requires that the universe should have a kind of centre
in which the density of the stars is a maximum, and that as we proceed
outwards from this centre the group-density of the stars should
diminish, until finally, at great distances, it is succeeded by an
infinite region of emptiness. The stellar universe ought to be a finite
island in the infinite ocean of space.[24]


 [24] _Proof_—According to the theory of Newton, the number of “lines
 of force” which come from infinity and terminate in a mass m is
 proportional to the mass _m_. If, on the average, the mass density ρ0
 is constant throughout the universe, then a sphere of volume _V_ will
 enclose the average mass ρ0_V_. Thus the number of lines of force
 passing through the surface _F_ of the sphere into its interior is
 proportional to ρ0_V_. For unit area of the surface of the sphere the
 number of lines of force which enters the sphere is thus proportional
 to ρ0_V/F_ or to ρ0_R_. Hence the intensity of the field at the
 surface would ultimately become infinite with increasing radius _R_ of
 the sphere, which is impossible.


This conception is in itself not very satisfactory. It is still less
satisfactory because it leads to the result that the light emitted by
the stars and also individual stars of the stellar system are
perpetually passing out into infinite space, never to return, and
without ever again coming into interaction with other objects of
nature. Such a finite material universe would be destined to become
gradually but systematically impoverished.

In order to escape this dilemma, Seeliger suggested a modification of
Newton’s law, in which he assumes that for great distances the force of
attraction between two masses diminishes more rapidly than would result
from the inverse square law. In this way it is possible for the mean
density of matter to be constant everywhere, even to infinity, without
infinitely large gravitational fields being produced. We thus free
ourselves from the distasteful conception that the material universe
ought to possess something of the nature of a centre. Of course we
purchase our emancipation from the fundamental difficulties mentioned,
at the cost of a modification and complication of Newton’s law which
has neither empirical nor theoretical foundation. We can imagine
innumerable laws which would serve the same purpose, without our being
able to state a reason why one of them is to be preferred to the
others; for any one of these laws would be founded just as little on
more general theoretical principles as is the law of Newton.


XXXI.

THE POSSIBILITY OF A “FINITE” AND YET “UNBOUNDED” UNIVERSE

But speculations on the structure of the universe also move in quite
another direction. The development of non-Euclidean geometry led to the
recognition of the fact, that we can cast doubt on the _infiniteness_
of our space without coming into conflict with the laws of thought or
with experience (Riemann, Helmholtz). These questions have already been
treated in detail and with unsurpassable lucidity by Helmholtz and
Poincaré, whereas I can only touch on them briefly here.

In the first place, we imagine an existence in two dimensional space.
Flat beings with flat implements, and in particular flat rigid
measuring-rods, are free to move in a _plane_. For them nothing exists
outside of this plane: that which they observe to happen to themselves
and to their flat “things” is the all-inclusive reality of their plane.
In particular, the constructions of plane Euclidean geometry can be
carried out by means of the rods _e.g._ the lattice construction,
considered in Section XXIV. In contrast to ours, the universe of these
beings is two-dimensional; but, like ours, it extends to infinity. In
their universe there is room for an infinite number of identical
squares made up of rods, _i.e._ its volume (surface) is infinite. If
these beings say their universe is “plane,” there is sense in the
statement, because they mean that they can perform the constructions of
plane Euclidean geometry with their rods. In this connection the
individual rods always represent the same distance, independently of
their position.

Let us consider now a second two-dimensional existence, but this time
on a spherical surface instead of on a plane. The flat beings with
their measuring-rods and other objects fit exactly on this surface and
they are unable to leave it. Their whole universe of observation
extends exclusively over the surface of the sphere. Are these beings
able to regard the geometry of their universe as being plane geometry
and their rods withal as the realisation of “distance”? They cannot do
this. For if they attempt to realise a straight line, they will obtain
a curve, which we “three-dimensional beings” designate as a great
circle, _i.e._ a self-contained line of definite finite length, which
can be measured up by means of a measuring-rod. Similarly, this
universe has a finite area that can be compared with the area, of a
square constructed with rods. The great charm resulting from this
consideration lies in the recognition of the fact that _the universe of
these beings is finite and yet has no limits._

But the spherical-surface beings do not need to go on a world-tour in
order to perceive that they are not living in a Euclidean universe.
They can convince themselves of this on every part of their “world,”
provided they do not use too small a piece of it. Starting from a
point, they draw “straight lines” (arcs of circles as judged in three
dimensional space) of equal length in all directions. They will call
the line joining the free ends of these lines a “circle.” For a plane
surface, the ratio of the circumference of a circle to its diameter,
both lengths being measured with the same rod, is, according to
Euclidean geometry of the plane, equal to a constant value π, which is
independent of the diameter of the circle. On their spherical surface
our flat beings would find for this ratio the value

image036


_i.e._ a smaller value than π, the difference being the more
considerable, the greater is the radius of the circle in comparison
with the radius _R_ of the “world-sphere.” By means of this relation
the spherical beings can determine the radius of their universe
(“world”), even when only a relatively small part of their worldsphere
is available for their measurements. But if this part is very small
indeed, they will no longer be able to demonstrate that they are on a
spherical “world” and not on a Euclidean plane, for a small part of a
spherical surface differs only slightly from a piece of a plane of the
same size.

Thus if the spherical surface beings are living on a planet of which
the solar system occupies only a negligibly small part of the spherical
universe, they have no means of determining whether they are living in
a finite or in an infinite universe, because the “piece of universe” to
which they have access is in both cases practically plane, or
Euclidean. It follows directly from this discussion, that for our
sphere-beings the circumference of a circle first increases with the
radius until the “circumference of the universe” is reached, and that
it thenceforward gradually decreases to zero for still further
increasing values of the radius. During this process the area of the
circle continues to increase more and more, until finally it becomes
equal to the total area of the whole “world-sphere.”

Perhaps the reader will wonder why we have placed our “beings” on a
sphere rather than on another closed surface. But this choice has its
justification in the fact that, of all closed surfaces, the sphere is
unique in possessing the property that all points on it are equivalent.
I admit that the ratio of the circumference _c_ of a circle to its
radius _r_ depends on _r_, but for a given value of _r_ it is the same
for all points of the “worldsphere”; in other words, the “world-sphere”
is a “surface of constant curvature.”

To this two-dimensional sphere-universe there is a three-dimensional
analogy, namely, the three-dimensional spherical space which was
discovered by Riemann. its points are likewise all equivalent. It
possesses a finite volume, which is determined by its “radius”
(2π2_R_3). Is it possible to imagine a spherical space? To imagine a
space means nothing else than that we imagine an epitome of our “space”
experience, _i.e._ of experience that we can have in the movement of
“rigid” bodies. In this sense we _can_ imagine a spherical space.

Suppose we draw lines or stretch strings in all directions from a
point, and mark off from each of these the distance _r_ with a
measuring-rod. All the free end-points of these lengths lie on a
spherical surface. We can specially measure up the area (_F_) of this
surface by means of a square made up of measuring-rods. If the universe
is Euclidean, then _F_ = 4π_r_2; if it is spherical, then _F_ is always
less than 4π_r_2. With increasing values of _r, F_ increases from zero
up to a maximum value which is determined by the “world-radius,” but
for still further increasing values of _r_, the area gradually
diminishes to zero. At first, the straight lines which radiate from the
starting point diverge farther and farther from one another, but later
they approach each other, and finally they run together again at a
“counter-point” to the starting point. Under such conditions they have
traversed the whole spherical space. It is easily seen that the
three-dimensional spherical space is quite analogous to the
two-dimensional spherical surface. It is finite (_i.e._ of finite
volume), and has no bounds.

It may be mentioned that there is yet another kind of curved space:
“elliptical space.” It can be regarded as a curved space in which the
two “counter-points” are identical (indistinguishable from each other).
An elliptical universe can thus be considered to some extent as a
curved universe possessing central symmetry.

It follows from what has been said, that closed spaces without limits
are conceivable. From amongst these, the spherical space (and the
elliptical) excels in its simplicity, since all points on it are
equivalent. As a result of this discussion, a most interesting question
arises for astronomers and physicists, and that is whether the universe
in which we live is infinite, or whether it is finite in the manner of
the spherical universe. Our experience is far from being sufficient to
enable us to answer this question. But the general theory of relativity
permits of our answering it with a moderate degree of certainty, and in
this connection the difficulty mentioned in Section XXX finds its
solution.


XXXII.

THE STRUCTURE OF SPACE ACCORDING TO THE GENERAL THEORY OF RELATIVITY

According to the general theory of relativity, the geometrical
properties of space are not independent, but they are determined by
matter. Thus we can draw conclusions about the geometrical structure of
the universe only if we base our considerations on the state of the
matter as being something that is known. We know from experience that,
for a suitably chosen co-ordinate system, the velocities of the stars
are small as compared with the velocity of transmission of light. We
can thus as a rough approximation arrive at a conclusion as to the
nature of the universe as a whole, if we treat the matter as being at
rest.

We already know from our previous discussion that the behaviour of
measuring-rods and clocks is influenced by gravitational fields, _i.e._
by the distribution of matter. This in itself is sufficient to exclude
the possibility of the exact validity of Euclidean geometry in our
universe. But it is conceivable that our universe differs only slightly
from a Euclidean one, and this notion seems all the more probable,
since calculations show that the metrics of surrounding space is
influenced only to an exceedingly small extent by masses even of the
magnitude of our sun. We might imagine that, as regards geometry, our
universe behaves analogously to a surface which is irregularly curved
in its individual parts, but which nowhere departs appreciably from a
plane: something like the rippled surface of a lake. Such a universe
might fittingly be called a quasi-Euclidean universe. As regards its
space it would be infinite. But calculation shows that in a
quasi-Euclidean universe the average density of matter would
necessarily be _nil_. Thus such a universe could not be inhabited by
matter everywhere; it would present to us that unsatisfactory picture
which we portrayed in Section XXX.

If we are to have in the universe an average density of matter which
differs from zero, however small may be that difference, then the
universe cannot be quasi-Euclidean. On the contrary, the results of
calculation indicate that if matter be distributed uniformly, the
universe would necessarily be spherical (or elliptical). Since in
reality the detailed distribution of matter is not uniform, the real
universe will deviate in individual parts from the spherical, _i.e._
the universe will be quasi-spherical. But it will be necessarily
finite. In fact, the theory supplies us with a simple connection[25]
between the space-expanse of the universe and the average density of
matter in it.


 [25] For the radius _R_ of the universe we obtain the equation


image037


The use of the C.G.S. system in this equation gives 2/k = 1.08 x 1027;
ρ is the average density of the matter and _k_ is a constant connected
with the Newtonian constant of gravitation.


APPENDICES


APPENDIX I

SIMPLE DERIVATION OF THE LORENTZ TRANSFORMATION

(SUPPLEMENTARY TO SECTION XI)

For the relative orientation of the co-ordinate systems indicated in
Fig. 2, the _x_-axes of both systems permanently coincide. In the
present case we can divide the problem into parts by considering first
only events which are localised on the _x_-axis. Any such event is
represented with respect to the co-ordinate system _K_ by the abscissa
_x_ and the time _t_, and with respect to the system _K′_ by the
abscissa _x′_ and the time _t′_. We require to find _x′_ and _t′_ when
_x_ and _t_ are given.

A light-signal, which is proceeding along the positive axis of _x_, is
transmitted according to the equation

_x_ = _ct_


or

_x_ – _ct_ = 0 . . . . . (1).


Since the same light-signal has to be transmitted relative to _K′_ with
the velocity _c_, the propagation relative to the system _K′_ will be
represented by the analogous formula

_x′_ – _ct′_ = 0 . . . . . (2)


Those space-time points (events) which satisfy (1) must also satisfy
(2). Obviously this will be the case when the relation

(_x′_ – _ct′_) = λ(_x_ – _ct_) . . . (3).


is fulfilled in general, where λ indicates a constant; for, according
to (3), the disappearance of (_x_ – _ct_) involves the disappearance of
(_x′_ – _ct′_).

If we apply quite similar considerations to light rays which are being
transmitted along the negative _x_-axis, we obtain the condition

(_x′_ + _ct′_) = (_x + ct_) . . . (4).


By adding (or subtracting) equations (3) and (4), and introducing for
convenience the constants _a_ and _b_ in place of the constants λ and μ
where

image038


and

image039


we obtain the equations

image040


We should thus have the solution of our problem, if the constants _a_
and _b_ were known. These result from the following discussion.

For the origin of _K′_ we have permanently _x′_ = 0, and hence
according to the first of the equations (5)

image041


If we call _v_ the velocity with which the origin of _K′_ is moving
relative to _K_, we then have

image042


The same value _v_ can be obtained from equations (5), if we calculate
the velocity of another point of _K′_ relative to _K_, or the velocity
(directed towards the negative _x_-axis) of a point of _K_ with respect
to _K′_. In short, we can designate _v_ as the relative velocity of the
two systems.

Furthermore, the principle of relativity teaches us that, as judged
from K, the length of a unit measuring-rod which is at rest with
reference to _K′_ must be exactly the same as the length, as judged
from _K′_, of a unit measuring-rod which is at rest relative to _K_. In
order to see how the points of the _x′_-axis appear as viewed from _K_,
we only require to take a “snapshot” of _K′_ from _K_; this means that
we have to insert a particular value of _t_ (time of _K_), _e.g._ _t_ =
0. For this value of _t_ we then obtain from the first of the equations
(5)

_x′_ = _ax_


Two points of the _x′_-axis which are separated by the distance Δ_x′_ =
1 when measured in the _K′_ system are thus separated in our
instantaneous photograph by the distance

image043


But if the snapshot be taken from _K′_(_t′_ = 0), and if we eliminate
_t_ from the equations (5), taking into account the expression (6), we
obtain

image044


From this we conclude that two points on the _x_-axis separated by the
distance 1 (relative to _K_) will be represented on our snapshot by the
distance

image045


But from what has been said, the two snapshots must be identical; hence
Δ_x_ in (7) must be equal to Δ_x′_ in (7_a_), so that we obtain

image046


The equations (6) and (7_b_) determine the constants _a_ and _b_. By
inserting the values of these constants in (5), we obtain the first and
the fourth of the equations given in Section XI.

image047


Thus we have obtained the Lorentz transformation for events on the
_x_-axis. It satisfies the condition

_x′_2 – _c_2_t′_2 = _x_2 – _c_2_t_2 . . . . . . (8a).


The extension of this result, to include events which take place
outside the _x_-axis, is obtained by retaining equations (8) and
supplementing them by the relations

image048


In this way we satisfy the postulate of the constancy of the velocity
of light _in vacuo_ for rays of light of arbitrary direction, both for
the system _K_ and for the system _K′_. This may be shown in the
following manner.

We suppose a light-signal sent out from the origin of _K_ at the time
_t_ = 0. It will be propagated according to the equation

image049


or, if we square this equation, according to the equation

_x_2 + _y_2 + _z_2 – _c_2_t_2 = 0 . . . . . (10).


It is required by the law of propagation of light, in conjunction with
the postulate of relativity, that the transmission of the signal in
question should take place—as judged from _K′_—in accordance with the
corresponding formula

_r′_ = _ct′_


or,

_x′_2 + _y′_2 + _z′_2 – _c_2_t′_2 = 0 . . . . . . (10_a_).


In order that equation (10_a_) may be a consequence of equation (10),
we must have

_x′_2 + _y′_2 + _z′_2 – _c_2_t′_2 = σ (_x_2 + _y_2 + _z_2 – _c_2_t_2)
(11).


Since equation (8_a_) must hold for points on the _x_-axis, we thus
have σ = 1. It is easily seen that the Lorentz transformation really
satisfies equation (11) for σ = 1; for (11) is a consequence of (8_a_)
and (9), and hence also of (8) and (9). We have thus derived the
Lorentz transformation.

The Lorentz transformation represented by (8) and (9) still requires to
be generalised. Obviously it is immaterial whether the axes of _K′_ be
chosen so that they are spatially parallel to those of _K_. It is also
not essential that the velocity of translation of _K′_ with respect to
_K_ should be in the direction of the _x_-axis. A simple consideration
shows that we are able to construct the Lorentz transformation in this
general sense from two kinds of transformations, viz. from Lorentz
transformations in the special sense and from purely spatial
transformations. which corresponds to the replacement of the
rectangular co-ordinate system by a new system with its axes pointing
in other directions.

Mathematically, we can characterise the generalised Lorentz
transformation thus:

It expresses _x′, y′, x′, t′_, in terms of linear homogeneous functions
of _x, y, x, t_, of such a kind that the relation

_x′_2 + _y′_2 + _z′_2 – _c_2_t′_2 = _x_2 + _y_2 + _z_2 – _c_2_t_2
(11_a_).


is satisficd identically. That is to say: If we substitute their
expressions in _x, y, x, t_, in place of _x′, y′, x′, t′_, on the
left-hand side, then the left-hand side of (11_a_) agrees with the
right-hand side.


APPENDIX II

MINKOWSKI’S FOUR-DIMENSIONAL SPACE (“WORLD”)

(SUPPLEMENTARY TO SECTION XVII)

We can characterise the Lorentz transformation still more simply if we
introduce the imaginary

image031


in place of _t_, as time-variable. If, in accordance with this, we
insert

image050


and similarly for the accented system _K′_, then the condition which is
identically satisfied by the transformation can be expressed thus:

_x_1′2 + _x_2′2 + _x_3′2 + _x_4′2 = _x_12 + _x_22 + _x_32 + _x_42 (12).


That is, by the afore-mentioned choice of “coordinates,” (11_a_) [see
the end of Appendix II] is transformed into this equation.

We see from (12) that the imaginary time co-ordinate _x_4, enters into
the condition of transformation in exactly the same way as the space
co-ordinates _x_1, _x_2, _x_3. It is due to this fact that, according
to the theory of relativity, the “time” _x_4, enters into natural laws
in the same form as the space co ordinates _x_1, _x_2, _x_3.

A four-dimensional continuum described by the “co-ordinates” _x_1,
_x_2, _x_3, _x_4, was called “world” by Minkowski, who also termed a
point-event a “world-point.” From a “happening” in three-dimensional
space, physics becomes, as it were, an “existence” in the
four-dimensional “world.”

This four-dimensional “world” bears a close similarity to the
three-dimensional “space” of (Euclidean) analytical geometry. If we
introduce into the latter a new Cartesian co-ordinate system (_x′_1,
_x′_2, _x′_3) with the same origin, then _x′_1, _x′_2, _x′_3, are
linear homogeneous functions of _x_1, _x_2, _x_3 which identically
satisfy the equation

_x_1′2 + _x_2′2 + _x_3′2 = _x_12 + _x_22 + _x_32


The analogy with (12) is a complete one. We can regard Minkowski’s
“world” in a formal manner as a four-dimensional Euclidean space (with
an imaginary time coordinate); the Lorentz transformation corresponds
to a “rotation” of the co-ordinate system in the four-dimensional
“world.”


APPENDIX III

THE EXPERIMENTAL CONFIRMATION OF THE GENERAL THEORY OF RELATIVITY

From a systematic theoretical point of view, we may imagine the process
of evolution of an empirical science to be a continuous process of
induction. Theories are evolved and are expressed in short compass as
statements of a large number of individual observations in the form of
empirical laws, from which the general laws can be ascertained by
comparison. Regarded in this way, the development of a science bears
some resemblance to the compilation of a classified catalogue. It is,
as it were, a purely empirical enterprise.

But this point of view by no means embraces the whole of the actual
process; for it slurs over the important part played by intuition and
deductive thought in the development of an exact science. As soon as a
science has emerged from its initial stages, theoretical advances are
no longer achieved merely by a process of arrangement. Guided by
empirical data, the investigator rather develops a system of thought
which, in general, is built up logically from a small number of
fundamental assumptions, the so-called axioms. We call such a system of
thought a _theory_. The theory finds the justification for its
existence in the fact that it correlates a large number of single
observations, and it is just here that the “truth” of the theory lies.

Corresponding to the same complex of empirical data, there may be
several theories, which differ from one another to a considerable
extent. But as regards the deductions from the theories which are
capable of being tested, the agreement between the theories may be so
complete that it becomes difficult to find any deductions in which the
two theories differ from each other. As an example, a case of general
interest is available in the province of biology, in the Darwinian
theory of the development of species by selection in the struggle for
existence, and in the theory of development which is based on the
hypothesis of the hereditary transmission of acquired characters.

We have another instance of far-reaching agreement between the
deductions from two theories in Newtonian mechanics on the one hand,
and the general theory of relativity on the other. This agreement goes
so far, that up to the present we have been able to find only a few
deductions from the general theory of relativity which are capable of
investigation, and to which the physics of pre-relativity days does not
also lead, and this despite the profound difference in the fundamental
assumptions of the two theories. In what follows, we shall again
consider these important deductions, and we shall also discuss the
empirical evidence appertaining to them which has hitherto been
obtained.

(_a_) Motion of the Perihelion of Mercury

According to Newtonian mechanics and Newton’s law of gravitation, a
planet which is revolving round the sun would describe an ellipse round
the latter, or, more correctly, round the common centre of gravity of
the sun and the planet. In such a system, the sun, or the common centre
of gravity, lies in one of the foci of the orbital ellipse in such a
manner that, in the course of a planet-year, the distance sun-planet
grows from a minimum to a maximum, and then decreases again to a
minimum. If instead of Newton’s law we insert a somewhat different law
of attraction into the calculation, we find that, according to this new
law, the motion would still take place in such a manner that the
distance sun-planet exhibits periodic variations; but in this case the
angle described by the line joining sun and planet during such a period
(from perihelion—closest proximity to the sun—to perihelion) would
differ from 360°. The line of the orbit would not then be a closed one
but in the course of time it would fill up an annular part of the
orbital plane, viz. between the circle of least and the circle of
greatest distance of the planet from the sun.

According also to the general theory of relativity, which differs of
course from the theory of Newton, a small variation from the
Newton-Kepler motion of a planet in its orbit should take place, and in
such away, that the angle described by the radius sun-planet between
one perhelion and the next should exceed that corresponding to one
complete revolution by an amount given by

image051


(_N.B._—One complete revolution corresponds to the angle 2π in the
absolute angular measure customary in physics, and the above expression
given the amount by which the radius sun-planet exceeds this angle
during the interval between one perihelion and the next.) In this
expression _a_ represents the major semi-axis of the ellipse, _e_ its
eccentricity, _c_ the velocity of light, and _T_ the period of
revolution of the planet. Our result may also be stated as follows:
According to the general theory of relativity, the major axis of the
ellipse rotates round the sun in the same sense as the orbital motion
of the planet. Theory requires that this rotation should amount to 43
seconds of arc per century for the planet Mercury, but for the other
Planets of our solar system its magnitude should be so small that it
would necessarily escape detection.[26]


 [26] Especially since the next planet Venus has an orbit that is
 almost an exact circle, which makes it more difficult to locate the
 perihelion with precision.


In point of fact, astronomers have found that the theory of Newton does
not suffice to calculate the observed motion of Mercury with an
exactness corresponding to that of the delicacy of observation
attainable at the present time. After taking account of all the
disturbing influences exerted on Mercury by the remaining planets, it
was found (Leverrier: 1859; and Newcomb: 1895) that an unexplained
perihelial movement of the orbit of Mercury remained over, the amount
of which does not differ sensibly from the above mentioned +43 seconds
of arc per century. The uncertainty of the empirical result amounts to
a few seconds only.

(_b_) Deflection of Light by a Gravitational Field

image052


In Section XXII it has been already mentioned that according to the
general theory of relativity, a ray of light will experience a
curvature of its path when passing through a gravitational field, this
curvature being similar to that experienced by the path of a body which
is projected through a gravitational field. As a result of this theory,
we should expect that a ray of light which is passing close to a
heavenly body would be deviated towards the latter. For a ray of light
which passes the sun at a distance of Δ sun-radii from its centre, the
angle of deflection (α) should amount to

image053


It may be added that, according to the theory, half of this deflection
is produced by the Newtonian field of attraction of the sun, and the
other half by the geometrical modification (“curvature”) of space
caused by the sun.

This result admits of an experimental test by means of the photographic
registration of stars during a total eclipse of the sun. The only
reason why we must wait for a total eclipse is because at every other
time the atmosphere is so strongly illuminated by the light from the
sun that the stars situated near the sun’s disc are invisible. The
predicted effect can be seen clearly from the accompanying diagram. If
the sun (_S_) were not present, a star which is practically infinitely
distant would be seen in the direction _D_1, as observed front the
earth. But as a consequence of the deflection of light from the star by
the sun, the star will be seen in the direction _D_2, _i.e._ at a
somewhat greater distance from the centre of the sun than corresponds
to its real position.

In practice, the question is tested in the following way. The stars in
the neighbourhood of the sun are photographed during a solar eclipse.

In addition, a second photograph of the same stars is taken when the
sun is situated at another position in the sky, _i.e._ a few months
earlier or later. As compared with the standard photograph, the
positions of the stars on the eclipse-photograph ought to appear
displaced radially outwards (away from the centre of the sun) by an
amount corresponding to the angle _a_.

We are indebted to the [British] Royal Society and to the Royal
Astronomical Society for the investigation of this important deduction.
Undaunted by the [first world] war and by difficulties of both a
material and a psychological nature aroused by the war, these societies
equipped two expeditions—to Sobral (Brazil), and to the island of
Principe (West Africa)—and sent several of Britain’s most celebrated
astronomers (Eddington, Cottingham, Crommelin, Davidson), in order to
obtain photographs of the solar eclipse of 29th May, 1919. The relative
discrepancies to be expected between the stellar photographs obtained
during the eclipse and the comparison photographs amounted to a few
hundredths of a millimetre only. Thus great accuracy was necessary in
making the adjustments required for the taking of the photographs, and
in their subsequent measurement.

The results of the measurements confirmed the theory in a thoroughly
satisfactory manner. The rectangular components of the observed and of
the calculated deviations of the stars (in seconds of arc) are set
forth in the following table of results:

image054


(_c_) Displacement of Spectral Lines Towards the Red

In Section XXIII it has been shown that in a system _K′_ which is in
rotation with regard to a Galileian system _K_, clocks of identical
construction, and which are considered at rest with respect to the
rotating reference-body, go at rates which are dependent on the
positions of the clocks. We shall now examine this dependence
quantitatively. A clock, which is situated at a distance r from the
centre of the disc, has a velocity relative to _K_ which is given by

_v_ = ω_r_,


where ω represents the angular velocity of rotation of the disc _K′_
with respect to _K_. If _v_0, represents the number of ticks of the
clock per unit time (“rate” of the clock) relative to _K_ when the
clock is at rest, then the “rate” of the clock (_v_) when it is moving
relative to _K_ with a velocity _v_, but at rest with respect to the
disc, will, in accordance with Section XII, be given by

image055


or with sufficient accuracy by

image056


This expression may also be stated in the following form:

image057


If we represent the difference of potential of the centrifugal force
between the position of the clock and the centre of the disc by φ,
_i.e._ the work, considered negatively, which must be performed on the
unit of mass against the centrifugal force in order to transport it
from the position of the clock on the rotating disc to the centre of
the disc, then we have

image058


From this it follows that

image059


In the first place, we see from this expression that two clocks of
identical construction will go at different rates when situated at
different distances from the centre of the disc. This result is also
valid from the standpoint of an observer who is rotating with the disc.

Now, as judged from the disc, the latter is in a gravitational field of
potential φ, hence the result we have obtained will hold quite
generally for gravitational fields. Furthermore, we can regard an atom
which is emitting spectral lines as a clock, so that the following
statement will hold:

_An atom absorbs or emits light of a frequency which is dependent on
the potential of the gravitational field in which it is situated._

The frequency of an atom situated on the surface of a heavenly body
will be somewhat less than the frequency of an atom of the same element
which is situated in free space (or on the surface of a smaller
celestial body).

Now φ = – _K (M/r)_, where _K_ is Newton’s constant of gravitation, and
_M_ is the mass of the heavenly body. Thus a displacement towards the
red ought to take place for spectral lines produced at the surface of
stars as compared with the spectral lines of the same element produced
at the surface of the earth, the amount of this displacement being

image060


For the sun, the displacement towards the red predicted by theory
amounts to about two millionths of the wave-length. A trustworthy
calculation is not possible in the case of the stars, because in
general neither the mass _M_ nor the radius _r_ are known.

It is an open question whether or not this effect exists, and at the
present time (1920) astronomers are working with great zeal towards the
solution. Owing to the smallness of the effect in the case of the sun,
it is difficult to form an opinion as to its existence. Whereas Grebe
and Bachem (Bonn), as a result of their own measurements and those of
Evershed and Schwarzschild on the cyanogen bands, have placed the
existence of the effect almost beyond doubt, while other investigators,
particularly St. John, have been led to the opposite opinion in
consequence of their measurements.

Mean displacements of lines towards the less refrangible end of the
spectrum are certainly revealed by statistical investigations of the
fixed stars; but up to the present the examination of the available
data does not allow of any definite decision being arrived at, as to
whether or not these displacements are to be referred in reality to the
effect of gravitation. The results of observation have been collected
together, and discussed in detail from the standpoint of the question
which has been engaging our attention here, in a paper by E. Freundlich
entitled “Zur Prüfung der allgemeinen Relativitäts-Theorie” (_Die
Naturwissenschaften_, 1919, No. 35, p. 520: Julius Springer, Berlin).

At all events, a definite decision will be reached during the next few
years. If the displacement of spectral lines towards the red by the
gravitational potential does not exist, then the general theory of
relativity will be untenable. On the other hand, if the cause of the
displacement of spectral lines be definitely traced to the
gravitational potential, then the study of this displacement will
furnish us with important information as to the mass of the heavenly
bodies.[27]


 [27] The displacement of spectral lines towards the red end of the
 spectrum was definitely established by Adams in 1924, by observations
 on the dense companion of Sirius, for which the effect is about thirty
 times greater than for the Sun. R.W.L.—translator


APPENDIX IV

THE STRUCTURE OF SPACE ACCORDING TO THE GENERAL THEORY OF RELATIVITY

(SUPPLEMENTARY TO SECTION XXXII)

Since the publication of the first edition of this little book, our
knowledge about the structure of space in the large (“cosmological
problem”) has had an important development, which ought to be mentioned
even in a popular presentation of the subject.

My original considerations on the subject were based on two hypotheses:

(1) There exists an average density of matter in the whole of space
which is everywhere the same and different from zero.

(2) The magnitude (“radius”) of space is independent of time.

Both these hypotheses proved to be consistent, according to the general
theory of relativity, but only after a hypothetical term was added to
the field equations, a term which was not required by the theory as
such nor did it seem natural from a theoretical point of view
(“cosmological term of the field equations”).

Hypothesis (2) appeared unavoidable to me at the time, since I thought
that one would get into bottomless speculations if one departed from
it.

However, already in the ’twenties, the Russian mathematician Friedman
showed that a different hypothesis was natural from a purely
theoretical point of view. He realized that it was possible to preserve
hypothesis (1) without introducing the less natural cosmological term
into the field equations of gravitation, if one was ready to drop
hypothesis (2). Namely, the original field equations admit a solution
in which the “world radius” depends on time (expanding space). In that
sense one can say, according to Friedman, that the theory demands an
expansion of space.

A few years later Hubble showed, by a special investigation of the
extra-galactic nebulae (“milky ways”), that the spectral lines emitted
showed a red shift which increased regularly with the distance of the
nebulae. This can be interpreted in regard to our present knowledge
only in the sense of Doppler’s principle, as an expansive motion of the
system of stars in the large—as required, according to Friedman, by the
field equations of gravitation. Hubble’s discovery can, therefore, be
considered to some extent as a confirmation of the theory.

There does arise, however, a strange difficulty. The interpretation of
the galactic line-shift discovered by Hubble as an expansion (which can
hardly be doubted from a theoretical point of view), leads to an origin
of this expansion which lies “only” about 109 years ago, while physical
astronomy makes it appear likely that the development of individual
stars and systems of stars takes considerably longer. It is in no way
known how this incongruity is to be overcome.

I further want to remark that the theory of expanding space, together
with the empirical data of astronomy, permit no decision to be reached
about the finite or infinite character of (three-dimensional) space,
while the original “static” hypothesis of space yielded the closure
(finiteness) of space.

_K_ = co-ordinate system

_x, y_ = two-dimensional co-ordinates

_x, y, z_ = three-dimensional co-ordinates

_x, y, z, t_ = four-dimensional co-ordinates

_t_ = time

_I_ = distance

_v_ = velocity

_F_ = force

_G_ = gravitational field