Part I
The Special Theory of Relativity
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 form 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.
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.
Notes
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.
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. 1)
Every description of the scene of an event or of the position of an
object in space is based on the specification of the point on a rigid body
(body of reference) with which that event or object coincides. This
applies not only to scientific description, but also to everyday life. If
I analyse the place specification " Times Square, New York,"
[A] I arrive at the
following result. The earth is the rigid body to which the specification
of place refers; " Times Square, New York," is a well-defined point, to
which a name has been assigned, and with which the event coincides in
space.2)
This primitive method of place specification deals only with places on
the surface of rigid bodies, and is dependent on the existence of points
on this surface which are distinguishable from each other. But we can free
ourselves from both of these limitations without altering the nature of
our specification of position. If, for instance, a cloud is hovering over
Times Square, then we can determine its position relative to the surface
of the earth by erecting a pole perpendicularly on the Square, so that it
reaches the cloud. The length of the pole measured with the standard
measuring-rod, combined with the specification of the position of the foot
of the pole, supplies us with a complete place specification. On the basis
of this illustration, we are able to see the manner in which a refinement
of the conception of position has been developed.
(a) We imagine the rigid body, to which the place
specification is referred, supplemented in such a manner that the object
whose position we require is reached by. the completed rigid body.
(b) In locating the position of the object, we make
use of a number (here the length of the pole measured with the
measuring-rod) instead of designated points of reference.
(c) We speak of the height of the cloud even when
the pole which reaches the cloud has not been erected. By means of optical
observations of the cloud from different positions on the ground, and
taking into account the properties of the propagation of light, we
determine the length of the pole we should have required in order to reach
the cloud.
From this consideration we see that it will be advantageous if, in the
description of position, it should be possible by means of numerical
measures to make ourselves independent of the existence of marked
positions (possessing names) on the rigid body of reference. In the
physics of measurement this is attained by the application of the
Cartesian system of co-ordinates.
This consists of three plane surfaces perpendicular to each other and
rigidly attached to a rigid body. Referred to a system of co-ordinates,
the scene of any event will be determined (for the main part) by the
specification of the lengths of the three perpendiculars or co-ordinates (x,
y, z) which can be dropped from the scene of the event to those three
plane surfaces. The lengths of these three perpendiculars can be
determined by a series of manipulations with rigid measuring-rods
performed according to the rules and methods laid down by Euclidean
geometry.
In practice, the rigid surfaces which constitute the system of
co-ordinates are generally not available ; furthermore, the magnitudes of
the co-ordinates are not actually determined by constructions with rigid
rods, but by indirect means. If the results of physics and astronomy are
to maintain their clearness, the physical meaning of specifications of
position must always be sought in accordance with the above
considerations. 3)
We thus obtain the following result: Every description of events in
space involves the use of a rigid body to which such events have to be
referred. The resulting relationship takes for granted that the laws of
Euclidean geometry hold for "distances;" the "distance" being represented
physically by means of the convention of two marks on a rigid body.
Notes
1) Here
we have assumed that there is nothing left over i.e. that the
measurement gives a whole number. This difficulty is got over by the use
of divided measuring-rods, the introduction of which does not demand any
fundamentally new method.
[A]
Einstein used "Potsdamer Platz, Berlin" in the original text. In the
authorised translation this was supplemented with "Tranfalgar Square,
London". We have changed this to "Times Square, New York", as this is the
most well known/identifiable location to English speakers in the present
day. [Note by the janitor.]
2) 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.
3) A
refinement and modification of these views does not become necessary until
we come to deal with the general theory of relativity, treated in the
second part of this book.
Space and Time in Classical Mechanics
The purpose of mechanics is to describe how bodies change their
position in space with "time." I should load my conscience with grave sins
against the sacred spirit of lucidity were I to formulate the aims of
mechanics in this way, without serious reflection and detailed
explanations. Let us proceed to disclose these sins.
It is not clear what is to be understood here by "position" and
"space." I stand at the window of a railway carriage which is travelling
uniformly, and drop a stone on the embankment, without throwing it. Then,
disregarding the influence of the air resistance, I see the stone descend
in a straight line. A pedestrian who observes the misdeed from the
footpath notices that the stone falls to earth in a parabolic curve. I now
ask: Do the "positions" traversed by the stone lie "in reality" on a
straight line or on a parabola? Moreover, what is meant here by motion "in
space" ? From the considerations of the previous section the answer is
self-evident. In the first place we entirely shun the vague word "space,"
of which, we must honestly acknowledge, we cannot form the slightest
conception, and we replace it by "motion relative to a practically rigid
body of reference." The positions relative to the body of reference
(railway carriage or embankment) have already been defined in detail in
the preceding section. If instead of " body of reference " we insert "
system of co-ordinates," which is a useful idea for mathematical
description, we are in a position to say : The stone traverses a straight
line relative to a system of co-ordinates rigidly attached to the
carriage, but relative to a system of co-ordinates rigidly attached to the
ground (embankment) it describes a parabola. With the aid of this example
it is clearly seen that there is no such thing as an independently
existing trajectory (lit. "path-curve"
1)), but only a trajectory relative to a
particular body of reference.
In order to have a complete description of the motion, we must
specify how the body alters its position with time ; i.e. for
every point on the trajectory it must be stated at what time the body is
situated there. These data must be supplemented by such a definition of
time that, in virtue of this definition, these time-values can be regarded
essentially as magnitudes (results of measurements) capable of
observation. If we take our stand on the ground of classical mechanics, we
can satisfy this requirement for our illustration in the following manner.
We imagine two clocks of identical construction ; the man at the
railway-carriage window is holding one of them, and the man on the
footpath the other. Each of the observers determines the position on his
own reference-body occupied by the stone at each tick of the clock he is
holding in his hand. In this connection we have not taken account of the
inaccuracy involved by the finiteness of the velocity of propagation of
light. With this and with a second difficulty prevailing here we shall
have to deal in detail later.
Notes
1) That
is, a curve along which the body moves.
The Galileian System of Co-ordinates
As is well known, the fundamental law of the mechanics of Galilei-Newton,
which is known as the law of inertia, can be stated thus: A body
removed sufficiently far from other bodies continues in a state of rest or
of uniform motion in a straight line. This law not only says something
about the motion of the bodies, but it also indicates the reference-bodies
or systems of coordinates, permissible in mechanics, which can be used in
mechanical description. The visible fixed stars are bodies for which the
law of inertia certainly holds to a high degree of approximation. Now if
we use a system of co-ordinates which is rigidly attached to the earth,
then, relative to this system, every fixed star describes a circle of
immense radius in the course of an astronomical day, a result which is
opposed to the statement of the law of inertia. So that if we adhere to
this law we must refer these motions only to systems of coordinates
relative to which the fixed stars do not move in a circle. A system of
co-ordinates of which the state of motion is such that the law of inertia
holds relative to it is called a " Galileian system of co-ordinates." The
laws of the mechanics of Galelei-Newton can be regarded as valid only for
a Galileian system of co-ordinates.
The Principle of Relativity
(in the restricted sense)
In order to attain the greatest possible clearness, let us return to
our example of the railway carriage supposed to be travelling uniformly.
We call its motion a uniform translation ("uniform" because it is of
constant velocity and direction, " translation " because although the
carriage changes its position relative to the embankment yet it does not
rotate in so doing). Let us imagine a raven flying through the air in such
a manner that its motion, as observed from the embankment, is uniform and
in a straight line. If we were to observe the flying raven from the moving
railway carriage. we should find that the motion of the raven would be one
of different velocity and direction, but that it would still be uniform
and in a straight line. Expressed in an abstract manner we may say : If a
mass m is moving uniformly in a straight line
with respect to a co-ordinate system
K, then it
will also be moving uniformly and in a straight line relative to a second
co-ordinate system K1 provided that
the latter is executing a uniform translatory motion with respect to
K. In accordance with the discussion contained in
the preceding section, it follows that:
If
K is a Galileian co-ordinate
system. then every other co-ordinate system
K' is
a Galileian one, when, in relation to
K, it is in
a condition of uniform motion of translation. Relative to
K1 the mechanical laws of Galilei-Newton
hold good exactly as they do with respect to
K.
We advance a step farther in our generalisation when we express the
tenet thus: If, relative to K,
K1 is a uniformly moving co-ordinate system devoid of
rotation, then natural phenomena run their course with respect to
K1 according to exactly the same
general laws as with respect to K. This statement
is called the principle of relativity (in the restricted sense).
As long as one was convinced that all natural phenomena were capable of
representation with the help of classical mechanics, there was no need to
doubt the validity of this principle of relativity. But in view of the
more recent development of electrodynamics and optics it became more and
more evident that classical mechanics affords an insufficient foundation
for the physical description of all natural phenomena. At this juncture
the question of the validity of the principle of relativity became ripe
for discussion, and it did not appear impossible that the answer to this
question might be in the negative.
Nevertheless, there are two general facts which at the outset speak
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,
K1, K2, 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 organ pipe 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.
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 be 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.
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 cut 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 incompatibility
between the principle of relativity and the law of propagation of light,
and that by systematically holding fast to both these laws a logically
rigid theory could be arrived at. This theory has been called the
special theory of relativity to distinguish it from the extended
theory, with which we shall deal later. In the following pages we shall
present the fundamental ideas of the special theory of relativity.
On the Idea of Time in Physics
Lightning has struck the rails on our railway embankment at two places
A and
B far distant from
each other. I make the additional assertion that these two lightning
flashes occurred simultaneously. If I ask you whether there is sense in
this statement, you will answer my question with a decided "Yes." But if I
now approach you with the request to explain to me the sense of the
statement more precisely, you find after some consideration that the
answer to this question is not so easy as it appears at first sight.
After some time perhaps the following answer would occur to you: "The
significance of the statement is clear in itself and needs no further
explanation; of course it would require some consideration if I were to be
commissioned to determine by observations whether in the actual case the
two events took place simultaneously or not." I cannot be satisfied with
this answer for the following reason. Supposing that as a result of
ingenious considerations an able meteorologist were to discover that the
lightning must always strike the places
A and
B simultaneously, then we should be faced with
the task of testing whether or not this theoretical result is in
accordance with the reality. We encounter the same difficulty with all
physical statements in which the conception " simultaneous " plays a part.
The concept does not exist for the physicist until he has the possibility
of discovering whether or not it is fulfilled in an actual case. We thus
require a definition of simultaneity such that this definition supplies us
with the method by means of which, in the present case, he can decide by
experiment whether or not both the lightning strokes occurred
simultaneously. As long as this requirement is not satisfied, I allow
myself to be deceived as a physicist (and of course the same applies if I
am not a physicist), when I imagine that I am able to attach a meaning to
the statement of simultaneity. (I would ask the reader not to proceed
farther until he is fully convinced on this point.)
After thinking the matter over for some time you then offer the
following suggestion with which to test simultaneity. By measuring along
the rails, the connecting line AB should be
measured up and an observer placed at the mid-point
M
of the distance AB. This observer should be
supplied with an arrangement (e.g. two mirrors inclined at 900)
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
1) (here the railway embankment). We are
thus led also to a definition of " time " in physics. For this purpose we
suppose that clocks of identical construction are placed at the points
A,
B and
C of the railway line (co-ordinate system) and
that they are set in such a manner that the positions of their pointers
are simultaneously (in the above sense) the same. Under these conditions
we understand by the " time " of an event the reading (position of the
hands) of that one of these clocks which is in the immediate vicinity (in
space) of the event. In this manner a time-value is associated with every
event which is essentially capable of observation.
This stipulation contains a further physical hypothesis, the validity
of which will hardly be doubted without empirical evidence to the
contrary. It has been assumed that all these clocks go at the same
rate if they are of identical construction. Stated more exactly: When
two clocks arranged at rest in different places of a reference-body are
set in such a manner that a particular position of the pointers
of the one clock is simultaneous (in the above sense) with the
same position, of the pointers of the other clock, then identical "
settings " are always simultaneous (in the sense of the above definition).
Footnotes
1)
We suppose further, that, when three events
A,
B and
C occur in
different places in such a manner that
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 the 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.
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 :
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 M1 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
M1 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 M1 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 7) disappears.
We were led to that conflict by the considerations of
Section 6, 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 6 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.
On the Relativity of the Conception of Distance
Let us consider two particular points on the train
1) 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.
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
A1 and
B1
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
A1 and
B1 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 8. The distance between these points
A
and B is then measured by repeated application of
thee 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 6. 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.
Footnotes
1)
e.g. the middle of the first and of the hundredth carriage.
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
7) 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 7 disappears, because the theorem of the addition of
velocities derived in Section 6 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 6 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 6 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 2 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 K1, the same event
would be fixed in respect of space and time by corresponding values
x1, y1, z1, t1,
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.
Obviously our problem can be exactly formulated in the following
manner. What are the values x1, y1,
z1, t1, of an event with respect to
K1, 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
K1.
For the relative orientation in space of the co-ordinate systems indicated
in the diagram  , this problem is solved by means of the equations :
y1 = y
z1 = z
This system of equations is known as the " Lorentz transformation."
1)
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:
x1 = x - vt
y1 = y
z1 = z
t1 = 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
K1.
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 x1 and
t1. In point of fact, if we substitute
for x the value
ct in
the first and fourth equations of the Lorentz transformation, we obtain:
from which, by division, the expression
x1 = ct1
immediately follows. If referred to the system
K1, the propagation of light takes
place according to this equation. We thus see that the velocity of
transmission relative to the reference-body
K1
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.
Footnotes
1)
A simple derivation of the Lorentz transformation is given in
Appendix I.
The Behaviour of Measuring-Rods and Clocks in Motion
Place a metre-rod in the
x1-axis of
K1 in such a manner that one end (the
beginning) coincides with the point
x1=0
whilst the other end (the end of the rod) coincides with the point
x1=I. 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
the distance between the points being
 .
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
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
 ,
and for stiII 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
K1 would have been
;
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 (x1=0) of
K1.
t1=0 and
t1=I are two successive ticks of this
clock. The first and fourth equations of the Lorentz transformation give
for these two ticks :
t = 0
and
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
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.
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 6 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 horn the Galilei
transformation (Section
11). In place of the man walking inside the carriage, we introduce a
point moving relatively to the co-ordinate system
K1
in accordance with the equation
x1 = wt1
By means of the first and fourth equations of the Galilei
transformation we can express x1 and
t1 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 6,
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
x1 = wt1 B)
we must then express
x1and
t1 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
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 axe 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,
 ) when the liquid above mentioned is flowing through the tube
with a velocity v ?
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 6 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 K1, 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. Experiment1)
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.
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.
Footnotes
1)
Fizeau found  ,
where 
is the index of refraction of the liquid. On the other
hand, owing to the smallness of
as compared with I,
we can replace (B) in the first place by
 ,
or to the same order of approximation by
 ,
which agrees with Fizeau's result.
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 x1, y1, z1, t1
of a co-ordinate system K1. 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.
General Results of the Theory
It is clear from our previous considerations that the (special) theory
of relativity has grown out of electrodynamics and optics. In these fields
it has not appreciably altered the predictions of theory, but it has
considerably simplified the theoretical structure, i.e. the
derivation of laws, and — what is incomparably more important — it has
considerably reduced the number of independent hypothese forming the basis
of theory. The special theory of relativity has rendered the Maxwell-Lorentz
theory so plausible, that the latter would have been generally accepted by
physicists even if experiment had decided less unequivocally in its favour.
Classical mechanics required to be modified before it could come into
line with the demands of the special theory of relativity. For the main
part, however, this modification affects only the laws for rapid motions,
in which the velocities of matter
v are not very
small as compared with the velocity of light. We have experience of such
rapid motions only in the case of electrons and ions; for other motions
the variations from the laws of classical mechanics are too small to make
themselves evident in practice. We shall not consider the motion of stars
until we come to speak of the general theory of relativity. In accordance
with the theory of relativity the kinetic energy of a material point of
mass m is no longer given by the well-known
expression
but by the expression
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
When 
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 mc2 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 K1 which is in a state of
uniform motion of translation relative to
K, or,
briefly, relative to every " Galileian " system of co-ordinates. In
contrast to classical mechanics; the Lorentz transformation is the
deciding factor in the transition from one such system to another.
By means of comparatively simple considerations we are led to draw the
following conclusion from these premises, in conjunction with the
fundamental equations of the electrodynamics of Maxwell: A body moving
with the velocity v, which absorbs
1) an amount of energy
E0 in the form of radiation without
suffering an alteration in velocity in the process, has, as a consequence,
its energy increased by an amount
In consideration of the expression given above for the kinetic energy
of the body, the required energy of the body comes out to be
Thus the body has the same energy as a body of mass
moving with the velocity
v. Hence
we can say: If a body takes up an amount of energy
E0,
then its inertial mass increases by an amount
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
we see that the term
mc2, which has
hitherto attracted our attention, is nothing else than the energy
possessed by the body 2)
before it absorbed the energy E0.
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
E0
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.
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.
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.
Footnotes
1)
E0 is the energy taken up, as judged
from a co-ordinate system moving with the body.
2) As
judged from a co-ordinate system moving with the body.
[Note]
The equation E = mc2 has been
thoroughly proved time and again since this time.
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.1)
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
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.
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 13 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 5 that all attempts of this nature led to a negative result.
Before the theory of relativity was put forward, it was difficult to
become reconciled to this negative result, for reasons now to be
discussed. The inherited prejudices about time and space did not allow any
doubt to arise as to the prime importance of the Galileian transformation
for changing over from one body of reference to another. Now assuming that
the Maxwell-Lorentz equations hold for a reference-body
K, we then find that they do not hold for a reference-body
K1 moving uniformly with respect to
K, if we assume that the relations of the
Galileian transformation exist between the co-ordinates of
K and
K1. It
thus appears that, of all Galileian co-ordinate systems, one (K)
corresponding to a particular state of motion is physically unique. This
result was interpreted physically by regarding
K
as at rest with respect to a hypothetical æther of space. On the other
hand, all coordinate systems K1 moving
relatively to K were to be regarded as in motion
with respect to the æther. To this motion of
K1
against the æther ("æther-drift " relative to
K1)
were attributed the more complicated laws which were supposed to hold
relative to K1. Strictly speaking,
such an æther-drift ought also to be assumed relative to the earth, and
for a long time the efforts of physicists were devoted to attempts to
detect the existence of an æther-drift at the earth's surface.
In one of the most notable of these attempts Michelson devised a method
which appears as though it must be decisive. Imagine two mirrors so
arranged on a rigid body that the reflecting surfaces face each other. A
ray of light requires a perfectly definite time
T
to pass from one mirror to the other and back again, if the whole system
be at rest with respect to the æther. It is found by calculation, however,
that a slightly different time T1 is
required for this process, if the body, together with the mirrors, be
moving relatively to the æther. And yet another point: it is shown by
calculation that for a given velocity
v with
reference to the æther, this time
T1
is different when the body is moving perpendicularly to the planes of the
mirrors from that resulting when the motion is parallel to these planes.
Although the estimated difference between these two times is exceedingly
small, Michelson and Morley performed an experiment involving interference
in which this difference should have been clearly detectable. But the
experiment gave a negative result — a fact very perplexing to physicists.
Lorentz and FitzGerald rescued the theory from this difficulty by assuming
that the motion of the body relative to the æther produces a contraction
of the body in the direction of motion, the amount of contraction being
just sufficient to compensate for the difference in time mentioned above.
Comparison with the discussion in Section 11 shows that also from the standpoint of the theory of
relativity this solution of the difficulty was the right one. But on the
basis of the theory of relativity the method of interpretation is
incomparably more satisfactory. According to this theory there is no such
thing as a "specially favoured" (unique) co-ordinate system to occasion
the introduction of the æther-idea, and hence there can be no æther-drift,
nor any experiment with which to demonstrate it. Here the contraction of
moving bodies follows from the two fundamental principles of the theory,
without the introduction of particular hypotheses; and as the prime
factor involved in this contraction we find, not the motion in itself, to
which we cannot attach any meaning, but the motion with respect to the
body of reference chosen in the particular case in point. Thus for a
co-ordinate system moving with the earth the mirror system of Michelson
and Morley is not shortened, but it is shortened for a co-ordinate system
which is at rest relatively to the sun.
Footnotes
1) The
general theory of relativity renders it likely that the electrical masses
of an electron are held together by gravitational forces.
Minkowski's Four-Dimensional Space
The non-mathematician is seized by a mysterious shuddering when he
hears of "four-dimensional" things, by a feeling not unlike that awakened
by thoughts of the occult. And yet there is no more common-place statement
than that the world in which we live is a four-dimensional space-time
continuum.
Space is a three-dimensional continuum. By this we mean
that it is possible to describe the position of a point (at rest) by means
of three numbers (coordinales) 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 role, as compared with the space
coordinates. It is for this reason that we have been in the habit of
treating time as an independent continuum. As a matter of fact, according
to classical mechanics, time is absolute, i.e. it is independent
of the position and the condition of motion of the system of co-ordinates.
We see this expressed in the last equation of the Galileian transformation
(t1 = t)
The four-dimensional mode of consideration of the "world" is natural on
the theory of relativity, since according to this theory time is robbed of
its independence. This is shown by the fourth equation of the Lorentz
transformation:
Moreover, according to this equation the time difference Δt1
of two events with respect to K1 does
not in general vanish, even when the time difference Δt1
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.1)
In order to give due prominence to this relationship, however, we must
replace the usual time co-ordinate
t by an
imaginary magnitude 
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.
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 2.
Footnotes
1) Cf.
the somewhat more detailed discussion in
Appendix II.
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