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.”