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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
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.* 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
*) 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. *
Every description of the scene of an event or of the position of an
object in space is based on the specification of the point on a rigid
body (body of reference) with which that event or object coincides.
This applies not only to scientific description, but also to everyday
life. If I analyse the place specification " Times Square, New York,"
**A I arrive at the following result. The earth is the rigid body
to which the specification of place refers; " Times Square, New York,"
is a well-defined point, to which a name has been assigned, and with
which the event coincides in space.**B
This primitive method of place specification deals only with places on
the surface of rigid bodies, and is dependent on the existence of
points on this surface which are distinguishable from each other. But
we can free ourselves from both of these limitations without altering
the nature of our specification of position. If, for instance, a cloud
is hovering over Times Square, then we can determine its position
relative to the surface of the earth by erecting a pole
perpendicularly on the Square, so that it reaches the cloud. The
length of the pole measured with the standard measuring-rod, combined
with the specification of the position of the foot of the pole,
supplies us with a complete place specification. On the basis of this
illustration, we are able to see the manner in which a refinement of
the conception of position has been developed.
(a) We imagine the rigid body, to which the place specification is
referred, supplemented in such a manner that the object whose position
we require is reached by. the completed rigid body.
(b) In locating the position of the object, we make use of a number
(here the length of the pole measured with the measuring-rod) instead
of designated points of reference.
(c) We speak of the height of the cloud even when the pole which
reaches the cloud has not been erected. By means of optical
observations of the cloud from different positions on the ground, and
taking into account the properties of the propagation of light, we
determine the length of the pole we should have required in order to
reach the cloud.
From this consideration we see that it will be advantageous if, in the
description of position, it should be possible by means of numerical
measures to make ourselves independent of the existence of marked
positions (possessing names) on the rigid body of reference. In the
physics of measurement this is attained by the application of the
Cartesian system of co-ordinates.
This consists of three plane surfaces perpendicular to each other and
rigidly attached to a rigid body. Referred to a system of
co-ordinates, the scene of any event will be determined (for the main
part) by the specification of the lengths of the three perpendiculars
or co-ordinates (x, y, z) which can be dropped from the scene of the
event to those three plane surfaces. The lengths of these three
perpendiculars can be determined by a series of manipulations with
rigid measuring-rods performed according to the rules and methods laid
down by Euclidean geometry.
In practice, the rigid surfaces which constitute the system of
co-ordinates are generally not available ; furthermore, the magnitudes
of the co-ordinates are not actually determined by constructions with
rigid rods, but by indirect means. If the results of physics and
astronomy are to maintain their clearness, the physical meaning of
specifications of position must always be sought in accordance with
the above considerations. ***
We thus obtain the following result: Every description of events in
space involves the use of a rigid body to which such events have to be
referred. The resulting relationship takes for granted that the laws
of Euclidean geometry hold for "distances;" the "distance" being
represented physically by means of the convention of two marks on a
rigid body.
Notes
* Here we have assumed that there is nothing left over i.e. that
the measurement gives a whole number. This difficulty is got over by
the use of divided measuring-rods, the introduction of which does not
demand any fundamentally new method.
**A Einstein used "Potsdamer Platz, Berlin" in the original text.
In the authorised translation this was supplemented with "Tranfalgar
Square, London". We have changed this to "Times Square, New York", as
this is the most well known/identifiable location to English speakers
in the present day. [Note by the janitor.]
**B It is not necessary here to investigate further the significance
of the expression "coincidence in space." This conception is
sufficiently obvious to ensure that differences of opinion are
scarcely likely to arise as to its applicability in practice.
*** A refinement and modification of these views does not become
necessary until we come to deal with the general theory of relativity,
treated in the second part of this book.
SPACE AND TIME IN CLASSICAL MECHANICS
The purpose of mechanics is to describe how bodies change their
position in space with "time." I should load my conscience with grave
sins against the sacred spirit of lucidity were I to formulate the
aims of mechanics in this way, without serious reflection and detailed
explanations. Let us proceed to disclose these sins.
It is not clear what is to be understood here by "position" and
"space." I stand at the window of a railway carriage which is
travelling uniformly, and drop a stone on the embankment, without
throwing it. Then, disregarding the influence of the air resistance, I
see the stone descend in a straight line. A pedestrian who observes
the misdeed from the footpath notices that the stone falls to earth in
a parabolic curve. I now ask: Do the "positions" traversed by the
stone lie "in reality" on a straight line or on a parabola? Moreover,
what is meant here by motion "in space" ? From the considerations of
the previous section the answer is self-evident. In the first place we
entirely shun the vague word "space," of which, we must honestly
acknowledge, we cannot form the slightest conception, and we replace
it by "motion relative to a practically rigid body of reference." The
positions relative to the body of reference (railway carriage or
embankment) have already been defined in detail in the preceding
section. If instead of " body of reference " we insert " system of
co-ordinates," which is a useful idea for mathematical description, we
are in a position to say : The stone traverses a straight line
relative to a system of co-ordinates rigidly attached to the carriage,
but relative to a system of co-ordinates rigidly attached to the
ground (embankment) it describes a parabola. With the aid of this
example it is clearly seen that there is no such thing as an
independently existing trajectory (lit. "path-curve"*), but only
a trajectory relative to a particular body of reference.
In order to have a complete description of the motion, we must specify
how the body alters its position with time ; i.e. for every point on
the trajectory it must be stated at what time the body is situated
there. These data must be supplemented by such a definition of time
that, in virtue of this definition, these time-values can be regarded
essentially as magnitudes (results of measurements) capable of
observation. If we take our stand on the ground of classical
mechanics, we can satisfy this requirement for our illustration in the
following manner. We imagine two clocks of identical construction ;
the man at the railway-carriage window is holding one of them, and the
man on the footpath the other. Each of the observers determines the
position on his own reference-body occupied by the stone at each tick
of the clock he is holding in his hand. In this connection we have not
taken account of the inaccuracy involved by the finiteness of the
velocity of propagation of light. With this and with a second
difficulty prevailing here we shall have to deal in detail later.
Notes
*) That is, a curve along which the body moves.
THE GALILEIAN SYSTEM OF CO-ORDINATES
As is well known, the fundamental law of the mechanics of
Galilei-Newton, which is known as the law of inertia, can be stated
thus: A body removed sufficiently far from other bodies continues in a
state of rest or of uniform motion in a straight line. This law not
only says something about the motion of the bodies, but it also
indicates the reference-bodies or systems of coordinates, permissible
in mechanics, which can be used in mechanical description. The visible
fixed stars are bodies for which the law of inertia certainly holds to
a high degree of approximation. Now if we use a system of co-ordinates
which is rigidly attached to the earth, then, relative to this system,
every fixed star describes a circle of immense radius in the course of
an astronomical day, a result which is opposed to the statement of the
law of inertia. So that if we adhere to this law we must refer these
motions only to systems of coordinates relative to which the fixed
stars do not move in a circle. A system of co-ordinates of which the
state of motion is such that the law of inertia holds relative to it
is called a " Galileian system of co-ordinates." The laws of the
mechanics of Galflei-Newton can be regarded as valid only for a
Galileian system of co-ordinates.
THE PRINCIPLE OF RELATIVITY
(IN THE RESTRICTED SENSE)
In order to attain the greatest possible clearness, let us return to
our example of the railway carriage supposed to be travelling
uniformly. We call its motion a uniform translation ("uniform" because
it is of constant velocity and direction, " translation " because
although the carriage changes its position relative to the embankment
yet it does not rotate in so doing). Let us imagine a raven flying
through the air in such a manner that its motion, as observed from the
embankment, is uniform and in a straight line. If we were to observe
the flying raven from the moving railway carriage. we should find that
the motion of the raven would be one of different velocity and
direction, but that it would still be uniform and in a straight line.
Expressed in an abstract manner we may say : If a mass m is moving
uniformly in a straight line with respect to a co-ordinate system K,
then it will also be moving uniformly and in a straight line relative
to a second co-ordinate system K1 provided that the latter is
executing a uniform translatory motion with respect to K. In
accordance with the discussion contained in the preceding section, it
follows that:
If K is a Galileian co-ordinate system. then every other co-ordinate
system K' is a Galileian one, when, in relation to K, it is in a
condition of uniform motion of translation. Relative to K1 the
mechanical laws of Galilei-Newton hold good exactly as they do with
respect to K.
We advance a step farther in our generalisation when we express the
tenet thus: If, relative to K, K1 is a uniformly moving co-ordinate
system devoid of rotation, then natural phenomena run their course
with respect to K1 according to exactly the same general laws as with
respect to K. This statement is called the principle of relativity (in
the restricted sense).
As long as one was convinced that all natural phenomena were capable
of representation with the help of classical mechanics, there was no
need to doubt the validity of this principle of relativity. But in
view of the more recent development of electrodynamics and optics it
became more and more evident that classical mechanics affords an
insufficient foundation for the physical description of all natural
phenomena. At this juncture the question of the validity of the
principle of relativity became ripe for discussion, and it did not
appear impossible that the answer to this question might be in the
negative.
Nevertheless, there are two general facts which at the outset speak
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hypothetical system K[0] 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 observa- tions 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 theref
ore more inclined to reject the principle of relativity, in spite of
the fact that no empirical data had been found which were
contradictory to this principle.
At this juncture the theory of relativity entered the arena. As a
result of an analysis of the physical conceptions of time and space,
it became evident that in realily there is not the least
incompatibilitiy between the principle of relativity and the law of
propagation of light, and that by systematically holding fast to both
these laws a logically rigid theory could be arrived at. This theory
has been called the special theory of relativity to distinguish it
from the extended theory, with which we shall deal later. In the
following pages we shall present the fundamental ideas of the special
theory of relativity.
ON THE IDEA OF TIME IN PHYSICS
Lightning has struck the rails on our railway embankment at two places
A and B far distant from each other. I make the additional assertion
that these two lightning flashes occurred simultaneously. If I ask you
whether there is sense in this statement, you will answer my question
with a decided "Yes." But if I now approach you with the request to
explain to me the sense of the statement more precisely, you find
after some consideration that the answer to this question is not so
easy as it appears at first sight.
After some time perhaps the following answer would occur to you: "The
significance of the statement is clear in itself and needs no further
explanation; of course it would require some consideration if I were
to be commissioned to determine by observations whether in the actual
case the two events took place simultaneously or not." I cannot be
satisfied with this answer for the following reason. Supposing that as
a result of ingenious considerations an able meteorologist were to
discover that the lightning must always strike the places A and B
simultaneously, then we should be faced with the task of testing
whether or not this theoretical result is in accordance with the
reality. We encounter the same difficulty with all physical statements
in which the conception " simultaneous " plays a part. The concept
does not exist for the physicist until he has the possibility of
discovering whether or not it is fulfilled in an actual case. We thus
require a definition of simultaneity such that this definition
supplies us with the method by means of which, in the present case, he
can decide by experiment whether or not both the lightning strokes
occurred simultaneously. As long as this requirement is not satisfied,
I allow myself to be deceived as a physicist (and of course the same
applies if I am not a physicist), when I imagine that I am able to
attach a meaning to the statement of simultaneity. (I would ask the
reader not to proceed farther until he is fully convinced on this
point.)
After thinking the matter over for some time you then offer the
following suggestion with which to test simultaneity. By measuring
along the rails, the connecting line AB should be measured up and an
observer placed at the mid-point M of the distance AB. This observer
should be supplied with an arrangement (e.g. two mirrors inclined at
90^0) which allows him visually to observe both places A and B at the
same time. If the observer perceives the two flashes of lightning at
the same time, then they are simultaneous.
I am very pleased with this suggestion, but for all that I cannot
regard the matter as quite settled, because I feel constrained to
raise the following objection:
"Your definition would certainly be right, if only I knew that the
light by means of which the observer at M perceives the lightning
flashes travels along the length A arrow M with the same velocity as
along the length B arrow M. But an examination of this supposition
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me -- and rightly so -- and you declare:
"I maintain my previous definition nevertheless, because in reality it
assumes absolutely nothing about light. There is only one demand to be
made of the definition of simultaneity, namely, that in every real
case it must supply us with an empirical decision as to whether or not
the conception that has to be defined is fulfilled. That my definition
satisfies this demand is indisputable. That light requires the same
time to traverse the path A arrow M as for the path B arrow M is in
reality neither a supposition nor a hypothesis about the physical
nature of light, but a stipulation which I can make of my own freewill
in order to arrive at a definition of simultaneity."
It is clear that this definition can be used to give an exact meaning
not only to two events, but to as many events as we care to choose,
and independently of the positions of the scenes of the events with
respect to the body of reference * (here the railway embankment).
We are thus led also to a definition of " time " in physics. For this
purpose we suppose that clocks of identical construction are placed at
the points A, B and C of the railway line (co-ordinate system) and
that they are set in such a manner that the positions of their
pointers are simultaneously (in the above sense) the same. Under these
conditions we understand by the " time " of an event the reading
(position of the hands) of that one of these clocks which is in the
immediate vicinity (in space) of the event. In this manner a
time-value is associated with every event which is essentially capable
of observation.
This stipulation contains a further physical hypothesis, the validity
of which will hardly be doubted without empirical evidence to the
contrary. It has been assumed that all these clocks go at the same
rate if they are of identical construction. Stated more exactly: When
two clocks arranged at rest in different places of a reference-body
are set in such a manner that a particular position of the pointers of
the one clock is simultaneous (in the above sense) with the same
position, of the pointers of the other clock, then identical "
settings " are always simultaneous (in the sense of the above
definition).
Notes
*) 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 SIMULATNEITY
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
Fig. 01: file fig01.gif
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 arrow 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 arrow 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
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introduction of the theory of relativity.
ON THE RELATIVITY OF THE CONCEPTION OF DISTANCE
Let us consider two particular points on the train * 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
A^1 and B^1 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^1 and B^1 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.
Notes
*) 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. Fig. 2 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 ([7]Fig. 2), this problem is solved by means of the
equations :
t/Relativity.test view on Meta::CPAN
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:
eq. 3: file eq03.gif
eq. 4: file eq04.gif
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.
Notes
*) 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
eq. 05a: file eq05a.gif
eq. 05b: file eq05b.gif
the distance between the points being eq. 06 .
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 eq. 06 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 eq. 06a ,
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 eq. 06 ;
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
eq. 07: file eq07.gif
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
eq. 08: file eq08.gif
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
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moving with the body.
**) As judged from a co-ordinate system moving with the body.
***[Note] The equation E = mc^2 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 b-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.* 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
eq. 05: file eq05.gif
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 transformstion exist between the
co-ordinates of K and K1. It thus appears that, of all Galileian
co-ordinate systems, one (K) corresponding to a particular state of
motion is physically unique. This result was interpreted physically by
regarding K as at rest with respect to a hypothetical æther of space.
On the other hand, all coordinate systems K1 moving relatively to K
were to be regarded as in motion with respect to the æther. To this
motion of K1 against the æther ("æther-drift " relative to K1) were
attributed the more complicated laws which were supposed to hold
relative to K1. Strictly speaking, such an æther-drift ought also to
be assumed relative to the earth, and for a long time the efforts of
physicists were devoted to attempts to detect the existence of an
æther-drift at the earth's surface.
In one of the most notable of these attempts Michelson devised a
method which appears as though it must be decisive. Imagine two
mirrors so arranged on a rigid body that the reflecting surfaces face
each other. A ray of light requires a perfectly definite time T to
pass from one mirror to the other and back again, if the whole system
be at rest with respect to the æther. It is found by calculation,
however, that a slightly different time T1 is required for this
process, if the body, together with the mirrors, be moving relatively
to the æther. And yet another point: it is shown by calculation that
for a given velocity v with reference to the æther, this time T1 is
different when the body is moving perpendicularly to the planes of the
mirrors from that resulting when the motion is parallel to these
planes. Although the estimated difference between these two times is
exceedingly small, Michelson and Morley performed an experiment
involving interference in which this difference should have been
clearly detectable. But the experiment gave a negative result -- a
fact very perplexing to physicists. Lorentz and FitzGerald rescued the
theory from this difficulty by assuming that the motion of the body
relative to the æther produces a contraction of the body in the
direction of motion, the amount of contraction being just sufficient
to compensate for the differeace in time mentioned above. Comparison
with the discussion in Section 11 shows that also from the
standpoint of the theory of relativity this solution of the difficulty
was the right one. But on the basis of the theory of relativity the
method of interpretation is incomparably more satisfactory. According
to this theory there is no such thing as a " specially favoured "
(unique) co-ordinate system to occasion the introduction of the
æther-idea, and hence there can be no æther-drift, nor any experiment
with which to demonstrate it. Here the contraction of moving bodies
follows from the two fundamental principles of the theory, without the
introduction of particular hypotheses ; and as the prime factor
involved in this contraction we find, not the motion in itself, to
which we cannot attach any meaning, but the motion with respect to the
body of reference chosen in the particular case in point. Thus for a
co-ordinate system moving with the earth the mirror system of
Michelson and Morley is not shortened, but it is shortened for a
co-ordinate system which is at rest relatively to the sun.
Notes
*) The general theory of relativity renders it likely that the
electrical masses of an electron are held together by gravitational
forces.
MINKOWSKI'S FOUR-DIMENSIONAL SPACE
The non-mathematician is seized by a mysterious shuddering when he
hears of "four-dimensional" things, by a feeling not unlike that
awakened by thoughts of the occult. And yet there is no more
common-place statement than that the world in which we live is a
four-dimensional space-time continuum.
Space is a three-dimensional continuum. By this we mean that it is
possible to describe the position of a point (at rest) by means of
three numbers (co-ordinales) x, y, z, and that there is an indefinite
number of points in the neighbourhood of this one, the position of
which can be described by co-ordinates such as x[1], y[1], z[1], which
may be as near as we choose to the respective values of the
co-ordinates x, y, z, of the first point. In virtue of the latter
t/Relativity.test view on Meta::CPAN
squares. The arrangement is such, that each side of a square belongs
to two squares and each corner to four squares.
It is a veritable wonder that we can carry out this business without
getting into the greatest difficulties. We only need to think of the
following. If at any moment three squares meet at a corner, then two
sides of the fourth square are already laid, and, as a consequence,
the arrangement of the remaining two sides of the square is already
completely determined. But I am now no longer able to adjust the
quadrilateral so that its diagonals may be equal. If they are equal of
their own accord, then this is an especial favour of the marble slab
and of the little rods, about which I can only be thankfully
surprised. We must experience many such surprises if the construction
is to be successful.
If everything has really gone smoothly, then I say that the points of
the marble slab constitute a Euclidean continuum with respect to the
little rod, which has been used as a " distance " (line-interval). By
choosing one corner of a square as " origin" I can characterise every
other corner of a square with reference to this origin by means of two
numbers. I only need state how many rods I must pass over when,
starting from the origin, I proceed towards the " right " and then "
upwards," in order to arrive at the corner of the square under
consideration. These two numbers are then the " Cartesian co-ordinates
" of this corner with reference to the " Cartesian co-ordinate system"
which is determined by the arrangement of little rods.
By making use of the following modification of this abstract
experiment, we recognise that there must also be cases in which the
experiment would be unsuccessful. We shall suppose that the rods "
expand " by in amount proportional to the increase of temperature. We
heat the central part of the marble slab, but not the periphery, in
which case two of our little rods can still be brought into
coincidence at every position on the table. But our construction of
squares must necessarily come into disorder during the heating,
because the little rods on the central region of the table expand,
whereas those on the outer part do not.
With reference to our little rods -- defined as unit lengths -- the
marble slab is no longer a Euclidean continuum, and we are also no
longer in the position of defining Cartesian co-ordinates directly
with their aid, since the above construction can no longer be carried
out. But since there are other things which are not influenced in a
similar manner to the little rods (or perhaps not at all) by the
temperature of the table, it is possible quite naturally to maintain
the point of view that the marble slab is a " Euclidean continuum."
This can be done in a satisfactory manner by making a more subtle
stipulation about the measurement or the comparison of lengths.
But if rods of every kind (i.e. of every material) were to behave in
the same way as regards the influence of temperature when they are on
the variably heated marble slab, and if we had no other means of
detecting the effect of temperature than the geometrical behaviour of
our rods in experiments analogous to the one described above, then our
best plan would be to assign the distance one to two points on the
slab, provided that the ends of one of our rods could be made to
coincide with these two points ; for how else should we define the
distance without our proceeding being in the highest measure grossly
arbitrary ? The method of Cartesian coordinates must then be
discarded, and replaced by another which does not assume the validity
of Euclidean geometry for rigid bodies.* The reader will notice
that the situation depicted here corresponds to the one brought about
by the general postitlate of relativity (Section 23).
Notes
*) Mathematicians have been confronted with our problem in the
following form. If we are given a surface (e.g. an ellipsoid) in
Euclidean three-dimensional space, then there exists for this surface
a two-dimensional geometry, just as much as for a plane surface. Gauss
undertook the task of treating this two-dimensional geometry from
first principles, without making use of the fact that the surface
belongs to a Euclidean continuum of three dimensions. If we imagine
constructions to be made with rigid rods in the surface (similar to
that above with the marble slab), we should find that different laws
hold for these from those resulting on the basis of Euclidean plane
geometry. The surface is not a Euclidean continuum with respect to the
rods, and we cannot define Cartesian co-ordinates in the surface.
Gauss indicated the principles according to which we can treat the
geometrical relationships in the surface, and thus pointed out the way
to the method of Riemman of treating multi-dimensional, non-Euclidean
continuum. Thus it is that mathematicians long ago solved the formal
problems to which we are led by the general postulate of relativity.
GAUSSIAN CO-ORDINATES
According to Gauss, this combined analytical and geometrical mode of
handling the problem can be arrived at in the following way. We
imagine a system of arbitrary curves (see Fig. 4) drawn on the surface
of the table. These we designate as u-curves, and we indicate each of
them by means of a number. The Curves u= 1, u= 2 and u= 3 are drawn in
the diagram. Between the curves u= 1 and u= 2 we must imagine an
infinitely large number to be drawn, all of which correspond to real
numbers lying between 1 and 2. fig. 04 We have then a system of
u-curves, and this "infinitely dense" system covers the whole surface
of the table. These u-curves must not intersect each other, and
through each point of the surface one and only one curve must pass.
Thus a perfectly definite value of u belongs to every point on the
surface of the marble slab. In like manner we imagine a system of
v-curves drawn on the surface. These satisfy the same conditions as
the u-curves, they are provided with numbers in a corresponding
manner, and they may likewise be of arbitrary shape. It follows that a
value of u and a value of v belong to every point on the surface of
the table. We call these two numbers the co-ordinates of the surface
of the table (Gaussian co-ordinates). For example, the point P in the
diagram has the Gaussian co-ordinates u= 3, v= 1. Two neighbouring
points P and P1 on the surface then correspond to the co-ordinates
P: u,v
P1: u + du, v + dv,
where du and dv signify very small numbers. In a similar manner we may
indicate the distance (line-interval) between P and P1, as measured
with a little rod, by means of the very small number ds. Then
according to Gauss we have
ds2 = g[11]du2 + 2g[12]dudv = g[22]dv2
where g[11], g[12], g[22], are magnitudes which depend in a perfectly
definite way on u and v. The magnitudes g[11], g[12] and g[22],
determine the behaviour of the rods relative to the u-curves and
v-curves, and thus also relative to the surface of the table. For the
case in which the points of the surface considered form a Euclidean
continuum with reference to the measuring-rods, but only in this case,
it is possible to draw the u-curves and v-curves and to attach numbers
to them, in such a manner, that we simply have :
ds2 = du2 + dv2
Under these conditions, the u-curves and v-curves are straight lines
t/Relativity.test view on Meta::CPAN
dx2 + dy2 + dz2 - c^2dt2 = dx1 2 + dy1 2 + dz1 2 - c^2dt1 2.
The validity of the Lorentz transformation follows from this
condition. We can express this as follows: The magnitude
ds2 = dx2 + dy2 + dz2 - c^2dt2,
which belongs to two adjacent points of the four-dimensional
space-time continuum, has the same value for all selected (Galileian)
reference-bodies. If we replace x, y, z, sq. rt. -I . ct , by x[1],
x[2], x[3], x[4], we also obtaill the result that
ds2 = dx[1]^2 + dx[2]^2 + dx[3]^2 + dx[4]^2.
is independent of the choice of the body of reference. We call the
magnitude ds the " distance " apart of the two events or
four-dimensional points.
Thus, if we choose as time-variable the imaginary variable sq. rt. -I
. ct instead of the real quantity t, we can regard the space-time
contintium -- accordance with the special theory of relativity -- as a
", Euclidean " four-dimensional continuum, a result which follows from
the considerations of the preceding section.
Notes
*) Cf. Appendixes I and 2. The relations which are derived
there for the co-ordlnates themselves are valid also for co-ordinate
differences, and thus also for co-ordinate differentials (indefinitely
small differences).
THE SPACE-TIME CONTINUUM OF THE GENERAL THEORY OF REALTIIVTY IS NOT A
ECULIDEAN CONTINUUM
In the first part of this book we were able to make use of space-time
co-ordinates which allowed of a simple and direct physical
interpretation, and which, according to Section 26, can be regarded
as four-dimensional Cartesian co-ordinates. This was possible on the
basis of the law of the constancy of the velocity of tight. But
according to Section 21 the general theory of relativity cannot
retain this law. On the contrary, we arrived at the result that
according to this latter theory the velocity of light must always
depend on the co-ordinates when a gravitational field is present. In
connection with a specific illustration in Section 23, we found
that the presence of a gravitational field invalidates the definition
of the coordinates and the ifine, which led us to our objective in the
special theory of relativity.
In view of the resuIts of these considerations we are led to the
conviction that, according to the general principle of relativity, the
space-time continuum cannot be regarded as a Euclidean one, but that
here we have the general case, corresponding to the marble slab with
local variations of temperature, and with which we made acquaintance
as an example of a two-dimensional continuum. Just as it was there
impossible to construct a Cartesian co-ordinate system from equal
rods, so here it is impossible to build up a system (reference-body)
from rigid bodies and clocks, which shall be of such a nature that
measuring-rods and clocks, arranged rigidly with respect to one
another, shaIll indicate position and time directly. Such was the
essence of the difficulty with which we were confronted in Section
23.
But the considerations of Sections 25 and 26 show us the way to
surmount this difficulty. We refer the fourdimensional space-time
continuum in an arbitrary manner to Gauss co-ordinates. We assign to
every point of the continuum (event) four numbers, x[1], x[2], x[3],
x[4] (co-ordinates), which have not the least direct physical
significance, but only serve the purpose of numbering the points of
the continuum in a definite but arbitrary manner. This arrangement
does not even need to be of such a kind that we must regard x[1],
x[2], x[3], as "space" co-ordinates and x[4], as a " time "
co-ordinate.
The reader may think that such a description of the world would be
quite inadequate. What does it mean to assign to an event the
particular co-ordinates x[1], x[2], x[3], x[4], if in themselves these
co-ordinates have no significance ? More careful consideration shows,
however, that this anxiety is unfounded. Let us consider, for
instance, a material point with any kind of motion. If this point had
only a momentary existence without duration, then it would to
described in space-time by a single system of values x[1], x[2], x[3],
x[4]. Thus its permanent existence must be characterised by an
infinitely large number of such systems of values, the co-ordinate
values of which are so close together as to give continuity;
corresponding to the material point, we thus have a (uni-dimensional)
line in the four-dimensional continuum. In the same way, any such
lines in our continuum correspond to many points in motion. The only
statements having regard to these points which can claim a physical
existence are in reality the statements about their encounters. In our
mathematical treatment, such an encounter is expressed in the fact
that the two lines which represent the motions of the points in
question have a particular system of co-ordinate values, x[1], x[2],
x[3], x[4], in common. After mature consideration the reader will
doubtless admit that in reality such encounters constitute the only
actual evidence of a time-space nature with which we meet in physical
statements.
When we were describing the motion of a material point relative to a
body of reference, we stated nothing more than the encounters of this
point with particular points of the reference-body. We can also
determine the corresponding values of the time by the observation of
encounters of the body with clocks, in conjunction with the
observation of the encounter of the hands of clocks with particular
points on the dials. It is just the same in the case of
space-measurements by means of measuring-rods, as a litttle
consideration will show.
The following statements hold generally : Every physical description
resolves itself into a number of statements, each of which refers to
the space-time coincidence of two events A and B. In terms of Gaussian
co-ordinates, every such statement is expressed by the agreement of
their four co-ordinates x[1], x[2], x[3], x[4]. Thus in reality, the
description of the time-space continuum by means of Gauss co-ordinates
completely replaces the description with the aid of a body of
reference, without suffering from the defects of the latter mode of
description; it is not tied down to the Euclidean character of the
continuum which has to be represented.
EXACT FORMULATION OF THE GENERAL PRINCIPLE OF RELATIVITY
We are now in a position to replace the pro. visional formulation of
the general principle of relativity given in Section 18 by an exact
formulation. The form there used, "All bodies of reference K, K1,
etc., are equivalent for the description of natural phenomena
(formulation of the general laws of nature), whatever may be their
state of motion," cannot be maintained, because the use of rigid
reference-bodies, in the sense of the method followed in the special
theory of relativity, is in general not possible in space-time
description. The Gauss co-ordinate system has to take the place of the
body of reference. The following statement corresponds to the
fundamental idea of the general principle of relativity: "All Gaussian
co-ordinate systems are essentially equivalent for the formulation of
the general laws of nature."
We can state this general principle of relativity in still another
form, which renders it yet more clearly intelligible than it is when
in the form of the natural extension of the special principle of
relativity. According to the special theory of relativity, the
equations which express the general laws of nature pass over into
equations of the same form when, by making use of the Lorentz
transformation, we replace the space-time variables x, y, z, t, of a
(Galileian) reference-body K by the space-time variables x1, y1, z1,
t1, of a new reference-body K1. According to the general theory of
relativity, on the other hand, by application of arbitrary
substitutions of the Gauss variables x[1], x[2], x[3], x[4], the
equations must pass over into equations of the same form; for every
transformation (not only the Lorentz transformation) corresponds to
the transition of one Gauss co-ordinate system into another.
If we desire to adhere to our "old-time" three-dimensional view of
things, then we can characterise the development which is being
undergone by the fundamental idea of the general theory of relativity
as follows : The special theory of relativity has reference to
Galileian domains, i.e. to those in which no gravitational field
exists. In this connection a Galileian reference-body serves as body
of reference, i.e. a rigid body the state of motion of which is so
chosen that the Galileian law of the uniform rectilinear motion of
"isolated" material points holds relatively to it.
Certain considerations suggest that we should refer the same Galileian
domains to non-Galileian reference-bodies also. A gravitational field
of a special kind is then present with respect to these bodies (cf.
Sections 20 and 23).
In gravitational fields there are no such things as rigid bodies with
Euclidean properties; thus the fictitious rigid body of reference is
of no avail in the general theory of relativity. The motion of clocks
is also influenced by gravitational fields, and in such a way that a
physical definition of time which is made directly with the aid of
clocks has by no means the same degree of plausibility as in the
special theory of relativity.
For this reason non-rigid reference-bodies are used, which are as a
whole not only moving in any way whatsoever, but which also suffer
alterations in form ad lib. during their motion. Clocks, for which the
law of motion is of any kind, however irregular, serve for the
definition of time. We have to imagine each of these clocks fixed at a
point on the non-rigid reference-body. These clocks satisfy only the
one condition, that the "readings" which are observed simultaneously
on adjacent clocks (in space) differ from each other by an
indefinitely small amount. This non-rigid reference-body, which might
appropriately be termed a "reference-mollusc", is in the main
equivalent to a Gaussian four-dimensional co-ordinate system chosen
arbitrarily. That which gives the "mollusc" a certain
comprehensibility as compared with the Gauss co-ordinate system is the
(really unjustified) formal retention of the separate existence of the
space co-ordinates as opposed to the time co-ordinate. Every point on
the mollusc is treated as a space-point, and every material point
which is at rest relatively to it as at rest, so long as the mollusc
is considered as reference-body. The general principle of relativity
requires that all these molluscs can be used as reference-bodies with
equal right and equal success in the formulation of the general laws
of nature; the laws themselves must be quite independent of the choice
of mollusc.
The great power possessed by the general principle of relativity lies
in the comprehensive limitation which is imposed on the laws of nature
in consequence of what we have seen above.
THE SOLUTION OF THE PROBLEM OF GRAVITATION ON THE BASIS OF THE GENERAL
PRINCIPLE OF RELATIVITY
If the reader has followed all our previous considerations, he will
have no further difficulty in understanding the methods leading to the
solution of the problem of gravitation.
We start off on a consideration of a Galileian domain, i.e. a domain
in which there is no gravitational field relative to the Galileian
reference-body K. The behaviour of measuring-rods and clocks with
reference to K is known from the special theory of relativity,
likewise the behaviour of "isolated" material points; the latter move
uniformly and in straight lines.
Now let us refer this domain to a random Gauss coordinate system or to
a "mollusc" as reference-body K1. Then with respect to K1 there is a
gravitational field G (of a particular kind). We learn the behaviour
of measuring-rods and clocks and also of freely-moving material points
with reference to K1 simply by mathematical transformation. We
interpret this behaviour as the behaviour of measuring-rods, docks and
material points tinder the influence of the gravitational field G.
Hereupon we introduce a hypothesis: that the influence of the
gravitational field on measuringrods, clocks and freely-moving
material points continues to take place according to the same laws,
even in the case where the prevailing gravitational field is not
derivable from the Galfleian special care, simply by means of a
transformation of co-ordinates.
The next step is to investigate the space-time behaviour of the
gravitational field G, which was derived from the Galileian special
case simply by transformation of the coordinates. This behaviour is
formulated in a law, which is always valid, no matter how the
reference-body (mollusc) used in the description may be chosen.
This law is not yet the general law of the gravitational field, since
the gravitational field under consideration is of a special kind. In
order to find out the general law-of-field of gravitation we still
require to obtain a generalisation of the law as found above. This can
be obtained without caprice, however, by taking into consideration the
t/Relativity.test view on Meta::CPAN
theory rather requires that the universe should have a kind of centre
in which the density of the stars is a maximum, and that as we proceed
outwards from this centre the group-density of the stars should
diminish, until finally, at great distances, it is succeeded by an
infinite region of emptiness. The stellar universe ought to be a
finite island in the infinite ocean of space.*
This conception is in itself not very satisfactory. It is still less
satisfactory because it leads to the result that the light emitted by
the stars and also individual stars of the stellar system are
perpetually passing out into infinite space, never to return, and
without ever again coming into interaction with other objects of
nature. Such a finite material universe would be destined to become
gradually but systematically impoverished.
In order to escape this dilemma, Seeliger suggested a modification of
Newton's law, in which he assumes that for great distances the force
of attraction between two masses diminishes more rapidly than would
result from the inverse square law. In this way it is possible for the
mean density of matter to be constant everywhere, even to infinity,
without infinitely large gravitational fields being produced. We thus
free ourselves from the distasteful conception that the material
universe ought to possess something of the nature of a centre. Of
course we purchase our emancipation from the fundamental difficulties
mentioned, at the cost of a modification and complication of Newton's
law which has neither empirical nor theoretical foundation. We can
imagine innumerable laws which would serve the same purpose, without
our being able to state a reason why one of them is to be preferred to
the others ; for any one of these laws would be founded just as little
on more general theoretical principles as is the law of Newton.
Notes
*) Proof -- According to the theory of Newton, the number of "lines
of force" which come from infinity and terminate in a mass m is
proportional to the mass m. If, on the average, the Mass density p[0]
is constant throughout tithe universe, then a sphere of volume V will
enclose the average man p[0]V. Thus the number of lines of force
passing through the surface F of the sphere into its interior is
proportional to p[0] V. For unit area of the surface of the sphere the
number of lines of force which enters the sphere is thus proportional
to p[0] V/F or to p[0]R. Hence the intensity of the field at the
surface would ultimately become infinite with increasing radius R of
the sphere, which is impossible.
THE POSSIBILITY OF A "FINITE" AND YET "UNBOUNDED" UNIVERSE
But speculations on the structure of the universe also move in quite
another direction. The development of non-Euclidean geometry led to
the recognition of the fact, that we can cast doubt on the
infiniteness of our space without coming into conflict with the laws
of thought or with experience (Riemann, Helmholtz). These questions
have already been treated in detail and with unsurpassable lucidity by
Helmholtz and Poincaré, whereas I can only touch on them briefly here.
In the first place, we imagine an existence in two dimensional space.
Flat beings with flat implements, and in particular flat rigid
measuring-rods, are free to move in a plane. For them nothing exists
outside of this plane: that which they observe to happen to themselves
and to their flat " things " is the all-inclusive reality of their
plane. In particular, the constructions of plane Euclidean geometry
can be carried out by means of the rods e.g. the lattice construction,
considered in Section 24. In contrast to ours, the universe of
these beings is two-dimensional; but, like ours, it extends to
infinity. In their universe there is room for an infinite number of
identical squares made up of rods, i.e. its volume (surface) is
infinite. If these beings say their universe is " plane," there is
sense in the statement, because they mean that they can perform the
constructions of plane Euclidean geometry with their rods. In this
connection the individual rods always represent the same distance,
independently of their position.
Let us consider now a second two-dimensional existence, but this time
on a spherical surface instead of on a plane. The flat beings with
their measuring-rods and other objects fit exactly on this surface and
they are unable to leave it. Their whole universe of observation
extends exclusively over the surface of the sphere. Are these beings
able to regard the geometry of their universe as being plane geometry
and their rods withal as the realisation of " distance " ? They cannot
do this. For if they attempt to realise a straight line, they will
obtain a curve, which we " three-dimensional beings " designate as a
great circle, i.e. a self-contained line of definite finite length,
which can be measured up by means of a measuring-rod. Similarly, this
universe has a finite area that can be compared with the area, of a
square constructed with rods. The great charm resulting from this
consideration lies in the recognition of the fact that the universe of
these beings is finite and yet has no limits.
But the spherical-surface beings do not need to go on a world-tour in
order to perceive that they are not living in a Euclidean universe.
They can convince themselves of this on every part of their " world,"
provided they do not use too small a piece of it. Starting from a
point, they draw " straight lines " (arcs of circles as judged in
three dimensional space) of equal length in all directions. They will
call the line joining the free ends of these lines a " circle." For a
plane surface, the ratio of the circumference of a circle to its
diameter, both lengths being measured with the same rod, is, according
to Euclidean geometry of the plane, equal to a constant value p, which
is independent of the diameter of the circle. On their spherical
surface our flat beings would find for this ratio the value
eq. 27: file eq27.gif
i.e. a smaller value than p, the difference being the more
considerable, the greater is the radius of the circle in comparison
with the radius R of the " world-sphere." By means of this relation
the spherical beings can determine the radius of their universe ("
world "), even when only a relatively small part of their worldsphere
is available for their measurements. But if this part is very small
indeed, they will no longer be able to demonstrate that they are on a
spherical " world " and not on a Euclidean plane, for a small part of
a spherical surface differs only slightly from a piece of a plane of
the same size.
Thus if the spherical surface beings are living on a planet of which
the solar system occupies only a negligibly small part of the
spherical universe, they have no means of determining whether they are
living in a finite or in an infinite universe, because the " piece of
universe " to which they have access is in both cases practically
plane, or Euclidean. It follows directly from this discussion, that
for our sphere-beings the circumference of a circle first increases
with the radius until the " circumference of the universe " is
reached, and that it thenceforward gradually decreases to zero for
still further increasing values of the radius. During this process the
area of the circle continues to increase more and more, until finally
it becomes equal to the total area of the whole " world-sphere."
Perhaps the reader will wonder why we have placed our " beings " on a
sphere rather than on another closed surface. But this choice has its
justification in the fact that, of all closed surfaces, the sphere is
unique in possessing the property that all points on it are
equivalent. I admit that the ratio of the circumference c of a circle
to its radius r depends on r, but for a given value of r it is the
same for all points of the " worldsphere "; in other words, the "
world-sphere " is a " surface of constant curvature."
To this two-dimensional sphere-universe there is a three-dimensional
analogy, namely, the three-dimensional spherical space which was
discovered by Riemann. its points are likewise all equivalent. It
possesses a finite volume, which is determined by its "radius"
(2p2R3). Is it possible to imagine a spherical space? To imagine a
space means nothing else than that we imagine an epitome of our "
space " experience, i.e. of experience that we can have in the
movement of " rigid " bodies. In this sense we can imagine a spherical
space.
Suppose we draw lines or stretch strings in all directions from a
point, and mark off from each of these the distance r with a
measuring-rod. All the free end-points of these lengths lie on a
spherical surface. We can specially measure up the area (F) of this
surface by means of a square made up of measuring-rods. If the
universe is Euclidean, then F = 4pR2 ; if it is spherical, then F is
always less than 4pR2. With increasing values of r, F increases from
zero up to a maximum value which is determined by the " world-radius,"
but for still further increasing values of r, the area gradually
diminishes to zero. At first, the straight lines which radiate from
the starting point diverge farther and farther from one another, but
later they approach each other, and finally they run together again at
a "counter-point" to the starting point. Under such conditions they
have traversed the whole spherical space. It is easily seen that the
three-dimensional spherical space is quite analogous to the
two-dimensional spherical surface. It is finite (i.e. of finite
volume), and has no bounds.
It may be mentioned that there is yet another kind of curved space: "
elliptical space." It can be regarded as a curved space in which the
two " counter-points " are identical (indistinguishable from each
other). An elliptical universe can thus be considered to some extent
as a curved universe possessing central symmetry.
It follows from what has been said, that closed spaces without limits
are conceivable. From amongst these, the spherical space (and the
elliptical) excels in its simplicity, since all points on it are
equivalent. As a result of this discussion, a most interesting
question arises for astronomers and physicists, and that is whether
the universe in which we live is infinite, or whether it is finite in
the manner of the spherical universe. Our experience is far from being
sufficient to enable us to answer this question. But the general
theory of relativity permits of our answering it with a moduate degree
of certainty, and in this connection the difficulty mentioned in
Section 30 finds its solution.
THE STRUCTURE OF SPACE ACCORDING TO THE GENERAL THEORY OF RELATIVITY
According to the general theory of relativity, the geometrical
properties of space are not independent, but they are determined by
matter. Thus we can draw conclusions about the geometrical structure
of the universe only if we base our considerations on the state of the
matter as being something that is known. We know from experience that,
for a suitably chosen co-ordinate system, the velocities of the stars
are small as compared with the velocity of transmission of light. We
can thus as a rough approximation arrive at a conclusion as to the
nature of the universe as a whole, if we treat the matter as being at
rest.
We already know from our previous discussion that the behaviour of
measuring-rods and clocks is influenced by gravitational fields, i.e.
by the distribution of matter. This in itself is sufficient to exclude
the possibility of the exact validity of Euclidean geometry in our
universe. But it is conceivable that our universe differs only
slightly from a Euclidean one, and this notion seems all the more
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