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The Principle of Relativity in restricted sense 1st arg.
In order to attain the greatest possible clearness, let us return to
our example of the railway carriage supposed to be travelling
uniformly. We call its motion a uniform translation ("uniform" because
it is of constant velocity and direction, " translation " because
although the carriage changes its position relative to the embankment
yet it does not rotate in so doing). Let us imagine a raven flying
through the air in such a manner that its motion, as observed from the
embankment, is uniform and in a straight line. If we were to observe
the flying raven from the moving railway carriage. we should find that
the motion of the raven would be one of different velocity and
direction, but that it would still be uniform and in a straight line.
Expressed in an abstract manner we may say : If a mass m is moving uniformly in a straight line with respect to a co-ordinate system K, then it will also be moving uniformly and in a straight line relative to a second co-ordinate system K1 provided that the latter is executing a uniform translatory motion with respect to K. In accordance with the discussion contained in the preceding section, it follows that:
If K is a Galileian co-ordinate system. then every other co-ordinate system K' is a Galileian one, when, in relation to K, it is in a condition of uniform motion of translation. Relative to K1 the mechanical laws of Galilei-Newton hold good exactly as they do with respect to K.
We advance a step farther in our generalisation when we express the tenet thus: If, relative to K, K1 is a uniformly moving co-ordinate system devoid of rotation, then natural phenomena run their course with respect to K1 according to exactly the same general laws as with respect to K. This statement is called the principle of relativity (in the restricted sense).
As long as one was convinced that all natural phenomena were capable of
representation with the help of classical mechanics, there was no need
to doubt the validity of this principle of relativity. But in view of
the more recent development of electrodynamics and optics it became
more and more evident that classical mechanics affords an insufficient
foundation for the physical description of all natural phenomena. At
this juncture the question of the validity of the principle of
relativity became ripe for discussion, and it did not appear impossible
that the answer to this question might be in the negative.
Nevertheless, there are two general facts which at the outset speak
very much in favour of the validity of the principle of relativity.
Even though classical mechanics does not supply us with a sufficiently
broad basis for the theoretical presentation of all physical phenomena,
still we must grant it a considerable measure of " truth," since it
supplies us with the actual motions of the heavenly bodies with a
delicacy of detail little short of wonderful. The principle of
relativity must therefore
apply with great accuracy in the domain of mechanics. But
that a principle of such broad generality should hold with such
exactness in one domain of phenomena, and yet should be invalid for
another, is a priori not very probable.
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The System of Co-ordinates
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.2)
This primitive method of place specification deals only with places on
the surface of rigid bodies, and is dependent on the existence of
points on this surface which are distinguishable from each other. But
we can free ourselves from both of these limitations without altering
the nature of our specification of position. If, for instance, a cloud
is hovering over Times Square, then we can determine its position
relative to the surface of the earth by erecting a pole perpendicularly
on the Square, so that it reaches the cloud. The length of the pole
measured with the standard measuring-rod, combined with the
specification of the position of the foot of the pole, supplies us with
a complete place specification. On the basis of this illustration, we
are able to see the manner in which a refinement of the conception of
position has been developed.
(a) We imagine the rigid body, to which the
place specification is referred, supplemented in such a manner that the
object whose position we require is reached by. the completed rigid
(b) In locating the position of the object, we
make use of a number (here the length of the pole measured with the
measuring-rod) instead of designated points of reference.
(c) We speak of the height of the cloud even
when the pole which reaches the cloud has not been erected. By means of
optical observations of the cloud from different positions on the
ground, and taking into account the properties of the propagation of
light, we determine the length of the pole we should have required in
order to reach the cloud.
From this consideration we see that it will be advantageous if, in
the description of position, it should be possible by means of
numerical measures to make ourselves independent of the existence of
marked positions (possessing names) on the rigid body of reference. In
the physics of measurement this is attained by the application of the
Cartesian system of co-ordinates.
This consists of three plane surfaces perpendicular to each other
and rigidly attached to a rigid body. Referred to a system of
co-ordinates, the scene of any event will be determined (for the main
part) by the specification of the lengths of the three perpendiculars
or co-ordinates (x, y, z)
which can be dropped from the scene of the event to those three plane
surfaces. The lengths of these three perpendiculars can be determined
by a series of manipulations with rigid measuring-rods performed
according to the rules and methods laid down by Euclidean geometry.
In practice, the rigid surfaces which constitute the system of
co-ordinates are generally not available ; furthermore, the magnitudes
of the co-ordinates are not actually determined by constructions with
rigid rods, but by indirect means. If the results of physics and
astronomy are to maintain their clearness, the physical meaning of
specifications of position must always be sought in accordance with the
above considerations. 3)
We thus obtain the following result: Every description of events in
space involves the use of a rigid body to which such events have to be
referred. The resulting relationship takes for granted that the laws of
Euclidean geometry hold for "distances;" the "distance" being
represented physically by means of the convention of two marks on a
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Relativity - The Special Theory Part 1
The Special Theory of Relativity
Physical Meaning of Geometrical Propositions
In your schooldays most of you who read this book made acquaintance
with the noble building of Euclid's geometry, and you remember —
perhaps with more respect than love — the magnificent structure, on the
lofty staircase of which you were chased about for uncounted hours by
conscientious teachers. By reason of our past experience, you would
certainly regard everyone with disdain who should pronounce even the
most out-of-the-way proposition of this science to be untrue. But
perhaps this feeling of proud certainty would leave you immediately if
some one were to ask you: "What, then, do you mean by the assertion
that these propositions are true?" Let us proceed to give this question
a little consideration.
Geometry sets out form certain conceptions such as "plane," "point,"
and "straight line," with which we are able to associate more or less
definite ideas, and from certain simple propositions (axioms) which, in
virtue of these ideas, we are inclined to accept as "true." Then, on
the basis of a logical process, the justification of which we feel
ourselves compelled to admit, all remaining propositions are shown to
follow from those axioms, i.e.
they are proven. A proposition is then correct ("true") when it has
been derived in the recognised manner from the axioms. The question of
"truth" of the individual geometrical propositions is thus reduced to
one of the "truth" of the axioms. Now it has long been known that the
last question is not only unanswerable by the methods of geometry, but
that it is in itself entirely without meaning. We cannot ask whether it
is true that only one straight line goes through two points. We can
only say that Euclidean geometry deals with things called "straight
lines," to each of which is ascribed the property of being uniquely
determined by two points situated on it. The concept "true" does not
tally with the assertions of pure geometry, because by the word "true"
we are eventually in the habit of designating always the correspondence
with a "real" object; geometry, however, is not concerned with the
relation of the ideas involved in it to objects of experience, but only
with the logical connection of these ideas among themselves.
It is not difficult to understand why, in spite of this, we feel
constrained to call the propositions of geometry "true." Geometrical
ideas correspond to more or less exact objects in nature, and these
last are undoubtedly the exclusive cause of the genesis of those ideas.
Geometry ought to refrain from such a course, in order to give to its
structure the largest possible logical unity. The practice, for
example, of seeing in a "distance" two marked positions on a
practically rigid body is something which is lodged deeply in our habit
of thought. We are accustomed further to regard three points as being
situated on a straight line, if their apparent positions can be made to
coincide for observation with one eye, under suitable choice of our
place of observation.
If, in pursuance of our habit of thought, we now supplement the
propositions of Euclidean geometry by the single proposition that two
points on a practically rigid body always correspond to the same
distance (line-interval), independently of any changes in position to
which we may subject the body, the propositions of Euclidean geometry
then resolve themselves into propositions on the possible relative
position of practically rigid bodies.1)
Geometry which has been supplemented in this way is then to be treated
as a branch of physics. We can now legitimately ask as to the "truth"
of geometrical propositions interpreted in this way, since we are
justified in asking whether these propositions are satisfied for those
real things we have associated with the geometrical ideas. In less
exact terms we can express this by saying that by the "truth" of a
geometrical proposition in this sense we understand its validity for a
construction with rule and compasses.
Of course the conviction of the "truth" of geometrical propositions
in this sense is founded exclusively on rather incomplete experience.
For the present we shall assume the "truth" of the geometrical
propositions, then at a later stage (in the general theory of
relativity) we shall see that this "truth" is limited, and we shall
consider the extent of its limitation.
Next: The System of Co-ordinates
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.
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Relativity - 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!
Relativity - The Special and General Theory
Albert Einstein Reference Archive
The Special and General Theory
Source: Relativity: The Special and General Theory © 1920
Publisher: Methuen & Co Ltd
First Published: December, 1916
Translated: Robert W. Lawson (Authorised translation)
Transcription/Markup: Brian Basgen
Thanks to The Project Gutenberg
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