Relativity. The Special and General
Theory, Part I. The Special Theory of
Relativity
Relativity: The Special and General Theory
by Albert Einstein
Part I: The Special Theory of Relativity
Section I: Physical Meaning of Geometrical Propositions
1.01.1 - answerIn 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.
Relativity: The Special and General Theory
by Albert Einstein
Part I: The Special Theory of Relativity
Section I: Physical Meaning of Geometrical Propositions
1.01.2
Geometry sets out from certain conceptions such as "plane," "point," and "straight line,"
with which we are able to associate more or less definite ideas, and from certain simple
propositions (axioms) which, in virtue of these ideas, we are inclined to accept as "true."
Then, on the basis of a logical process, the justification of which we feel ourselves
compelled to admit, all remaining propositions are shown to follow from those axioms,
i.e. they are proven. A proposition is then correct ("true") when it has been derived in
the recognised manner from the axioms. - answerThe 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.
Relativity: The Special and General Theory
by Albert Einstein
Part I: The Special Theory of Relativity
Section I: Physical Meaning of Geometrical Propositions
1.01.3
It is not difficult to understand why, in spite of this, we feel constrained to call the
propositions of geometry "true." - answerGeometrical 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.
Relativity: The Special and General Theory
by Albert Einstein
Part I: The Special Theory of Relativity
Section I: Physical Meaning of Geometrical Propositions
1.01.4 - answerIf, 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.
Relativity: The Special and General Theory
by Albert Einstein
Part I: The Special Theory of Relativity
Section I: Physical Meaning of Geometrical Propositions
1.01.5 - answerOf 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.
Relativity: The Special and General Theory
, by Albert Einstein
Part I: The Special Theory of Relativity
Section II: The System of Co-ordinates
1.02.1
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. - answerFor 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.
Relativity: The Special and General Theory
by Albert Einstein
Part I: The Special Theory of Relativity
Section II: The System of Co-ordinates
1.02.2 - answerEvery 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 "Potsdamer Platz, Berlin," I
arrive at the following result. The earth is the rigid body to which the specification of
place refers; "Potsdamer Platz, Berlin," is a well-defined point, to which a name has
been assigned, and with which the event coincides in space.
Relativity: The Special and General Theory
by Albert Einstein
Part I: The Special Theory of Relativity
Section II: The System of Co-ordinates
1.02.3
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. - answerBut 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 Potsdamer Platz, 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.
Relativity: The Special and General Theory
by Albert Einstein
Part I: The Special Theory of Relativity
Section II: The System of Co-ordinates
1.02.4
