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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.

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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

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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.

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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

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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

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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 :

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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

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                       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

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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

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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

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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.

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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

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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

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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

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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

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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,"