introduction
A review of Einstein's famous book "Relativity, The Special and
General Theory". For a long time I have wanted to write a detailed
analysis, review, and "interpretation" of Einstein's book, "Relativity".
Personally, I think it might be the greatest scientific book ever written, not
so much because of advanced scientific content, but instead because Einstein was
able to explain it in a way that made it accessible to a wide layperson
audience. Even so, I thought there was room for a more mainstream
interpretation of these topics. Reading a number of interpretations and
explanations of Einstein's theory from a number of other authors helped me to
better understand these concepts. This will be an ongoing
effort that I expect to take a year or more, but I hope
when I am done this might help others (and myself) to better understand what I believe to be the
greatest scientific accomplishment in history.
1.
Physical Meaning of Geometrical Proportions
2.
The System Of Coordinates
3.
Space And Time In Classical Mechanics
4.
The Galileian System of Coordinates
5.
The Principle Of Relativity (In The Restricted Sense)

Chapter One 
Analysis 
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 your past
experience, you would certainly regard every one with disdain who
should pronounce even the most outoftheway 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. 
1 
Geometry sets out from certain conceptions such as “plane,”
“point,” and “straight line,” with which we are
able to associate more or less definite ideas, and from certain
simple propositions (axioms) which, in virtue of these ideas, we are
inclined to accept as “true.” Then, on the basis of a logical
process, the justification of which we feel ourselves compelled to
admit, all remaining propositions are shown to follow from those
axioms, i.e. they are proven. A proposition is then correct
(“true”) when it has been derived in the recognised manner from the
axioms. The question of the “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 line,” 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. 
2 
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. 
3 
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 (lineinterval),
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 ruler and compasses. 
4 
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. 

Chapter Two 
Analysis 
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. 1 
1 
Every description of the scene of
an event or of the position of an object in space is based on the
specification of the point on a rigid body (body of reference) with
which that event or object coincides. This
applies not only to scientific description, but also to everyday
life. If I analyse the place specification “Trafalgar Square,
London,” 2
I arrive at the following result. The earth is the rigid body to
which the specification of place refers; “Trafalgar Square, London”
is a welldefined point, to which a name has been assigned, and with
which the event coincides in space. 3 
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 Trafalgar Square, then we can
determine its position relative to the surface of the earth by
erecting a pole perpendicularly on the Square, so that it reaches
the cloud. The length of the pole measured with the standard
measuringrod, 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. 
3 
(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. 
4 
(b)
In locating the position of the object, we make use of a number
(here the length of the pole measured with the measuringrod)
instead of designated points of reference. 
5 
(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. 
6 
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 coordinates. 
7 
This consists of three plane surfaces perpendicular to each other
and rigidly attached to a rigid body. Referred
to a system of coordinates, the scene of any event will be
determined (for the main part) by the specification of the lengths
of the three perpendiculars or coordinates (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 measuringrods
performed according to the rules and methods laid down by Euclidean
geometry. 
8 
In practice, the rigid surfaces
which constitute the system of coordinates are generally not
available; furthermore, the magnitudes of the coordinates 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. 4 
9 
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. 
10 
Chapter Three 
Analysis 
“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 selfevident. 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 coordinates,” 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 coordinates
rigidly attached to the carriage, but relative to a system of
coordinates 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. “pathcurve” 1),
but only a trajectory relative to a particular body of reference. 
I stand at the window
of a railway carriage which is travelling uniformly, and drop a
stone on the embankment, without throwing it. 
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 timevalues 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 railwaycarriage window is
holding one of them, and the man on the footpath the other. Each of
the observers determines the position on his own referencebody
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. 

Note 1. That
is, a curve along which the body moves. [back]
Chapter Four 
Analysis 
AS is
well known, the fundamental law of the mechanics of GalileiNewton,
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 referencebodies 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
coordinates 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 coordinates of which the
state of motion is such that the law of inertia holds relative to it
is called a “Galileian system of coordinates.” The laws of the
mechanics of GalileiNewton can be regarded as valid only for a
Galileian system of coordinates. 

Chapter Five 
Analysis 
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 coordinate
system K,
then it will also be moving uniformly and in a straight line
relative to a second coordinate system
K',
provided that the latter is executing a
uniform translatory motion with respect to
K.
In accordance with the discussion contained in the preceding
section, it follows that: 
1 
If K is a
Galileian coordinate system, then every other coordinate system
K'
is a Galileian one, when, in relation to
K,
it is in a condition of uniform motion of translation. Relative to
K'
the mechanical laws of GalileiNewton hold good exactly as they do
with respect to
K. 
2 
We advance a step
farther in our generalisation when we express the tenet thus: If,
relative to
K, K' is a
uniformly moving coordinate system devoid of rotation, then natural
phenomena run their course with respect to K' according to exactly
the same general laws as with respect to
K.
This statement is called the principle of relativity (in the
restricted sense). 
3 
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. 
4 
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. 
5 
We now proceed to
the second argument, to which, moreover, we shall return later. If
the principle of relativity (in the restricted sense) does not hold,
then the Galileian coordinate systems
K, K', K'',
etc., which are moving uniformly relative to each other, will not be
equivalent for the description of natural phenomena. In this
case we should be constrained to believe that natural laws are
capable of being formulated in a particularly simple manner, and of
course only on condition that, from amongst all possible Galileian
coordinate systems, we should have chosen one (K_{0})
of a particular state of motion as our body of reference. We should
then be justified (because of its merits for the description of
natural phenomena) in calling this system “absolutely at rest,” and
all other Galileian systems K
“in motion.” If, for instance, our embankment were the system
K_{0},
then our railway carriage would be a system
K,
relative to which less simple laws would hold than with respect to
K_{0}.
This diminished simplicity would be due to the fact that the
carriage K
would be in motion (i.e. “really”) with respect to
K_{0}.
In the general laws of natural which have been formulated with
reference to
K,
the magnitude and direction of the velocity of the carriage would
necessarily play a part. We should expect, for instance, that the
note emitted by an organpipe placed with its axis parallel to the
direction of travel would be different from that emitted if the axis
of the pipe were placed perpendicular to this direction. Now in
virtue of its motion in an orbit round the sun, our earth is
comparable with a railway carriage travelling with a velocity of
about 30 kilometres per second. If the principle of relativity were
not valid we should therefore expect that the direction of motion of
the earth at any moment would enter into the laws of nature, and
also that physical systems in their behaviour would be dependent on
the orientation in space with respect to the
earth. For owing to the alteration in direction of the velocity of
rotation of the earth in the course of a year, the earth cannot be
at rest relative to the 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 nonequivalence of different
directions. This is a very powerful argument in favour of the
principle of relativity. 

