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(00:00):
This is a LibriVox recording. AllLibriVox recordings are in the public domain.
For more information or to volunteer,please visit LibriVox dot org. Recording by
Kelly Bicher of Mattapoisett, Massachusetts.Relativity The Special and General Theory by Albert
(00:21):
Einstein, Part one, The SpecialTheory of Relativity, Sections one through three.
Section one Physical meaning of geometrical propositions. In your school days, most
of you who read this book madeacquaintance with the noble building of Euclid's geometry,
(00:42):
and you remember, perhaps with morerespect than love, the magnificent structure
on the lofty staircase of which youwere chased about for uncounted hours by conscientious
teachers. By reason of our pastexperience, you would certainly regard every one
with disdain who should pronounce even themost out of the way proposition of this
science to be untrue. But perhapsthis feeling of proud certainty would leave you
(01:06):
immediately if someone were to ask you, what, then, do you mean
by the assertion that these propositions aretrue? Let us proceed to give this
question a little consideration. Geometry setsout from certain conceptions, such as plain
point in straight line with which weare able to associate more or less definite
ideas, and from certain simple propositionsor axioms, which and virtue of these
(01:30):
ideas we are inclined to accept astrue. Then, on the basis of
a logical process, the justification ofwhich we feel ourselves compelled to admit,
all remaining propositions are shown to followfrom those axioms, i e. They
are proven. Our proposition is thencorrect or true when it has been derived
(01:51):
in the recognized manner from the axioms. The question of truth of the individual
geometrical propositions is thus reduced to oneof the true truth of the axioms.
Now, it has long been knownthat the last question is not only unanswerable
by the methods of geometry, butthat it is in itself entirely without meaning.
We cannot ask whether it is truethat only one straight line goes through
(02:14):
two points. We can only saythat Euclidean geometry deals with things called straight
lines, to each of which isascribed the property of being uniquely determined by
two points situated on it. Theconcept true does not tally with the assertions
of pure geometry, because by theword true, we are eventually in the
habit of designating always the correspondence witha real object. Geometry, however,
(02:38):
is not concerned with the relation ofthe ideas involved in it to objects of
experience, but only with the logicalconnection 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
(02:59):
exact objects in nature, and theselast are undoubtedly the exclusive cause of the
genesis of those ideas. Geometry oughtto refrain from such a course in order
to give its structure the largest possiblelogical unity. The practice, for example,
of seeing in a distance two marketpoints on a practically rigid body is
(03:19):
something which is lodged deeply in ourhabit of thought. We are accustomed further
to regard three points as being situatedon a straight line, if their apparent
positions can be made to coincide forobservation with one eye under suitable choice of
our place of observation. If inpursuance of our habit of thought, we
now supplement the propositions of Euclidean geometryby the single proposition that two points on
(03:42):
a practically rigid body always correspond tothe same distance line interval, independently of
any changes in position to which wemay subject the body. The propositions of
Euclidean geometry then resolve themselves into propositionson the possible relative position of rigid bodies.
Begin footnote, it follows that anatural object is associated also with a
(04:06):
straight line. Three points A,B, and C on a rigid body
thus lie in a straight line whenthe points A and C being given B
is chosen such that the sum ofthe distances A, B and bc is
as short as possible. This incompletesuggestion will suffice for the present purpose and
(04:30):
a footnote. Geometry, which hasbeen supplemented in this way, is then
to be treated as a branch ofphysics. We can now legitimately ask as
to the truth of geometrical propositions interpretedin this way, since we are justified
in asking whether these propositions are satisfiedfor those real things we have associated with
the geometrical ideas. In less exactterms, we can express this by saying
(04:55):
that by the truth of a geometricalproposition in this sense, we understand its
fellodity for construction with ruler and compasses. Of course, the conviction of the
truth of geometrical propositions in this senseis founded exclusively on rather incomplete experience.
For the present we shall assume thetruth of the geometrical propositions. Then,
(05:15):
at a later stage in the generaltheory of relativity we shall see that this
truth is limited, and we shallconsider the extent of its limitation. End
of section one. Section two thesystem of co ordinates. On the basis
of the physical interpretation of distance whichhas been indicated, we are also in
(05:41):
a position to establish the distance betweentwo points on a rigid body by means
of measurements. For this purpose werequire a distance broad S, which is
to be used once and for all, and which we employ as a standard
measure. If now A and Bare two points on a rigid body,
we can construct the line joining themaccording to the rules of geometry. Then,
(06:03):
starting from A, we can markoff the distance S time after time
until we reach B. A numberof these operations required is the numerical measure
of the distance A B. Thisis the basis of all measurement of length.
Begin footnote. Here we have assumedthat there is nothing left over,
(06:24):
i e. That the measurement givesa whole number. This difficulty is got
over by the use of divided measuringrods, the introduction of which does not
demand any fundamentally new method en footnote. Every description of the scene, of
an event, or of the positionof an object in space is based on
the specification of the point on arigid body body of reference with which that
(06:48):
event or object coincides. This impliesnot only to scientific description, but also
to everyday life. If I analyzethe place specification time square, New York.
Begin footnote. Einstein used Potsdamer PlatzBerlin in the original text. In
the authorized translation, this was supplementedwith Traffalger Square, London. We have
(07:12):
changed this to time square, NewYork, as this is the most well
known identifiable location to English speakers inthe present day. Note by the janitor
and footnote, I arrive at thefollowing result. The Earth is the rigid
body to which the specification of placerefers. Time square New York is a
well defined point to which a namehas been assigned and with which the event
(07:34):
coincides in space. Begin footnote.It is not necessary here to investigate further
the significance of the expression coincidence inspace. This conception is sufficiently obvious to
ensure that the differences of opinion arescarcely likely to arise as to its applicability
in practice. End footnote. Thisprimitive method of place bestiali vacation deals only
(08:01):
with places on the surface of rigidbodies, and is dependent on the existence
of points on the surface which aredistinguishable from each other. But we can
free ourselves from both of these limitationswithout altering the nature of our specification of
physition. If, for instance,a cloud is hovering over time square,
then we can determine its position relativeto the surface of the Earth by erecting
(08:24):
a pole perpendicularly on the square sothat it reaches the cloud. The length
of the pole, measured with thestandard measuring rod, combined with the specification
of the position of the foot ofthe pole, supplies us with a complete
place specification. On the basis ofthis illustration, we are able to see
the manner in which a refinement ofthe conception of position has been developed.
(08:48):
A we imagine the rigid body towhich the place specification is referred supplemented in
such a manner that the object whoseposition we require is completed by the completed
rigid body. B. In locatingthe position of the object, we make
use of a number here the lengthof the pole measured with the measuring rod,
instead of designated points of reference.See we speak of the height of
(09:11):
the cloud even when the pole whichreaches the cloud has not been erected.
By means of optical observations of thecloud from different positions on the ground,
and taking into account the properties ofthe propagation of light, we determine the
length of the pole we should haverequired in order to reach the cloud.
From this consideration we see that itwill be advantageous if, in the description
of position, it should be possible, by means of numerical measure, to
(09:35):
make ourselves independent of the existence ofmarked positions those possessing names on the rigid
body of reference. In the physicsof measurement, this is attained by the
application of the Cartesian system of coordinates. This consists of three plane services perpendicular
to each other and rigidly attached toa rigid body referred to a system of
(09:58):
coordinates. The scene of any eventwill be determined for the main part by
the specification of the lengths of thethree perpendiculars or coordinates X, y,
z, which can be dropped fromthe scene of the event to those three
plane surfaces. The lengths of thesethree perpendiculars can be determined by a series
of manipulations with rigid measuring rods performedaccording to the rules and methods laid down
(10:22):
by Euclidean geometry. 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 the constructions with rigidrods, but by indirect means. If
the result of physics and astronomy areto maintain their clearness, the physical meaning
of specifications of position must always besought in accordance with the above considerations.
(10:48):
Begin footnote. A refinement and modificationof these views does not become necessary until
we come to deal with the generaltheory of relativity treated in the second part
of this book and footnote. Wethus obtain the following result. Every description
of events in space involves the useof a rigid body to which such events
(11:09):
have to be referred. The resultingrelationship takes for granted that the laws of
Euclidean geometry hold for distances, thedistance being represented physically by means of the
convention of tu marks on a rigidbody and of section two Section three,
Space and time. In classical mechanics, the purpose of mechanics is to describe
(11:35):
how bodies change their position in spacewith time. I should load my conscience
with grave sins against the sacred spiritof lucidity, where I to formulate the
aims of mechanics in this way withoutserious reflection and detailed explanations. Let us
proceed to disclose these sins. Itis not clear what is to be understood
(11:56):
here by a position in space.I stand at the window of a railway
carriage which is traveling uniformly, anddrop a stone on the embankment without throwing
it. Then, disregarding the influenceof 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 towardsin a parabolic curve. I now ask
(12:20):
do the positions traversed by the stonelie in reality on a straight line or
on a parabola. Moreover, whatis meant here by motion in space?
From the consideration of the previous section, The answer is self evident. In
the first place, we entirely shunthe vague word space, of which we
must honestly acknowledge we cannot form theslightest conception, and we replace it by
(12:46):
motion relative to a practically rigid bodyof reference. The positions relative to the
body of reference railway, carriage orembankment have already been defined in detail and
preceding section. If instead of bodyof reference we insert system of coordinates,
which is a useful idea for mathematicaldescription, we are in a position to
(13:07):
say the stone traverses a straight linerelative to a system of coordinates rigidly attached
to the carriage, but relative toa 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 isno such thing as an independently existing trajectory
(13:28):
literally path curve begin footnote. Thatis a curve along which the body moves
en footnote, but only a trajectoryrelative to a particular body of reference.
In order to have a complete descriptionof the motion, we must specify how
the body alters its position with time. How e, for every point on
(13:50):
the trajectory, it must be statedat what time the body is situated there.
These data must be supplemented by sucha definition of time that, in
virtue of this definition, these timevalues can be regarded essentially as magnitudes results
of measurements capable of observation. Ifwe take our stand on the ground of
classical mechanics, we can satisfy thisrequirement. For our illustration in the following
(14:16):
manner, we imagine two clocks ofidentical 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 determinesthe position on his own reference body occupied
by the stone at each tick ofthe clock he is holding in his hand.
In this connection, we have nottaken account of the inaccuracy involved by
(14:37):
the finiteness of the velocity of propagationof light. With this, and with
a second difficulty prevailing here, weshall have to deal in detail later.
End of Section three
This is a LibriVox recording. AllLibriVox recordings are in the public domain.
For more information or to volunteer,please visit LibriVox dot org. Recording by
Kelly Bicher of Mattapoisett, Massachusetts.Relativity The Special and General Theory by Albert
(00:21):
Einstein, Part one, The SpecialTheory of Relativity, Sections one through three.
Section one Physical meaning of geometrical propositions. In your school days, most
of you who read this book madeacquaintance with the noble building of Euclid's geometry,
(00:42):
and you remember, perhaps with morerespect than love, the magnificent structure
on the lofty staircase of which youwere chased about for uncounted hours by conscientious
teachers. By reason of our pastexperience, you would certainly regard every one
with disdain who should pronounce even themost out of the way proposition of this
science to be untrue. But perhapsthis feeling of proud certainty would leave you
(01:06):
immediately if someone were to ask you, what, then, do you mean
by the assertion that these propositions aretrue? Let us proceed to give this
question a little consideration. Geometry setsout from certain conceptions, such as plain
point in straight line with which weare able to associate more or less definite
ideas, and from certain simple propositionsor axioms, which and virtue of these
(01:30):
ideas we are inclined to accept astrue. Then, on the basis of
a logical process, the justification ofwhich we feel ourselves compelled to admit,
all remaining propositions are shown to followfrom those axioms, i e. They
are proven. Our proposition is thencorrect or true when it has been derived
(01:51):
in the recognized manner from the axioms. The question of truth of the individual
geometrical propositions is thus reduced to oneof the true truth of the axioms.
Now, it has long been knownthat the last question is not only unanswerable
by the methods of geometry, butthat it is in itself entirely without meaning.
We cannot ask whether it is truethat only one straight line goes through
(02:14):
two points. We can only saythat Euclidean geometry deals with things called straight
lines, to each of which isascribed the property of being uniquely determined by
two points situated on it. Theconcept true does not tally with the assertions
of pure geometry, because by theword true, we are eventually in the
habit of designating always the correspondence witha real object. Geometry, however,
(02:38):
is not concerned with the relation ofthe ideas involved in it to objects of
experience, but only with the logicalconnection 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
(02:59):
exact objects in nature, and theselast are undoubtedly the exclusive cause of the
genesis of those ideas. Geometry oughtto refrain from such a course in order
to give its structure the largest possiblelogical unity. The practice, for example,
of seeing in a distance two marketpoints on a practically rigid body is
(03:19):
something which is lodged deeply in ourhabit of thought. We are accustomed further
to regard three points as being situatedon a straight line, if their apparent
positions can be made to coincide forobservation with one eye under suitable choice of
our place of observation. If inpursuance of our habit of thought, we
now supplement the propositions of Euclidean geometryby the single proposition that two points on
(03:42):
a practically rigid body always correspond tothe same distance line interval, independently of
any changes in position to which wemay subject the body. The propositions of
Euclidean geometry then resolve themselves into propositionson the possible relative position of rigid bodies.
Begin footnote, it follows that anatural object is associated also with a
(04:06):
straight line. Three points A,B, and C on a rigid body
thus lie in a straight line whenthe points A and C being given B
is chosen such that the sum ofthe distances A, B and bc is
as short as possible. This incompletesuggestion will suffice for the present purpose and
(04:30):
a footnote. Geometry, which hasbeen supplemented in this way, is then
to be treated as a branch ofphysics. We can now legitimately ask as
to the truth of geometrical propositions interpretedin this way, since we are justified
in asking whether these propositions are satisfiedfor those real things we have associated with
the geometrical ideas. In less exactterms, we can express this by saying
(04:55):
that by the truth of a geometricalproposition in this sense, we understand its
fellodity for construction with ruler and compasses. Of course, the conviction of the
truth of geometrical propositions in this senseis founded exclusively on rather incomplete experience.
For the present we shall assume thetruth of the geometrical propositions. Then,
(05:15):
at a later stage in the generaltheory of relativity we shall see that this
truth is limited, and we shallconsider the extent of its limitation. End
of section one. Section two thesystem of co ordinates. On the basis
of the physical interpretation of distance whichhas been indicated, we are also in
(05:41):
a position to establish the distance betweentwo points on a rigid body by means
of measurements. For this purpose werequire a distance broad S, which is
to be used once and for all, and which we employ as a standard
measure. If now A and Bare two points on a rigid body,
we can construct the line joining themaccording to the rules of geometry. Then,
(06:03):
starting from A, we can markoff the distance S time after time
until we reach B. A numberof these operations required is the numerical measure
of the distance A B. Thisis the basis of all measurement of length.
Begin footnote. Here we have assumedthat there is nothing left over,
(06:24):
i e. That the measurement givesa whole number. This difficulty is got
over by the use of divided measuringrods, the introduction of which does not
demand any fundamentally new method en footnote. Every description of the scene, of
an event, or of the positionof an object in space is based on
the specification of the point on arigid body body of reference with which that
(06:48):
event or object coincides. This impliesnot only to scientific description, but also
to everyday life. If I analyzethe place specification time square, New York.
Begin footnote. Einstein used Potsdamer PlatzBerlin in the original text. In
the authorized translation, this was supplementedwith Traffalger Square, London. We have
(07:12):
changed this to time square, NewYork, as this is the most well
known identifiable location to English speakers inthe present day. Note by the janitor
and footnote, I arrive at thefollowing result. The Earth is the rigid
body to which the specification of placerefers. Time square New York is a
well defined point to which a namehas been assigned and with which the event
(07:34):
coincides in space. Begin footnote.It is not necessary here to investigate further
the significance of the expression coincidence inspace. This conception is sufficiently obvious to
ensure that the differences of opinion arescarcely likely to arise as to its applicability
in practice. End footnote. Thisprimitive method of place bestiali vacation deals only
(08:01):
with places on the surface of rigidbodies, and is dependent on the existence
of points on the surface which aredistinguishable from each other. But we can
free ourselves from both of these limitationswithout altering the nature of our specification of
physition. If, for instance,a cloud is hovering over time square,
then we can determine its position relativeto the surface of the Earth by erecting
(08:24):
a pole perpendicularly on the square sothat it reaches the cloud. The length
of the pole, measured with thestandard measuring rod, combined with the specification
of the position of the foot ofthe pole, supplies us with a complete
place specification. On the basis ofthis illustration, we are able to see
the manner in which a refinement ofthe conception of position has been developed.
(08:48):
A we imagine the rigid body towhich the place specification is referred supplemented in
such a manner that the object whoseposition we require is completed by the completed
rigid body. B. In locatingthe position of the object, we make
use of a number here the lengthof the pole measured with the measuring rod,
instead of designated points of reference.See we speak of the height of
(09:11):
the cloud even when the pole whichreaches the cloud has not been erected.
By means of optical observations of thecloud from different positions on the ground,
and taking into account the properties ofthe propagation of light, we determine the
length of the pole we should haverequired in order to reach the cloud.
From this consideration we see that itwill be advantageous if, in the description
of position, it should be possible, by means of numerical measure, to
(09:35):
make ourselves independent of the existence ofmarked positions those possessing names on the rigid
body of reference. In the physicsof measurement, this is attained by the
application of the Cartesian system of coordinates. This consists of three plane services perpendicular
to each other and rigidly attached toa rigid body referred to a system of
(09:58):
coordinates. The scene of any eventwill be determined for the main part by
the specification of the lengths of thethree perpendiculars or coordinates X, y,
z, which can be dropped fromthe scene of the event to those three
plane surfaces. The lengths of thesethree perpendiculars can be determined by a series
of manipulations with rigid measuring rods performedaccording to the rules and methods laid down
(10:22):
by Euclidean geometry. 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 the constructions with rigidrods, but by indirect means. If
the result of physics and astronomy areto maintain their clearness, the physical meaning
of specifications of position must always besought in accordance with the above considerations.
(10:48):
Begin footnote. A refinement and modificationof these views does not become necessary until
we come to deal with the generaltheory of relativity treated in the second part
of this book and footnote. Wethus obtain the following result. Every description
of events in space involves the useof a rigid body to which such events
(11:09):
have to be referred. The resultingrelationship takes for granted that the laws of
Euclidean geometry hold for distances, thedistance being represented physically by means of the
convention of tu marks on a rigidbody and of section two Section three,
Space and time. In classical mechanics, the purpose of mechanics is to describe
(11:35):
how bodies change their position in spacewith time. I should load my conscience
with grave sins against the sacred spiritof lucidity, where I to formulate the
aims of mechanics in this way withoutserious reflection and detailed explanations. Let us
proceed to disclose these sins. Itis not clear what is to be understood
(11:56):
here by a position in space.I stand at the window of a railway
carriage which is traveling uniformly, anddrop a stone on the embankment without throwing
it. Then, disregarding the influenceof 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 towardsin a parabolic curve. I now ask
(12:20):
do the positions traversed by the stonelie in reality on a straight line or
on a parabola. Moreover, whatis meant here by motion in space?
From the consideration of the previous section, The answer is self evident. In
the first place, we entirely shunthe vague word space, of which we
must honestly acknowledge we cannot form theslightest conception, and we replace it by
(12:46):
motion relative to a practically rigid bodyof reference. The positions relative to the
body of reference railway, carriage orembankment have already been defined in detail and
preceding section. If instead of bodyof reference we insert system of coordinates,
which is a useful idea for mathematicaldescription, we are in a position to
(13:07):
say the stone traverses a straight linerelative to a system of coordinates rigidly attached
to the carriage, but relative toa 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 isno such thing as an independently existing trajectory
(13:28):
literally path curve begin footnote. Thatis a curve along which the body moves
en footnote, but only a trajectoryrelative to a particular body of reference.
In order to have a complete descriptionof the motion, we must specify how
the body alters its position with time. How e, for every point on
(13:50):
the trajectory, it must be statedat what time the body is situated there.
These data must be supplemented by sucha definition of time that, in
virtue of this definition, these timevalues can be regarded essentially as magnitudes results
of measurements capable of observation. Ifwe take our stand on the ground of
classical mechanics, we can satisfy thisrequirement. For our illustration in the following
(14:16):
manner, we imagine two clocks ofidentical 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 determinesthe position on his own reference body occupied
by the stone at each tick ofthe clock he is holding in his hand.
In this connection, we have nottaken account of the inaccuracy involved by
(14:37):
the finiteness of the velocity of propagationof light. With this, and with
a second difficulty prevailing here, weshall have to deal in detail later.
End of Section three
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