Episode Transcript
Available transcripts are automatically generated. Complete accuracy is not guaranteed.
Speaker 1 (00:00):
Correction for this chapter in mathematical formulae instead of I
here one. This is a LibriVox recording. All LibriVox recordings
are in the public domain. For more information or to volunteer,
please visit LibriVox dot org. Relativity The Special and General
(00:23):
Theory by Albert Einstein, Continuing Part one The Special Theory
of relativity, sections ten to twelve. Section ten on the
relativity of the conception of distance. Let us consider two
particular points on the train, for example, the middle of
(00:45):
the first and of the one hundredth carriage traveling along
the embankment with the velocity V, and inquire as to
their distance apart. We already know that it is necessary
to have a body of reference for the measurement of
a day distance. With respect to which body the distance
can be measured up, it is the simplest plan to
(01:06):
use the train itself as reference body or coordinate system.
An observer in the train measures the interval by marking
off his measuring rod in a straight line, for example,
along the floor of the carriage, as many times as
is necessary to take him from the one marked point
to the other. Then the number which tells us how
(01:28):
often the rod has to be laid down is the
required distance. It is a different matter when the distance
has to be judged from the railway line. Here, the
following method suggests itself. If we call A prime and
B prime the two points on the train whose distance
apart is required, then both of these points are moving
(01:52):
with the velocity V along the embankment. In the first place,
we require to determine the points A and B of
the embankment, which are just being passed by the two
points A prime and B prime at a particular time
T judged from the embankment. These points A and B
(02:13):
on the embankment can be determined by applying the definition
of time given in section eight. The distance between these
points A and B is then measured by repeated application
of the measuring rod along the embankment a priori. It
is by no means certain that this last measurement will
(02:34):
supply us with the same result as the first. Thus,
the length of the train as measured from the embankment
may be different from that obtained by measuring in the
train itself. This circumstance leads us to a second objection
which must be raised against the apparently obvious consideration of
section six. Namely, if the man in the carriage covers
(02:59):
the distance W in a unit of time measured from
the train, then this distance as measured from the embankment
is not necessarily also equal to W Section eleven the
Lorentz transformation. The results of the last three sections show
(03:21):
that the apparent incompatibility of the law of propagation of
light with the principle of relativity section seven has been
derived by means of a consideration which borrowed two unjustifiable
hypotheses from classical mechanics. These are as follows. One the
(03:42):
time interval or time between two events is independent of
the condition of motion of the body of reference, and two,
the space interval or distance between two points of a
rigid body is independent of the condition of motion of
the reference. If we drop these hypotheses, then the dilemma
(04:06):
of section seven disappears, because the theorem of the addition
of velocities derived in section six 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
(04:29):
considerations of section six 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 section six,
we have to do with places and times relative both
to the train and to the embankment. How are we
(04:51):
to find the place and time of an event in
relation to the train when we know the place and
time of the event with respect to the railway embankment.
Is there a thinkable answer to this question of such
a nature that the law of transmission of light in
vacuo does not contradict the principle of relativity? In other words,
(05:14):
can we conceive of a relation between place and time
of the individual events relative to both reference bodies such
that every ray of light possesses the velocity of transmissions
c relative to the embankment and relative to the train.
This question leads to a quite definite positive answer, and
(05:36):
to a perfectly definite transformation law for the space time
magnitudes of an event when changing over from one body
of reference to another. Before we deal with this, we
shall introduce the following incidental consideration. Up to the present,
we have only considered events taking place along the embankment,
(05:59):
which had mathematic to assume the function of a straight
line in the manner indicated in section two. We can
imagine this reference body supplemented laterally and in a vertical
direction by means of a framework of rods, so that
an event which takes place anywhere can be localized with
reference to this framework. Similarly, we can imagine the train
(06:23):
traveling with the velocity V to be continued across the
whole of space, so that every event, no matter how
far off it may be, could also be localized with
respect to the second framework without committing any fundamental error.
We can disregard the fact that in reality, these frameworks
(06:44):
would continually interfere with each other, owing to the impenetrability
of solid bodies. In every such framework, we imagine three
surfaces perpendicular to each other, marked out and designated as
coordinate plane or coordinate system. A coordinate system K then
(07:06):
corresponds to the embankment, and a coordinate system K prime
to the train. An event, wherever it may have taken place,
would be fixed in space with respect to K by
the three perpendiculars x, y, and z on the coordinate planes,
and with regard to time by a time value T
(07:29):
relative to k prime. The same event would be fixed
in respect of space and time by corresponding values x prime,
y prime, z prime, and t prime, which of course
are not identical with x, y, z, and t. It
has already been set forth in detail how these magnitudes
(07:51):
are to be regarded as results of physical measurements. Obviously,
our problem can be exactly formulated in the following manner.
What are the values x prime, y prime, z prime,
and t prime of an event with respect to k prime.
When the magnitudes x, y, z, and t of the
(08:11):
same event with respect to k are given. The relations
must be so chosen that the law of the transmission
of light in vacuo is satisfied for one and the
same ray of light, and of course for every ray
with respect to K and k prime. For the relative
(08:32):
orientation in space of the coordinate systems indicated in the diagram.
This problem is solved by means of the equations x
prime equals x minus vt over the square root of
I minus V squared over C squared, y prime equals y,
(08:55):
z prime equals z and t prime eq t minus
v over c squared times x over the square root
of I minus v squared over C squared. This system
of equations is known as the Lorentz transformation. Footnote. A
(09:20):
simple derivation of the Lorentz transformation is given in Appendix one.
If in place of the law of transmission of light,
we had taken as our basis the tacit assumptions of
the older mechanics as to the absolute character of times
and lengths, then instead of the above we should have
(09:40):
obtained the following equations X prime equals x minus v t,
y prime equals y, z prime equals z, t prime
equals t. This system of equations is often termed the
Galilee transformation. The Galilee transformation can be obtained from the
(10:03):
Lorentz transformation by substituting an infinitely large value for the
velocity of light c in the latter transformation. Aided by
the following illustration, we can readily see that, in accordance
with the Lorentz transformation, the law of the transmission of
light in vacuo is satisfied both for the reference body
(10:26):
K and for the reference body k prime. A light
signal is sent along the positive x axis, and this
light stimulus advances in accordance with the equation x equals
c t i e with the velocity c. According to
the equations of the Lorentz transformation, this simple relation between
(10:50):
x and t involves a relation between x prime and
t prime. In point of fact, if we substitute for
x the value c s e t in the first
and fourth equations of the Lorentz transformation, we obtain x
prime equals c minus v times t over the square
(11:13):
root of I minus v squared over C squared, and
T prime equals i minus v over c multiplied by
t over the square root of i minus v squared
over c squared, from which by division the expression x
(11:36):
prime equals c t prime immediately follows. If referred to
the system k prime, the propagation of light takes place.
According to this equation. We thus see that the velocity
of transmission relative to the reference body k prime is
also equal to c. The same result is obtained for
(11:58):
rays of light advance in any other direction whatsoever. Of course,
this is not surprising, since the equations of the Lorentz
transformation were derived conformably to this point of view. Section twelve.
The behavior of measuring rods and clocks in motion place
(12:22):
a meter rod in the x prime axis of k
prime in such a manner that one end the beginning,
coincides with the point x prime equals zero, while the
other end, the end of the rod, coincides with the
point x prime equals i. What is the length of
the meta rod relatively to the system K. In order
(12:46):
to learn this, we need only ask where the beginning
of the rod and the end of the rod lie
with respect to K at a particular time T of
the system K. By means of the first equation of
the Lorentz transformation, the values of these two points at
the time T equals zero can be shown to be X.
(13:08):
Beginning of rod equals zero over the square root of
I minus v squared over C squared. X end of
rod equals i over the square root of I minus
v squared over C squared, the distance between the points
(13:29):
being the square root of I minus v squared over
C squared. But the meta rod is moving with the
velocity V relative to K. It therefore follows that the
length of a rigid meter rod moving in the direction
of its length with a velocity V is the square
(13:50):
root of I minus v squared over C squared of
a meter 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 equals c, we should have the square root of
(14:11):
I minus V squared over C squared equals zero, and
for still greater velocities the square root becomes imaginary. For
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.
(14:35):
Of course, this feature of the velocity C as a
limiting velocity also clearly follows from the equations of the
Lorentz transformation, for these become meaningless if we choose values
of v greater than C. If, on the contrary, we
had considered a meter rod at rest in the x
(14:55):
axis with respect to k, then we should have found
that the length of the rod, as judged from k prime,
would have been the square root of i v squared
over C squared. This is quite in accordance with the
principle of relativity, which forms the basis of our considerations.
(15:16):
A priori, it is quite clear that we must be
able to learn something about the physical behavior of measuring
rods and clocks from the equations of transformation. For the
magnitudes z, y, x, and t are nothing more nor
less than the results of measurements obtainable by means of
measuring rods and clocks. If we had based our considerations
(15:40):
on the Galilean transformation, we should not have obtained a
contraction of the rod as a consequence of its motion.
Let us now consider a seconds clock which is permanently
situated at the origin x prime equals zero of k prime,
T prime equals zero and t prime equals i. Are
(16:02):
two successive ticks of this clock. The first and fourth
equations of the Lorentz transformation give For these two ticks,
t equals zero and t prime equals i divided by
the square root of I minus v squared over c squared.
(16:25):
As judged from k. The clock is moving with the
velocity v as judged from this reference body, The time
which elapses between two strokes of the clock is not
one second, but i divided by the square root of
I minus v squared over c squared seconds i e
(16:46):
a somewhat larger time. As a consequence of its motion,
the clock goes more slowly than when at rest. Here
also the velocity C plays the part of an unattainable
limit smitting velocity. End of Section twelve.
Correction for this chapter in mathematical formulae instead of I
here one. This is a LibriVox recording. All LibriVox recordings
are in the public domain. For more information or to volunteer,
please visit LibriVox dot org. Relativity The Special and General
(00:23):
Theory by Albert Einstein, Continuing Part one The Special Theory
of relativity, sections ten to twelve. Section ten on the
relativity of the conception of distance. Let us consider two
particular points on the train, for example, the middle of
(00:45):
the first and of the one hundredth carriage traveling along
the embankment with the velocity V, and inquire as to
their distance apart. We already know that it is necessary
to have a body of reference for the measurement of
a day distance. With respect to which body the distance
can be measured up, it is the simplest plan to
(01:06):
use the train itself as reference body or coordinate system.
An observer in the train measures the interval by marking
off his measuring rod in a straight line, for example,
along the floor of the carriage, as many times as
is necessary to take him from the one marked point
to the other. Then the number which tells us how
(01:28):
often the rod has to be laid down is the
required distance. It is a different matter when the distance
has to be judged from the railway line. Here, the
following method suggests itself. If we call A prime and
B prime the two points on the train whose distance
apart is required, then both of these points are moving
(01:52):
with the velocity V along the embankment. In the first place,
we require to determine the points A and B of
the embankment, which are just being passed by the two
points A prime and B prime at a particular time
T judged from the embankment. These points A and B
(02:13):
on the embankment can be determined by applying the definition
of time given in section eight. The distance between these
points A and B is then measured by repeated application
of the measuring rod along the embankment a priori. It
is by no means certain that this last measurement will
(02:34):
supply us with the same result as the first. Thus,
the length of the train as measured from the embankment
may be different from that obtained by measuring in the
train itself. This circumstance leads us to a second objection
which must be raised against the apparently obvious consideration of
section six. Namely, if the man in the carriage covers
(02:59):
the distance W in a unit of time measured from
the train, then this distance as measured from the embankment
is not necessarily also equal to W Section eleven the
Lorentz transformation. The results of the last three sections show
(03:21):
that the apparent incompatibility of the law of propagation of
light with the principle of relativity section seven has been
derived by means of a consideration which borrowed two unjustifiable
hypotheses from classical mechanics. These are as follows. One the
(03:42):
time interval or time between two events is independent of
the condition of motion of the body of reference, and two,
the space interval or distance between two points of a
rigid body is independent of the condition of motion of
the reference. If we drop these hypotheses, then the dilemma
(04:06):
of section seven disappears, because the theorem of the addition
of velocities derived in section six 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
(04:29):
considerations of section six 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 section six,
we have to do with places and times relative both
to the train and to the embankment. How are we
(04:51):
to find the place and time of an event in
relation to the train when we know the place and
time of the event with respect to the railway embankment.
Is there a thinkable answer to this question of such
a nature that the law of transmission of light in
vacuo does not contradict the principle of relativity? In other words,
(05:14):
can we conceive of a relation between place and time
of the individual events relative to both reference bodies such
that every ray of light possesses the velocity of transmissions
c relative to the embankment and relative to the train.
This question leads to a quite definite positive answer, and
(05:36):
to a perfectly definite transformation law for the space time
magnitudes of an event when changing over from one body
of reference to another. Before we deal with this, we
shall introduce the following incidental consideration. Up to the present,
we have only considered events taking place along the embankment,
(05:59):
which had mathematic to assume the function of a straight
line in the manner indicated in section two. We can
imagine this reference body supplemented laterally and in a vertical
direction by means of a framework of rods, so that
an event which takes place anywhere can be localized with
reference to this framework. Similarly, we can imagine the train
(06:23):
traveling with the velocity V to be continued across the
whole of space, so that every event, no matter how
far off it may be, could also be localized with
respect to the second framework without committing any fundamental error.
We can disregard the fact that in reality, these frameworks
(06:44):
would continually interfere with each other, owing to the impenetrability
of solid bodies. In every such framework, we imagine three
surfaces perpendicular to each other, marked out and designated as
coordinate plane or coordinate system. A coordinate system K then
(07:06):
corresponds to the embankment, and a coordinate system K prime
to the train. An event, wherever it may have taken place,
would be fixed in space with respect to K by
the three perpendiculars x, y, and z on the coordinate planes,
and with regard to time by a time value T
(07:29):
relative to k prime. The same event would be fixed
in respect of space and time by corresponding values x prime,
y prime, z prime, and t prime, which of course
are not identical with x, y, z, and t. It
has already been set forth in detail how these magnitudes
(07:51):
are to be regarded as results of physical measurements. Obviously,
our problem can be exactly formulated in the following manner.
What are the values x prime, y prime, z prime,
and t prime of an event with respect to k prime.
When the magnitudes x, y, z, and t of the
(08:11):
same event with respect to k are given. The relations
must be so chosen that the law of the transmission
of light in vacuo is satisfied for one and the
same ray of light, and of course for every ray
with respect to K and k prime. For the relative
(08:32):
orientation in space of the coordinate systems indicated in the diagram.
This problem is solved by means of the equations x
prime equals x minus vt over the square root of
I minus V squared over C squared, y prime equals y,
(08:55):
z prime equals z and t prime eq t minus
v over c squared times x over the square root
of I minus v squared over C squared. This system
of equations is known as the Lorentz transformation. Footnote. A
(09:20):
simple derivation of the Lorentz transformation is given in Appendix one.
If in place of the law of transmission of light,
we had taken as our basis the tacit assumptions of
the older mechanics as to the absolute character of times
and lengths, then instead of the above we should have
(09:40):
obtained the following equations X prime equals x minus v t,
y prime equals y, z prime equals z, t prime
equals t. This system of equations is often termed the
Galilee transformation. The Galilee transformation can be obtained from the
(10:03):
Lorentz transformation by substituting an infinitely large value for the
velocity of light c in the latter transformation. Aided by
the following illustration, we can readily see that, in accordance
with the Lorentz transformation, the law of the transmission of
light in vacuo is satisfied both for the reference body
(10:26):
K and for the reference body k prime. A light
signal is sent along the positive x axis, and this
light stimulus advances in accordance with the equation x equals
c t i e with the velocity c. According to
the equations of the Lorentz transformation, this simple relation between
(10:50):
x and t involves a relation between x prime and
t prime. In point of fact, if we substitute for
x the value c s e t in the first
and fourth equations of the Lorentz transformation, we obtain x
prime equals c minus v times t over the square
(11:13):
root of I minus v squared over C squared, and
T prime equals i minus v over c multiplied by
t over the square root of i minus v squared
over c squared, from which by division the expression x
(11:36):
prime equals c t prime immediately follows. If referred to
the system k prime, the propagation of light takes place.
According to this equation. We thus see that the velocity
of transmission relative to the reference body k prime is
also equal to c. The same result is obtained for
(11:58):
rays of light advance in any other direction whatsoever. Of course,
this is not surprising, since the equations of the Lorentz
transformation were derived conformably to this point of view. Section twelve.
The behavior of measuring rods and clocks in motion place
(12:22):
a meter rod in the x prime axis of k
prime in such a manner that one end the beginning,
coincides with the point x prime equals zero, while the
other end, the end of the rod, coincides with the
point x prime equals i. What is the length of
the meta rod relatively to the system K. In order
(12:46):
to learn this, we need only ask where the beginning
of the rod and the end of the rod lie
with respect to K at a particular time T of
the system K. By means of the first equation of
the Lorentz transformation, the values of these two points at
the time T equals zero can be shown to be X.
(13:08):
Beginning of rod equals zero over the square root of
I minus v squared over C squared. X end of
rod equals i over the square root of I minus
v squared over C squared, the distance between the points
(13:29):
being the square root of I minus v squared over
C squared. But the meta rod is moving with the
velocity V relative to K. It therefore follows that the
length of a rigid meter rod moving in the direction
of its length with a velocity V is the square
(13:50):
root of I minus v squared over C squared of
a meter 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 equals c, we should have the square root of
(14:11):
I minus V squared over C squared equals zero, and
for still greater velocities the square root becomes imaginary. For
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.
(14:35):
Of course, this feature of the velocity C as a
limiting velocity also clearly follows from the equations of the
Lorentz transformation, for these become meaningless if we choose values
of v greater than C. If, on the contrary, we
had considered a meter rod at rest in the x
(14:55):
axis with respect to k, then we should have found
that the length of the rod, as judged from k prime,
would have been the square root of i v squared
over C squared. This is quite in accordance with the
principle of relativity, which forms the basis of our considerations.
(15:16):
A priori, it is quite clear that we must be
able to learn something about the physical behavior of measuring
rods and clocks from the equations of transformation. For the
magnitudes z, y, x, and t are nothing more nor
less than the results of measurements obtainable by means of
measuring rods and clocks. If we had based our considerations
(15:40):
on the Galilean transformation, we should not have obtained a
contraction of the rod as a consequence of its motion.
Let us now consider a seconds clock which is permanently
situated at the origin x prime equals zero of k prime,
T prime equals zero and t prime equals i. Are
(16:02):
two successive ticks of this clock. The first and fourth
equations of the Lorentz transformation give For these two ticks,
t equals zero and t prime equals i divided by
the square root of I minus v squared over c squared.
(16:25):
As judged from k. The clock is moving with the
velocity v as judged from this reference body, The time
which elapses between two strokes of the clock is not
one second, but i divided by the square root of
I minus v squared over c squared seconds i e
(16:46):
a somewhat larger time. As a consequence of its motion,
the clock goes more slowly than when at rest. Here
also the velocity C plays the part of an unattainable
limit smitting velocity. End of Section twelve.
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