Section ten on the relativity of the conception of distance. Let us consider two
particular points on the train, for example, the middle of the first and
of the one hundredth carriage, traveling along the embankment with the velocity V,
and the enquire as to their distance apart. We already know that it is
necessary to have a body of reference for the measurement of a distance. With
respect to each body, the distance can be measured up. It is the
simplest plan to use the train itself as reference body or co ordinate 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 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 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 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 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 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 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
considerable which borrowed two unjustifiable hypotheses from classical mechanics. These are as follows one,
the time interval or time between two events is independent of the condition of
motion of the body of reference, and the space interval or 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 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 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 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, 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 ce relative to the embankment and relative to the
train. This question leads to a quite definite positive answer, and 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, which had mathematically 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 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 all be localized with
respect to the second framework without committing any fundamental error. We can disregard the
fact that in reality these frameworks 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 planes or coordinate system.
A coordinate system k then 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 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 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 same event with respect to k are given. The
relations must be so chosen that the law of the transmission of light in vacuum
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 orientation in
base of the coordinate systems indicated in the diagram. This problem is solved by
means of the equations x prime equals x vt over the square root of i
v squared over C squared, why prime equals y, z prime equals z,
and t prime equals t v overc squared times x over the square root
of i v squared over C squared. This system of equations is known as
the Lorentz transformation. Footnote. A 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 obtained the following equations. X prime equals x vt, y
prime equals y, z prime equals z, t prime equals t. This
system of equations is often termed the Galileay transformation. The Galileay transformation can be
obtained from the 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 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 x and t involves a relation between x prime
and t prime. In point of fact, if we substitute for x the
value ct in the first and fourth equations of the Lorentz transformation, we obtain
x prime equals c v times t. Over the square root of i v
squared overse squared, and T prime equals I minus v overse multiplied by t
over the square root of i v squared oversea squared, from which by division
the expression x prime equals ct 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 rays of light
advancing 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 a
meter rod in the x prime axis of K prime in such a manner.
At 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 meter rod relatively to the system
K. In order 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. Beginning of rod equals zero
over the square root of i v squared over C squared. X end of
rod equals i over the square root of i v squared over C squared,
the distance between the points being the square root of i v square oversea squared.
But the meter 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 root of I v squared
oversea 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 i v squared oversea 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. 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 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. 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 result of measurements obtainable by means of measuring rods
and clocks. If we had based our considerations 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 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 v squared over C squared.
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 v squared over C squared seconds i e 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 limiting velocity.
End of section twelve.