Relativity - Part 10-12 Einstein

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Correction for this chapter in mathematical formulaeinstead of I here one. This is
a LibriVox recording. All LibriVox recordingsare in the public domain. For more
information or to volunteer, please visitLibriVox dot org. Relativity The Special and

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General Theory by Albert Einstein, ContinuingPart one The Special Theory of relativity,
sections ten to twelve. Section tenon the relativity of the conception of distance.
Let us consider two particular points onthe train, for example, the

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middle of the first and of theone hundredth carriage traveling along the embankment with
the velocity V, and inquire asto their distance apart. We already know
that it is necessary to have abody of reference for the measurement of a
day distance. With respect to whichbody the distance can be measured up,
it is the simplest plan to usethe train itself as reference body or coordinate

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system. An observer in the trainmeasures the interval by marking off his measuring
rod in a straight line, forexample, along the floor of the carriage,
as many times as is necessary totake him from the one marked point
to the other. Then the numberwhich tells us how often the rod has
to be laid down is the requireddistance. It is a different matter when

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the distance has to be judged fromthe railway line. Here, the following
method suggests itself. If we callA prime and B prime the two points
on the train whose distance apart isrequired, then both of these points are
moving with the velocity V along theembankment. In the first place, we

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require to determine the points A andB of the embankment, which are just
being passed by the two points Aprime and B prime at a particular time
T judged from the embankment. Thesepoints A and B on the embankment can
be determined by applying the definition oftime given in section eight. The distance

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between these points A and B isthen 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 thesame result as the first. Thus,
the length of the train as measuredfrom the embankment may be different from that

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obtained by measuring in the train itself. This circumstance leads us to a second
objection which must be raised against theapparently obvious consideration of section six. Namely,
if the man in the carriage coversthe distance W in a unit of
time measured from the train, thenthis distance as measured from the embankment is

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not necessarily also equal to W Sectioneleven the Lorentz transformation. The results of
the last three sections show that theapparent incompatibility of the law of propagation of
light with the principle of relativity sectionseven has been derived by means of a

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consideration which borrowed two unjustifiable hypotheses fromclassical mechanics. These are as follows.
One the time interval or time betweentwo events is independent of the condition of
motion of the body of reference,and two, the space interval or distance

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between two points of a rigid bodyis independent of the condition of motion of
the reference. If we drop thesehypotheses, then the dilemma of section seven
disappears, because the theorem of theaddition of velocities derived in section six becomes

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invalid. The possibility presents itself thatthe law of the propagation of light in
vacuo may be compatible with the principleof relativity, and the question arises,
how have we to modify the considerationsof section six in order to remove the
apparent disagreement between these two fundamental resultsof experience. This question leads to a

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general one. In the discussion ofsection six, we have to do with
places and times relative both to thetrain and to the embankment. How are
we to find the place and timeof an event in relation to the train
when we know the place and timeof the event with respect to the railway
embankment. Is there a thinkable answerto this question of such a nature that

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the law of transmission of light invacuo does not contradict the principle of relativity?
In other words, can we conceiveof a relation between place and time
of the individual events relative to bothreference bodies such that every ray of light
possesses the velocity of transmissions c relativeto the embankment and relative to the train.

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This question leads to a quite definitepositive answer, and to a perfectly
definite transformation law for the space timemagnitudes 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. Upto the present, we have only considered

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events taking place along the embankment,which had mathematic to assume the function of
a straight line in the manner indicatedin section two. We can imagine this
reference body supplemented laterally and in avertical direction by means of a framework of
rods, so that an event whichtakes place anywhere can be localized with reference

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to this framework. Similarly, wecan imagine the train traveling with the velocity
V to be continued across the wholeof space, so that every event,
no matter how far off it maybe, could also be localized with respect
to the second framework without committing anyfundamental error. We can disregard the fact

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that in reality, these frameworks wouldcontinually interfere with each other, owing to
the impenetrability of solid bodies. Inevery such framework, we imagine three surfaces
perpendicular to each other, marked outand designated as coordinate plane or coordinate system.
A coordinate system K then corresponds tothe embankment, and a coordinate system

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K prime to the train. Anevent, wherever it may have taken place,
would be fixed in space with respectto K by the three perpendiculars x,
y, and z on the coordinateplanes, and with regard to time
by a time value T relative tok prime. The same event would be

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fixed in respect of space and timeby corresponding values x prime, y prime,
z prime, and t prime,which of course are not identical with
x, y, z, andt. It has already been set forth
in detail how these magnitudes are tobe regarded as results of physical measurements.

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Obviously, our problem can be exactlyformulated 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 sameevent with respect to k are given.

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The relations must be so chosen thatthe law of the transmission of light in
vacuo is satisfied for one and thesame ray of light, and of course
for every ray with respect to Kand k prime. For the relative orientation
in space of the coordinate systems indicatedin the diagram. This problem is solved

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by means of the equations x primeequals x minus vt over the square root
of I minus V squared over Csquared, y prime equals y, z
prime equals z and t prime eqt minus v over c squared times x

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over the square root of I minusv squared over C squared. This system
of equations is known as the Lorentztransformation. Footnote. A simple derivation of
the Lorentz transformation is given in Appendixone. If in place of the law

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of transmission of light, we hadtaken as our basis the tacit assumptions of
the older mechanics as to the absolutecharacter of times and lengths, then instead
of the above we should have obtainedthe following equations X prime equals x minus
v t, y prime equals y, z prime equals z, t prime

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equals t. This system of equationsis often termed the Galilee transformation. The
Galilee transformation can be obtained from theLorentz transformation by substituting an infinitely large value
for the velocity of light c inthe latter transformation. Aided by the following

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illustration, we can readily see that, in accordance with the Lorentz transformation,
the law of the transmission of lightin vacuo is satisfied both for the reference
body K and for the reference bodyk prime. A light signal is sent
along the positive x axis, andthis light stimulus advances in accordance with the

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equation x equals c t i ewith the velocity c. According to the
equations of the Lorentz transformation, thissimple 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 c cs e t in the first and fourth

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equations of the Lorentz transformation, weobtain x prime equals c minus v times
t over the square root of Iminus v squared over C squared, and
T prime equals i minus v overc multiplied by t over the square root

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of i minus v squared over csquared, from which by division the expression
x prime equals c t prime immediatelyfollows. If referred to the system k
prime, the propagation of light takesplace. According to this equation. We

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thus see that the velocity of transmissionrelative to the reference body k prime is
also equal to c. The sameresult is obtained for rays of light advance
in any other direction whatsoever. Ofcourse, this is not surprising, since
the equations of the Lorentz transformation werederived conformably to this point of view.

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Section twelve. The behavior of measuringrods and clocks in motion place a meter
rod in the x prime axis ofk prime in such a manner that one
end the beginning, coincides with thepoint x prime equals zero, while the

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other end, the end of therod, coincides with the point x prime
equals i. What is the lengthof the meta rod relatively to the system
K. In order to learn this, we need only ask where the beginning
of the rod and the end ofthe rod lie with respect to K at
a particular time T of the systemK. By means of the first equation

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of the Lorentz transformation, the valuesof these two points at the time T
equals zero can be shown to beX. Beginning of rod equals zero over
the square root of I minus vsquared over C squared. X end of

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rod equals i over the square rootof I minus v squared over C squared,
the distance between the points being thesquare root of I minus v squared
over C squared. But the metarod is moving with the velocity V relative
to K. It therefore follows thatthe length of a rigid meter rod moving

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in the direction of its length witha velocity V is the square root of
I minus v squared over C squaredof a meter The rigid rod is thus
shorter when in motion than when atrest, and the more quickly it is
moving, the shorter is the rod. For the velocity V equals c,

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we should have the square root ofI minus V squared over C squared equals
zero, and for still greater velocitiesthe square root becomes imaginary. For this
we conclude that in the theory ofrelativity, the velocity C plays the part
of a limiting velocity, which canneither be reached nor exceeded by any real

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body. Of course, this featureof the velocity C as a limiting velocity
also clearly follows from the equations ofthe Lorentz transformation, for these become meaningless
if we choose values of v greaterthan C. If, on the contrary,
we had considered a meter rod atrest in the x axis with respect

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to k, then we should havefound that the length of the rod,
as judged from k prime, wouldhave been the square root of i v
squared over C squared. This isquite in accordance with the principle of relativity,
which forms the basis of our considerations. A priori, it is quite

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clear that we must be able tolearn something about the physical behavior of measuring
rods and clocks from the equations oftransformation. For the magnitudes z, y,
x, and t are nothing morenor less than the results of measurements
obtainable by means of measuring rods andclocks. If we had based our considerations

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on the Galilean transformation, we shouldnot 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 xprime equals zero of k prime, T
prime equals zero and t prime equalsi. Are two successive ticks of this

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clock. The first and fourth equationsof the Lorentz transformation give For these two
ticks, t equals zero and tprime equals i divided by the square root
of I minus v squared over csquared. As judged from k. The

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clock is moving with the velocity vas judged from this reference body, The
time which elapses between two strokes ofthe clock is not one second, but
i divided by the square root ofI minus v squared over c squared seconds
i e a somewhat larger time.As a consequence of its motion, the

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clock goes more slowly than when atrest. Here also the velocity C plays
the part of an unattainable limit mittingvelocity. End of Section twelve.