Relativity - The Special and General Theory, Albert Einstein [best ereader for academics txt] 📗

we to modify the considerations of Section 6 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 6 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 transmission c 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 2 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 localised with reference to this framework. Fig. 2 Similarly, we can imagine the train travelling 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 localised 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 ” (” coordinate system “). A coordinate system K then corresponds to the embankment, and a coordinate system K’ 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, z on the coordinate planes, and with regard to time by a time value t. Relative to K1, the same event would be fixed in respect of space and time by corresponding values x1, y1, z1, t1, which of course are not identical with x, y, z, 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 x1, y1, z1, t1, of an event with respect to K1, when the magnitudes x, y, z, 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 vacuo is satisfied for one and the same ray of light (and of course for every ray) with respect to K and K1. For the relative orientation in space of the coordinate systems indicated in the diagram ([7]Fig. 2), this problem is solved by means of the equations :

eq. 1: file eq01.gif

y1 = y z1 = z

eq. 2: file eq02.gif

This system of equations is known as the ” Lorentz transformation.” *

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:

x1 = x - vt y1 = y z1 = z t1 = t

This system of equations is often termed the ” Galilei transformation.” The Galilei 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 K1. A light-signal is sent along the positive x-axis, and this light-stimulus advances in accordance with the equation

x = ct,

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 x1 and t1. In point of fact, if we substitute for x the value ct in the first and fourth equations of the Lorentz transformation, we obtain:

eq. 3: file eq03.gif

 

eq. 4: file eq04.gif

from which, by division, the expression

x1 = ct1

immediately follows. If referred to the system K1, the propagation of light takes place according to this equation. We thus see that the velocity of transmission relative to the reference-body K1 is also equal to c. The same result is obtained for rays of light advancing in any other direction whatsoever. Of cause this is not surprising, since the equations of the Lorentz transformation were derived conformably to this point of view.

 

Notes

*) A simple derivation of the Lorentz transformation is given in Appendix I.

 

THE BEHAVIOUR OF MEASURINGRODS AND CLOCKS IN MOTION

 

Place a metre-rod in the x1-axis of K1 in such a manner that one end (the beginning) coincides with the point x1=0 whilst the other end (the end of the rod) coincides with the point x1=I. What is the length of the metre-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 = 0 can be shown to be

eq. 05a: file eq05a.gif

 

eq. 05b: file eq05b.gif

 

the distance between the points being eq. 06 .

But the metre-rod is moving with the velocity v relative to K. It therefore follows that the length of a rigid metre-rod moving in the direction of its length with a velocity v is eq. 06 of a metre.

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=c we should have eq. 06a ,

and for stiII greater velocities the square-root becomes imaginary. From 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 became meaningless if we choose values of v greater than c.

If, on the contrary, we had considered a metre-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 K1 would have been eq. 06 ;

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 behaviour of measuringrods and clocks from the equations of transformation, for the magnitudes z, y, x, t, are nothing more nor less than the results of measurements obtainable by means of measuringrods and clocks. If we had based our considerations on the Galileian 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 (x1=0) of K1. t1=0 and t1=I are two successive ticks of this clock. The first and fourth equations of the Lorentz transformation give for these two ticks :

t = 0

and

eq. 07: file eq07.gif

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

eq. 08: file eq08.gif

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.

 

THEOREM OF THE ADDITION OF VELOCITIES. THE EXPERIMENT OF FIZEAU

 

Now in practice we can move clocks and measuringrods only with velocities that are small compared with the velocity of light; hence we shall hardly be able to compare the results of the previous section directly with the reality. But, on the other hand, these results must strike you as being very singular, and for that reason I shall now draw another conclusion from the theory, one which can easily be derived from the foregoing considerations, and which has been most elegantly confirmed by experiment.

In Section 6 we derived the theorem of the addition of velocities in one direction in the form which also results from the hypotheses of classical mechanics-This theorem can also be deduced readily horn the Galilei transformation (Section 11). In place of the man walking inside the carriage, we introduce a point moving relatively to the coordinate system K1 in accordance with the equation

x1 = wt1

By means of the first and fourth equations of the Galilei transformation we can express x1 and t1 in terms of x and t, and we then obtain

x = (v + w)t

This equation expresses nothing else than the law of motion of the point with reference to the system K (of the man with reference to the embankment). We denote this velocity by the symbol W, and we then obtain, as in Section 6,

W=v+w A)

But we can carry out this consideration just as well on the basis of the theory of relativity. In the equation

x1 = wt1 B)

we must then express x1and t1 in terms of x and t, making use of the first and fourth equations of the Lorentz transformation. Instead of the equation (A) we then obtain the equation

eq. 09: file eq09.gif

 

which corresponds to the theorem of addition for velocities in one direction according to the theory of relativity. The question now arises as to which of these two theorems is the better in accord with experience. On this point we axe enlightened by a most important experiment which the brilliant physicist Fizeau performed more than half a century ago, and which has been repeated since then by some of the best experimental physicists, so that there can be no doubt about its result. The experiment is concerned with the following question. Light travels in a motionless liquid with a particular velocity w. How quickly does it travel in the direction of the arrow in the tube T (see the accompanying diagram, Fig. 3) when the liquid above mentioned is flowing through the tube with a velocity v ?

In accordance with the principle of relativity we shall certainly have to take for granted that the propagation of light always takes place with the same velocity w with respect to the liquid, whether the latter is in motion with reference to other bodies or not. The velocity of light relative to the liquid and the velocity of the latter relative to the tube are thus known, and we require the velocity of light relative to the tube.

It is clear that we have the problem of Section 6 again before us. The tube plays the part