Relativity - The Special and General Theory by Albert Einstein

Author:Albert Einstein [Einstein, Albert]
Language: eng
Format: epub
Published: 2011-02-17T05:00:00+00:00

PART II

30

The observer performs experiments on his circular disc with clocks and measuring−rods. In doing so, it is his intention to arrive at exact definitions for the signification of time− and space−data with reference to the circular disc K1, these definitions being based on his observations. What will be his experience in this enterprise ?

To start with, he places one of two identically constructed clocks at the centre of the circular disc, and the other on the edge of the disc, so that they are at rest relative to it. We now ask ourselves whether both clocks go at the same rate from the standpoint of the non−rotating Galileian reference−body K. As judged from this body, the clock at the centre of the disc has no velocity, whereas the clock at the edge of the disc is in motion relative to K in consequence of the rotation. According to a result obtained in Section 12, it follows that the latter clock goes at a rate permanently slower than that of the clock at the centre of the circular disc, i.e. as observed from K. It is obvious that the same effect would be noted by an observer whom we will imagine sitting alongside his clock at the centre of the circular disc. Thus on our circular disc, or, to make the case more general, in every gravitational field, a clock will go more quickly or less quickly, according to the position in which the clock is situated (at rest). For this reason it is not possible to obtain a reasonable definition of time with the aid of clocks which are arranged at rest with respect to the body of reference. A similar difficulty presents itself when we attempt to apply our earlier definition of simultaneity in such a case, but I do not wish to go any farther into this question.

Moreover, at this stage the definition of the space co−ordinates also presents insurmountable difficulties. If the observer applies his standard measuring−rod (a rod which is short as compared with the radius of the disc) tangentially to the edge of the disc, then, as judged from the Galileian system, the length of this rod will be less than I, since, according to Section 12, moving bodies suffer a shortening in the direction of the motion.

On the other hand, the measaring−rod will not experience a shortening in length, as judged from K, if it is applied to the disc in the direction of the radius. If, then, the observer first measures the circumference of the disc with his measuring−rod and then the diameter of the disc, on dividing the one by the other, he will not obtain as quotient the familiar number p = 3.14 . . ., but a larger number,[4]** whereas of course, for a disc which is at rest with respect to K, this operation would yield p exactly. This proves that the propositions of Euclidean geometry cannot hold exactly on the rotating disc, nor in general in a

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