Einstein 1920: Conception of Distance

Reference: Einstein’s 1920 Book

This paper presents Part 1, Chapter 10 from the book RELATIVITY: THE SPECIAL AND GENERAL THEORY by A. EINSTEIN. The contents are from the original publication of this book by Henry Holt and Company, New York (1920).

The paragraphs of the original material (in black) are accompanied by brief comments (in color) based on the present understanding.  Feedback on these comments is appreciated.

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On the Relativity of the Conception of Distance

Let us consider two particular points on the train1 travelling 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 distance, with respect to which body the distance can be measured up. It is the simplest plan to use the train itself as the reference-body (co-ordinate system). An observer in the train measures the interval by marking off his measuring-rod in a straight line (e.g. 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.

1 e.g. the middle of the first and of the hundredth carriage.

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’ and B’ 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’ and B’ at a particular time t—judged from the embankment. These points A and B of the embankment can be determined by applying the definition of time given in Section 8. 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 6. 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.

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