Einstein on the relativity of distance.

[zaaron]

I've only just begun reading about relativity, and am finding Einstein's popular level book "Relativity: The Special and General Theory" quite satisfying. This particular example, from Section X (ON THE RELATIVITY OF THE CONCEPTION OF DISTANCE) has me a bit confused, though:

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 VIII. 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 the same result as the first¹.

¹ the first measurement being the distance between the points A' and B' on the train, judged from the train as reference body, simply by the "repeated application of the measuring-rod."

Any hints as to why it is "by no means certain" that the distances would be the same? I'm trying to picture the setup of the example, but am having a hard time. The definition of time of an event was given as "the reading (position of the hands) of that one of these clocks² which is in the immediate vicinity (in space) of the event."

² "clocks of identical construction placed at points D, E, and F of the railway line (co-ordinate system)" which were started with identical settings, simultaneously.

I'm picturing something like an A clock and a B clock, which are stopped as points A' and B', on the train, pass by, and would have identical settings (identical times) when stop. Is that even close? I guess you'd need plenty of clocks. In any case, the distance between the could be measured, as he says, but why wouldn't they necessarily be the same distance apart as A' and B', as measured from the train as reference body?

 

[PhanthomJay]

Keep on reading...the chapter on the Lorentz Transformation MIGHT make it a bit more clear, but there are no guarantees, as the concept is quite difficult to understand. Essentially, Einstein shows that as a consequence of the constancy of the speed of light in all reference frames, regardless of their state of motion, then it follows that each person must have their own measure of distance and time as a function of their speed with respect to another reference frame. As the speed of the moving observer approaches lightspeed, the distance between and A and B on the 'stationary' embankment, which might be measured as say 1000 meters by the stationary observer, contracts to near zero as measured by the moving observer. I'll be honest: I've read that book once every year for the past 10 years, and just when I think I've understood it, I don't !

 

[JesseM]

Any hints as to why it is "by no means certain" that the distances would be the same? I'm trying to picture the setup of the example, but am having a hard time.

Because each observer is measuring the distance using their own rulers, and you shouldn't rule out a priori that one observer's ruler might seem shrunk or expanded as measured by the other (as it turns out each does measure the other's ruler to be shrunk relative to their own--take at the illustration of two ruler/clock systems moving alongside each other which I posted here)

The definition of time of an event was given as "the reading (position of the hands) of that one of these clocks² which is in the immediate vicinity (in space) of the event."

² "clocks of identical construction placed at points D, E, and F of the railway line (co-ordinate system)" which were started with identical settings, simultaneously.

I'm picturing something like an A clock and a B clock, which are stopped as points A' and B', on the train, pass by, and would have identical settings (identical times) when stop. Is that even close?

There's a set of clocks at rest on the rails, and a different set of clocks at rest on the train. All the clocks on the train are set so they show the same reading "simultaneously" as defined by an observer at rest on the track; in section 8 he explains that he wants to define "simultaneity" for each observer in terms of the idea that they should assume that light travels at the same speed in all directions relative to themselves, so if they are standing at the exact midpoint of two clocks, the light from each clock showing a given reading (say, 1 minute 22 seconds) should reach their eyes at the same moment. Of course, as he explains in section 9, if different observers use this definition to synchronize their own clocks, then they will disagree about simultaneity--the observer on the tracks will observe the clocks on the train to be out-of-sync (when he is equidistant from two train clocks he will see them showing different times) and vice versa.

Simultaneity is also involved in measurements of the length of moving objects--the idea is to look at the position of the front and back end of an object "at a single moment" in your frame, so if the back end is next to a clock at the 0-meter mark on your ruler when your clock attached to that mark reads 30 seconds, and the front end is next to a clock at the 10-meter mark on your ruler when the clock attached to that mark reads 30 seconds, then in your frame the moving object is 10 meters long.