# Concerned on the relativity of lengths

1. Sep 5, 2006

### myoho.renge.kyo

A. Einstein writes the following on The Principle of Relativity, p. 41:

"Let there be given a stationary rigid rod;and let its length be L as measured by a measuring-rod which is also stationary. We now imagine the axis of the rod lying along the axis of x of the stationary system of co-ordinates, and that a uniform motion of paralell translation with velocity v along the axis of x in the direction of increasing x is then imparted to the rod. We now inquire as to the length of the moving rod, and imagine its length to be ascertain by the following two operations:-"

my concern is the following operation (b):

"(b) by means of stationary clocks set up in the stationary system and synchronizing in accordance with chapter 1, the observer ascertains at what points of the stationary system the two ends of the rod to be measured are located at a definite time. The distance between these two points, measured by the measuring-rod already employed, which in this case is at rest, is also a length which may be designated 'the length of the rod.'"

the present period of time is from 2:00 am thru 3:00 am, 9/5/2006, in burbank, california.

let the observer ascertain that the points of the stationary system (corresponding to the two ends of the rod to be measured) are located at 2:00 am and then at 3:00 am.

the present period of time then is the definite time at which the observer ascertains at what points of the stationary system the two ends of the rod to be measured are located.

shouldn't the "length of the rod" be implied in (2:00 am thru 3:00 am) = ("length of the rod") / v? if not, why? thanks! (2:00 am thru 3:00 am, 9/5/2006)

2. Sep 5, 2006

### HallsofIvy

Staff Emeritus
No, the two ends of the rod are located at two different positions at a single time, not at two different times.

Yes, and again the "location" is position not time

There is a single time, two different positions, not two different times.

3. Sep 5, 2006

### myoho.renge.kyo

thank you for helping me understand. i really appreciate it.

let's say that at t1 (9:00 am) a uniform motion of paralell translation with velocity v along the axis of x in the direction of increasing x is then imparted to the rod, and that at t2 (10:00 am) the observer ascertains that at x1 and at x2 of the stationary system the two ends of the rod to be measured are located.

does the following then imply the "lenght of the rod?":

(9:00 am thru 10:00 am) = (x2 - x1) / v

or

if t1 = 0, then

t2 = (x2 - x1) / v

thank you again. (9:00 am thru 10:00 am, 9/5/2006, in burbank, california)

Last edited: Sep 5, 2006
4. Sep 6, 2006

### Ich

The "length" is x2-x1. No need for t1.

5. Sep 10, 2006

### myoho.renge.kyo

thanks!

but i am confused. at some point in time (let's call it t1) a uniform motion of parallel translation with velocity v along the axis of x in the direction of increasing x is imparted to the rod. and at a later point in time (let's call it t2) the observer ascertains that at x1 and at x2 of the stationary system the two ends of the rod to be measured are located.

why is it that there is no need for t1?

if t1 = 0, and the "length" is x2 - x1, does that mean that t2 = (t2 - t1) = (x2 - x1) / v?

thanks again! (9/10/2006, 8:00 am thru 9:00 am in Burbank, California)

Last edited: Sep 10, 2006