Question concerning the relativity of length

1. Oct 30, 2006

myoho.renge.kyo

let K' = a stationary rigid rod. let its length be AB (the length 299,792,458 meters between the two ends A and B of K'). the axis of K' is lying along the x-axis of a stationary system of coordinates K.

let there be a stationary red clock and an observer stationed at A.

let there be a stationary blue clock and another observer at the origin of K.

let the red clock and the blue clock synchronize.

let t = the time values of events as measured by the observer stationed at the origin of K using the blue clock.

let T = the time values of events as measured by the observer stationed at A using the red clock.

let a ray of light depart from A at t0 = T0 in the direction of B, and let the ray of light be reflected at B at t1 = T1.

the observer stationed at A measures the time (T1 - T0 = 1 s) required by the ray of light to travel from A to B.

the observer stationed at the origin of K measures the time (t1 - t0 = 1 s) required by the ray of light to travel from A to B.

c = AB/(T1 - T0) = AB/(t1 - t0).

c is the velocity of the ray of light as measured by the observer stationed at the origin of K, whether K' is stationary or moving at a constant speed v in the direction of increasing x.

let a constant speed v in the direction of increasing x be imparted to K'.

let a ray of light depart again from A at t0 = T0 in the direction of B, and let the ray of light be reflected at B at T1 and at t1.

the observer stationed at A measures again the time (T1 - T0 = 1 s) required by the ray of light to travel from A to B.

according to the relativity of time, the observer stationed at the origin of K is expected to measure the time [t1 - t0 = (1 s)/sqrt(1 - v^2/c^2)] required by the ray of light to travel from A to B. therefore, T1 is not equal to t1.

since c is the velocity of the ray of light as measured by the observer stationed at the origin of K, whether K' is stationary or moving at a constant speed v in the direction of increasing x, c = rAB/[(1 s)/sqrt(1 - v^2/c^2)], where rAB is not equal to AB.

What is the value of rAB, such that c = AB/(T1 -T0) = rAB/[(1 s)/sqrt(1 - v^2/c^2)]? thanks!

Last edited: Oct 30, 2006