# Definition of simultaneity

1. Sep 19, 2006

### myoho.renge.kyo

in this thread, i have quoted A. Einstein in The Principle of Relativity, pp 38 - 42. this thread is really a reply to the thread "relativity of simultaneity". i am posting it here hoping that the one as a reply will be removed and this one kept as a new topic. thanks!

all our judgments in which time plays a part are always judgments of simultaneous events. for example, if I say that at 11:00 am, 9/19/2006, in Burbank, California, a train arrived, I am saying that the pointing of the small hand of my watch to 11:00 am and the arrival of the train are simultaneous events.

let A = a point of space where there is a clock and an observer who determines the time value of an event in the immediate proximity of A by finding the position of the hands (of the clock) which are simultaneous with this event.

let B = a point of space where there is another clock (in all respects resembling the one at A) and another observer who determines the time value of an event in the immediate proximity of B by finding the position of the hands (of the clock) which are simultaneous with this event.

let A-time = the time value of the event in the immediate proximity of A (determined by the observer with the clock at A)

let B-time = the time value of the event in the immediate proximity of B (determined by the observer with the clock at B)

let the common A-and-B-time = the time value of the event in the immediate proximity of either A or B (either one determined by either the obsever with the clock at A or the observer with the clock at B). a common A-and-B-time cannot be defined unless we establish that the time required by light to travel from A to B equals the time required by light to travel from B to A. thus to say that there is now a common A-and-B-time is to say that the clock at A and the clock at B synchronize.

let the stationary system (K) = a system of coordinates in which the equations of Newtonian mechanics hold good.

let the stationary rigid rod (K') = a stationary rigid rod lying along the axis of x of K. the length of K' is L as measured by a measuring-rod which is also stationary. let a uniform motion of parallel translation with velocity v along the axis of x of K be imparted to K'.

let the length of K' in the moving system (the moving system in this case is K' itself) = L = the length of K' ascertain by the following operation: an observer moving together with the given measuring-rod and K' measures the length of K' directly by superposing the measuring-rod, in just the same way as if all three were at rest.

let the length of the moving system K' in the stationary system K = rAB = the length of K' ascertain by the following operation: by means of stationary clocks set up in the stationary system K and synchronizing, an observer ascertains at what points (x1 and x2) of the stationary system K the two ends (A and B) of the moving system K' are located respectively at a definite time. the observer then measures the distance between these two points by the measuring-rod already employed, which in this case is at rest.

at the end A of the moving system K' there is a clock, and at the end B there is another clock. these two clocks and the stationary clocks in the stationary system K synchronize. with the clock at A there is an observer, and with the clock at B there is another observer.

at tA (the time value determined by the observer with the clock at A) a ray of light departs from A towards B. at tB (the time value determined by the observer with the clock at B) the ray of light is reflected from B. and at t'A (the time value determined again by the observer with the clock at A) the ray of light reaches A again.

observers moving with K' would find that in K

c = [rAB / (tB - tA)] + v = [rAB / (t'A - tB)] - v

but observers in the stationary system K' declare that in K

c = L / (tB - tA) = L / (t'A - tB)

for example, let's say that v = 0.9*c, and that at the definte time tA (11:00 am, 9/19/2006), a ray of light departs from A towards B, and at the definite time tB (11:30 am, 9/19/2006), the ray of light is reflected back to A, and at the definite time t'A (12:00 pm, 9/19/2006), the ray of light reaches A again.

observers moving with K' would find that in K

299792458 m/s = (rAB / 1800 s) + 0.9*299792458 m/s
rAB = 53962642440 m
and
299792458 m/s = [53962642440 m / (t'A - tB)] - 0.9*299792458 m/s
(t'A - tB) = 94.7368421053 s

but observers in the stationary sytem K' declare that in K

299792458 m/s = L / 1800 s (tB - tA = t'A - tB = 1800 s)
L = 539626424400 m

so in the interval of time from 11:00 am thru 11:30 am, the observers moving with K' see that in K, rAB = 53962642440 m, and the observers in the stationary K' see that in K, L = 539626424400 m.

and in the interval of time from 11:30 am thru 12:00 pm, the observers moving with K' see that in K, the interval of time is 11:30 am thru 11:31:34.4 am, and the observers in the stationary K' see that in K, the interval of time is 11:30 am thru 12:00 pm.

so we see that we cannot attach any absolute signification to the concept of simultaneity. for when two events are viewed as simultaneous events from a stationary system of coordinates, the same two events can no longer be looked upon as simultaneous events when envisaged from a system which is in motion relatively to that system.

in other words, when the position of the hands of the clock at A and the position of the hands of the clock at B are viewed as 12:00 pm in K from the stationary system K', the position of the hands of the clock at A and the position of the hands of the clock at B are looked upon as 11:31:34.4 in K when they are envisaged from the moving K'.

Last edited: Sep 20, 2006