I propose a new look at lenghts and times relative to S' and S (at least new to me).(adsbygoogle = window.adsbygoogle || []).push({});

Let S' be an x'-coordinate system. Let the x'-axis of S' coincide with the x-axis of an x-coordinate system S, and let S' move along the x-axis of S with velocity v in the direction of increasing x.

Let t' = the time, with respect to S', a ray of light emitted by S' takes to move along the x'-axis of S'.

Let t = the time, with respect to S, a ray of light emitted by S takes to move along the x-axis of S.

Let T = the time, with respect to S, the ray of light emitted by S' takes to move along the x'-axis of S'.

At t' = 0s, the ray of light emitted by S' is located at x' = 0m.

At t' = t'1, the ray of light emitted by S' is located at x' = c*t'1.

At t' = 2*t'1, the ray of light emitted by S' is located at x' = 0m.

At t = 0s, (1) the ray of light emitted by S is located at x = 0m, (2) the origin of S' is located at x = 0m, and (3) t' = t.

At t = t1, (1) the ray of light emitted by S is located at x = c*t1, (2) the origin of S' is located at x = v*t1, and (3) t' = t.

At = t = 2*t1, (1) the ray of light emitted by S is located at x = 0m, (2) the origin of S' is located at x = 2*v*t1, and (3) t' = t.

At T = 0s, the ray of light emitted by S' is located at x = 0m.

At T = T1, the ray of light emitted by S' is located at x = c*T1 = v*t1 + c*t'1 = c*t'1 + v*t'1 = t'1*(c + v).

Note: t1 = t'1. It is given that the length of the path of the ray of light emitted by S' is equal to the length of the path of the ray of light emitted by S.

At T = T2, the ray of light emitted by S' is located at x = 2*v*t1.

T1 = t'1(c + v)/c.

c*(T2 - T1) = |2*v*t1 - (c*t'1 + v*t'1)| = c*t'1 + v*t'1 - 2*v*t'1 = c*t'1 - v*t'1 = t'1*(c - v).

T2 - T1 = t'1*(c - v)/c.

Note: T1 is not equal to T2 - T1, but this inequality does not mean that the clock located at x = 0m and the one located at x = c*T1 are not synchronous. The clock that marks the time T = 0s is located at x = 0m, the one that marks the time T = T1 is located at x = c*T1, and the one that marks the time T = T2 is located at x = 2*v*t'1 (or x = v*T2).

T = T1 + (T2 - T1) = T2 = 2*t'1.

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# Lengths and times relative to S' and S

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