Time dilation

Let A'B' be a segment of non-zero length in K', perpendicular to the x'-axis. We saw before that A'B' is perceived in K as being perpendicular to the x-axis, traveling at velocity v.

In K', let's send a light signal from A' toward B'. It will arrive at B' after some time Δt' > 0. Let AB and CD denote the segments in K that correspond to A'B' when the light signal is sent and arrives, respectively. Moreover, let e1 and e2 be two events, both happening at B', the first when the light signal is sent and the second when it arrives. That is, viewed from K, e1 happens at B and e2 at D. Clearly, Δt' is the time difference between e1 and e2 in K'. For the corresponding time difference in K, Δt > 0 must hold due to causality. (Imagine there is a particle at B' to which both e1 and e2 happen.) And since B' is moving at v relative to K, the x-coordinate of CD is greater than that of AB.

Again, as the light signal travels at a constant velocity relative to K too, it has to travel in K along a straight line from A toward D, at the constant speed c. It was shown before that d(A, B) = d(C, D) = d'(A', B'). Let d denote this common distance. In K', the light signal then needs

Δt' = d / c

time to travel from A' to B'. We also know that:

d(A, D) = c · Δt
d(B, D) = v · Δt

Due to the Pythagorean theorem:

d2 = (c · Δt)2 – (v · Δt)2

Replacing d with c · Δt' and rearranging:

Δt' = Δt · √1 – v2   / c2   

, which means from the perspective of an observer at B' that:

(Theorem) If an observer is at rest for Δt' time in K', they observe a corresponding Δt = Δt' / √1 – v2   / c2    time elapsed in K.

So the duration between events e1 and e2 is different in K and K', or more generally, the (coordinate) time that elapses between two events depends on the inertial frame in which it is measured.

As a byproduct, we have just determined the value of λ as well:

(Corollary) λ = 1 / √1 – v2   / c2   .

The effect is called time dilation since λ > 1, and thus Δt > Δt' for any v. In other words, a resting observer in K' will always see as if in K time was passing faster.

Note: time dilation is unnoticeable in our everyday life.

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