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 e_{1} and e_{2} 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, e_{1} happens at B and e_{2} at D. Clearly, Δt' is the time difference between e_{1} and e_{2} 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 e_{1} and e_{2} 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:

d^{2} = (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 e_{1} and e_{2} 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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