I Is there any way to derive the time dilation formula?

[sweet springs]

I've already read the derivation in which we use the light pulse clock kept in a spacecraft such that the light pulse follows a zig-zag motion due to motion of the spacecraft being perpendicular to motion of the light pulse. Then, we apply Pythagoras theorem to derive the formula.
BUT this seems like derivation in a special case, not a universal derivation.

Further to my post #30, go-return or your "a zig-zag" path of light scheme seems essential to define time.
You prepare a measure and emit light and accept light that was reflected at the other end of the measure.
This procedure defines proper time interval of the observer, I think.

I think further
-Time cannot be defined on mathematical point of spacetime. We need space for light to go-return that we can set as small as we like theoretically.
-QM might say something on fundamental uncertainty of deciding time and also space that matters setting length and position of the measure used for time decision.

 

[Sorcerer]

It is enough - @pixel mentions this possibility in #22 above.

Funny thing, the light clock and the Pythagorean theorem is exactly how I derived the Lorentz factor for the very first time. But back then I thought special relativity was just taking a Newtonian physics relationship and multiplying it by the Lorentz factor, lol. Works for 3-momentum, though. ;). γ(u)mu