Question

can anyone derive time dilation without using lorentz transformation?
(if yes, please tell me)

Answer 1

Yes, you can.

The only thing you have to say, is that celerity of light is independent of frame.
You have to consider light bouncing on a mirror from two different frames, one in which the mirror is stationary, the other where it is moving.

I will not type the derivation here, as it can be found in:
http://en.wikipedia.org/wiki/Time_dilation#Simple_inference_of_time_dilation_due_to_relative_velocity

Answer 2

Lot of Thanks for it. But a professor of IUCAA showed a derivation using completely different methode which I have not found anywhere. And I have remembered that he used:
\[(\Delta s)^2=c^2(\Delta t)^2-(\Delta x)^2\] and I have also remembered that he used Taylor series expantion

Answer 3

This method is invariance of the magnitude of the space-time 4-vector.

Answer 4

can you tell me the detail

Answer 5

Take a light clock, with two facing mirrors a distance d apart, and put it on a spaceship moving at a high speed. Photon, which moves at a constant speed independent of one's frame of reference, takes time d = ct => t = d/c to go from one mirror to the next. Person watching spaceship go by at speed v sees photon moving diagonally, so that it must move r^2 = (vt)^2 + d^2, where time t is time it takes photon to move r = ct. Replace r with ct => (ct)^2 = (vt)^2 + d^2 => t^2(c^2 - v^2) = d^2 => t = sqrt(d^2 / (c^2 - v^2))

Answer 6

So, if v = c, we divide by zero => time dilation approaches infinity. (timeless)

Answer 7

I don't know if that's what you were asking for, or if it's even correct... but it was based on the simple observation that photons move at a constant speed regardless of one's frame of reference and some basic trig.

Answer 8

@queelius is right
from\[t=\sqrt{d^2\over c^2-v^2}\]we can rewrite this as\[t={\frac dc\over\sqrt{1-(v/c)^2}}\] call \(\frac dc=t_r\) the rest time that it takes for the light to move from one mirror to the next and we get relative to an observer on the ship and we get\[t={t_r\over\sqrt{1-(v/c)^2}}\] which is the formula for time dilation as it is usually written.

Answer 9

That's exactly true, I know about this. But I want to know another process which is theoretical and Lorentz transformation has not been used there