# Confusion about time dilation proof using Lorentz Transformation

For Length Contraction using Lorentz Transformation, we use the condition that from the 'Fixed frame' (in the following image) the length of the object was measured simultaneously (i.e. $$t_1 = t_2$$ ). I understand that if this condition is not fulfilled, an observer at 'Fixed frame' will make error in measuring the object length (since the object is moving with respect to the observer in 'Fixed frame').

Now for time dilation proof, a similar condition is used. Here,for the 'Moving frame', the time difference has been measured for such two events that took place in the same position (i.e. $$x_1' = x_2'$$). But I cannot understand why this condition must be fulfilled.

Would you please explain this? Thanks.

(image source: http://hyperphysics.phy-astr.gsu.edu/hbase/Relativ/tdil.html)

They're assuming that the "moving frame" is the frame following the clock itself. That is, it is the frame where the clock isn't moving at all, so $$x_1' = x_2'$$.
• @NafisSadik Textbooks focus on the case $x_1' = x_2'$ because it's simpler, so it's less confusing for students if they introduce one new effect at a time. Commented Feb 19, 2020 at 20:35