A standard light clock has two mirrors, say one metre apart, and a light pulse. When the clock is stationary the light path is perpendicular. When moving, diagonal. The clock ticks over each time the light hits the mirrors. The light takes longer to travel the diagonal path, so the clock takes longer to tick over, so we get time dilation. What if the clock ticked over each time the light travelled one metre? The light would still travel the same paths, perpendicular or diagonal,except those paths would be divided up into one metre lengths. Is one metre the same in both frames? It has to be, it is the distance the mirrors are apart in both frames. If you are having trouble visualising this,draw a light clock and just mark the light's path in one metre lengths, as opposed to where the light hits the mirrors.With one metre ( or whatever the mirrors perpendicular distance apart is ) the clock will tick over at the same rate in both frames. This contradicts time dilation. Can you explain why? Thanks. DAC.
Is one metre the same in both frames? It has to be, it is the distance the mirrors are apart in both frames.
No, a one meter light path is most assuredly not the same in both frames. If the mirrors are 1 m apart then in one frame the pulse of light travels 1 m but in another frame the same pulse of light travels more than 1 m. The point where the light has traveled 1 m in the “moving” frame will be less than 1 m in the stationary frame.
Note that the relevant distance is the length of the light path, not the distance between the mirrors
Well the difficulty is - how is an observer going to measure those 1m lengths?
The Moving Observer
The observer moving with the clock doesn't notice light taking a longer path, all interactions in the observer's body and equipment will suffer the same effect. That's a symptom of them and their equipment being constructed of energy, the same as light. Their distance measurements in their frame of reference will be the same for them regardless of any relative speed to someone else.
The Equivalence Principle
Relativity really is founded on the idea that (aprox wording here) "the laws of physics are the same in all frames of reference". It's called the equivalence principle.
The Stationary Observer
The observer who is not moving can mark the light path into lengths, but they must use a physical tool to do it (probably it will utilise light), and their view is also that the speed of light cannot change. As the back and front of an item pass such a detector the speed of the item reduces the result. They therefore measure lengths in the moving frame as contracted in the direction of motion, and maintain c as constant.
The stationary observer sees time tick slower for the subject, and also gets a length contracted view of objects moving with the subject.
For effects due to relative velocity, it's the same the other way around too.
Effect due to gravitational acceleration are a bit more one sided.
This is an good page https://en.wikipedia.org/wiki/Ladder_paradox
(Not wishing to complicate things I ignore that either observer can do relativistic math, and calculate the other's view of the system if they know their relative speed.)