Confusion about time shift in special relativity

Question

I have never really found a way to comfortably comprehend the idea of time shift even though I know its not the hard part of relativity theory. In that light, can someone point out what is wrong or right about the following logic I'm thinking of here:

The scenario is:

There are four things in my universe: a wallclock, a flashlight, my wrist watch and myself. I shine the flashlight at my wrist which is in the same line of sight as the wallclock and see they two clocks are synchronized. Now my wristwatch, my flashlight and myself begin traveling away from the wallclock along the line of sight.

It seems to me that logically the light I'm running away from will take longer to reach my eye, so that means I would see the wall clock run slower, no?

But according to what I think I know, when a minute passes for me, more than a minute passes for the wall clock.

I just feel like I must be looking at it wrong. Any pointers are appreciated. This is just recreational thinking...

Answer 1

This is the situation you describe seen from the perspective of the wall clock i.e. the wall clock is stationary and you are moving.

We'll choose time zero to be when you pass the wall clock i.e. the spacing between you and the wall clock can be taken to be zero. At this point you synchronise your watch with the clock. At this point both you and the clock agree about the time and your relative speed.

From the perspective of the wall clock your time is running slowly i.e. when the wall clock measures 1 hour your writwatch will have measured less than one hour.

However from your perspective you are stationary and it's the wall clock that's moving. So when you measure one hour on your wristwatch less than one hour will pass for the wall clock. This is the opposite of what you say in your question:

But according to what I think I know, when a minute passes for me, more than a minute passes for the wall clock.

The situation is symmetric. Both frames see time running more slowly in the other frame. It must be this way otherwise you could say which frame was moving and which was stationary, and this would contradict the basic principle of relativity.

It is very important to be clear that this is not due to the time the light from your flashlight takes to get to the clock and back. Because you know how fast the wall clock is moving (both frames agree the relative velocity is $v$, and you know how much time has elapsed since you separated) you can calculate the flight time of the light and correct for it. If you do this you will still find that time is running more slowly for the wall clock.