Time Dilation Confusion: Need Help Understanding

[DarkStalker]

Hello.
I need some help understanding time dilation.

So the idea is that moving clocks tick slowly. Say there was an observer A who, by his frame of reference, is stationary, and he sees an object B moving past it at high velocity. From what I understand, to A it would appear as if B's clocks are ticking slowly compared to his. In other words, to B it would appear as if A's clocks are ticking faster.

But that's from the point of view of A. If you look at it from the point of view of B, it is he who is stationary, and A is the one who is moving at high velocity. So A's clock should be ticking slower compared to B's.

There's seemingly a conflict between these two conclusions.
Where's the mistake in my understanding?

 

[Orodruin]

You are not considering the relativity of simultaneity.

 

[DarkStalker]

You are not considering the relativity of simultaneity.

I'm familiar with the concept, but I do not see the link between that and my scenario. Could you please elaborate, or perhaps link me to some texts that help me understand it a little better?

 

[Orodruin]

When you are saying that a clock runs slow when moving, you are just saying that the time elapsed on the clock is less than the coordinate time difference. The coordinate time difference represents the difference in the time coordinate between to surfaces of simultaneity. These surfaces are going to be different depending on which system you consider the situation in.

 

[Nugatory]

I'm familiar with the concept, but I do not see the link between that and my scenario. Could you please elaborate, or perhaps link me to some texts that help me understand it a little better?

While the two clocks are side by side we set them to both to 12:00 noon. A looks at his clock one hour later and sees that it reads 1:00 PM. He also sees that at the same time that his clock reads 1:00 PM B's clock reads 12:30 PM so he concludes that B's clock is running slow. But because of the relativity of simultaneity, it does not follow that B will find that A's clock reads 1:00 PM at the same time that B's clock reads 12:30; in fact, for B "my clock reads 12:30" happens at the same time as "A's clock reads 12:15" and B will conclude that A's clock is the one that's slow.

I've bolded all the occurrences of that phrase "at the same time" to underscore the extent to which simultaneity is being assumed whenever we compare physically separated clocks.