Time dilation paradox between two moving observers

[krylea]

I only just started learning about special relativity in my physics class, and I have been running into a problem that none of my teachers have satisfactorily explained.

Objects X and Y are moving relative to one another at some significant fraction of the speed of light, where ɣ is the Lorentz factor for their relative motion.
From the perspective of Object X, Object Y is moving. Thus, for each one second that passes from Object Y’s perspective, ɣ seconds pass from Object X’s perspective
From the perspective of Object Y, Object X is moving. Thus, for each one second that passes from Object X’s perspective, ɣ seconds pass from Object Y’s perspective.
Thus, if ɣ seconds pass for Object Y, ɣ*ɣ seconds pass for Object X, by the first premise. By the second premise, however, ɣ*ɣ*ɣ seconds pass for Object Y. The time that passes for Object Y cannot be both ɣ and ɣ^3 seconds.

I am sure I am making some sort of error at some point in this reasoning, I just don't quite see what it is...

 

[Janus]

You'll have to study the Relativity of Simultaneity for this to make sense. By rights, you should have started with this.

So for instance, if v is 0.866c then after 1 sec according to X, .5 sec has passed for Y.

And after 0.5 sec according to Y, 0.25 sec has passed for X. This does not however mean that the time that passes for X is both 0.5 and 0.25 at the same time, because X and Y will not agree as to what "the same time" means.

 

[ghwellsjr]

I only just started learning about special relativity in my physics class, and I have been running into a problem that none of my teachers have satisfactorily explained.

Objects X and Y are moving relative to one another at some significant fraction of the speed of light, where ɣ is the Lorentz factor for their relative motion.
From the perspective of Object X, Object Y is moving. Thus, for each one second that passes from Object Y’s perspective, ɣ seconds pass from Object X’s perspective
From the perspective of Object Y, Object X is moving. Thus, for each one second that passes from Object X’s perspective, ɣ seconds pass from Object Y’s perspective.
Thus, if ɣ seconds pass for Object Y, ɣ*ɣ seconds pass for Object X, by the first premise. By the second premise, however, ɣ*ɣ*ɣ seconds pass for Object Y. The time that passes for Object Y cannot be both ɣ and ɣ^3 seconds.

I am sure I am making some sort of error at some point in this reasoning, I just don't quite see what it is...

Hopefully you have studied how Einstein constructs a Frame of Reference by synchronizing a series of clocks at measured intervals of distance apart. Each Object has its own set of synchronized clocks. These clocks provide Co-ordinate Time at different distances from the Object.

Einstein shows that as all these clocks are traveling past each other, each Observer can look at the other Observer's clocks, whichever ones are just passing him and he will see that as his own clock ticks off one second, the other clocks are ticking off more seconds, in fact, ɣ times as many seconds.

It would be like if you were traveling down the highway and looking at clocks on different buildings or signs as you pass by them and seeing that these clocks tick faster than your own. But once a clock passes you, you're not going to look at that clock again, correct? You always have new clocks to look at that you have never seen before.

Now, at the same time that Object Y is looking at the passing clocks that were synchronized by Object X, Object X is looking at the passing clocks that were synchronized by Object Y and they both will come to the same conclusion, that the other Object's clocks are ticking faster than their own clock.

Once you understand how each Object compares its own clock to the multitude of the other Object's clocks, I think you can see that you can't recursively repeat the process. What would that mean?