# Lorentz Transformation/Time Dilation

1. ### bon

561
1. The problem statement, all variables and given/known data

Ok so I know how to prove Time Dilation using the Lorentz transformation...

My problem is that I don't see why one couldn't use the Inverse Lorentz Transform and come to the opposite result as follows:

Time period in rest frame S: T = t2 - t1

ILT: t = $(t1' + B/c x') where$ is gamma

So: t2-t1 = $(t2'-t1') so T' = T/$ which is the wrong result...

I.e. how does one know that one must use the LT rather than ILT?

Thanks

2. Relevant equations

3. The attempt at a solution

2. ### vela

12,780
Staff Emeritus
Why is that wrong? γ is greater than 1, so T' < T, as expected. The time elapsed in the moving frame is shorter than the time elapsed in the rest frame.

3. ### bon

561
No it should be the other way round. "Time Dilation" - time measured in any frame moving wrt rest frame should be dilated (i.e. longer time period) than in rest frame..

4. ### CompuChip

4,297
I didn't look at it too closely, but I suspect that by using ILT you have implicitly inverted the rest frame and the moving frame.
Basically, you have relabelled T to T' and vice versa.

5. ### bon

561
Thanks, yes I agree. I just can't see where the inconsistency comes in in the maths... :S

561
anyone?

7. ### vela

12,780
Staff Emeritus
OK, so you're assuming the clock is in the rest frame and trying to calculate what someone in the moving frame would see?

8. ### vela

12,780
Staff Emeritus
Your inconsistency stems from ignoring x' in the Lorentz transformation. It looks like you just erased it going from the ILT to your conclusion.

9. ### bon

561
but i though x1' = x2' since the two times were measured at the same position in S'?

10. ### vela

12,780
Staff Emeritus
If the clock is at rest in S, it's moving in S', so x1' and x2' won't be equal.

11. ### bon

561
Ahh this is getting confusing now. Ok so its measured in S at a position x to have time interval t2-t1

Now what happens when the measurement is made in S' - what stays constant? My understanding was that the time interval is measured to be t2'-t1' made at a fixed position in S' - call this x'..

What have i don wrong?

12. ### bon

561
Basically what i don't understand is the asymmetry:

time dilation: clock at rest in S. moving in S' therefore x1' and x2' aren't equal..
length contraction: clock at rest in S. moving in S'. Therefore t1' and t2' aren't equal..

but for length contraction t1' and t2' are equal - we use the inverse LT:

x1 = γ(x1'+Bct')
x2 = γ(x2' + Bct')

i.e. t1'=t2'=t
which gives the right result: x2-x1 = γ(x2'-x1')

13. ### vela

12,780
Staff Emeritus
For time dilation, you're comparing the time elapsed between the same two events in different frames.

When measuring length, you're finding the spatial separation of, say, the two ends of a stick at the same instant. Since simultaneity is frame dependent, you're looking at different pairs of events in different frames.

14. ### bon

561
Thanks ok - so what went wrong in my last post? (no. 12)..

Thanks

561
?
thanks

16. ### vela

12,780
Staff Emeritus
Just when you said this: "length contraction: clock at rest in S. moving in S'. Therefore t1' and t2' aren't equal." When you measure the length in S', you have to have t1'=t2', right? You have to measure where the ends are at the same time (in S').

17. ### bon

561
but then by parity of argument, when we measure time in S', we have to measure at x1'=x2', right? You have to measure the time at the same point in space?

18. ### vela

12,780
Staff Emeritus
No. The reason you have to measure where the endpoints are simultaneously is just an operational requirement. If the stick is moving, you can't measure where one end is at one time, measure where the other end is a second later, and subtract the two to get the length. You have to see where both ends are at the same time, and only then can you subtract those numbers to get the length.

With time dilation, you have two spacetime events, and different observers will say different amounts of time separate them, depending on their relative motion.

The symmetry you're looking for isn't there. Time dilation compares the same two events in different frames. Length contraction compares a pair of events in one frame to a different pair in another.