Deriving Lorentz Factor: An Analysis of Jimmy360's Method

[DatLemonDoe]

Hello!

This is my first post on this forum, so make sure to tell me if I am doing something wrong :)

I was trying to derive the Lorentz factor today, and I used the following page as a guide. The top answer by Jimmy360 is what I followed.
https://physics.stackexchange.com/questions/173268/deriving-the-lorentz-factor-gamma

I was unable to solve for t' as they did, and it occurred to me that there was perhaps an issue with the derivation. Now I am in no way knowledgeable nor talented in derivations so I wanted to have someone look at my thought process. I have a hard time imagining that the derivation I linked is wrong, but then again I made it work for me with a small change that seems to make sense.

I realize that I am most likely wrong but I can't figure out why. That's why I need your help. If I'm misunderstanding something I want to find what that something is.So, it seems that Jimmy360 set t' to be the coordinate time, and t to be the proper time.
Hence l=ct (that's an observation made by an observer in the reference frame of the rods), and d=ct' (that's an observation made by an observer in the other reference frame, which doesn't follow the rods)

Now when they find that d²= l²+(vt)², I would have expected to see d²= l²+(vt')² instead. The distance that the rods travel in the x-axis is not measured at all by an observer in the rods' reference frame, is it? That's why I expected d to be expressed in terms of f' rather than f

After that change is made we get
c²t'²=c²t²+v²t'²

and then
c²t²=c²t'²-v²t'²

From there I am able to solve for t and in doing so find the Lorentz factor.

Is my thought process correct? If not, can you guide me towards what I don't understand?

Thank you very much for your time, I appreciate it :)

 

[Ibix]

There is a mistake in the derivation, yes. It should say $l=ct'$ (i.e. in the frame where the light clock is at rest, it takes time $t'$ for the light pulse to travel one way up the clock) and $d=ct$ (i.e. in the frame where the light clock is moving it takes time $t$ to go up the diagonal distance $d$) not the other way around. The expression as (incorrectly) derived makes $t'/t=\gamma$, but the LHS should be the other way up: $t/t'=\gamma$. If you put the definitions of l and d in correctly the correct answer drops out.

An observation I've made about relativity is that it seems to be quite easy to get correct-looking (but wrong) expressions when you make a mistake.

 

[DatLemonDoe]

yes, I agree that it makes sense to have t' in the frame where the clock is at rest, and t in the frame where the clock is moving.

However the choice of variables is not the core mistake, is it? I agree that the proper time should be t' , and the coordinate time t, but realistically what stops us from doing the entire derivation with the variables the other way around? we would simply get the end result
t/t'=γ

What I'm getting at is that in Jimmy's derivation, vt should instead be vt'. It seems to me that that's the core mistake, it's an actual physics mistake.

If we let t' be the proper time as you suggested (which I totally agree is the way to go), then we would have vt instead of vt'

Is that correct?

Thank you once again, I know that I'm being picky but I want to make sure that I understand it all

 

[Ibix]

I think that's just another way of saying the same thing.

It's not clear to me whether the original derivation intended to use $t$ for the moving frame or the rest frame. The problem is that it used one in the expression $d=ct'$ and the other in $vt$, which is inconsistent. Both of these should use the same one of $t$ or $t'$, and the expression for $l$ should use the other.

You seem to have fixed the problem by changing $vt$ to $vt'$. I fixed it by switching $t$ and $t'$ in the other two expressions. Either approach is self consistent. Since the original derivation doesn't specify which frame is supposed to be using $t$ and which $t'$, either approach is also consistent with the full problem specification (such as it is).

I'm not sure whether it's a physics problem or a "confused by their own notation" problem. Either way, the maths is inconsistent and you seem to have found a valid fix.

 

[DatLemonDoe]

Ok, I get it now. Indeed I was really saying the same thing as you were, just differently.

Thank you, have a great day.