Lorentz transformation, mistake but right formula (for light) ?

[digi99]

I was thinking when I take the Lorentz formula for a location γ.(x – v.t) as an observer in S and find the ratio compared with the location for an observer within the inertial system S’ it selves: 1/γ . Δx

But I made a mistake and took 1/γ. x

When I use the found ratio (for derivation, see attachment) for the time as an observer within the inertial system it selves: 1/γ. t, I find the Lorentz formula for time as an observer in S, but only for light x=c.t

Why is this accidently right ?

 

[Fredrik]

I'm fascinated by the fact that not a single word you wrote is even close to legible. I can't even guess what any of them are. I'm not sure I recognize a single letter (in the comments to the calculation). One of them looks like a g, but I have no idea what most of them are.

Please start using LaTeX so that you no longer have a reason to post hand-written stuff. Link to FAQ post. If you're going to post hand-written stuff, at least clean it up so that it's possible to read it.

I can read the symbols in the calculation, but I still don't understand what you're asking.

 

[digi99]

Thanks for the answer Frederik, I go to rewrite this, I did it in a hurry late and of course I was thinking today between in, what I am writing and I confess it is not clear :) ...

During rewriting and I did with itex, I saw the problem and don't want to spend to much time to it.

If I could delete this topic, I did (too late).

Ok, this was just found in my expressions:

You have a inertial system S1 with a lightwave 1 emitted, a system S2 where you see this light wave with Doppler effect and a second light wave 2 emitted.

The ratio traveling distance lightwave with Doppler effect in S2, and traveling distance lightwave 2, at the same time in S1, is the same ratio as for the related times in S2 for both lightwaves.
So nothing special to report/ask, but took a long time to analyse this ... I thought it was a mistake, but it was not ...