Why Does the Simultaneity Equation Use Gamma Squared?

[skynelson]

I have poured over my old college textbook (Mould) in its description of simultaneity. I came across one sticking point that I don't understand.

The proper equation to describe the relative time between two different points is:

Eqn 1) t' = gamma (t - Lv/c^2)

This equation makes sense to me because I thought I understood simultaneity (the offset factor of Lv/c^2) as well as time dilation (the factor of gamma when changing into the moving/primed frame).

However, the book then inserts another factor of gamma (ostensibly from the t'). Didn't we already account for time dilation with the first gamma?

I know the book is right, since the following reduces properly to an identity:
t/gamma = gamma (t - Lv/c^2)

I just don't understand the logic/physics of it. Yes, of course t' = t/gamma, but we already included that gamma in the right half of equation 1, right?

Why do we have gamma^2? What does that MEAN?
I've pummeled my brain for 6 months on this issue ;-) I think it's time to ask.
Thanks!
Sky

 

[Ich]

Yes, of course t' = t/gamma

No, that is a derived result and only valid for L=vt, i.e. at the position of a moving clock. You should write x instead of L to make clear that we're talking about a space coordinate here.
Eq. 1) is the one and only equation to use (of course in combination with x'=gamma(x-vt/c²)), not t'=t/gamma.