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.

 

[Entropia]

The equation for simultaneity, as described in your textbook, accounts for both time dilation and the offset factor of Lv/c^2. The additional factor of gamma in the equation is necessary because it represents the time dilation for the moving frame, not just the stationary frame. In other words, it takes into account the fact that time is passing at a different rate for the moving frame compared to the stationary frame.

To understand this better, let's break down the equation: t' = gamma (t - Lv/c^2)

The first part, gamma, represents the time dilation factor for the moving frame. This is necessary because in the moving frame, time is passing at a different rate compared to the stationary frame. This is due to the fact that the moving frame is experiencing a change in velocity, and as a result, time is passing differently for objects in that frame.

The second part, (t - Lv/c^2), represents the offset factor of Lv/c^2. This accounts for the difference in time between the two points due to the distance (L) between them and the speed of light (c).

So, in short, the gamma^2 factor in the equation is necessary to account for the time dilation in the moving frame, while the gamma factor on its own only accounts for time dilation in the stationary frame. I hope this helps clarify the logic and physics behind the equation for you. Keep up the good work in trying to understand this complex concept!