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Suppose that in the reference frame O, the reference frame O' is moving at a speed v in the x-direction, with their origins coinciding at t = 0. An event E occurs at (x, y, z, t) in the O frame. It's straightforward to show that y' = y and z' = z. Next I considered x'. In the O frame, the distance between O' and E is x - vt. Because the ruler O' uses is shortened by a factor of γ, she will then measure the distance x - vt as being greater and O measures it, by a factor of γ. Thus, x' = γ(x - vt) (if this approach is incorrect, let me know!).

However, I'm having trouble with t'. I know that t' = γ(t - vx/c2). I assume that the -vx/c2 term comes from that, because O' believes that she's at rest, when the light emitted from the event reaches her, she doesn't treat herself as moving into the light, and thus there's a discrepancy as to how long before the light reaches O' did the event actually occur. Unfortunately, I can't arrive algebraically at this term. Finally, the gamma factor. I assume this comes from time dilation, but the wouldn't O' 's clock be running slower? So wouldn't the term have to be 1/γ (because less time transpires on her clock)? Can someone please tell me how to get the final Lorentz term this way? I know there are probably other, easier routes, but for personal reasons I would like to know how to do it this way. Thanks a lot!