# Lorentz transformation for time - why the 'x' term?

chipotleaway
I have two questions having to do with the Lorentz transformation for the time...some preamble first:

The Lorentz transformation for time along the x-axis is
$$t'=\frac{t-\frac{ux}{c^2}}{\sqrt{1-\frac{u^2}{c^2}}}$$, where u is the relative velocity of S'.

Why is there a dependence on x there? I'm interpreting x to be the distance between the origin of S and some event. At t=0, the origins of S' and S cross and their clocks are synced. S' is moving to the right at velocity u relative to S. When an event occurs at (x,t) according to S, it occurs at (x', t') for S'.

i) Let's say t=2s in the frame of S. Is t' the time that S' sees the event according to S? So in her frame, S' really sees it happening at 2 seconds but when she alerts S, t' has passed for S (t'>t)?

ii) And the dependence on x - why should the distance of S to the event affect how time is affected in S' frame? This distance dependence doesn't appear in the time dilation formulas I see in all the textbooks I've flipped through!

Mentor
2021 Award
Why is there a dependence on x there?
The dependence on x in the Lorentz transform is the mathematical expression of the relativity of simultaneity. Two events which happen at the same time in frame A will happen at different times in frame B if they have different positions in the x direction.

This distance dependence doesn't appear in the time dilation formulas I see in all the textbooks I've flipped through!
The Lorentz transform is the general formula and should be the one that you memorize. The simplified time dilation formula is a special case of the Lorentz transform for use only when calculating the time between two events which are at the same location in one frame. In that case Δx=0 and that term drops out of the equation.

Do NOT use the simplified formula if you do not fully understand its limitations. Instead, use the full Lorentz transform which will automatically simplify when appropriate.