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!

 

[Dale]

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.

 

[ghwellsjr]

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)?

No, the coordinates of an event in each frame apply only to that frame and have nothing to do with any other frame.

It's not enough to say t=2s in frame S when talking about an event. There are an infinite number of events at different locations with t=2s. You have to also specify its location to identify an event in frame S. Once you do that, you can transform those time and location coordinates into the time and location coordinates for any other frame moving at any speed with respect to S.

Then you should say that the given event happens at x,t in S and at x',t' in S'. It has nothing to do with anything that any observer really sees. The coordinates that are applied to any particular frame have nothing to do with what any particular observer really sees, not even the frame in which a particular observer is at rest. You can analyze what all observers really see using any frame and transform to any other frame and they will still see the same things. All frames are equally valid. None is more real than any other. None is preferred over any other.