How does the Special Theory of Relativity work?

 

The Lorentz transformation

Recall that in the previous section, we tried to model space and time in a way that would be consistent with the observed constancy of the speed of light in Nature. The resulting model for space and time measurements has observers measuring time and space differently when they are moving relatively to one another. The two basic differences between what relatively-moving observers measure can be summarized as:

Relativistic time dilation: The process that occurred in the blue driver's rest frame with in time Tb was perceived by the red driver to have occurred in time Tr = Tb / (1 - (U/c)2)1/2 . This means that a clock appears to tick more slowly to an observer who perceives that the clock is moving than it does to an observer who is in the rest frame of the clock.

Relativistic length contraction: The blue car measured to have length Lb in the blue driver's rest frame was measured by the red driver to have have the length Lr = Lb (1 - (U/c)2)1/2 . This means that the length of some object appears to have a shorter length to an observer who perceives that the object is moving than it does to an observer who is in the rest frame of the object. (This applies to the length parallel to the direction of motion only.)

We will now state (without going through the grunge work of proving) that the mathematical model that best encompasses the observed constancy of the speed of light in Nature (pretending for now that gravity doesn't exist) is the one where the spacetime coordinates of two such observers as described above are related through what is now called a Lorentz transformation:

c T = (1 - U2/c2)-1/2 c T' + (U/c)(1 - U2/c2)-1/2 X' X = (U/c)(1 - U2/c2)-1/2 c T' + (1 - U2/c2)-1/2 X'

This transformation tells us how some observer O, who sees another observer O' moving at velocity U, translates measurements made by O' into measurements valid in the O frame of reference. In our specific case, we'd say this is how the red driver sees things that are happening in the frame of reference of the blue driver.
For example, in time dilation: in the blue car's frame of reference, the laser pulse did not travel in the X_b-direction at all, so X' = X_b = 0, leaving T = T' /(1 - U²/c²)^1/2.
In length contraction: in the red car's frame of reference, the red driver must measure the length of the blue car at a single moment of the red driver's time. This means we have T_r = T = 0 and so c T' = - (U/c) X', leading to X = X' (1 - U²/c²)^1/2.
The Lorentz transformation is the foundation of relativistic geometry, which we will examine next.