13 - The Lorentz transformations

Summary

From youth onwards, my entire scientific effort was directed to deepening the foundations of physics.

Albert Einstein, interview on 6 February 1924

Deriving the Lorentz transformations

Relativity theory is provocative. It incites an urge to show that the theory is inconsistent or that one can beat the speed of light. In this chapter we examine both issues. As a prelude, we need to study how Alice and Bob assign coordinates to an event, that is, how they specify the location and time of an event. Then we need to learn how those assignments are related. Once we have those relations, we can compute how differently Alice and Bob perceive the velocity of some object. And that is a major step toward trying to beat the speed of light.

Alice and Bob specify the location of an event – a firecracker going off, say – by measuring distances from the origins of their respective reference frames. Figure 13.1 displays this. The distance Bob assigns is denoted x; the distance Alice assigns is called x. Alice specifies the time by using a clock that is present at the event, is at rest in her frame, and is synchronized with the master clock at the origin of her frame. Bob does the same, using a clock at rest in his frame. This procedure – using clocks at the event itself – is a matter of convenience and is something we discussed earlier, in section 9.5. The times are denoted t for Bob's value and t' for Alice's.