Question on Einstein's Simple derivation of Lorentz Transformation.

[Peeter]

In appendix 1 of Einstein's "Relativity, The Special and General Theory", a book intended for the general public, he gives a derivation of the Lorentz transformation.

The math involved is pretty straightforward, but I wonder if anybody can clarify his reasoning for his equation 3 (below)

He has two frames, with motion along x and x’ coordinates. The path of light shined along the positive direction of these axis is described by:


$$\begin{align*} x -ct = 0 & \ \ \ (1) \end{align*}$$

$$\begin{align*} x' -ct' = 0 & \ \ \ (2) \end{align*}$$

He states:

“Those space-time points (events) which satisfy (1) must also satisfy (2). Obviously this will be the case when the relation “

$$(x' -ct') = \lambda(x -ct)$$

is fulfilled in general, where $\lambda$ indicates a constant; for, according to (3), the disappearance of (x – ct) involves the disappearance of (x' – ct')”

His “Obviously” isn’t so obvious to me. Given what he described I don’t see how the concurrent disappearance implies that these are linearly related by a constant. This step isn't terribly suprising given that the whole point of the appendix is to find the linear transformation between these (ie: the Lorentz tx.).

If I pretend that I didn't know that such a linear relationship was being looked for, I don't follow is argument of why to expect these should be linearly related. Is this obvious to anybody else?

--
ps. For reference I found an online version of this appendix here:

http://www.bartleby.com/173/a1.html

(I didn’t look to see if the whole book is there … I’ve got a copy from the public library).

 

[Fredrik]

It's not obvious, and it's not really easy to prove. It's also not really difficult, but you have to be very careful about what your assumptions are when you try to do it rigorously. My advice is: don't bother. Einstein's "postulates" aren't well-defined enough to be used as a starting point of a rigorous derivation. Think of them as a list of properties you want the theory you're looking for to have. You're trying to find a theory that contains something that looks like Einstein's postulates.

You can use any methods you want as long as you're just looking for a candidate theory. Once you have found it (i.e. once you have the definition of Minkowski space), you can take that as a mathematical axiom and derive everything rigorously from that.

 

[Fredrik]

One more thing: To derive the linearity, we have to make more assumptions (or rather make implicit assumptions explicit). In particular we have to assume that every function that describes a coordinate change from one inertial frame to another is smooth (differentiable as many times as you'd like) and takes straight lines to straight lines.