Trouble following the derivation of Scott and Viner 1965

[achmorrison]

I've been reading the Scott and Viner AJP paper from 1965 "The Geometrical Appearance of Large Objects Moving at Relativistic Speeds" and I am having a little trouble following their derivation of the expression for x in the appendix of the paper.

I understand how they get the expression for t:

$$t = -[(x^2+y^2+(z-d)^2)^{1/2}-d]/c$$

Which they put into the Lorentz transformation for x':

$$x'=\gamma(x-vt)=\gamma[x+\beta[(x^2+y^2+(z-d)^2)^{1/2}-d]]$$

But, then they say that they want to get an expression for x, y, z in terms of x',y',z' which implies that they just solve the above expression for x. But, when I do that, I don't get anything at all like what they get.

I have looked at dozens of papers that reference this one and they all just start with Scott and Viner's results with no discussion of how they get there.

I feel like I'm missing something very simple, but I just don't see it.

Thanks!

 

[Fredrik]

I haven't read the paper, but it sounds to me like you're trying to find the inverse of a matrix (or equivalently: solve a system of equations) by only looking at one of the rows (one of the equations).

 

[achmorrison]

Right, that's what I'm trying to do, but I don't get their result:

$$x= \gamma((x'+\gamma\beta d)-\beta[(x'+\gamma\beta d)^2+y'^2+(z'-d)^2]^{1/2})$$