Lorentz Transformation in One-Dimensional Space

[the_emi_guy]

If space only had one dimension would Einstein's speed of light postulate still lead to Lorentz transformation for motion along that one dimension?
Relativity of simultaneity can obviously be demonstrated in one dimension (lightning bolts hitting opposite ends of stationary and moving train). But all derivations of the Lorentz transformation seem to require at least a second space dimension (i.e. the familiar light clock and Einsteins original 1905 paper) in order to obtain the Lorentz factor. Also, description of light cone:
c²dt²=dx²+dy²+dz² reduces to
cdt=dx so space-time intervals would no longer have the square roots of squares involved.

 

[Nugatory]

But all derivations of the Lorentz transformation seem to require at least a second space dimension

It's easy to do a one-dimensional derivation; there's one by Einstein in the appendix of his book "Relativity: The special and general theory".

Basically we're looking for coordinate transformations such that $x\pm{c}t=0$ implies $x'\pm{c}t'=0$, which is to say the speed of light is $c$ in both frames.

 

[Orodruin]

Also, description of light cone:
c2dt2=dx2+dy2+dz2 reduces to
cdt=dx so space-time intervals would no longer have the square roots of squares involved.

Yes it would. You are missing one root by asserting c dt = dx.

 

[the_emi_guy]

Thanks, this is what I was looking for.

 

[vanhees71]

It's easy to do a one-dimensional derivation; there's one by Einstein in the appendix of his book "Relativity: The special and general theory".

Basically we're looking for coordinate transformations such that $x\pm{c}t=0$ implies $x'\pm{c}t'=0$, which is to say the speed of light is $c$ in both frames.

Then you are let even to a larger group of transformations, namely the whole conformel group!