Proving invariance of $ds^2$ from the invariance of the speed of light I've started today the book of Landau "Field theory". He starts from the invariance of the speed of light, expresses it as the fact that $c^2(\Delta t)^2-(\Delta x)^2-(\Delta y)^2-(\Delta z)^2=0$ is preserved when we change inertial frame, so he considers $ds^2=c^2dt^2-dx^2-dy^2-dz^2$, and says 
"We have observed that if $ds=0$ in one frame then $ds'=0$ in another frame. But $ds$ and $ds'$ are infinitesimal of the same order. So it follows that $ds^2$ and $ds'^2$ have to be proportional that is $ds^2=ads'^2$..." and he goes on to prove that $a=1$.
How to translate this argument in a rigorous one? I'm really interested in this, both to understand this deduction and also to be able in future to make similar ones.
Thanks to everyone who will help
Bye!
 A: I don't think the handwavy argument in Landau's book "The classical theory of fields" (pg. 4) can be made mathematically rigorous. From the Newtonian equation x=vt and the light speed invariance postulate he obtains  $c^2(\Delta t)^2=(\Delta x)^2+(\Delta y)^2+(\Delta z)^2 $ and consequently the equation (1) $c^2(\Delta t)^2-(\Delta x)^2-(\Delta y)^2-(\Delta z)^2=0$. 
But the jump from this equation to a 4 dimensional invariant interval $ds^2=c^2dt^2-dx^2-dy^2-dz^2$ for $ds^2=0$ is not mathematically justified, (then again Landau calls the 4-space fictitious up to three times in that page and maybe that's why he doesn't intend to do a rigorous derivation of the Minkowskian interval).
It is quite obvious from the way it is derived  that in equation (1) t is a parameter in 3-dimensional space as used in Newtonian physics and in Einstein's Special relativity previous to Minkowski formalism, not a fourth coordinate in 4-space.
The moral here is that one cannot derive $ds^2=0$ from the invariance of the speed of light, they are the same statement that comes by postulate in special relativity, cannot be deduced. 
