Invariance of spacetime interval in special relativity: linearity

Question

I'm trying to understand which assumption are necessary to prove the invariance of the spacetime interval $$\Delta s^2=c^2\Delta t^2-\Delta \mathbf{x}^2$$ in special relativity. The postulates of special relativity are:

• Principle of relativity

• speed of light is constant in every frame of reference

From the second postulate is evident that $$ds'^2=0 \iff ds^2=0.$$ Then from the answer Invariance of spacetime interval directly from postulate we may see how this, and the fact that the two infinitesimal are of the same order, leads to $$ds'^2=ads^2.$$ In some other answers (e.g.Proving invariance of $ds^2$ from the invariance of the speed of light) it is pointed out that $ds'^2=ads^2$ arises from the fact that my coordinate transformation is linear.

My question is: what is the assumption I use to prove $ds'^2=ads^2$? Do I need linearity and if so where do I use it? And what about the principle of relativity?

Answer 1

I think there are two ways to argue this:

1. via the principle of relativity: the metric should be the same in all inertial frames, thus $ds^2 = ds'^2$ immediately. (However it's not immediate that the metric should be Minkowski)

2. via the constancy of the speed of light: $ds^2=0$ if and only if $ds'^2=0$, then using linearity as in Valter Moretti's linked answer we must have $ds'^2 = a ds^2$. One can then argue that $a=1$.

Note that linearity of Lorentz transformations is required in the second path. One can show that homogeneity implies that Lorentz transformations are affine. In this case this is sufficient for linearity because we only care about differences $\Delta x$, which are insensitive to an added constant.

As to the question: where do I use linearity? the answer is in Valter Moretti's proof to the theorem linked above.