In this paper as well as in Landau's textbook on classical field theory, there is a proof of the conservation of spacetime interval in which authors deduce, that the differentials of the interval in two different inertial reference frames are infinitesimals of the same order and therefore must be proportional. What does it mean and how can you prove it without refering to the properties of Lorentz transform, which is to be derived from conservation of spacetime interval.
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1That link is broken. When you say the interval, do you mean $ds^2 = dx^2 +dy^2 +dz^2-c^2 dt^2$? – CDCM Jun 07 '17 at 18:12
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2Possible duplicate of Proving invariance of $ds^2$ from the invariance of the speed of light – Kyle Kanos Jun 07 '17 at 20:09
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$ds'$ is some function of $ds$ which we want to find:
$ds'=ds'(ds)$
We can taylor expand this function:
$ds'(ds)=ds'(0)+ads + \mathcal{O}(ds^2)$
for some $a$.
Since $ds'$ and $ds$ are both of first order, we neglect higher order terms.
From the propagation of light, we know that if $ds=0$ then also $ds'=0$ and therefore
$ds'(0)=0$
This leaves us with
$ds'=ads$
for some $a$.
Photon
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How do we know, that a is nonzero? If it is, we need to take higher order terms into account, and the two infinitesimals wouldn't be of the same order. Also, how can an ifinitesimal value be function of another infinitesimal value? – Jakub Skórka Jun 07 '17 at 20:12
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I think, you should rather go with the explanation linked in the second comment to your question which is much more rigorous than my sketch. – Photon Jun 08 '17 at 06:49