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If I postulate the principle of relativity and the constancy of the speed of light for every inertial observer can I then prove all SR? Or do I need some other postulate?

For example: do I need to also postulate the structure of the Lorentz transformations or the Lorentz transformations derive completely only from this two basic postulates. (Do I have to also postulate, for example, that the transformation are linear to prove them from the two starting postulates?)

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You can not prove all of SR. You can derive the Lorentz transformation using those two postulates plus linearity. The Lorentz transformation then gives you time dilation, length contraction and relativity of simultaneity. But this is not all of SR. You can not get the relativistic formula for momentum and the well-known formula $E=mc^2$ without also postulating conservation of momentum and energy, and using some definition of momentum and energy.

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    $\begingroup$ I'd just add that linearity is a consequence of assuming space homogeneity $\endgroup$
    – daydreamer
    Commented Oct 15, 2020 at 19:03
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By more modern standards, if you are looking for a proof (and not just handwaving arguments), you'll need to clearly state assumptions about the "space[time]" and other mathematical structures that model the physics that you are using as your starting point, and formulate your postulates in precise terms with those structures.

(Don't assume that "we all know that THIS [term] means THAT".
Clearly state the assumptions... A successful proof rests on the details.)

You might find enlightening this diagram describing various pathways to the Lorentz Transformations. (Sorry I don't have a nicer scan.)

Lucas and Hodgson p 152
(from "Spacetime and electromagnetism : an essay on the philosophy of the special theory of relativity"
by J R Lucas & P E Hodgson, Oxford University Press, 1990.)

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No, the two starting postulates are the only required thing. Lorentz's transformations are also derived from them.

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  • $\begingroup$ But for example on Morin's textbook is stated that we have to postulate the linearity of LT. $\endgroup$
    – Noumeno
    Commented Aug 3, 2020 at 13:47
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    $\begingroup$ This would be a more useful answer if you provided a bit more info on how Lorentz equations derive from those axioms. $\endgroup$
    – Carl Witthoft
    Commented Aug 3, 2020 at 13:47
  • $\begingroup$ I know but I have little time now. I just posted the answer because some users strongly dislike comments which kind of answer hthe question haha, but I must said that I found a lot of "how SR is derived" in this site, just search for it. $\endgroup$
    – FGSUZ
    Commented Aug 3, 2020 at 13:49
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    $\begingroup$ Problem is I am not sure your answer is correct, based on what I read. What is your reasoning\your source? $\endgroup$
    – Noumeno
    Commented Aug 3, 2020 at 13:55
  • $\begingroup$ what have you read which contradicts FGSUZ. use Minkowsky diagrams and you can derive everything. what do you mean with linearity of LT? $\endgroup$
    – trula
    Commented Aug 3, 2020 at 17:22
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Your two postulates would work if you were to build the whole of relativity, including GR. For special Relativity, you would have to include that two objects or frames or reference should be moving without any acceleration and in a straight line.

Once you have the constancy of light postulate, you find out that Galilean transformations do not work. So, you need to derive up a new type of transformations - the Lorentz one; which can bed one by considering some symmetries and space-time diagrams.

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  • $\begingroup$ Ok, but to prove LT I Indeed need, on top of the two principles, to postulate simmetries or linearity, right? $\endgroup$
    – Noumeno
    Commented Aug 4, 2020 at 12:07
  • $\begingroup$ Yes, you do. But it is not an explicit postulate. Like for example if you are proving a theorem, you need not specify every instance where Euclid's axioms are used. In special relativity, as the name suggests, you only deal with linear motion, so is an assumed postulate. $\endgroup$
    – PNS
    Commented Aug 4, 2020 at 12:16
  • $\begingroup$ It seems to me that “including GR” is too strong a statement... the two postulates do not not imply spacetime curvature or anything about the causal structure and the global topology of the spacetime. (GR admits solutions that do not have the topology of $R^4$.) $\endgroup$
    – robphy
    Commented Aug 5, 2020 at 14:33

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