Question on Einstein original paper

[DrSirius]

Hi, I am reading the Einstein original paper on Special Relativity, in this link (English translation), I am having difficulties on this sentence:

"We have now to show, that every ray of light moves in the moving system with a velocity c (when measured in the moving system), in case, as we have actually assumed, c is also the velocity in the stationary system ; for we have not as yet adduced any proof in support of the assumption that the principle of relativity is reconcilable with the principle of constant light-velocity."

Why does he need to show that c is constant also in the moving system if that is a principle? Is it necessary? Is it done for completeness or really he does need to show to complete the demonstration?

 

[PeterDonis]

Why does he need to show that c is constant also in the moving system if that is a principle?

Because, as he says at the end of the quote, he needs to prove that the principle of relativity--by which he means here basically the rules of how to transform quantities from the stationary system to the moving system--is consistent ("reconcilable") with the principle of $c$ being the same in all frames. Assuming the constancy of $c$ as a principle does not automatically guarantee that it is consistent with other principles that are assumed.

 

[PeterDonis]

[Moderator's note: adjusted thread title to be more descriptive of the actual topic.]

 

[DrSirius]

I see, so it is a kind of "sanity check". But then all kind of paradoxes happen and you would need to check them one by one, so, once he proves that the proposition ( c being constant ) is satisfied, he can be sure that the theory is self-consistent? Could that be proved by simply mapping the entire spacetime? I mean, when you get the Lorentz relationships, you see that for every point (x, y, z, t) there exists a set of coordinates (x', y', z', t') which describe that very same point. Wouldn't it be sufficient? For example, if you say that it is c/2 which is constant, and you maintain c as the fastest speed, used in the synchronization of clocks, you would surely arrive to a logical inconsistency when trying to establish the relationships of coordinates. If you don't arrive to any inconsistency, you are done with everything, including any apparent paradox. In fact, if a paradox arises, then you are surely discovering a new physical effect: for example, the transversal Doppler effect. It is predicted solely on Einsteinian relativity grounds, and does not exists in classical electrodynamics.

 

[PeterDonis]

all kind of paradoxes happen and you would need to check them one by one

I'm not sure what you mean here.

once he proves that the proposition ( c being constant ) is satisfied, he can be sure that the theory is self-consistent?

He didn't "prove that the proposition is satisfied"; he proved that assuming it to be true is consistent with the principle of relativity, by showing that if all light rays move at $c$ in one inertial frame, they also move at $c$ in any other inertial frame. That is sufficient to show that the two postulates are consistent with each other.

 

[DrSirius]

I'm not sure what you mean here.
I mean that, IMHO, all the paradoxes, like "barn and pole","twin paradox", "Bell spaceship", etc., are all apparent paradoxes which have only educational interest, but that they cannot introduce any real paradox, as the theory is self-consistent. In fact, I think that it has been demonstrated, mathematically rigourously, this self-consistency. ( I mean, extending my previous reasoning to this part in Einstein's original paper ).He didn't "prove that the proposition is satisfied"; he proved that assuming it to be true is consistent with the principle of relativity, by showing that if all light rays move at $c$ in one inertial frame, they also move at $c$ in any other inertial frame. That is sufficient to show that the two postulates are consistent with each other.

Yes, I see that and I think that I understand your point of view. However, if we go to the part of velocity composition, we could also check that "c + c = c" ( "+" is, in fact, the composition sign), but my point in this commentary is that it is not really necesary.

With the paragraph that I was discussing, I think that the same principle could be applied: namely, that because we have started by assuming that we can have a speed "c = constant", (indeed, a maximum speed - being this speed physically that of light, but this point is not so necesary to the theory from a theoretical-mathematical point of view), then we only need to assure that we obtain a transformation or mapping from (x,y,z,t) to new coordinates (x',y',z',t') and that we, indeed, don't need to check that c = constant, because the only fact of obtaining those coordinates, having started with the assumption of c = constant is just enough to assure that. (I am not sure if I am explaining right here).

 

[PeterDonis]

we only need to assure that we obtain a transformation or mapping from (x,y,z,t) to new coordinates (x',y',z',t') and that we, indeed, don't need to check that c = constant, because the only fact of obtaining those coordinates, having started with the assumption of c = constant is just enough to assure that

The assumption that c = constant in the moving system was not used to derive the transformation from the stationary system to the moving system; only the assumption that light moves at c in the stationary system was used. So you still need to check that light moves at c in the moving system. That's what Einstein is doing.