I found that editing these pages was frustrating, as you can see in the discussion at Talk:History of special relativity. The articles explain who did what, but then deny credit to Lorentz and Poincare because some of their terminology was allegedly defective.
Here are examples of Poincare quotes:
"some day, no doubt, the aether will be thrown aside as useless." Science and hypothesis Historians use these quotes to argue that Poincare did not really believe in relativity. The argument is that "true time" must have meant time as measured by an observer who is stationary with respect to the aether, and that belief in the aether is contrary to relativity.
"The watches adjusted in that manner do not mark, therefore, the true time; they mark what one may call the local time, so that one of them goes slow on the other." The Principles of Mathematical Physics 
"since the fictitious electromagnetic mass depends upon this velocity, the total apparent mass, alone observable, must depend upon it, though the real mass does not depend upon it and may be constant." The New Mechanics 
They also argue that terms like "apparent mass" indicate that Poincare did not believe that relativistic effects were real. Poincare does say that the apparent mass is what is observable, but they say that the reference to "real mass" suggests that there is something not real about the (observed) apparent mass.
These arguments are purely terminological, and have no substance. Poincare's "apparent mass" and "real mass" are the same as what some textbooks call "relativistic mass" and "rest mass". There is not even a consensus today about the best terms for these quantities.
The argument about "true time" is even stranger. Poincare is plainly denying true time and the aether, not endorsing them. It is as if he said, "I went to the zoo and I saw a gorilla, not bigfoot." Such a statement would not imply a belief in bigfoot.
A NY Times column answers this question:
Q. When I read that “the universe is 13.7 billion years old,” I wonder: Don’t scientists use some more universal measurement than years, something not tied to the orbit of one tiny planet?The answer is true time, of course.
While some people claim that Lorentz's understanding of local time was deficient, Poincare clearly and correctly understood Lorentz's local time as the time observed on clocks. As Michel Janssen explains in chap. 3 of his thesis:
The passage I have in mind occurs in the section “The principle of relativity” in Poincaré’s famous 1904 lecture in St. Louis. Poincaré vividly describes the situation in ether theory around the turn of the century. While the dominant theories posit a stationary ether, the experiments aimed at detecting the earth’s presumed motion through this medium consistently give negative results. The task of explaining these experimental findings theoretically, Poincaré writes “was not easy, and if Lorentz has got through it, it is only by accumulating hypotheses” (Poincaré 1904, p. 99). Starting a new paragraph, he continues: “The most ingenious idea was that of local time” (ibid.). Poincaré proceeds to explain that if an observer in uniform motion through the ether synchronizes his clocks using what a modern reader immediately recognizes as the light signaling method from Einstein 1905a, these clocks will not read the true Newtonian time, but Lorentz’s local time. Poincaré does not indicate in any way that this interpretation is entirely his own and is not to be found in any of Lorentz’s writings up to this point. For Poincaré, the notion of local time clearly involves a physical assumption. Poincaré assumes that the local time is, in effect, the time registered by moving observers, which helps to account for the fact that such observers do not detect ether drift. After making this point (ibid, pp. 99–100), Poincaré starts his next paragraph saying: “Unhappily, that does not suffice, and complementary hypotheses are necessary; it is necessary to admit that bodies in motion undergo a uniform contraction in the sense of the motion” (ibid., p. 100). So, for Poincaré the notion of local time and the contraction hypothesis (to be discussed in detail in sections 3.2 and 3.3) have essentially the same status. They are both physical assumptions. This is a far cry from Lorentz’s understanding of the situation. He obviously looked upon the contraction hypothesis as a physical assumption, but local time for him is no more than a convenient purely mathematical auxiliary quantity.Einstein wrote in 1907 that local time was the crucial idea for relativity:
Surprisingly, however, it turned out that it was only necessary to grasp the concept of time sharply enough in order to get around the above difficulty. It required only the recognition that the auxiliary quantity introduced by H.A. Lorentz, and called by him "local time," can be defined sas simply "time." If one adheres to the indicated definition of time, then the basic equations of Lorentz's theory accord with the principle of relativity, provided only the above transformation equations are replaced by transformation equations that agree with the new time concept.Whether or not Lorentz had this idea, Poincare published it in 1900 and 1904. Einstein acts as if it is his own idea, but does not explicitly say so. My guess is that he did not dare claim that the idea was original to him, because too many people knew that it was Poincare's idea. But Einstein does not reference Poincare's papers either.
I don't even see why it makes any difference whether Lorentz and Poincare believed in the aether. The aether played no part in their theories. Lorentz refused to express assumptions about the nature of the aether. What used to be called Lorentz electron theory is now called Lorentz aether theory, but it hardly has anything to do with the aether. It is only in Kuhnian paradigm shift philosophy that terminology is so important, and not in science. None of these terminology differences had any difference to how observations were explained.
Wikipedia guidelines are to prefer secondary sources over primary sources. Most of the historians do indeed say that Poincare believed in the aether and in true time, based on the above quotes. So that is what Wikipedia says. Wikipedia is great but sometimes you have to read the quotes for yourself, and parse the article statements carefully.
For another view of Poincare contributions to relativity, read my my book.
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