Poincare's Special Theory of Relativity Part 1: Discovery of the Lorentz Group

|Author: Jin Xiaofeng

(Department of Physics, Fudan University)

This article is selected from the 3rd issue of Physics in 2022

"Isn't this another clickbait? We've only heard of Einstein's special theory of relativity, and never heard of Poincare's special theory of relativity!" It's not surprising that people in and outside the physics community have the same doubts. Even I would have thought the same thing if I had seen such a title a year and a half ago. So, how did this 180-degree change happen?

The reason is simple. The following two books, especially the second one, completely changed my original idea: (1) O. Darrigol's Electrodynamics from Ampere to Einstein [1], (2) AA Logunov's Henri Poincare and the Theory of Relativity [2].

The former book and its references provide a relatively detailed introduction to the early historical contributions of Poincaré (Figure 1) to the special theory of relativity (before 1905); the latter book, especially Logunov’s introduction to Poincaré’s original article (Figure 2)[3], convincingly proves that Poincaré had fully established all the basic contents of the special theory of relativity in 1905.

Figure 1 Henri Poincaré (1854-1912)

It was precisely because of this transformation that the report “The Road to Special Relativity: A Belated Tribute to Henri Poincare” was delivered at the Faculty Lunch Salon of the Department of Physics at Fudan University on October 16, 2020. Because the “Kouxiang Science” platform broadcast live online [4], in addition to the on-site discussion, a lot of feedback was received after the meeting. Among them, a friend suggested whether the oral report could be converted into written text and formally published in a Chinese journal. However, due to the complexity and sensitivity of this issue, turning an oral report into written language means heavy and arduous work. After more than a year of hesitation, I finally made up my mind to spend some effort to write it down for my colleagues to criticize and correct. The abstract prepared for the oral report a year ago was directly transferred to the abstract of this article with the last sentence added.

Figure 2 Logunov’s book On the articles by Henri Poincaré “On the dynamics of the electron” is an introduction to Poincaré’s original article

In 1921, the 21-year-old Pauli said in the opening chapter of his famous book The Theory of Relativity, “Historical Background (Lorentz, Poincare, Einstein)”[5]:

"We now come to the contributions of Lorentz, Poincare, and Einstein, which contain the line of reasoning and the development which underlies the theory of relativity. Lorentz was the first in chronological terms, and he proved the covariance of the passive Maxwell equations under the following coordinate transformations:

(Author's note: here l is a function of v l (v)) ... We should emphasize that the principle of relativity was unfamiliar to Lorentz. Some deficiencies left by Lorentz's work were made up by Poincare. Poincare stated that the principle of relativity is universal and strictly valid. ... He further corrected the errors in Lorentz's formula in the presence of charge density and current, thus obtaining the complete covariance of the active Maxwell equations. ... In the end, it was Einstein who completed the basic theoretical construction of this new field... His 1905 article not only contains the main results of the previous two, but also shows a completely new and much deeper understanding of the whole problem. "

In 1955, to commemorate the 50th anniversary of the birth of the theory of relativity, Pauli, who was 55 years old at the time, wrote a short article entitled “Relativity and Science”[6], in which he said:

"The special theory of relativity is closely connected with the mathematical concept of the group, which, if only hinted at in the mechanics of Galileo and Newton, is now firmly established by experimental facts. ... The development of electrodynamics in the 19th century ended with a set of partial differential equations of Maxwell and Lorentz. It is obvious that this set of equations does not have the symmetry of the previous classical mechanical group ... The question now becomes: Is the claim that the laws of nature have group symmetries only an approximate property and must be abandoned, or is it that the classical mechanical group is perhaps only approximately valid and should be replaced by a more general group, valid for both classical mechanics and electrodynamics? The final decision leans towards the second option. This hypothesis can be There are two ways to achieve this. One is to examine, by purely mathematical methods, what is the most general group of transformations satisfied by the covariance of the Maxwell-Lorentz equations. This is the way of the mathematician Poincare. The other is to determine, by critical analysis, the physical assumptions that led to the special group adopted by Galileo and Newtonian mechanics. This is the way of Einstein. ... Both Einstein and Poincare based their results on the previous work of Lorentz, who had come quite close but not really reached the final result. From the independent and different paths of Einstein and Poincare, I realized the deeper meaning of the harmony between the mathematical method and the thought experiment (Gedankenexperimente) (based on the basic characteristics of physical experience). "

It is important to compare the different comments Pauli wrote about the same event at the age of 21 and 55 in the above two paragraphs, or at least worth noting. Obviously, he previously put Poincare on a par with Lorentz and highlighted Einstein's contribution; later he put Poincare on a par with Einstein, pointing out that both of them were based on Lorentz's previous work, but from two different paths, they reached the same destination of special relativity (Figure 3). Einstein's special theory of relativity is well known, while Poincare's special theory of relativity is little known. Even in the field of public cognition, people are left with an illusion that without Einstein, there would be no birth of special relativity. This is obviously inconsistent with historical facts. Therefore, it is necessary to let more people know about Poincare's special theory of relativity.

Figure 3 A group photo of Lorentz, Poincare and Einstein at the first Solvay Conference in 1911

A question that must be asked is: If Poincare really made such a great contribution to the special theory of relativity, how could his name disappear without a trace in so many modern textbooks and monographs on relativity? Obviously, in addition to introducing Poincare's special theory of relativity, the author must also answer this unavoidable question. The reasons for this century-level phenomenon seem to be complicated and intricate, but in fact they are so simple that they can be fully summarized in one sentence: In fact, it is not history that played a joke on Poincare, on the contrary, Poincare played a joke on history! Next, we will try our best to sort out the outline of the whole story. Through the vivid "jokes" played by Poincare, we will appreciate his unique way of doing research and his scientific philosophy, and admire his noble character.

For more details, please read:

https://mp.weixin.qq.com/s?__biz=MjM5NjYwNzM1Ng==&mid=2651692043&idx=1&sn=a176fd9ecbeb14c75b8005bdc2a0c785&ch ksm=bd1f98188a68110e5af06091aad9ef1065cbe2cf79ca297ddba803d70b7bc7d17e69d0933274&token=954467291&lang=zh_CN#rd

PDF reading: http://www.wuli.ac.cn/

To subscribe to Physics magazine, please contact editor Li at 82649029