Cover of the book
Cover of the book
This tale, which is arguably one of the best on the success of mathematics as a language to understand nature, features four major characters. Out of these four, three are mathematicians; Gregorio Ricci-Curbastro (Italian), Tullio Levi-Civita (Italian), Marcel Grossmann (Swiss) and a physicist; Albert Einstein (German-born).
This is a story of two desperate researchers; Ricci and Einstein. Ricci wanted his mathematical discovery, the absolute differential calculus, to see the light of application in physics and be accepted as an important work by mathematicians. Einstein, on the other hand, was looking for a mathematical formalism for his general theory of relativity. They both had the tenacity of the bulldog on their respective fields. Ricci nurtured and developed his mathematics, detail by detail, for more than a decade. Einstein did the same to his theory; philosophically advancing but with attempts and failure to model it mathematically. It was when they crossed path, the fundamental perception of the nature would never be the same again.
This book starts with the vivid portrait of the life of Ricci in Italy which was virtually spent in two towns Lugo and Padua, his hometown and university where he taught respectively. He devoted most of his active career in developing the absolute differential calculus which aimed at representing the equations in such a way that same equation would be valid for any choice of coordinate system. This independence in the choice of the coordinate system is what suggests the word 'absolute' in the absolute differential calculus. In modern terms this calculus is called tensor calculus.
In his working life, Ricci did not enjoy good amount of respect and acknowledgement for his discovery of absolute differential calculus. It was mainly because the mathematics looked promising, but lacked application which could demonstrate its effectiveness. Obviously Ricci applied his calculus on some problems of physics but it did not demonstrate much insight as compared to the mathematical methods that were already in use.
His absolute differential calculus was in dark, but this was soon to be changed; and this is where Levi-Civita, one of his brightest students, later collaborator, and life long friend, comes into the play. In 1900 they published a joint review paper titled Methods of the Absolute Differential Calculus and Their Applications which presented tutorial of absolute differential calculus with all necessary mathematical foundations lucidly explained. Applications to Mechanics and Applications to Physics were included as final two chapters in the hope that the theory be applied to wide range of more important problems by other researchers.
" Grossmann, you must help me or else I'll go crazy." - Einstein to Grossmann, summer of 1912
In 1912 Einstein went to Zürich as a professor of theoretical physics at the ETH Zürich under the recommendation of his old friend and a brilliant mathematician specializing on non-Euclidian geometry; Marcel Grossmann. Einstein presented the problem he had been grappling for last five years to Grossmann. The problem in question was to find a suitable mathematical construction for his general theory of relativity. Grossmann suggested that a differential calculus that is independent to the coordinate system is something Einstein should go for. Grossmann knew that it was the very same mathematics presented in the 1900 paper by Ricci and Levi-Civita.
"Taking Christoffel's results as their starting point, Ricci and Levi-Civita developed their methods of the absolute differential calculus - i.e, a differential calculus that is independent of the coordinate system - which permit our giving an invariant form to the differential equations of mathematical physics." - Marcel Grossmann on Outline of a Generalized Theory of Relativity and of a Theory of Gravitation (1913)
There is no doubt that Einstein was a first rate physicist and great philosopher but when it came to mathematics, he was but a humble man. It was clear that he needed help with absolute differential calculus which Grossmann was happy to offer. They published a joint paper Outline of a Generalized Theory of Relativity and of a Theory of Gravitation in 1913. In this paper, Einstein expounded the physical aspects of his theory and Grossmann did same in mathematical language using the tools of absolute differential calculus. (The paper been translated to English and is present in The Collected Papers of Albert Einstein. The clarity in the line of reasoning of Grossmann as he uses the mathematics to explain gravity is evident to the readers with mathematical inclination.) This paper, as it is, had issues with some important physical considerations which would be solved by Einstein in November of 1915. However, the complete paper on theory by Einstein in 1916 has the same mathematical architecture which Grossmann drafted in 1913.
"I admire the elegance of your method of computation; it must be nice to ride through these fields upon the horse of true mathematics while the like of us have to make our way laboriously on foot." - Einstein to Levi-Civita
Einstein left Zürich and moved to Berlin in 1914. From 1912 to 1914, he had equipped himself with the necessary mathematical tools of absolute differential calculus with the help of Grossmann. Levi-Civita got in touch with Einstein with regards to what he saw as a fault in the use of absolute differential calculus in his paper of November 1914 (which Einstein wrote after he moved to Berlin). They had lively and respectful correspondence for six month till the fall of 1915.
In the November of 1915 Einstein gave set of four lectures to the members of the Prussian Academy in Berlin. In the course of these lectures, for four weeks, Einstein wrestled with a succession of mathematical presentations, corrections, and updates, before presenting his final version of the gravitational field equations on November 25th... and the rest, as they say, is history.
With the success of general theory of relativity, Ricci witnessed the grandeur of his mathematical discovery (a pleasure not many mathematicians witness in their lifetime). He died at the age of 73 in 1925.
Levi-Civita went on to contribute on various problems of general relativity and popularized the field in Italy. The ending, however, was not so happy for him. In spite of his mathematical fame, he could not survive the Italian Fascist government due to his Jewish heritage. His professorship and of his membership to all scientific societies were stripped. Tullio Levi-Civita, one of the best mathematicians of his time, died in 1941, aged 68 in his home country, as if he were nobody.
I loved the narrative for its graphic portrayal of Italian academics (mathematics) and its rigidity during 1800s and the struggles Ricci went through to establish himself as competent mathematician. There are picturesque accounts of how the university system operated at those times and the politics involved. Promotions in academics did not come easily to Ricci which is perhaps best portrayed by following excerpt from the book:
"In effect, the ministry of public instruction, which typically deferred to its Consiglio superiore - an advisory board known as the high council, on such matters [promotions in academics]- could, and did, play dice with the fortunes of Italian academicians, a situation that would profoundly affect Ricci's career."
However, as I went through the book, I missed the amount of detailed mathematical exploration on the works of Ricci and Levi-Civita which I had originally expected. All this changed when I reached APPENDIX-A which came as a beautiful surprise to me. The appendix provides good amount of mathematical insights to absolute differential calculus (with reference to the 1900 paper) and its use in general theory of relativity. This book, in my view, is equally accessible to both general readers (skipping APPENDIX-A) and those who are into mathematics.
Personally, the book would have been more exciting if the line of mathematical reasoning had extended from works of Gauss to Riemann to Christoffel to Ricci. It sure does mentions the link but I would love to read more about the work of first three mathematicians and how they were useful to the latter. But again, the book is not titled On the Complete History of Tensor Calculus, so maybe I should limit my expectations.
I got interested in this book as the source of historical background on differential geometry. I enjoyed it a lot and would certainly recommend the book to interested readers.
[Thank you Sriza for editing this review.]