## Friday, August 10, 2012

### Tensors in relativity

A couple of Italians posted On the relativistic unification of electricity and magnetism, a few months ago.
On the relativistic unification of electricity and magnetism

The unification of electricity and magnetism achieved by special relativity has remained for decades a model of unification in theoretical physics. We discuss the way the four main relevant authors (Lorentz, Poincaré, Einstein, Minkowski) dealt with this issue and show that Poincaré's derivation of the transformation laws for the potentials and the fields was definitely less arbitrary than those of the other cited authors, in contrast with the fact that here, as in other cases, Poincaré's contribution to relativity was systematically belittled by authoritative German physicists in the first two decades. In the course of the historical analysis a number of questions which are of contemporary foundational interest concerning relativistic electromagnetism are are also examined.

The culmination of special relativity theory was the Lorentz covariance of the electromagnetic field tensor. This was proved by Poincare in his long 1905 paper and popularized by Minkowski in this famous 1908 paper.

This paper explains the details. In particular, it shows how Poincare proved it, how Minkowski got it from Poincare, how Minkowski was stingy about crediting Poincare, and how Einstein did not have the concept at all, or even anything on that level.

This was the biggest relativity breakthru in 1905. None of the Einstein books even mention it, except for my book.

The word tensor was coined in 1898 by Voigt. The first decent mathematical treatment of the concept (without using the word) was by the Italians Ricci and Levi-Civita in 1901. The above paper says:

Poincaré does not refer to the concept of a tensor and does notuse the absolute calculus introduced by Ricci-Curbastro following a remark, by Christoffel, and devel- oped together with his pupil, Levi-Civita ([45]). This is particularly intriguing, since in the preface of their joint work they stressed the value of the tensor formalism citing Poincaré himself, who had written that “a good notation has the same philosophical importance as a good classification in the natural sciences”. However there is only a linguistic divide between what Poincaré does in his paper and the formal recognition that the electric and magnetic fields have been proven to be ‘parts’ of a double tensor in 4-dimensional space. Notice that in contrast to Lorentz’s style, Poincaré does not use the vector formalism either (he always deals with components), so on the same ground one might suggest that he ignores vectors as well, which is obviously absurd.15 What is missing in his treatment, with respect to what Minkowski will do, is the use of 4-dimensional differential operators to reformulate the Maxwell equations.
I did not know that 1901 Ricci-Levi-Civita paper credited Poincare, and I did not know that Poincare had written anything about the concept.

Poincare's long 1905 paper does not define a tensor or say that the electromagnetic field is a tensor. What he does say is that the field is the exterior derivative of a spacetime 4-vector, whose components transform according to Lorentz transformation. This is essentially the same as saying that the field is a tensor.

Grossmann explained tensors to Einstein, and they used the term in their joint 1913 paper. Einstein still did not fully accept the concept, and wrote papers against tensors with his fallacious hole argument. Levi-Civita and Hilbert eventually convinced him of tensors, and his famous 1916 general relativity paper relied heavily on tensors.

Most historians do not agree with me about how relativity credit should be distributed. They are entitled to their opinions, of course. But if they credit Einstein for everything, look to see what they say about electromagnetic covariance. If they don't mention it, then they are not telling you the whole story. If they mention it, look to see where and when Poincare had the concept, and where and when Einstein had it. They usually just talk about Einstein having the concept of Lorentz transformations in 1905, but of course the transofrmations are named after Lorentz because he had them earlier.