#Covariance or not covariance, that is the question

Covariance in Physics - by Alberto Vecchiato

The Principle of Covariance is "traditionally" a source of controversy in Physics. Some physicists believe that this principle is trivial and basically insignificant, because nothing useful can be deduced from it. Others (Albert Einstein, among others) claim that covariance is a fundamental ingredient to build a physics theory, and that general covariance was the greatest achievement of General Relativity.

Some years ago, this controversy triggered my curiosity and, among others, it was one of the reasons that pushed me to write a book about Gravity Field Theories.

In this book I gave a great deal of importance to the discussion of how the so-called "fundamental principles" can be used to build a physics theory.

I think that the question is also tightly connected with another famous debate, namely Eugene Wigner's headache with "The Unreasonable Effectiveness of Mathematics in the Natural Sciences".

Certainly, I do neither claim to be the first nor a particularly authoritative source on this discussion, but the requirement of "covariance" is not specific of General Relativity, nor it is necessarily connected with the somewhat arcane and "modern" mathematical theory of differential geometry. Rather, I would say that the roots of this debate can be found in a much more remote past than that of Einstein, Kretschmann, Synge, or Wigner. Because of the inextricable connection between mathematics and physics, its origin might be even traced back to several thousands of years ago, in the way humans started to use and manipulate numbers for practical purposes. About 2500 years ago, this gave rise to Euclidean geometry, the first "real" mathematical and physical theory in the history.

It seems to me that, when seen from such a "constructivist perspective", things take a more familiar look; more reassuring and less extraordinary, or "magical", so to speak, so, I hope you won't judge me too badly if I conclude this few lines by citing my book:

"[...] a covariance requirement is just a more formal way to ask that the mathematical model of a physics theory does not change with respect to some set of transformations between reference systems. This in practice ensures that any observer (represented by a specific reference system) would be able to deduce the same theory by observing the same physical phenomena, which is certainly a reasonable and very basic requisite for any useful description of the physical world."