Geometry and the Heathens

First Peter Woit of Not Even Wrong and now Lubos Motl have noted a recent exchange between Steven Weinberg and Friedrich Hehl in Physics Today. Weinberg had said, what's the big deal about torsion - it's just another tensor. Hehl disagreed. And Weinberg added: I still don't get it.

With his talent for seeing every fragment of reality as part of the the titanic struggle between good and evil, Lubos casts himself as the heroic defender of Weinberg - like Weinberg needed some defense, as almost everyone would call him as one of the half-dozen or fewer greatest living physicists. He then launches one of his patented invective streams against Hehl.

The thing is, Hehl has a point, and Weinberg chooses willfully to ignore it. The point is that all tensors are not equal, or at any rate, if they are, some tensors are more equal than others. To make an analogy with a different domain of physics, there are many vectors in fluid dynamics, but the fluid velocity has a priviledged and special role. Torsion is a candidate to be special in relativity because it has a geometric significance - of course that doesn't mean physics necessarily cares about that geometric significance.

As Lubos points out, Weinberg made a special isssue of taking an anti-geometric approach to General Relativity in his important textbook. In this regard he is very much an outlier in the history of physics. Descartes, Newton, Faraday, Hamilton, Einstein, and many others were famously geometric in their thinking. Peter Woit also adds this quote from Paul Ginsparg about Weinberg and geometry:

back to big steve w., when he wrote the gravitation book he was presumably just trying to get his own personal handle on it all by replacing any geometrical intuition with mechanial manipulation of tensor indices. but by the early 80’s he had effectively renounced this viewpoint in his work on kaluza-klein theories (i was there, and discussed all the harmonic analysis with him, so this isn’t conjecture…), one can look up his research papers from that period to see the change in viewpoint.

The comments to Peter's blog contain a number of important points, notably a typically lucid explanation by Sean Carroll of what he believes Weinberg's point was, and his comment to the effect that geometry is all very well, but physics "plays by its own rules." I don't know if Sean is right about what Weinberg was thinking, but even if not, Sean's point is still a good one.

There is a ironic twist in that in the comments to Lubos's article Bee take a whack at torsion and Lubos sets her straight by pointing out how torsion can or could be important.