I knew this was going to be fun, but Rovelli’s book is surpassing all my expectations. I’ve just finished Chapter 2 of his Quantum Gravity, which is devoted to (classical) General Relativity. It starts presenting the formalism, using Cartan/Ehresman connections over fiber bundles, so one needs to refresh some of the advanced topics covered in differential geometry, topology and group theory courses. At least, i had to. There are lots of books around, but if you have a good background, you may find Maximilian Kreuzer‘s lectures notes on Geometry, Topology and Physics (parts one and two) interesting (i definitely did). Rovelli presents the needed equations at a quick pace, mildly entertaining but hardly exciting… the real fun begins afterwards, in section 2.2 (aptly called The conceptual path to the theory) and up to the end of the chapter. Everyone studying General Relativity should read these sections. More than once (and, while you’re at it, take a look at the Kyoto University’s Genesis of General Relativity to keep on the fun). And if you think you already know a lot about GR, read them too, please. I have yet to digest them, but they are the best discussion on the conceptual underpinnings of Einstein’s theory i’ve read in many, many years. There you’ll find a discussion of the physical tenets of GR and their interpretation, including in what sense the theory makes (inertial) acceleration a relative phenomenon (revisiting Newton’s famous bucket experiment–see also here), and the meaning of general covariance (the paradigmatic gedanken here being Einstein’s hole argument) and the key distinction between active and passive transformations. The discussion is illuminating in many respects, and has given me food for thought for many weeks. And the best of all is that no sophisticated maths are needed to follow it; besides, there are excellent (and original) examples illustrating the arguments in the main text, based mostly in Newtonian physics. The chapter closes with notes on Mach’s principle, the philosophical interpretation of spacetime (relationalism vs. substantivism), Kretschmann’s objections against the physical content of general covariance (which i used to share, to some extent, in the past) and a detailed and delightful description of how GPS works.
A must read for anyone interested in General Relativity’s foundations.
I must say that it’s refreshing to find theoretical physicists interested in conceptual and philosophical issues in these calculate-and-shut-up times. I hope it’s a sign that they are a’changin’.
(I just noticed that the draft PDF of the book has disappeared from Rovelli’s webpage, but you can get it here.)