That's the spirit!

Good. Did I mention that my Ph.D. is in math? I learned everything I know about gtr from reading books/papers and making computations/cogitations, so it -is- possible to learn gtr entirely outside the classroom. (But I don't think I could have done it had I lacked the benefit of a fine formal training in mathematics.) Funnily enough, my diss was on generalized Penrose tilings; I sometimes wonder if Penrose himself might not be the only other entity which knows about both Penrose tilings and NP formalism.

But note well: I've never been able to muster the intestinal fortitude to seriously study quantum mechanics, much less QFT, even though I possess the mathematical prerequisites like operator theory and Hilbert spaces, because, you know, it Just Doesn't Make Sense. As a non-physicist I figure I am permitted such luxuries!--- I obviously couldn't recommend this option to physics students.

Yes, indeed. Even in the context of classical gravitation, consider a binary star system. It produces gravitational radiation, and using the weak-field approximation we can compute the energy, momentum, polarization &c. of this radiation, but if we ask linearized theory to tell us where exactly the radiation is -created-, as it were, we ask in vain!

(I know hundreds of exact solutions describing propagating gravitational waves and beams of gravitational radiation, some of which I have found myself, but I know of no convincingly physically realistic exact solutions describing production of same. Using Bondi formalism one can write down good approximations to isolated sources of radiation, but again I don't see how to use this to answer such questions.)

when you throw in the slow motion condition: sources of the field are varying slowly, test particles are moving slowly wrt sources. Weak field also covers stuff like linearized gravitational radiation, first order approximations to isolated massive objects, and so on.

Agreed. Certainly if we assume that Nature adores the quantum, then gtr cannot be telling us "how gravitation really works" for the simple reason that it is not a quantum theory. But I doubt we can really say that QED allows us to understand "how EM really works", however.

Did you see the recent arXiv eprint by Luminet on the supposed death of science? He argues that we are near, not the death of science, but the birth of science, and furthermore, we should expect to always be near the birth of science. My sentiments exactly.

I guess everyone here knows that a paradigm seems to be emerging in which viable gravitation theories must be constrained by thermodynamical considerations, and furthermore, many notions which were originally discovered in the context of gtr and thus thought of as "gravitational phenomena" may one day be seen as thermodynamical phenomena.

BTW, my training was in ergodic theory so I have the mathematical prerequisites for that discussion also, although I can hardly claim to be a true expert in the huge discipline of ergodic theory.

Dunno, but I suppose I could ask John Baez if he has any comment. Maybe we should try to refine the question first...

[EDIT: oh, I think I see, you meant: can we take some simple example of quantum mechanical Lagrangian (Legendre-Young dual of the Hamiltonian) and add a Newtonian gravitational potential?]

BTW, did I misunderstand what kind of correspondence you are looking for? Are you looking for a quantum-classical gravitation dictionary?

Sorry, I'm lost. Can you try to reexpress that?

Symmetry: these days I am thinking about "geometric figures" in finite Kleinian geometries. A rambling general comment, possibly irrelevant here: Cartanian geometry, which has recently undergone a modest revival in physics, has been described as the common generalization of Kleinian geometry and Riemannian geometry. Over at the n-category cafe, Urs Schreiber, John Baez, and other luminaries are discussing things like groupoidification and n-categorification of mathematics and thus, of physics. Exciting times--- and maybe a bit brain-baffling for those who don't yet appreciate the mental labor saving virtues of the categorical approach to mathematics. One aspect of this area which I am particularly interested in right now is how the ideas of Galois, Klein, Joyal, Fraisse, Cameron, and others intersect in first order logic, enumerative combinatorics via "structors" (or "combinatorial species"), graph theory (esp. "random graphs"), group actions, topology, information theory, and discrete dynamical systems. For some hints of a the idea behind structors, see the eprint by Baez and Dolan, "From Finite Sets to Feynman Diagrams", http://arxiv.org/abs/math/0004133, which also has hints of a quantum version. Kleinian geometry is also part of the background for the lovely theory of reflection groups which does much to explain the special role played in mathematical physics by "finite simple groups of Lie type", and John has posted some lovely material relating a computation of the cohomology of the Lie group associated to a root lattice to computations using q-deformed binomial numbers. Which brings us right to finite projective spaces and closes the circle with finite Kleinian geometries.

Evidently I am somewhat reluctant to start thinking about gtr right now, huh?!

(In another thread, someone took umbrage to my remark that I've spent some of this week "slumming" at BAUT while waiting upon computations to terminate. If nothing else, I just gave a bit more indication of what those computations involve.)