As an aside to my earlier direct reply, these are foundations-of-physics issues. Most working physicists don't really care what are in the foundations, to the great annoyance of many philosophers. (Tim Maudlin comes instantly to mind.)
But annoyed philosophers don't change the results from numerical relativity.
"The very foundation of general covariant physics is the idea that the notion of a simultaneity surface all over the universe is devoid of physical meaning. Seems to me that it is better not to found hamiltonian mechanics on a notion devoid of physical significance."
Sure, nobody should disagree that such a surface is unphysical. However, 3+1 decompositions are at the bottom of successful results -- in particular you will be hard-pressed to find numerical relativity projects that don't split spacetime into spaces organized by some arbitrary time coordinate, where each space is a large enough simultaneity surface to trigger his fundamental objection. Yet we have good matches to real astrophysical results, for example Ransom, Archibald et. al, https://arxiv.org/abs/1401.0535 ( http://www.astron.nl/~archibald/video.html )
A complete theory from which CQG emerges in weak gravity is a good goal, and Rovelli is pursuing it, while trying to keep the good bits of modern physics (in particular that we've made practically all laws of physics generally covariant or at least relativistic, and that this is not just useful, but reflects something real about the universe).
More power to him. But his success in his project has practically no impact on the success of modern uses of General Relativity (in various formulations), or its complete accord with all available evidence accumulated so far. Seriously, there is no counter-evidence. It is mathematically complete. The only thing left for it is to study the mechanisms that generate the metric and the microscopic details of sources. (Maybe we can combine the two to study what metric large quantum systems (~ milligram rest masses) actually source when prepared in superpositions of position, for example. We think there's some problem of some sort there because semiclassical gravity -- relativistic quantum field theory on a "background" of standard General Relativity -- predicts something nonsensical; maybe Rovelli's work in the foundations will find a way to make a reliable non-nonsensical prediction there by the time we can prepare such a large quantum system. FWIW, progress on that front is being made: http://sciencenordic.com/can-large-objects-exist-quantum-sta... ).
Wow I am just floored by the thoroughness of your answers. Just want to let you know that someone enjoyed your effort though tbh I may never fully digest all you said here (and I do plan to read what you wrote a couple more times).
"Most working physicists don't really care what are in the foundations, to the great annoyance of many philosophers."
I was in physics grad school over twenty years ago. Your statement is accurate as far as I can tell. I did and do care about certain foundation issues. In particular I care whether free will is even fundamentally possible. However I never find any philosophical discussion capable of advancing my understanding "meaningfully", as in "to a physicist".
> I did and do care about certain foundation issues.
I meant more that while many foundational questions are interesting, the absence of conclusive answers don't stop most working physicists from making progress in their areas of professional interest.
For example:
> I care whether free will is even fundamentally possible
We act like it is. Similarly, we predict our past with greater fidelity than we predict our future, and we have a sense of "now".
Are these behaviours tied to something physically fundamental, or are they emergent? Nobody knows.
It'd be nice if we could have a long chat with a grey parrot or an octopus and find out whether it has a different view of past and future. Maybe we'd learn that it remembers in both directions equally well, and so doesn't really distinguish between them. Perhaps our future-directed memory was just an anatomical sacrifice like many others during the food-energy bottlenecks that dot humanity's evolution, and the cost of survival of our ancestors is us having a peculiar attitude towards time (and sports, like Wayne Gretzky's line about skating to where the puck will be) compared to less closely related organisms on our planet. ("Squawk, the future-memory part of your brain is missing, like the part of your brain that would deal with your vomeronasal organ. You will pass me a cracker now, while convincing yourself that you decided to. Squawk."
Since we are unable to have such interspecies conversations right now, we can only discuss this among ourselves, and are all stuck with essentially the same anatomy and no fossil early human brains to compare to ourselves.
I can understand the temptation to think that the very common idea of a fixed past and an undetermined future is a feature of the universe as a whole rather than something like impacted wisdom teeth or an inflamed appendix.
On the other hand, I think outright working (as in a likely self-funded job) on answering free-will-or-not is somewhere between pretty unproductive philosophizing and early symptoms of the sort of obsessions which overtook Linus Pauling, Brian Josephson, William Shockley, or Jack Parsons (among sadly too many others). Yes, that's a dickishly prejudicial thing to admit to, whether or not the attitude was determined by the information on a slice of the universe coupled to physical laws that let one recover the information everywhere else in the universe. As almost all of the content of any cross-universe slice is practically inaccessible to us, blaming eternalism or determinism for my "hot take" seems like a cheat.
In practice, though, none of this is likely to even occur to me consciously as I grind through different field equations, or try to match results from a static (time-independent) spacetime to a mostly-similar dynamical (varying over time) one, for example.
In https://arxiv.org/abs/gr-qc/0202079v1 Rovelli wrote something striking that I'm glad this discussion took me to:
"The very foundation of general covariant physics is the idea that the notion of a simultaneity surface all over the universe is devoid of physical meaning. Seems to me that it is better not to found hamiltonian mechanics on a notion devoid of physical significance."
Sure, nobody should disagree that such a surface is unphysical. However, 3+1 decompositions are at the bottom of successful results -- in particular you will be hard-pressed to find numerical relativity projects that don't split spacetime into spaces organized by some arbitrary time coordinate, where each space is a large enough simultaneity surface to trigger his fundamental objection. Yet we have good matches to real astrophysical results, for example Ransom, Archibald et. al, https://arxiv.org/abs/1401.0535 ( http://www.astron.nl/~archibald/video.html )
Few disagree that 3+1 decompositions pose difficulties; Alcubierre wrote a book cataloguing a bunch of them ( https://www.researchgate.net/publication/228588827_Introduct... )
And of course, quantizing the Hamiltonian formulation of General Relativity gives us Canonical Quantum Gravity (CQG), which exposed the non-renormalizability (by power set counting, anyway) of gravity. That's the starting point for Rovelli's objections. However, CQG is a perfectly fine effective theory ( http://www.preposterousuniverse.com/blog/2013/06/20/how-quan... and http://backreaction.blogspot.com/2014/11/the-paradigm-shift-... form a good one-two punch on this topic).
A complete theory from which CQG emerges in weak gravity is a good goal, and Rovelli is pursuing it, while trying to keep the good bits of modern physics (in particular that we've made practically all laws of physics generally covariant or at least relativistic, and that this is not just useful, but reflects something real about the universe).
More power to him. But his success in his project has practically no impact on the success of modern uses of General Relativity (in various formulations), or its complete accord with all available evidence accumulated so far. Seriously, there is no counter-evidence. It is mathematically complete. The only thing left for it is to study the mechanisms that generate the metric and the microscopic details of sources. (Maybe we can combine the two to study what metric large quantum systems (~ milligram rest masses) actually source when prepared in superpositions of position, for example. We think there's some problem of some sort there because semiclassical gravity -- relativistic quantum field theory on a "background" of standard General Relativity -- predicts something nonsensical; maybe Rovelli's work in the foundations will find a way to make a reliable non-nonsensical prediction there by the time we can prepare such a large quantum system. FWIW, progress on that front is being made: http://sciencenordic.com/can-large-objects-exist-quantum-sta... ).