What is time according to Connes and Rovelli - Part 3
Connes and Rovelli continue:
One can still recover weaker notions of physical time: in GR, for instance, on any given solution of the Einstein equations one can distinguish timelike from spacelike directions and define proper time along timelike world lines. This notion of time is weaker in the sense that the full dynamics of the theory cannot be formulated as evolution in such a time.
For some reason they are restricting themselves to general relativity. Don’t they know about other generally covariant theories? Theories with torsion, Kaluza-Klein theories, generally covariant theories based on gauging of the Galilei group (K\"unzle, Duval)? And what about
"full dynamics of the theory cannot be formulated as evolution in such a time. "
When we are dealing with field theories, rather than theories of particles, the full dynamics is based on partial differential equations, not on "time". Now we have a footnote:
Footnote: Of course one should avoid the unfortunate and common confusion between a dynamical theory on a given curved geometry with the dynamical theory of the geometry, which is what full GR is about, and what we are concerned with here.
GR is not the dynamical theory of the geometry - as the authors would like it to have. It presupposes already certain geometry - it presupposes a manifold structure, it presupposes a particular gauge group - O(3,1), it presupposes a particular way of choosing the variables etc. There are many variations of general relativity, some of them with two metrics rather than one, some with no metric at all - they start with a general affine connection. The term "given curved geometry" is fuzzy, is undefined. What kind of geometry? The devil is always in the details.
In particular, notice that this notion of time is state dependent.
State dependent? State of what? What is state here? A particular geometry? What kind of geometry? A particular solution of Einstein’s field euqations? Which equations? Is state defined up to a diffeomorphism, or not? Are two geometries related by a diffeomorphism considered as defining the same state (as Souriau would like). The authors are not clear.
Furthermore, this weaker notion of time is lost as soon as one tries to include either thermodynamics or quantum mechanics into the physical picture, because, in the presence of thermal or quantum “superpositions" of geometries, the spacetime causal structure is lost.
Of course - if we consider families of spacetime causal structures - then one spacetime causal structure is lost. What a surprise! But things are not that easy and here again the authors are precise enough. The space-time metric, the dynamical variable of the standard, orthodox, general relativity can be naturally split into two parts: causal (or conformal) structure, and the lenght scale (or, equivalently, volume element). It is then possible to vary geometry by varying only the scale, but not the conformal structure. We still can have superpositions of geometries, while the causal structure is constant. Thus the above statement is another not precise enough.
This embarrassing situation of not knowing “what is time" in the context of quantum gravity has generated the debated issue of time of quantum gravity.
The really embarassing situation is in not understanding what quantum theory is about, and whether we really need to "quantize gravity." By the pure force of inertia we believe that once quantizing electromagnetism prooved to be largely successfull (and partly a disaster due to divergencies and inconsistencies), then we should quantize everything in view - just for fun of doing it, as we do not know what to do otherwise.
As emphasized in C. Rovelli, Class. and Quant. Grav. 10, 1549 (1993), the very same problem appears already at the level of the classical statistical mechanics of gravity, namely as soon as we take into account the thermal fluctuations of the gravitational field.
But what are the "thermal fluctuations" prior to any "time". What fluctuates, and how long it takes for something to fluctuate? What is "temperature" prior to time and prior to motion? I think the authors are using the popular jargon without giving much thought to it. If it would be Igor and Grichka rathen than Connes and Rovelli - they would be already attacted viciously for using the term "fluactuations":
Then we have a footnote:
Footnote: The remark of the previous note applies here as well. Thermodynamics in the context of dynamical theories on a given curved geometry is well understood - see R.C. Tolman, Relativity, Thermodynamics, and Cosmology, Clarendon Press, Oxford, 1934.
Well, Tolman is a good but rather old book. But even Tolman does not claim that "Thermodynamics in the context of dynamical theories on a given curved geometry is well understood." Tolman (ibid. Chapter V.55) writes, for example:
" The construction of a fundamentally satisfactory relativistic statistical mechanics would be in any case a complicated business and at present a somewhat dubious undertaking. Nevertheless, some progress in this direction has already been made using classical rather than quantum-mechanical statistics as a starting point.."
Now compare: on one hand we have Connes and Rovelli writing that according to Tolman the business is well understood, and on the other hand we have Tolman telling us that the business is complicated and dubious and that some progress have been made. Of course thermodynamics is not exactly the same as statistical mechanics. But here are getting into an even bigger trouble, because to derive thermodynamics from statistical mechanics is another serious issue - not solved satisfactorily till now.
J. M. Souriau in his "Structure of dynamical systems" uses a more precise language when he writes:
"The barycentric decomposition does not exist in relativistic mechanics. However, by using theorem (16.224) and the fact that the symplectic cohomology of the Poincare group is zero (13.73), one can show that an isolated system has no natural equilibria - except in the case, devoid of any physical interpretation, where the Poincare group leaves invariant every motion of the system."
And then he adds:
Footnote: "Of course one might call such a system "vacuum" and consider that it is a natural equilibrium for every Z. This remark ceases to be purely linguistic in quantum mechanics."
Back to Connes and Rovelli:
Thus, a basic open problem is to understand how the physical time flow that characterizes the world in which we live may emerge from the fundamental “timeless" general covariant quantum field theory - see A. Ashtekar, Lectures at Les Houches 1992 Summer School, to appear.

