To be precise, here we mean by an observable a Dirac-observable, i.e. a phase space function which Poisson commutes with the constraints. In other words, it is a gauge invariant function, where the gauge transformations are roughly the diffeomorphisms of spacetime when the field equations are satisfied. Gauge and dynamics is thus intermingled, which can lead to some confusion.

In this post, we don’t want to repeat in all detail the theory of observables and gauge redundancy, but only point out some important points, which sometimes at least partially, are not receiving enough attention.

To begin with, one sometimes reads that there is no time or dynamics in GR because observables are commuting with the Hamiltonian, which is just a sum of constraints. Real dynamics is thus expected to happen only at infinity. This point of view clearly seems to miss the main issue that dynamics has to be extracted directly form the Dirac observables themselves. A very instructive example of this is given by the harmonic oscillator in parametrised form, as e.g. discussed here. In a version slightly adapted to the canonical framework, the two relevant observables here are functions of the position and velocity of the oscillator, as well as the time that passes. These functions take two values, corresponding to the initial values of position and velocity of the oscillator. Thus, knowing the value of both observables specifies the physical system, and solving the two observables for the position and velocity yields the solution to the equation of motion.

For GR, the situation is of course more complicated, but conceptually the same. One picks a physical field included in the model that one considers, e.g. a matter field, but maybe also some curvature scalar. This field serves as a time and one can now in principle compute the observables and solve them for the other fields. Since this process corresponds to solving the equations of motion, it is of course not possible to carry out in detail, but it shows that conceptually there is no problem with local dynamics in GR.

It thus transpires that observables in GR should be relations between fields and one often reads about "relational observables". One might wonder whether this class of observables is generic, i.e. whether all observables are of this type. Clearly, depending on the boundary conditions one imposes, one might have additional boundary observables, as e.g. the ADM charges. Locally however it is clear that the relational observables are generic.

To see this, one can look more closely in the construction of them. We first pick a set of commuting clocks and rods. A preferred subalgebra of phase space functions is now given by those which Poisson commute with the clocks and rods. To any element of this subalgebra, we can associate a Dirac-observable which corresponds to this function at a certain given value of the clock and rod fields, inheriting the algebraic relations from the preferred subalgebra. It follows that at least locally, the set of Dirac observables in GR can be constructed relationally. Moreover, there is an explicit formula for those observables, given by an infinite series of nested Poisson brackets, and their algebra is known, see for example here, here, and here.

Still, some caveats about observables in GR should be mentioned. No clock and rod functions are known which are generally valid, e.g. everywhere in a generic spacetime Rather, clocks and rods should be carefully chosen depending on the applications that one has in mind. Also, there is an issue about a possible non-differentiability and chaos, see here.

To conclude, at a conceptual level, a lot is known about observables in GR and the no-observables picture that one often finds painted does not reflect the current state of the art. Especially for quantum gravity, the classical construction of relational observables is very useful since we have full control over their algebra if they are suitably constructed and we thus can quantise directly the reduced phase space. Numerous examples of this have been discussed in the literature, see e.g. the references in here.

To see this, one can look more closely in the construction of them. We first pick a set of commuting clocks and rods. A preferred subalgebra of phase space functions is now given by those which Poisson commute with the clocks and rods. To any element of this subalgebra, we can associate a Dirac-observable which corresponds to this function at a certain given value of the clock and rod fields, inheriting the algebraic relations from the preferred subalgebra. It follows that at least locally, the set of Dirac observables in GR can be constructed relationally. Moreover, there is an explicit formula for those observables, given by an infinite series of nested Poisson brackets, and their algebra is known, see for example here, here, and here.

Still, some caveats about observables in GR should be mentioned. No clock and rod functions are known which are generally valid, e.g. everywhere in a generic spacetime Rather, clocks and rods should be carefully chosen depending on the applications that one has in mind. Also, there is an issue about a possible non-differentiability and chaos, see here.

To conclude, at a conceptual level, a lot is known about observables in GR and the no-observables picture that one often finds painted does not reflect the current state of the art. Especially for quantum gravity, the classical construction of relational observables is very useful since we have full control over their algebra if they are suitably constructed and we thus can quantise directly the reduced phase space. Numerous examples of this have been discussed in the literature, see e.g. the references in here.

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