Observables need not be diffeomorphism invariant in Classical and Quantum Gravity

The problem of observables in classical and quantum gravity is a long-standing one. It is sometimes argued that observable quantities should be diffeomorphsm invariant, following the philosophy of Dirac. We argue that diffeomorphism invariance in classical and quantum gravity is not primarily a case of gauge invariance but rather an example of spontaneous symmetry breaking of the diffeomorphism group. As a consequence, observables fall into two classes, Dirac observables which are invariant under the full diffeomorphism group and more general observables which take on different (expectation) values in the different phases of the broken (diffeomorphims) symmetry group. To this latter class belong for example scalar functions of the metric tensor.