Observables for general relativity related to geometry

Abstract

We present a new scheme of defining invariant observables for general relativistic systems. The scheme is based on the introduction of an observer which endowes the construction with a straightforward physical interpretation. The observables are invariant with respect to spatial diffeomorphisms which preserve the observer. The limited residual spatial gauge freedom is studied and fully understood. A full canonical analysis of the observables is presented: we analyze their variations, Poisson algebra and discuss their dynamics. Lastly, the observables are used to solve the vector constraint, which triggers a possible considerable reduction of the degrees of freedom of general relativistic theories.

Observer’s observables. Residual diffeomorphisms

“…Note, that those conditions are satisfied not only at σ 0 but in the entire neighborhood of σ 0 in which the spherical adapted coordinates are well-defined. 2 In [19] this second constituent of an observer has been described as a fixed tangent frame e 0 I ∈ Tσ 0 Σ, from which a metric dependent frame is produced using a Gram-Schmidt orthonormalization process with respect to the given metric q. For our current purposes, however, we can leave the details of the choice of the orthonormal frame unspecified, as the results we obtain in what follows do not depend on that choice (for a discussion of this issue see section 4).…”

“…A recently proposed family of observables are the observer's observables [19]. They require an introduction of an object referred to as the observer (possessing a location and a way of distinguishing directions), but offer a treatment of the spatial diffeomorphism freedom.…”

“…The time reparametrizations are dealt with separately, e.g., with the use of an irrotational dust [6]. In [19] it has been shown that an important virtue of this proposal is that their variations and hence Poisson brackets are not only under explicit control, but their Poisson algebra is particularly simple. Namely, there is a canonical, local subalgebra of observables which can be used to parametrize the whole gauge invariant content of the theory [20].…”

“…We start by recalling the steps of the definition of observer's observables as they were defined in [19] crucial for the current purposes. We will work in a canonical formulation of a generally relativistic theory, i.e., in a phase space Γ with points labeled by the spatial metric tensor q of a 3-dimensional spatial slice Σ of the spacetime M, its canonically conjugate momentum p being a tensor density and possibly also some matter fields.…”

Symmetries of Trautman retarded radial coordinates

“…This is a geometric version of the idea of the relational observables [5]. In the previous papers we considered in that context radial coordinates in 3d Euclidean space [6], and Gauss radial coordinates in 4d spacetime, respectively [7]. Similar construction was considered for asymptotically AdS spacetimes [8].…”