Abstract
The notion of "reference frame" is a central theoretical construct for interpreting the physical implications of spacetime diffeomorphism invariance in General Relativity. However, the alternative formulation of classical General Relativity known as Shape Dynamics suggest that a subset of spacetime diffeomorphisms - namely hypersurface deformations - are, in a certain sense, dual to spatial conformal (or Weyl) invariance. Moreover, holographic gauge/gravity dualities suggest that bulk spacetime diffeomorphism invariance can be replaced by the properties of boundary CFTs. How can these new frameworks be compatible with the traditional notion of reference frame so fundamental to our interpretation of General Relativity? In this paper, we address this question by investigating the classical case of maximally symmetric spacetimes with a positive cosmological constant. We find that it is possible to define a notion of "Shape Observer" that represents a conformal reference frame that is dual to the notion of inertial reference frame in spacetime. We then provide a precise dictionary relating the two notions. These Shape Observers are holographic in the sense that they are defined on the asymptotic conformal boundaries of spacetime but know about bulk physics. This leads to a first principles derivation of an exact classical holographic correspondence that can easily be generalized to more complicated situations and may lead to insights regarding the interpretation of the conformal invariance manifest in Shape Dynamics.
Highlights
In this work, we will explicitly compute the classical dynamics of a conformally invariant theory of reparametrization invariant point particles propagating on a Euclidean plane
We have presented a model, inspired by the duality between Shape Dynamics and General Relativity, where a bulk theory of free particles in de Sitter (dS) spacetime can be mapped to a reparametrization and conformally invariant theory on the conformal boundaries of dS
The bulk dS ismoetries map to conformal symmetries in the dual theory. This map is interpreted as a correspondence between bulk inertial reference frames and boundary conformal reference frames, who only see the scale-invariant information about the instantaneous shape of the particle system
Summary
We will explicitly compute the classical dynamics of a conformally invariant theory of reparametrization invariant point particles propagating on a Euclidean plane. The dynamics will be obtained by constructing the Hamilton–Jacobi functional of this theory holographically using the action for free inertial observers in a bulk de Sitter (dS) spacetime and relies of the special properties of dS near its conformal boundary. We will show how this construction leads to a notion of Shape Observer in Euclidean space that is dual to the conventional notion of inertial observer in spacetime. We speculate how this new notion shape observer might be used to understand the relationship between two formulations of classical gravity: the conventional description in terms of General Relativity and a new conformal description in terms of Shape Dynamics
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