Holographic Special Relativity

Abstract

We reinterpret special relativity, or more precisely its de Sitter deformation, in terms of 3d conformal geometry, as opposed to (3+1)d spacetime geometry. An inertial observer, usually described by a geodesic in spacetime, becomes instead a choice of ways to reverse the conformal compactification of a Euclidean vector space up to scale. The observer's "current time," usually given by a point along the geodesic, corresponds to the choice of scale in the decompactification. We also show how arbitrary conformal 3-geometries give rise to "observer space geometries," as defined in recent work, from which spacetime can be reconstructed under certain integrability conditions. We conjecture a relationship between this kind of "holographic relativity" and the "shape dynamics" proposal of Barbour and collaborators, in which conformal space takes the place of spacetime in general relativity. We also briefly survey related pictures of observer space, including the AdS analog and a representation related to twistor theory.