Advances in Theoretical and Mathematical Physics

Volume 19 (2015)

Number 5

Holographic special relativity

Pages: 1017 – 1041

DOI: http://dx.doi.org/10.4310/ATMP.2015.v19.n5.a3

Author

Derek K. Wise (Department Mathematik, Universität Erlangen–Nürnberg, Erlangen, Germany; and Concordia University, Saint Paul, Minnesota, U.S.A.)

Abstract

We reinterpret special relativity, or more precisely its de Sitter deformation, in terms of 3d conformal geometry, as opposed to $(3+1)\textrm{d}$ 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. This inertial 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 3d conformal 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.

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Published 31 March 2016