Lifting General Relativity to Observer Space

Abstract

The `observer space' of a Lorentzian spacetime is the space of future-timelike unit tangent vectors. Using Cartan geometry, we first study the structure a given spacetime induces on its observer space, then use this to define abstract observer space geometries for which no underlying spacetime is assumed. We propose taking observer space as fundamental in general relativity, and prove integrability conditions under which spacetime can be reconstructed as a quotient of observer space. Additional field equations on observer space then descend to Einstein's equations on the reconstructed spacetime. We also consider the case where no such reconstruction is possible, and spacetime becomes an observer-dependent, relative concept. Finally, we discuss applications of observer space, including a geometric link between covariant and canonical approaches to gravity.

35 pages; v2: substantial improvements of sections 2.4 and 5, small changes and corrections throughout the rest of the paper; v3: minor changes to match published version

References in corpus (10)

• Quantum Gravity at a Lifshitz Point
• Very Special Relativity
• Einstein gravity as a 3D conformally invariant theory
• Symmetric Space Cartan Connections and Gravity in Three and Four Dimensions
• Teleparallel Gravity as a Higher Gauge Theory
• Deformed General Relativity and Torsion
• Spontaneously broken Lorentz symmetry for Hamiltonian gravity
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• Holographic Special Relativity

Cited by in corpus (22)

• Hamilton geometry: Phase space geometry from modified dispersion relations
• Relativistic kinetic gases as direct sources of gravity
• Foundations of Finsler spacetimes from the Observers' Viewpoint
• Spacetime and observer space symmetries in the language of Cartan geometry
• Extensions of Lorentzian spacetime geometry: From Finsler to Cartan and vice versa
• An axiomatic purification of gravity
• The General Linear Cartan Khronon
• An operational approach to spacetime symmetries: Lorentz transformations from quantum communication
• On the Mathematics of Coframe Formalism and Einstein-Cartan Theory -- A Brief Review
• Non-metric fluid dynamics and cosmology on Finsler spacetimes
• Action and Observer dependence in Euclidean quantum gravity
• Scale-invariant cosmology in de Sitter gauge theory
• Consistent first order action functional for gauge theories
• Linking Covariant and Canonical General Relativity via Local Observers
• Voigt transformations in retrospect: missed opportunities?
• Holographic Special Relativity
• Normal coordinates based on curved tangent space
• Contracted Lorentz invariance for gravity with a preferred foliation
• Geometrodynamics and Lorentz symmetry
• Spontaneous breaking of Lorentz symmetry for canonical gravity
• Hidden symmetries of the gravitational contact structure of the classical phase space of general relativistic test particle
• A local quantum Mach principle and the metricity of spacetime