Lifting General Relativity to Observer Space

TL;DR

This paper explores the idea of using observer space as a fundamental concept in general relativity, demonstrating how spacetime can be reconstructed from it or treated as observer-dependent, with implications for gravity theories.

Contribution

It introduces a framework for observer space geometry using Cartan geometry, allowing spacetime reconstruction or a purely observer-dependent perspective in gravity.

Findings

Spacetime can be reconstructed as a quotient of observer space under certain conditions.

Observer space geometry can encode Einstein's equations.

Observer-dependent models of spacetime are possible when reconstruction is not feasible.

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.