Linking Covariant and Canonical General Relativity via Local Observers

Abstract

Hamiltonian gravity, relying on arbitrary choices of "space," can obscure spacetime symmetries. We present an alternative, manifestly spacetime covariant formulation that nonetheless distinguishes between "spatial" and "temporal" variables. The key is viewing dynamical fields from the perspective of a field of observers -- a unit timelike vector field that also transforms under local Lorentz transformations. On one hand, all fields are spacetime fields, covariant under spacetime symmeties. On the other, when the observer field is normal to a spatial foliation, the fields automatically fall into Hamiltonian form, recovering the Ashtekar formulation. We argue this provides a bridge between Ashtekar variables and covariant phase space methods. We also outline a framework where the 'space of observers' is fundamental, and spacetime geometry itself may be observer-dependent.

arXiv comment8 pages; Essay written for the 2012 Gravity Research Foundation Awards for Essays on Gravitation

References in corpus (11)

• Quantum Gravity at a Lifshitz Point
• The Dynamics of General Relativity
• Real Ashtekar Variables for Lorentzian Signature Space-times
• Reconstructing the Universe
• On smooth Cauchy hypersurfaces and Geroch's splitting theorem
• Einstein gravity as a 3D conformally invariant theory
• Relative locality: A deepening of the relativity principle
• Momentum Maps and Classical Relativistic Fields. Part I: Covariant Field Theory
• Lifting General Relativity to Observer Space
• Spontaneously broken Lorentz symmetry for Hamiltonian gravity
• The geometric role of symmetry breaking in gravity

Cited by in corpus (6)

• Hamilton geometry: Phase space geometry from modified dispersion relations
• Relativistic kinetic gases as direct sources of gravity
• Lifting General Relativity to Observer Space
• Contracted Lorentz invariance for gravity with a preferred foliation
• Geometrodynamics and Lorentz symmetry
• Spontaneous breaking of Lorentz symmetry for canonical gravity