Research Article

General Relativity and Gravitation

, Volume 44, Issue 12, pp 3103-3109

First online:

Linking covariant and canonical general relativity via local observers

  • Steffen GielenAffiliated withPerimeter Institute for Theoretical Physics
  • , Derek K. WiseAffiliated withInstitute for Quantum Gravity, Universität Erlangen-Nürnberg Email author 

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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.


Lorentz covariance Lorentz symmetry breaking Covariant real Ashtekar variables Observer space Relative locality