Tetrad Gravity: II) Dirac's Observables

Luca Lusanna; Stefano Russo

paper

Tetrad Gravity: II) Dirac's Observables

arXiv:gr-qc/9807074

Abstract

After a study of the Hamiltonian group of gauge transformations of canonical tetrad gravity on globally hyperbolic, asymptotically flat at spatial infinity, spacetimes with Cauchy hypersurfaces $Σ_{τ}$ diffeomorphic to $R^{3}, w e f in d t h e d e p e n d e n ceo f t h eco t r ia d so n$ Σ_τ $an d o f t h e i r m o m e n t a o n t h es i x p a r am e t er s a ssoc ia t e d w i t h r o t a t i o n s an d s p a ce d i f f eo m or p hi s m s . T h ec h o i ceo f 3 - coor d ina t eso n$ Σ_τ $i se q u i v a l e n tt o t h e p a r am e t r i z a t i o n o f t h eco t r ia d s w i t h t h e l a s tt h r ee d e g r eeso f f r ee d o min d i v i d u a t in g t h e 3 - g eo m e t r i es . T h e S hanm ug a d ha s an c an o ni c a l t r an s f or ma t i o n, cor r es p o n d in g t o t h ec h o i ceo f 3 - or t h o g o na l coor d ina t eso n$ Σ_τ $an d a d a pt e d t o 13 o f t h e 14 f i r s t c l a ssco n s t r ain t s, an d t h e in t er p r e t a t i o n o f t h e g a ug e t r an s f or ma t i o n s a r e g i v e n . T h e g a ug e in t er p r e t a t i o n o f t e t r a d g r a v i t y ba se d o n co n s t r ain tt h eor y im pl i es t ha t a " H ami l t o niank in e ma t i c a l g r a v i t a t i o na l f i e l d " i s an e q u i v a l e n cec l a sso f p se u d o - R i e mannian s p a ce t im es m o d u l o t h eH ami l t o nian g r o u p o f g a ug e t r an s f or ma t i o n s : i t in c l u d es a co n f or ma l 3 - g eo m e t r y an d a l l t h e d i f f er e n t 4 - g eo m e t r i es (s t an d a r dd e f ini t i o n o f ak in e ma t i c a l g r a v i t a t i o na l f i e l d,$ Riem M^4/Diff M^4$) connected to it by the gauge transformations. A "Hamiltonian Einstein or dynamical gravitational field" is a kinematical one which satisfies the Hamilton-Dirac equations generated by the ADM energy: it coincides with the standard Einstein or dynamical gravitational field, namely a 4-geometry solution of Einstein's equations, since the Hilbert and ADM actions both generate Einstein's equations so that the kinematical Hamiltonian gauge transformations are dynamically restricted to the spacetime diffeomorphisms of the solutions of Einstein's equations.

101 pages, revtex, revised version with revised interpretational aspects