Quantum Gravity

THE 3+1 DECOMPOSITION OF GENERAL RELATIVITY 119modifications for the case of open spaces (where ‘open’ means ‘asymptoticallyflat’). The necessary details can be found in Regge and Teitelboim (1974) andBeig and Ó Murchadha (1987).Variation of the full Hamiltonian H g with respect to the canonical variablesh ab and p cd yields∫δH g = d 3 x ( A ab δh ab + B ab δp ab) − δC , (4.85)where δC denotes surface terms. Because H g must be a differentiable functionwith respect to h ab and p cd (otherwise Hamilton’s equations of motion wouldnot make sense), δC must be cancelled by introducing explicit surface terms toH g . For the derivation of such surface terms, one must impose fall-off conditionsfor the canonical variables. For the three-metric they readh abr→∞∼( ) 1δab + Or, h ab,cr→∞∼ O( 1r 2 ), (4.86)and analogously for the momenta. The lapse and shift, if again combined to thefour-vector N µ , are supposed to obeyN µ r→∞∼ α µ + β a µ xa , (4.87)where α µ describe space–time translations, β ab = −β ba spatial rotations, andβa⊥ boosts. Together, they form the Poincaré group of Minkowski space–time,which is a symmetry in the asymptotic sense. The procedure mentioned abovethen leads to the following expression for the total Hamiltonian:∫H g = d 3 x (NH g ⊥ + N a Ha g )+αE ADM − α a P a + 1 2 β µνJ µν , (4.88)where E ADM (also called ‘ADM’ energy; see Arnowitt et al. 1962), P a ,andJ µνare the total energy, the total momentum, and the total angular momentumplus the generators of boosts, respectively. Together they form the generators ofthe Poincaré group at infinity. They obey the standard commutation relations(2.34)–(2.36). For the ADM energy, in particular, one finds the expressionE ADM = 1 ∮d 2 σ a (h ab,b − h bb,a ) . (4.89)16πG r→∞Note that the total energy is defined by a surface integral over a sphere forr →∞and not by a volume integral. One can prove that E ADM ≥ 0.The integral in (4.88) is the same integral as in (4.67). Because of the constraints(4.69) and (4.70), H g is numerically equal to the surface terms. Forvanishing asymptotic shift and lapse equal to one, it is just given by the ADMenergy. We emphasize that the asymptotic Poincaré transformations must notbe interpreted as gauge transformations (otherwise E ADM , P a ,andJ µν wouldbe constrained to vanish), but as proper physical symmetries, see the remarksin the following subsection.