Phase Space for the Einstein Equations

A Hilbert manifold structure is described for the phase space F of asymptotically flat initial data for the Einstein equations. The space of solutions of the constraint equations forms a Hilbert submanifold C ⊂ F . The ADM energy-momentum defines a function which is smooth on this submanifold, but which is not defined in general on all of F . The ADM Hamiltonian defines a smooth function on F which generates the Einstein evolution equations only if the lapse-shift satisfies rapid decay conditions. However a regularised Hamiltonian can be defined on F which agrees with the Regge-Teitelboim Hamiltonian on C and generates the evolution for any lapse-shift appropriately asymptotic to a (time) translation at infinity. Finally, critical points for the total (ADM) mass, considered as a function on the Hilbert manifold of constraint solutions, arise precisely at initial data generating stationary vacuum spacetimes. 2000 Mathematics Subject Classification: 83C05, 58D17, 58J05