Communications in Analysis and Geometry
Volume 13 (2005)
Phase Space for the Einstein Equations
Pages: 845 – 885
A Hilbert manifold structure is described for the phase space $\cF$ of asymptotically flat initial data for the Einstein equations. The space of solutions of the constraint equations forms a Hilbert submanifold $\cC \subset \cF$. The ADM energy-momentum defines a function which is smooth on this submanifold, but which is not defined in general on all of $\cF$. The ADM Hamiltonian defines a smooth function on $\cF$ which generates the Einstein evolution equations only if the lapse-shift satisfies rapid decay conditions. However a regularised Hamiltonian can be defined on $\cF$ which agrees with the Regge-Teitelboim Hamiltonian on $\cC$ 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.