Communications in Analysis and Geometry

Volume 13 (2005)

Number 5

Phase Space for the Einstein Equations

Pages: 845 – 885



Robert Bartnik (School of Mathematical Sciences, Monash University, Clayton, Victoria, Australia)


A Hilbert manifold structure is described for the phase space $\mathcal{F}$ of asymptotically flat initial data for the Einstein equations. The space of solutions of the constraint equations forms a Hilbert submanifold $\mathcal{C} \subset \mathcal{F}$. The ADM energy-momentum defines a function which is smooth on this submanifold, but which is not defined in general on all of $\mathcal{F}$. The ADM Hamiltonian defines a smooth function on $\mathcal{F}$ which generates the Einstein evolution equations only if the lapse-shift satisfies rapid decay conditions. However a regularised Hamiltonian can be defined on $\mathcal{F}$ which agrees with the Regge–Teitelboim Hamiltonian on $\mathcal{C}$ 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.

This research was supported in part by the Australian Research Council.

Published 1 January 2005