On the total mass of closed universes

The total mass, the Witten type gauge conditions and the spectral properties of the Sen-Witten and the 3-surface twistor operators in closed universes are investigated. It has been proven that a recently suggested expression ${\tt M}$ for the total mass density of closed universes is vanishing if and only if the spacetime is flat with toroidal spatial topology; it coincides with the first eigenvalue of the Sen-Witten operator; and it is vanishing if and only if Witten's gauge condition admits a non-trivial solution. Here we generalize slightly the result above on the zero-mass configurations: ${\tt M}=0$ if and only if the spacetime is holonomically trivial with toroidal spatial topology. Also, we show that the multiplicity of the eigenvalues of the (square of the) Sen-Witten operator is at least two, and a potentially viable gauge condition is suggested. The monotonicity properties of ${\tt M}$ through the examples of closed Bianchi I and IX cosmological spacetimes are also discussed. A potential spectral characterization of these cosmological spacetimes, in terms of the spectrum of the Riemannian Dirac operator and the Sen-Witten and the 3-surface twistor operators, is also indicated.Comment: 14 pages, plenary talk at the `Spanish Relativity Meeting in Portugal 2012', Guimar\~aes 3-7 September; Final version, appearing in General Relativity and Gravitatio

On the role of conformal three-geometries in the dynamics of General Relativity

It is shown that the Chern-Simons functional, built in the spinor representation from the initial data on spacelike hypersurfaces, is invariant with respect to infinitesimal conformal rescalings if and only if the vacuum Einstein equations are satisfied. As a consequence, we show that in the phase space the Hamiltonian constraint of vacuum general relativity is the Poisson bracket of the imaginary part of this Chern-Simons functional and Misner's time (essentially the 3-volume). Hence the vacuum Hamiltonian constraint is the condition on the canonical variables that the imaginary part of the Chern- Simons functional be constant along the volume flow. The vacuum momentum constraint can also be reformulated in a similar way as a (more complicated) condition on the change of the imaginary part of the Chern-Simons functional along the flow of York's time.Comment: 15 pages, plain Te

On a class of 2-surface observables in general relativity

The boundary conditions for canonical vacuum general relativity is investigated at the quasi-local level. It is shown that fixing the area element on the 2- surface S (rather than the induced 2-metric) is enough to have a well defined constraint algebra, and a well defined Poisson algebra of basic Hamiltonians parameterized by shifts that are tangent to and divergence-free on $. The evolution equations preserve these boundary conditions and the value of the basic Hamiltonian gives 2+2 covariant, gauge-invariant 2-surface observables. The meaning of these observables is also discussed.Comment: 11 pages, a discussion of the observables in stationary spacetimes is included, new references are added, typos correcte

Stationary untrapped boundary conditions in general relativity

A class of boundary conditions for canonical general relativity are proposed and studied at the quasi-local level. It is shown that for untrapped or marginal surfaces, fixing the area element on the 2-surface (rather than the induced 2-metric) and the angular momentum surface density is enough to have a functionally differentiable Hamiltonian, thus providing definition of conserved quantities for the quasi-local regions. If on the boundary the evolution vector normal to the 2-surface is chosen to be proportional to the dual expansion vector, we obtain a generalization of the Hawking energy associated with a generalized Kodama vector. This vector plays the role for the stationary untrapped boundary conditions which the stationary Killing vector plays for stationary black holes. When the dual expansion vector is null, the boundary conditions reduce to the ones given by the non-expanding horizons and the null trapping horizons.Comment: 11 pages, improved discussion section, a reference added, accepted for publication in Classical and Quantum Gravit