On a global conformal invariant of initial data sets

Abstract

In the present paper a global conformal invariant $Y$ of a closed initial data set is constructed. A spacelike hypersurface $\Sigma$ in a Lorentzian spacetime naturally inherits from the spacetime metric a differentiation ${\mathcal{D}}_e$, the so-called real Sen connection, which turns out to be determined completely by the initial data $h_{ab}$ and ${\chi }_{ab}$ induced on $\Sigma$, and coincides, in the case of vanishing second fundamental form ${\chi }_{ab}$, with the Levi-Civita covariant derivation $D_e$ of the induced metric $h_{ab}$. $Y$ is built from the real Sen connection ${\mathcal{D}}_e$ in the similar way as the standard Chern-Simons invariant is built from $D_e$. The number $Y$ is invariant with respect to changes of $h_{ab}$ and ${\chi }_{ab}$ corresponding to conformal rescalings of the spacetime metric. In contrast the quantity $Y$ built from the complex Ashtekar connection is not invariant in this sense. The critical points of our $Y$ are precisely the initial data sets which are locally imbeddable into conformal Minkowski space.

arXiv comment17 pages, Plain Tex

Cited by in corpus (5)

• On the construction of a geometric invariant measuring the deviation from Kerr data
• 3+1 description of silent universes: a uniqueness result for the Petrov type I vacuum case
• On certain global conformal invariants and 3-surface twistors of initial data sets
• On the role of conformal three-geometries in the dynamics of General Relativity
• A note on the Hamiltonian constraint in canonical GR