On a global conformal invariant of initial data sets

TL;DR

This paper introduces a new global conformal invariant $Y$ for closed initial data sets in general relativity, constructed from the real Sen connection, which remains invariant under conformal rescalings of the spacetime metric.

Contribution

The paper defines a novel conformal invariant $Y$ based on the real Sen connection, extending the concept of invariants from the Levi-Civita connection and characterizing initial data embeddability.

Findings

$Y$ is invariant under conformal rescalings of initial data.

Critical points of $Y$ correspond to data embeddable in conformal Minkowski space.

$Y$ differs from invariants built from the complex Ashtekar connection.

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 ${\cal D}_e$, 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 ${\cal D}_e$ 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.