On a global conformal invariant of initial data sets

Details On a global conformal invariant of initial data sets On a global conformal invariant of initial data sets

1997-06-26

arxiv.org/abs/gr-qc/9706078v1

Class.Quant.Grav. 14 (1997) 3091-3107

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

17 pages, Plain Tex

Scientific paper

10.1088/0264-9381/14/11/010

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.

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