On certain global conformal invariants and 3-surface twistors of initial data sets

Abstract

The Chern-Simons functionals built from various connections determined by the initial data $h_{\mu \nu }$, ${\chi }_{\mu \nu }$ on a 3-manifold $\Sigma$ are investigated. First it is shown that for asymptotically flat data sets the logarithmic fall-off for $h_{\mu \nu }$ and $r{\chi }_{\mu \nu }$ is the necessary and sufficient condition of the existence of these functionals. The functional $Y_{k,l}$, built in the vector bundle corresponding to the irreducible representation of SL(2,C) labelled by (k,l), is shown to be determined by the Ashtekar-Chern-Simons functional and its complex conjugate. $Y_{k,l}$ is conformally invariant precisely in the l=k (i.e. tensor) representations. An unexpected connection with twistor theory is found: $Y_{k,k}$ can be written as the Chern-Simons functional built from the 3-surface twistor connection, and the not identically vanishing spinor parts of the 3-surface twistor curvature are given by the variational derivatives of $Y_{k,k}$ with respect to $h_{\mu \nu }$ and ${\chi }_{\mu \nu }$. The time derivative ${\dot{Y}}_{k,k}$ of $Y_{k,k}$ is another conformal invariant of the initial data set, and for vanishing ${\dot{Y}}_{k,k}$, in particular for all Petrov III and N spacetimes, the Chern-Simons functional is a conformal invariant of the whole spacetime.

18 pages, Plain TEX

References in corpus (1)

• 3+1 description of silent universes: a uniqueness result for the Petrov type I vacuum case

Cited by in corpus (3)

• On the roots of the Poincare structure of asymptotically flat spacetimes
• On the role of conformal three-geometries in the dynamics of General Relativity
• A note on the Hamiltonian constraint in canonical GR