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

TL;DR

This paper explores conformal invariants derived from Chern-Simons functionals on initial data sets in general relativity, revealing connections with twistor theory and conditions for invariance in asymptotically flat spacetimes.

Contribution

It establishes the conformal invariance of certain Chern-Simons functionals related to initial data and uncovers a novel link with 3-surface twistor theory.

Findings

Y_{k,k} can be expressed as a Chern-Simons functional from 3-surface twistors.

Y_{k,k} is conformally invariant when k=l, i.e., tensor representations.

The time derivative ot Y_{k,k} is a conformal invariant, especially in Petrov type III and N spacetimes.

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.