Bowen-York Tensors

TL;DR

This paper introduces a family of linear differential operators on conformally flat three-spaces that generate tracefree, symmetric tensors from vectors, including the well-known Bowen-York TT-tensors, with applications in gravitational physics.

Contribution

It derives a new class of differential operators that produce TT-tensors from vectors, generalizing the Bowen-York construction for conformally flat spaces.

Findings

Operators map source-free electric fields into TT-tensors

The divergence of the tensor depends only on the original vector's divergence

Reproduces Bowen-York TT-tensors from Coulomb fields

Abstract

There is derived, for a conformally flat three-space, a family of linear second-order partial differential operators which send vectors into tracefree, symmetric two-tensors. These maps, which are parametrized by conformal Killing vectors on the three-space, are such that the divergence of the resulting tensor field depends only on the divergence of the original vector field. In particular these maps send source-free electric fields into TT-tensors. Moreover, if the original vector field is the Coulomb field on $\mathbb{R}^3\backslash \lbrace0\rbrace$, the resulting tensor fields on $\mathbb{R}^3\backslash \lbrace0\rbrace$ are nothing but the family of TT-tensors originally written down by Bowen and York.