Bowen-York Tensors

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 ${\mathbb{R}}^3\mathrm{\backslash }\{0\}$, the resulting tensor fields on ${\mathbb{R}}^3\mathrm{\backslash }\{0\}$ are nothing but the family of TT-tensors originally written down by Bowen and York.