Potentials for transverse trace-free tensors

TL;DR

This paper derives scalar potential expressions for transverse trace-free tensors in flat 3-space, simplifying their representation and providing insights relevant for initial conditions in numerical relativity.

Contribution

It presents explicit scalar potential formulas for all TT tensors in flat space and reduces the problem to PDEs in more general slices, advancing the understanding of TT tensor parametrizations.

Findings

Explicit scalar potentials for all TT tensors in flat space.

Reduction of TT tensor characterization to PDEs in general slices.

Derived potentials for Bowen-York curvature tensor.

Abstract

In the initial conditions of the $3 + 1$ formalism for numerical relativity, the transverse and trace-free (TT) part of the extrinsic curvature plays a key role. We know that TT tensors possess two degrees of freedom per space point. However, finding an expression for a TT tensor depending on only two scalar functions is a non-trivial task. Assuming either axial or translational symmetry, expressions depending on two scalar potentials alone are derived here for \emph{all} TT tensors in flat $3$-space. In a more general spatial slice, only one of these potentials is found, the same potential given in \cite{BakerPuzio} and \cite{Dain}, with the remaining equations reduced to a partial differential equation, depending on boundary conditions for a solution. As an exercise, we also derive the potentials which give the Bowen-York curvature tensor in flat space.