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arXiv · 1306.1363

Potentials for transverse trace-free tensors

Rory Conboye·Niall Ó Murchadha
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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.

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Rory Conboye, Niall Ó Murchadha. 2013-06-06. Potentials for transverse trace-free tensors. https://doi.org/10.1088/0264-9381%2F31%2F8%2F085019

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