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The transverse and trace-free (TT) part of the extrinsic curvature representshalf of the dynamical degrees of freedom of the gravitational field in the $3 +1$ formalism. As such, it is part of the freely specifiable initial data fornumerical relativity. Though TT tensors in 3-space possess only two componentdegrees of freedom, they cannot ordinarily be given solely by two scalarpotentials. Such expressions have been derived however, in coordinate form, forall TT tensors in flat space which are also translationally or axiallysymmetric (Conboye and \'O Murchadha 2014 \emph{Class. Quantum Grav.} {\bf 31}085019). Since TT tensors are conformally covariant, these also give TT tensorsin conformally-flat space. In this article, the work above has been extended bygiving a coordinate-\emph{independent} expression for these TT tensors. Thetranslational and axial symmetry conditions have also been generalized toinvariance along any hypersurface orthogonal Killing vector.

Coordinate independent expression for transverse trace-free tensors