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We show that the so-called flat-space rotational Killing vector method formeasuring the Cartesian components of a black hole spin can be derived from thesurface integral of Weinberg's pseudotensor over the apparent horizon surfacewhen using Gaussian normal coordinates in the integration. Moreover, theintegration of the pseudotensor in this gauge yields the Komar angular momentumintegral in a foliation adapted to the axisymmetry of the spacetime. As aresult, the method does not explicitly depend on the evolved lapse $\alpha$ andshift $\beta^i$ on the respective timeslice, as they are fixed to Gaussiannormal coordinates, while leaving the coordinate labels of the spatial metric$\gamma_{ij}$ and the extrinsic curvature $K_{ij}$ unchanged. Such gauge fixingendows the method with coordinate invariance, which is not present in integralexpressions using Weinberg's pseudotensor, as they normally rely on theexplicit use of Cartesian coordinates.

Measuring the black hole spin direction in 3D Cartesian numerical relativity simulations