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

Understand

We show that the so-called flat-space rotational Killing vector method for measuring the Cartesian components of a black hole spin can be derived from the surface integral of Weinberg's pseudotensor over the apparent horizon surface when using Gaussian normal coordinates in the integration.

• Moreover, the integration of the pseudotensor in this gauge yields the Komar angular momentum integral in a foliation adapted to the axisymmetry of the spacetime.
• As a result, the method does not explicitly depend on the evolved lapse $\alpha$ and shift $\beta^i$ on the respective timeslice, as they are fixed to Gaussian normal coordinates, while leaving the coordinate labels of the spatial metric $\gamma_{ij}$ and the extrinsic curvature $K_{ij}$ unchanged.
• Such gauge fixing endows the method with coordinate invariance, which is not present in integral expressions using Weinberg's pseudotensor, as they normally rely on the explicit use of Cartesian coordinates.