Numerical Implementation

The implementation assumes that the numerical solution, on a Cartesian grid, is approximately Schwarzshild on the spheres of constant $r=\sqrt(x^2+y^2+z^2)$ where the waveforms are extracted. The general procedure is then:

• Project the required metric components, and radial derivatives of metric components, onto spheres of constant coordinate radius (these spheres are chosen via parameters).

• Transform the metric components and there derivatives on the 2-spheres from Cartesian coordinates into a spherical coordinate system.

• Calculate the physical metric on these spheres if a conformal factor is being used.

• Calculate the transformation from the coordinate radius to an areal radius for each sphere.

• Calculate the $S$ factor on each sphere. Combined with the areal radius This also produces an estimate of the mass.

• Calculate the six Regge-Wheeler variables, and required radial derivatives, on these spheres by integration of combinations of the metric components over each sphere.

• Contruct the gauge invariant quantities from these Regge-Wheeler variables.