In $ x^{\mu}= \{t,\rho,\theta,\phi\}$Boyer-Lindquist coordinates the Kerr solution takes a manageable form. Given some initial conditions $x^{\mu}(\lambda=0)$ and $\dfrac{dx^{\mu}}{d\lambda}(\lambda=0)$ one can numerically solve the geodesic equations for the parametrised curve $x^{\mu}(\lambda)$. My question is, how would one try to plot and animate the curves in some simple plot3d function? I have seen people “convert” the B-L coordinates to euclidean $x,y,z$ coordinates and animate the curve via the affine parameter $\lambda$.
This seems wrong to me, as the transformation $$x = \sqrt{\rho^2+a^2} \sin\theta \cos\phi$$ $$y = \sqrt{\rho^2+a^2} \sin\theta \sin\phi$$ $$ z =\rho \cos\phi$$ is only valid in the $m=0$ limit of Kerr, while the geodesics were calculated with some nonzero mass.
Obviously one can NOT globally establish euclidean flat coordinates on a curved Kerr manifold, so this scheme makes no sense from a theoretical view. What is the proper way of doing such a plot?
