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The equations of motion for a test particle in a Kerr spacetime $(M,\mathcal{O}, \mathcal{A},\nabla^{L.C.})$ are dictated by four degrees of freedom (i.e. invariant mass $m$ in $p^\mu g_{\mu\nu}p^{\nu}=-m^2,$ the energy $E$, the Carter constant $Q, $ and the orbital angular momentum $L_z=-p_{\phi}$ in the spin direction). On the wikipedia page for the Kerr metric, there is a simulation on the right-hand side of the trajectory equations section (see simulation here). By any chance, does anyone know of such a program (for Python) that I can use to simulate this, as shown in the aforementioned program implementation?

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You can use some common technical computing tools like Octave (free) or Matlab or even Mathematica. Using a simple explicit numerical integrator like ode45 you can simulate any ODE, also like system you are asking of

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