Trajectory Equations
The equations of the trajectory and the time dependence for a particle in the Kerr field are as follows.
In the Hamilton-Jacobi equation we write the action S in the form:
where, m, and L are the conserved energy, the rest mass and the component of the angular momentum (along the axis of symmetry of the field) of the particle consecutively, and carry out the separation of variables in the Hamilton Jacobi equation as follows:
where K is a fourth arbitrary constant (usually called Carter's constant). The equation of the trajectory and the time dependence of the coordinates along the trajectory (motion equation) can be found then easily and directly from these equations:
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