Schwarzschild Limit
The spherical symmetry of the Schwarzschild metric for non-spinning black holes allows one to reduce the problem of finding the trajectories of particles to three dimensions. In this case one only needs, and to determine the motion; however, the symmetry leading to Carter's constant still exists. Carter's constant for Schwarzschild space is:
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By a rotation of coordinates we can put any orbit in the plane so . In this case, the square of the orbital angular momentum.
Read more about this topic: Carter Constant
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