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So, to put it simply. Let's say we have a point mass "m" at rest on the event horizon of a black hole of mass "M" and we throw it directly toward the location of the black hole's singularity. The particles only does linear motion and does not orbit the black hole at any point. IT falls directly into the spacetime curvature caused by the black hole. This means it will follow the shortest path and only travel the distance represented by the black hole's Schwarzschild radius until it reaches the singularity.

Let's say that initially the black hole and the point mass are stationary relative to each other. There's no spinning of either the black hole or the point mass.

When this happens, it accelerates. My question is, will it come off from the other side of the singularity and execute simple harmonic motion?