To calculate the stable circular orbit radius for a test particle around a non-rotating black hole (Schwarzschild black hole), we can use the formula for the innermost stable circular orbit (ISCO) given by:

R_ISCO = 6GM/c^2,

where R_ISCO is the radius of the ISCO, G is the gravitational constant, M is the mass of the black hole, and c is the speed of light.

For a rapidly rotating black hole (Kerr black hole), the formula for the ISCO radius becomes more complicated and depends on the spin parameter of the black hole, denoted as a. However, we can approximate the ISCO radius for a rapidly rotating black hole using the formula derived by Bardeen, Press, and Teukolsky (BPT formula) as follows:

R_ISCO = 3GM/c^2 + Z2 ± sqrt((3GM/c^2)^2 - Z),

where Z = (3GM/c^2) * sqrt[1 - (a^2 / (GM/c^2))^2].

This approximation assumes that the black hole spin is large (a > GM/c^2) and that no spin-flip effect occurs (a < 0.95GM/c^2).

Comparing the results for the stable circular orbit radii of a test particle around a non-rotating and a rapidly rotating black hole, we see that the ISCO radius for a rapidly rotating black hole is smaller than that for a non-rotating black hole. This means that a test particle must be closer to the rapidly rotating black hole to maintain a stable circular orbit compared to a non-rotating black hole. The rotation of the black hole increases the gravitational attraction near the event horizon, making the orbit more tightly bound.

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