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Suppose that we have a non-rotating black hole with mass $M$. We know that the time dilation $\Delta t'$ at a distance $r$ to the center of the black hole is given by $\Delta t' = \Delta t \sqrt{1 - \frac{2GM}{rc^2}}$.

Now, we want to consider a rotating black hole with mass $M$ and specific angular momentum $a = 1 - \epsilon$. How can we obtain the time dilation $\Delta t'$ at a distance $r$ to the center of the black hole?

In a previous question, we already had the time dilation for the special case of the innermost stable circular orbit (ISCO) for a maximally rotating black hole, but I have been unable to find a solution to the more general problem. I would like to be able to work out, for instance, the time dilation of a ship in orbit of the fictional Gargantua when $M = 10^{8}\ M_\odot = 2 \times 10^{38}\ kg,\ \epsilon = 10^{-14},\ r - R = 2\ AU = 3 \times 10^{11}\ m$.

Kenneth Kho
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The first answer in the question you linked includes the solution to the more general problem you are looking for right there before the special case for the ISCO, it is the first formula in asperanz's answer. It's just a matter of algebra. You just need to change your $r$ to be in terms of the minimal ISCO for your black hole in question.

Anthony Khodanian
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