Suppose one spaceship is stably orbiting a rotating black hole and another is far away from the black hole. What is the maximum time dilation factor between the two ships? Can it be made arbitrarily large, and if so does that require the black hole to be maximally rotating?
The innermost stable circular orbit (ISCO) for a maximally rotating Kerr black hole is for a prograde orbit at $r=m$. This is supposed to be the event horizon, which would make the time dilation factor infinite. However, the text around equation (22) in these lecture notes says that this "is an artifact of the coordinate system." Does this mean that the time dilation factor is not unbounded?
Best Answer
You can get the time dilation factor by computing the redshift of a radial photon emitted by someone on a circular orbit, compared to the frequency measured by someone at rest at infinity. The derivation of this formula is a bit involved, but the answer is not too complicated: $$ \frac{\omega_{emit}}{\omega_\infty}=\frac{r^{3/2}+a}{\left(r^2(r-3)+2ar^{3/2}\right)^{1/2}} $$ This is for a prograde orbit, and I'm using units where $G=c=M=1$. For an extremal black hole, $a=1$ and the ISCO is at $r=1$, so you can see this factor diverges and you can get arbitrarily high redshifts coming from orbits near the horizon.
It's also interesting to consider the nearly extremal black hole, where $a=1-\epsilon$. In that case the ISCO is located at (again, from a somewhat involved calculation): $$r_{ISCO} \approx 1+(4\epsilon)^{1/3}$$ Using these formulas, we can compute the time dilation factor coming from the ISCO to lowest order: $$\frac{\omega_{emit}}{\omega_\infty}\approx\left(\frac{2}\epsilon\right)^{1/3}$$
So it is diverging as $\epsilon\rightarrow 0$, but it is doing so rather slowly. For example, say for some reason you wanted 1 hour in the orbit to correspond to 7 years at infinity, which is a factor of about $60000$. This requires an $\epsilon \approx 10^{-14}$, so it requires the black hole to be very close to extremal. Also if you consider Kip Thorne's bound that says astrophysical black holes can only ever reach $\epsilon\approx .002$, the maximum time dilation factor you can achieve is around $10$.