I've been trying to make progress on some of the smaller pieces of this question about the environment around a Kerr black hole. In order to calculate the effects of special relativistic Doppler shift on background radiation, I'm trying to find out how to calculate the effective velocity of a particle in orbit around a Kerr black hole with respect to radiation falling in from infinity, and the proper-time period of the orbit in the Zero Angular Velocity frame- i.e., the proper time it would take for an orbiting observer to see the background stars revolve once.

I have found this useful-looking web page which provides lots of formulae for things like the innermost stable orbit, horizon radius, ergosphere radius, and the Keplerian angular velocity and frame dragging angular velocity. Unfortunately, not being an expert in GR myself, I am having trouble extracting simple formulae for the actual orbital period or velocity in terms of orbital radius ($r$), black hole mass ($M$) and angular momentum ($a$).

My guess is that I should be able to use the Keplerian angular velocity to derive tangential orbital velocity with respect to radiation falling in from infinity, scaling by $r$ to convert angular velocity into true velocity. Is that correct? If so, my only remaining hurdle there is figuring out what conversions I need to insert to go from geometrized to MKS units.

In order to determine the orbital period in the ZAVO frame, I then assume that I would add the angular velocity of frame dragging to the Keplerian angular velocity, and then divide $2\pi r$ radians by that summed angular velocity to get the period.

Finally, in each case, I would then apply the time dilation formula to convert coordinate velocity and coordinate time into proper velocity and proper time.

So, am I on the right track here? If not, where am I going wrong, and what's the correct way to calculate these quantities (proper orbital period and proper orbital velocity) in terms of $r$, $M$, and $a$?

  • 1
    $\begingroup$ The equations you are looking for can be found in Bardeen at al (1972), page 14+ (the ZAVO with 0 angular velcotity in your example is the ZAMO with 0 angelar momentum in the NLRF locally nonrotating frame in the terminology of the article). Examples comparing the local to the delayed velcities can be found here. $\endgroup$ – Yukterez Jul 19 '17 at 3:10

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