I'm struggling with the concept of moving to and from reference frames in GR. I'm doing a problem in Spacetime and Geometry by S. Carroll (Chapter 6, Question 6 - the text for the question is very long) and it's the idea of iron in an accretion disk on geodesics emitting photons of known frequency $$\nu_{0}$$ and then, depending on where the photon is emitted and the direction of rotation of the accretion disk compared to the Kerr B-H, what is the frequency of the photon measured by an observer very far away. One method I tried was using, from earlier in the book, $$\omega = -g_{\mu \nu}U^{\mu}\frac{dx^{\nu}}{d\lambda}$$ then if you assume the observer is stationary, and in the equatorial plane (as given in the question) you get that $$\omega = (1 - \frac{2GM}{r})^{\frac{-1}{2}}E$$ from the Killing vector and the normalisation of U in the Kerr metric. Then I think that the energy will be conserved along the Killing vector so just take $$E = \nu_{0}$$ if hbar = 1. So you end up with the formula $$\frac{\omega}{\nu_{0}} = (1 - \frac{GM}{r})^{\frac{-1}{2}}$$ which is just the Schwarzchild redshift formula right?
So clearly this doesn't depend on anything to do with the accretion disk, or even the 'angular momentum' of the Kerr B-H, and I definitely feel like I'm doing something wrong. Furthermore, I have rearranged the Killing vectors for L and E then used them, and the fact the photon is on a null path in the equatorial plane, to get the full form for the dx/d(lambda) in terms of conserved quantities. I want to try doing the idea of moving between frames of a particle in the accretion disk but I'm really struggling with deriving the transformation between the particle's rest frame and then Black-Hole's frame. I'm thinking along the lines of the preservation of the line element and saying that, in a very small vicinity around the particle, spacetime looks like Minkowski spacetime so, using again that we're in the equatorial plane, $$ds^2 = -dt'^2 + dr'^2 + r^2d{\phi'}^2 = -(1 - \frac{2GM}{r})dt^2 + \frac{2GMa}{r}(dtd{\phi} + d{\phi}dt) + \frac{r^2}{\Delta}dr^2 + \frac{\Sigma^2}{r^2}d{\phi}^2$$ where $$\Sigma^2 = [(r^2 + a^2)^2 - a^2{\Delta} ] || \Delta = r^2 - 2GMr + a^2$$ then if you leave all the other components constant, you get the effects of, for example, $$\frac{dr'}{dr} = \frac{r}{\sqrt{\Delta}} $$ I'm really not sure if this is correct. I've also had some ideas like $$U^{\mu'} = \frac{\partial{x^{\mu'}}}{\partial{x^{\mu}}}U^{\mu}$$ where the particle frame is un-primed and the Black-Hole frame is primed, but still I'm struggling to find (convincingly) the partial derivatives.
Please can someone give me a push in the right direction? I'm self studying (lockdown and all that) and I feel I'm missing some little conceptual, well basic, ideas like this. I don't know if this is against Stack Exchange policy, but please can you not give a full answer to Carroll's question, I would still like to try and fill in the gaps!