I have a hopefully straightforward question. It is well known that in the Schwarzschild metric the gravitational redshift is given by [tex]1+z=(1-r_{s}/r)^{-1/2}[/tex]. Clearly this is just the ratio of observed to emitted frequencies (or energies). I understand this so far. However, for the case of the Kerr spacetime, in Boyer-Lindquist coordinates(adsbygoogle = window.adsbygoogle || []).push({});

[tex]ds^{2}=\bigg(1-\frac{2Mr}{\Sigma}\bigg)dt^{2}+\frac{4aMr \sin^{2}\theta}{\Sigma}dt d\phi-\frac{\Sigma}{\Delta}dr^{2}-\Sigma d\theta^{2}-\bigg(r^{2}+a^{2}+\frac{2a^{2}Mr \sin^{2}\theta}{\Sigma} \bigg)\sin^{2}\theta d\phi^{2},[/tex]

where

[tex]\Sigma \equiv r^{2}+a^{2}\cos^{2}\theta[/tex] and [tex]\Delta \equiv r^{2}-2Mr+a^{2}.[/tex]This asymptotes to the Schwarzschild case in the limit [tex]a\rightarrow 0[/tex]

For the Schwarzschild black hole [tex]1+z=(g_{tt})^{-1/2}[/tex]. I believe this is not the case for the Kerr spacetime (because of frame-dragging in the cross-term?).

What is the expression for the gravitational redshift in the Kerr spacetime for a photon (I can list the geodesic equations of motion if needed)? Or, how would one go about deriving such a formula? Presumably there would be some [tex]r[/tex] as well as [tex]\theta[/tex] -dependence in said expression (as well as spin, a)?

Thank you.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Gravitational Redshift for Photon around Kerr Black Hole

**Physics Forums | Science Articles, Homework Help, Discussion**