# Radial outgoing null geodetic in Kerr spacetime

According to Poisson's book A relativist's toolkit, pag 52, with the Schwarzschild metric I can define outgoing radial null geodesic as follows: $$u=t-\int f(r)^{-1} dr$$ where $$f(r)=1-\dfrac{2m}{r}$$.

Can I define an analogous geodetic in the Kerr spacetime, with $$u=t-\int g(r,\theta) dr$$ where $$g(r,\theta)=\dfrac{\Sigma}{\Delta} = \dfrac{r^2+a^2\cos\theta^2}{r^2-2m r +a^2}$$?

• You can certainly define that coordinate, but whether it's useful depends on what properties of the Schwarzschild version you want to carry over to the present case. Can you clarify what properties you want this new coordinate to have? – Michael Seifert Aug 31 '20 at 15:29
• Also, note that the first equation doesn't define a null geodesic, per se; it defines a coordinate that is constant along outgoing null geodesics, which is not quite the same thing. – Michael Seifert Aug 31 '20 at 15:29