The surface gravity (acceleration at event horizon, as measured by an observer at infinity, since the proper acceleration is infinite) of a Kerr-Newman black hole is given (e.g. here) as:$$ \kappa = \frac{r_+ - r_-}{2 \cdot (r_+^2 + a^2)} $$What is the corresponding generalization of this formula for any distance from the event horizon (i.e., what's $\kappa(r)$)?


This would be $\ddot{r}$, so you evaluate ${\rm d}^2r/{\rm d}\tau^2$ of a corotating particle by solving the geodesic equation (see here) and take the limit for $r \to r_{+}$ in case of the horizon, or any $r$ and $\theta$ since $\ddot{r}$ is not only a function of the radial but also of the poloidial coordinate.

The general equation for a black hole of charge $Q$ and spin $a$ for a particle with the charge q is in natural units and in Boyer Lindquist coordinates

$ \ddot{r} =(a^2 \dot{\theta} \sin (2 \theta ) \ \dot{r})/(a^2 \cos ^2 \theta +r^2)+\dot{r}^2 ((r-1)/(a^2+(r-2) \ r+Q ^2)-r/(a^2 \cos ^2 \theta +r^2))+ $ $ ((a^2+(r-2) \ r+Q ^2) (8 a \sin ^2 \theta \ \dot{\phi} \ (a^2 \cos ^2 \theta \ (q \ Q -2 \dot{t})+r (2 (r-Q ^2) \dot{t}-q \ Q \ r))+ $ $ 8 \dot{t} \ (a^2 \cos ^2 \theta \ (\dot{t}-q \ Q )+r \ (q \ Q \ r+(Q ^2-r) \ \dot{t}))+8 r \ \dot{\theta}^2 \ (a^2 \cos ^2 \theta +r^2)^2+ $ $ \sin ^2 \theta \ \dot{\phi}^2 \ (2 a^4 \sin ^2(2 \theta )+r \ (a^2 (a^2 \cos (4 \theta )+3 a^2+4 \ (a-Q ) (a+Q ) \cos (2 \theta )+4 Q ^2)+ $ $ 8 r \ (-a^2 \sin ^2 \theta +2 a^2 r \cos ^2 \theta +r^3)))))/(8 \ (a^2 \cos ^2 \theta +r^2)^3) $

so you have to set $\dot{r}=\dot{\theta}=q=0, \ \dot{\phi}=|g_{t \phi}/g_{\phi \phi}|/\sqrt{|g^{t t}|}$ and $\dot{t}=1/\sqrt{|g^{t t}|}$ to get $\ddot{r}$ for a locally stationary particle with zero angular momentum at any given $r$ and $\theta$.


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