# Effective potential for Kerr geometry

In the review Foundations of Black Hole Accretion Disk Theory, the authors defines an effective potential for Kerr geometry as (Chap. 2, eqn. 23) $$\mathcal{U}_{eff}=-\frac{1}{2}\ln\left|g^{tt}-2lg^{t\phi}+l^2g^{\phi\phi}\right|$$ where $$l=\dfrac{\mathcal{L}}{\mathcal{E}}=-\dfrac{u_\phi}{u_t}$$ is the specific angular momentum, $$\mathcal{L}=p_\phi$$ is the angular momentum and $$\mathcal{E}=-p_t$$ is the energy.

It is mentioned that this form of the potential is chosen because using the potential $$\mathcal{U}_{eff}$$, the rescaled energy $$\mathcal{E}^*=\ln\mathcal{E}$$ and $$V=u^ru_r+u^\theta u_\theta<, slightly non-circular motion can be characterized by the the equation $$\frac{1}{2}V^2=\mathcal{E}^*-\mathcal{U}_{eff}$$

The form of this equation is indeed similar to that of the Newtonian equation. But there were nothing mentioned in the paper regarding the derivation of the effective potential. Also, I couldn't understand why they re-scaled the energy as $$\mathcal{E}^*=\ln\mathcal{E}$$.

My Questions:

1. How to derive the effective potential $$\mathcal{U}_{eff}$$? Hints to the derivation would be sufficient.
2. What is the logic behind the scaling of the conserved energy $$\mathcal{E}^*=\ln\mathcal{E}$$?

Take the four-velocity normalization $$u^\mu u_\mu = -1$$ and write it out as $$1 + u^r u_r + u^\vartheta u_\vartheta = -g^{tt}u_t^2 - 2 g^{t \varphi} u_t u_\varphi - g^{\varphi \varphi} u_\varphi^2$$ Now rewrite this in terms of $$\mathcal{E} = -u_t, \ell = -u_\varphi/u_t$$, take a logarithm of the equation, and use the properties $$\ln(xy) = \ln(x) + \ln(y)$$ and $$\ln(1 + x) = x + \mathcal{O}(x^2)$$. The quantity $$\mathcal{E}^*$$ is used just because it is additive and is actually the specific Newtonian energy without the rest-mass term in the Newtonian limit.
I will leave it as an exercise to the dear reader that the minima of the potential $$\mathcal{U}_{\rm eff}$$ actually also correspond to circular orbits.