Possible erratum in Hartle's book "Gravity" in example 15.1

Question

I'm reading Hartle's book Gravity (2014), and I think there's an erratum in example 15.1. In this example, he says that for a massive particle at infinity at rest, which starts falling radially towards a Kerr black hole, the shape of the particle's orbit can be calculated by integrating

\begin{equation} \frac{\text{d} \varphi}{\text{d} r} = -\frac{2Ma}{r \Delta} \Bigg(\frac{2M}{r} \Bigg(1 - \frac{a^2}{r^2} \Bigg) \Bigg)^{-\frac{1}{2}} \end{equation}

However, I think this equation is wrong. In order to derive this equation, Hartle uses the following equations for equatorial geodesics

$$\begin{equation} \frac{\text{d} \varphi}{\text{d} \tau} = \frac{1}{\Delta} \Bigg(\Bigg(1 - \frac{2M}{r} \Bigg) \ell + \frac{2Ma}{r} e \Bigg) \end{equation}\tag{15.18b}$$

$$\begin{equation} \frac{e^2 - 1}{2} = \frac{1}{2} \Bigg(\frac{\text{d} r}{\text{d} \tau} \Bigg)^2 + V_{\text{eff}} (e, r, \ell) \end{equation}\tag{15.19}$$

where

$$\begin{equation} V_{\text{eff}} (e, r, \ell) = -\frac{M}{r} + \frac{\ell^2 - a^2 (e^2 - 1)}{2r^2} - \frac{M (\ell - ae)^2}{r^3} \end{equation}\tag{15.20}$$

Since our particle is initially at rest and moves radially, $e = 1$ and $\ell = 0$. Putting these conditions in equation 15.19 together with the fact that our particle is falling towards our black hole $\text{d} r / \text{d} \tau = -\sqrt{-2 V_{\text{eff}} (e = 1, r, \ell = 0)} = -\sqrt{2M / r + 2Ma^2 / r^3}$. Thus, using the chain rule and plugging equation 15.18b (with $e = 1$ and $\ell = 0$) we obtain

\begin{equation} \frac{\text{d} \varphi}{\text{d} r} = -\frac{2Ma}{r \Delta} \Bigg(\frac{2M}{r} \Bigg(1 + \frac{a^2}{r^2} \Bigg) \Bigg)^{-\frac{1}{2}} \end{equation}

Which differs from Hartle's result in the radicand. To check this, I also did this example using Wald's equations 12.3.23 (with $\theta = \pi / 2$), 12.3.24 and 12.3.25 (which is the same as equations 15.19, 15.20 put together) and I arrive at the same conclusion. Am I right?

Answer 1

Hartle's expression for $d\phi/dr$ diverges not just at $r_+$ and $r_-$ but also at $r=a$, which is inside the horizon. There is no coordinate singularity at $r=a$, so this would seem to be physically meaningful, indicating something like a turning point, which doesn't seem right physically. We expect a particle infalling from infinity to get all the way to the singularity, not stop at $r=a$ and turn around. The Wikipedia article "Kerr metric" has a section "Trajectory equations" that is more general and set up somewhat differently, but they don't seem to show any singular behavior of $dr/d\lambda$ or $d\phi/d\lambda$ at $r=a$. So it does look to me like this must be a mistake in Hartle, but I could be wrong.