## 3 Canonical Analogue Black Holes
### Klein-Gordon Equation and Field Modes  
The massless scalar field living in our analogue spacetime can be described in terms of its modes, found by solving the Klein-Gordon equation. For a stationary spherically symmetric spacetime in Painleve-Gullstrand coordinates, the equation to solve is  
\[0 = \frac{1}{\sqrt{g}}\partial_{\mu}\sqrt{g}g^{\mu\
u}\partial_{\
u} f\left(t,r,\Omega\right)= \tag{39}\] \[= \left(-\partial_{c_{s}t}^{2}+\sqrt{1-h}\partial_{r}\partial_{c_{ s}t}+\frac{1}{r^{2}}\partial_{r}r^{2}\sqrt{1-h}\partial_{c_{s}t}+\frac{1}{r^{2}} \partial_{r}r^{2}h\partial_{r}-\frac{L^{2}}{r^{2}}\right)f\left(t,r,\Omega \right)\,,\]where \(\Omega\) is a compact notation for the solid angle and \(L^{2}\) is the usual angular momentum operator  
\[L^{2}= -\tfrac{1}{\sin\theta}\partial_{\theta}\sin\theta\partial_{\theta}-\tfrac{1 }{\sin^{2}\theta}\partial_{\theta}^{2}\,. \tag{40}\]  
We can find a complete set of solutions looking for those which preserve the symmetries of the system [38; 39], i.e., the eigenfunctions of both the time derivative and the angular momentum operator  
\[f_{\omega lm}\left(t,r,\Omega\right) = e^{-i\omega t}\gamma_{lm}\left(\Omega\right)f_{\omega l}\left(r \right)\,, \tag{41}\]  
\[\partial_{t}f_{\omega lm}\left(t,r,\Omega\right) = -i\omega f_{\omega lm}\left(t,r,\Omega\right)\,, \tag{42}\] \[L^{2}f_{\omega lm}\left(t,r,\Omega\right) = l\left(l+1\right)f_{\omega lm}\left(t,r,\Omega\right)\,,\] (43) \[\partial_{\phi}f_{\omega lm}\left(t,r,\Omega\right) = imf_{\omega lm}\left(t,r,\Omega\right)\,, \tag{44}\]  
where the functions \(Y_{lm}\left(\Omega\right)\) above are the usual spherical harmonics.  
So Equation (39) becomes  
\[0 = \left(\frac{\omega^{2}}{c_{s}^{2}}-i\frac{\omega}{c_{s}}\sqrt{1- h}\partial_{r}-i\frac{\omega}{c_{s}}\frac{1}{r^{2}}\partial_{r}r^{2}\sqrt{1-h}+ \frac{1}{r^{2}}\partial_{r}r^{2}h\partial_{r}-\frac{l\left(l+1\right)}{r^{2}} \right)f_{\omega l}\left(r\right)= \tag{45}\] \[= \frac{e^{i\frac{\omega}{c_{s}}}\int^{r}\frac{\sqrt{1-h}}{h} \mathrm{d}r}{rh}\left(h\partial_{r}h\partial_{r}+\frac{\omega^{2}}{c_{s}^{2}} -\frac{h\left(\partial_{r}h\right)}{r}-\frac{hl\left(l+1\right)}{r^{2}}\right) re^{-i\frac{\omega}{c_{s}}}\int^{r}\frac{\sqrt{1-h}}{h}\mathrm{d}r}f_{\omega l} \left(r\right)\,.\]  
We have written Equation (45) reorganizing the various contributions to have a clearer view of the problem. The operator \(h\partial_{r}\) can be understood as the directional derivative with respect to the tortoise radial coordinate. The term \(h\partial_{r}h/r\) is a potential which induces mode mixing. This second order differential equation will have two linearly independent solutions for each set of eigenvalues, an outgoing mode and an ingoing mode. Splitting the solutions in norm and phase we get two coupled real differential equations  
\[f_{\omega l}\left(r\right) = \sqrt{n_{\omega l}\left(r\right)}e^{i\phi_{\omega l}\left(r \right)}e^{i\frac{\omega}{c_{s}}\int^{r}\frac{\sqrt{1-h}}{h}\mathrm{d}r}\,, \tag{46}\] \[0 = \frac{1}{r}\left(h\partial_{r}h\partial_{r}+\frac{\omega^{2}}{c_ {s}^{2}}-\frac{h\left(\partial_{r}h\right)}{r}-\frac{l\left(l+1\right)h}{r^{2 }}\right)r\sqrt{n_{\omega l}\left(r\right)}e^{i\phi_{\omega l}\left(r\right)}\,,\] (47) \[\Downarrow\] \[\partial_{r}\phi_{\omega l}\left(r\right) = \frac{C}{r^{2}ln_{\omega l}\left(r\right)}\,,\] (48) \[0 = \frac{1}{r}\left(h\partial_{r}h\partial_{r}+\frac{\omega^{2}}{c_ {s}^{2}}-\frac{h\left(\partial_{r}h\right)}{r}-\frac{l\left(l+1\right)h}{r^{2 }}\right)r\sqrt{n_{\omega l}}-\frac{C^{2}}{r^{4}}\frac{1}{\sqrt{n_{\omega l} ^{3}\left(r\right)}}\,, \tag{49}\]  
Different integrating constants \(C\) set the sign of the derivative of the phase and give the different solutions.