## 3 Canonical Analogue Black Holes  
Among the acoustic metrics simulated with Bose-Einstein condensates, it is convenient for our investigation to consider the canonical analogue black hole.  
A canonical analogue black hole could be obtained as the stationary solution of the Gross-Pitaevskii equation with spherical symmetry and with homogeneous atom number density. Due to the spherical symmetry, it presents a horizon at the radius where the velocity of the ingoing flow is equal to the speed of sound. The stationarity of the background metric, of the mean-field, means that the deviations from this configuration should be understood as the effect of the phenomenology previously neglected: the subleading effect of the quasi-particles and their back-reaction. The homogeneity of the atom density \(\left\langle\rho_{0}\right\rangle\) means that the conformal factor \(\left\langle\rho_{0}\right\rangle/c_{s}\) is a number, and can trivially be transformed away.  
Taking a constant and homogeneous number density \(\left\langle\rho_{0}\right\rangle\), and assuming \(\left\langle\theta_{0}\right\rangle\) to be a function of the radius only, from Equations (13) and (14) we obtain the stationary spherically symmetric solution of the Gross-Pitaevskii equation. The superfluid velocity and the external potential setting the corresponding velocity profile are respectively  
\[-\frac{\partial_{r}\left\langle\theta_{0}\right\rangle}{m} = c_{s}\frac{r_{H}^{2}}{r^{2}}=V_{0}\,, \tag{36}\] \[V_{\rm ext} = -mc_{s}^{2}\left(1+\frac{1}{2}\frac{r_{H}^{4}}{r^{4}}\right)\,, \tag{37}\]  
with an ingoing flux of atoms crossing every closed surface containing the origin at a rate of \(4\pi c_{s}r_{H}^{2}\left\langle\rho_{0}\right\rangle\) atoms per unit time, which must be pumped in the system and move towards the singularity in the origin. The radius of the horizon \(r_{H}\) is set by the external potential \(V_{\rm ext}\).  
Omitting the conformal factor, the analogue metric implied by Equation (36) is therefore  
\[\mathrm{d}s^{2}= -\left(1-\frac{r_{H}^{4}}{r^{4}}\right)c_{s}^{2}\mathrm{d}t^{2}+2 \frac{r_{H}^{2}}{r^{4}}c_{s}\mathrm{d}t\mathrm{d}r+\mathrm{d}r^{2}+r^{2} \mathrm{d}\theta^{2}+r^{2}\sin^{2}\theta\mathrm{d}\phi^{2}\,. \tag{38}\]  
The singularity in the origin poses a problem that would be unavoidable in devising an experiment: one should either remove the atoms reaching the origin and put them back in the system at large radius [42], or should consider only the region in which the hydrodynamical limit holds with a good approximation, at a distance from the origin and for a period of time such that it is not affected by atoms which flowed towards the origin. (To implement the latter, one could devise a \(3D\) hard wall trapping potential, possibly spherical, plus a variation in the central region of the scattering length and suitable external potential there added via a superimposed laser beam.)  
These are however practical concerns that we shall assume dealt with in our theoretical study. In addition to them, we observe that the possible removal of atoms from the system would affect the atomic correlation functions and introduce statistical mixture, but the dynamics of the quasi-particles in the near-horizon region, on which we focus on the rest of the paper, would not be sensibly affected.