## 5 Back-Reaction
### Dynamics of the Horizon  
The production of Hawking radiation emitted by a black hole formed by gravitational collapse is a process that transfers energy towards the future null infinity. The outgoing flux of Hawking quanta induces a loss of mass of the black hole, whose apparent horizon gradually shrinks, until its eventual evaporation. In this non-stationary process, the gravitational collapse triggers the production of Hawking radiation quanta, consequently exerting a back-reaction on the classical metric tensor. In the case of the canonical analogue black hole, we can study the dynamics of the horizon assuming the condensate to be initially in the stationary condensate configuration described in Section 3. Then, turning on the production of quasi-particles of the field \(\theta_{1}\), we observe a back-reaction on the wavefunction. In general, with spherical symmetry, the horizon radius \(r_{H}\) is defined as the radius at which the radial velocity of the condensate equals the speed of sound. In the unperturbed case we have  
\[0= \left[c_{s}^{2}-V_{0}^{2}\right]_{r_{H}}\,. \tag{92}\]  
If the condensate is in a state which is a small spherically symmetric perturbation of the stationary solution of the Gross-Pitaevskii equation, the density and the phase of the condensate are modified as described in Equations (88) and (89). (Notice that we are assuming that both the perturbed condensate wavefunction and the quasi-particles are in a globally spherically symmetric state, in which modes of opposite angular momenta have the same amplitude. The Hamiltonian driving the evolution ensures the conservation of the total angular momentum which is initially vanishing.) Consequently, the velocity of the condensate and the speed of sound change and, under this variation, Equation (92) is satisfied at a different radius \(r_{H}+\delta r_{H}\), where \(r_{H}\) is the unperturbed horizon radius:  
\[V_{0} \rightarrow V=V_{0}+\delta V=-\frac{\partial_{r}\left\langle\theta_{0} \right\rangle}{m}-\frac{\partial_{r}\left\langle\delta\theta\right\rangle}{m}\,, \tag{93}\] \[c_{s}^{2} \rightarrow c_{s}^{2}\left(1+\frac{\left\langle\delta\rho\right\rangle}{ \left\langle\rho_{0}\right\rangle}\right)\,,\] (94) \[r_{H} \rightarrow r_{H}+\delta r_{H}=r_{H}-\left[\frac{\delta\left(V^{2}/c_{s}^{ 2}\right)}{\partial_{r}\left(V_{0}^{2}/c_{s}^{2}\right)}\right]_{r_{H}}\,. \tag{95}\]  
The denominator in Equation (95) is different for each spherically symmetric system and is proportional to the surface gravity of the horizon, i.e., to the temperature of the black hole in Equation (78)  
\[\left[-\partial_{r}\left(V^{2}/c_{s}^{2}\right)\right]_{r_{H}}= 2\kappa>0\,. \tag{96}\]  
The radius of the perturbed horizon is smaller or larger than \(r_{H}\) depending on the sign of the variation \(\delta\left(V^{2}/c_{s}^{2}\right)\). For the canonical analogue black hole, we substitute the expression of \(V_{0}\) of Equation (36) in Equation (95) and we get  
\[\delta r_{H} = -\frac{r_{H}}{4}\left[\frac{\left\langle\delta\rho\right\rangle} {\left\langle\rho_{0}\right\rangle}-2\frac{\delta V}{c_{s}}\right]_{r_{H}}\,. \tag{97}\]  
Deriving the variation of the horizon radius with respect to time we observe which quantities determine the dynamics of \(\delta r_{H}\)  
\[\partial_{t}\delta r_{H}= -\left[\frac{\partial_{t}\delta\left(V^{2}/c_{s}^{2}\right)}{ \partial_{r}\left(V^{2}/c_{s}^{2}\right)}\right]_{r_{H}}-\delta r_{H}\left[ \partial_{t}\ln\partial_{r}\left(V^{2}/c_{s}^{2}\right)\right]_{r_{H}}\,. \tag{98}\]  
For a stationary spacetime the second term vanishes.