Energy flow in acoustic black holes

We present the results of an analysis of superradiant energy flow due to scalar fields incident on an acoustic black hole. In addition to providing independent confirmation of the recent results in [E. Berti, V. Cardoso, and J. P. S. Lemos, Phys. Rev. D 70, 124006 (2004).], we determine in detail the profile of energy flow everywhere outside the horizon. We confirm explicitly that in a suitable frame the energy flow is inward at the horizon and outward at infinity, as expected on physical grounds.

Figure 1

Graph of $\mathrm{V}({\mathrm{r}}^{*})$ vs ${\mathrm{r}}^{*}$ for values of $B$ shown in the legend.

Figure 2

Graphs of $|\mathrm{R}{|}^2$ vs $\omega$ for $m=1$ and various values of $B$ as shown in the legends.

Figure 3

Graphs of $|\mathrm{R}{|}^2$ vs $\omega$ for $m=2$ and values of $B$ shown in the legends.

Figure 4

The left panel is a graph of the maximum value of $|\mathrm{R}{|}^2$ versus $B$ for $m=1$ using units where $A=c=1$. The right panel is a graph of $|\mathrm{R}{|}^2$ versus $\omega$ for $m=2$ and $\delta =B/\omega =10$.

Figure 5

Left panel is a graph of $|\mathcal{R}{|}^2$ versus $\alpha$ with $2\alpha {\omega }_2=10$. From the top the curves have $\alpha =.5\beta$, $\alpha =.4\beta$, $\alpha =.3\beta$, and $\alpha =.2\beta$. Right panel is the same set of curves with $2\alpha {\omega }_2=5$.

Figure 6

Graph of the rate of outward energy flow across a circle of radius $r$ surrounding the acoustic black hole for $B=0.5$, $m=1$, and $\omega =1$. The energy flow is strictly negative indicating that net energy flow is inward everywhere.

Figure 7

Graph of the rate of outward energy flow across a circle of radius $r$ surrounding the acoustic black hole for $B=1$, $m=1$, and $\omega =1$. Far from the event horizon the rate of outward energy flow is positive which indicates superresonance. The energy flow near the horizon ($r=1$) is negative indicating that the energy flow is inward.

Figure 8

Graph of outward energy flow through a circle at $r=20$ as a function of $\omega$ for $m=1$ and different values of $B$. As $\omega$ increases, net outward energy flow ceases in each case very close to the cutoff frequency $\omega =mB$, for the superresonance effect.

Figure 9

Graph of the well function for $B=1.6$, $m=1$, and $\omega =1$ (solid line), and the rate of energy flow as a function of $r$ for same parameters (dotted line). The net energy flow becomes outward at a radius greater than the smallest value of $r$ for which the potential crosses zero.