Acoustic analogues of black hole singularities

We search for acoustic analogues of a spherical symmetric black hole with a pointlike source. We show that the gravitational system has a dynamical counterpart in the constrained, steady motion of a fluid with a planar source. The equations governing the dynamics of the gravitational system can be exactly mapped in those governing the motion of the fluid. The different meaning that singularities and sources have in fluid dynamics and in general relativity is also discussed. Whereas in the latter a pointlike source is always associated with a (curvature) singularity in the former the presence of sources does not necessarily imply divergences of the fields.

Figure 1

Behavior of the normalized parameters $\psi /a^{4/3}$ (bold black line), $v_0/v_0(r_h)$ (black line), ${\rho }_0/{\rho }_0(r_h)$ (dashed line) $c/c(r_r)$ (gray line) as a function of the variable $Y$ for a flux tube with constant section and nonhomogeneous external potential characterized by $\alpha =1,M/\lambda =3/4$ and $n=2$. The velocity, fluid density and speed of sound are normalized to their horizon values. The acoustic horizon is located at $Y=1$, the singularity at $Y=Y_s=1/2$.

Figure 2

Behavior of the normalized parameters $\psi /a^{2/3}$ (bold black line), $v_0/v_0(r_h)$ (black line), ${\rho }_0/{\rho }_0(r_h)$ (dashed line) $c/c(r_h)$ (gray line) as a function of the variable $Y$ for a flux tube with constant section and nonhomogeneous external potential characterized by $\alpha =1,M/\lambda =3/4$ and $n=1/2$. The velocity, fluid density and speed of sound are normalized to their horizon values. The acoustic horizon is located at $Y=1$, the singularity at $Y=Y_s=1/2$.

Figure 3

Behavior of the normalized parameters $A/A(r_h)$ (bold black line), $v_0/v_0(r_h)$ (black line), ${\rho }_0/{\rho }_0(r_h)$ (dashed line) $c/c(r_h)$ (gray line) as a function of the variable $Y$ for a flux tube with nonconstant section characterized by $\omega =1,M/\lambda =9/16$ and $n=2$. The flux tube section, velocity and fluid density and the speed of sound are normalized to their horizon values. The acoustic horizon is located at $Y=1$, the singularity at $Y=Y_s=1/8$. Notice that the range of the coordinate $Y$ is $Y_s=1/8\le Y<\sqrt{27/8}$. For $Y=\sqrt{27/8}$ $A$ diverges whereas $v_0=0$. The acoustic black hole does not describe the asymptotic region of the gravitational black hole.

Figure 4

Behavior of the normalized parameters $A/A(r_h)$ (bold black line), $v_0/v_0(r_h)$ (black line), ${\rho }_0/{\rho }_0(r_h)$ (dashed line) $c/c(r_h)$ (gray line) as a function of the variable $Y$ for a flux tube with nonconstant section characterized by $\omega =1,M/\lambda =9/64$ and $n=1/2$. The flux tube section, velocity and fluid density and the speed of sound are normalized to their horizon values. The acoustic horizon is located at $Y=1$, the singularity at $Y=Y_s=1/8$. Notice that the range of the coordinate $Y$ is $Y_s=1/8\le Y<8/3^{3/2}$. For $Y=8/3^{3/2}$ $A$ diverges whereas $v_0=0$. The acoustic black hole does not describe the asymptotic region of the gravitational black hole.

Figure 5

Pictorial representation of the acoustic black hole (the Laval nozzle), with singularity, horizon ($v_0=c$), interior ($v_0>c$) and exterior ($v_0<c$) region. The direction of the fluid flow is represented by the arrow. Notice that in the depicted situation we have a well (a source with negative flow) in the position of the singularity.