An exact solution for the Hawking effect in a dispersive fluid

We consider the wave equation for sound in a moving fluid with a fourth-order anomalous dispersion relation. The velocity of the fluid is a linear function of position, giving two points in the flow where the fluid velocity matches the group velocity of low-frequency waves. We find the exact solution for wave propagation in the flow. The scattering shows amplification of classical waves, leading to spontaneous emission when the waves are quantized. In the dispersionless limit the system corresponds to a $1+1$-dimensional black-hole or white-hole binary and there is a thermal spectrum of Hawking radiation from each horizon. Dispersion changes the scattering coefficients so that the quantum emission is no longer thermal. The scattering coefficients were previously obtained by Busch and Parentani in a study of dispersive fields in de Sitter space [Phys. Rev. D 86, 104033 (2012)]. Our results give further details of the wave propagation in this exactly solvable case, where our focus is on laboratory systems.

Figure 1

Top: The linear velocity profile (4) for $\alpha =1/2$. At $x=\pm 1/\alpha$ the flow speed matches the speed relative to the fluid of low-$k$ waves. Middle: Ray plots for waves in the flow with $\omega =1$, and with dispersion removed ($k_c\to \infty$). Red (blue) rays are for waves moving left (right) relative to the fluid. Solid (dotted) rays have positive (negative) frequency $\omega -v(x)k$ in a frame comoving with the fluid. This corresponds to a white-hole binary. Bottom: The ray plots when dispersion is included ($k_c=5$). The mode labels 1, 2, 3 and 4 refer to the four solutions $k(\omega )$ of the dispersion relation (3); these solutions are shown graphically in Fig. 2.

Figure 2

Plots of $\omega$ versus $k$ for the dispersion relation (3) in the linear profile (4), for three values of $x$ (with $\alpha =1/2$, $k_c=5$). The horizontal magenta line is $\omega =1$ and its intersection with the dispersion plots gives the propagating (real $k$) modes at this frequency; these mode solutions are labeled 1 to 4. Modes from an intersection with a red (blue) dispersion curve travel left (right) relative to the fluid. In addition, if the intersection is with a solid (dotted) dispersion curve, the mode has positive (negative) comoving frequency $\omega -v(x)k$. In the region between the horizons, such as in the middle plot for $x=0$, there are two propagating modes. Beyond the horizons ($x\ll 0$ and $x\gg 0$) modes 3 and 4 can propagate.

Figure 3

The integrand in (5) plotted in the complex $k$-plane, with $c_1=1$ and $c_2=0$ in (8). Color indicates the phase while brightness indicates the absolute value (brighter means bigger). The branch cut is placed along the positive imaginary axis. Parameter values are $\omega =1$, $\alpha =1/2$ and $k_c=5$. The top plot is for $x=4$, the bottom for $x=-4$. We take $C^{\prime }$ as the contour in (5) and thereby define an exact solution of (2).

Figure 4

The same as Fig. 3, but with the branch cut along the negative imaginary axis and a different contour $C^{\prime \prime }$. This defines another independent solution of (2).

Figure 5

A portion of the upper plot in Fig. 3, with the contour $C^{\prime }$ deformed to wrap around the singularity at $k=0$. The integrand decays exponentially along the positive imaginary axis. For the portion of the contour shown here, only the part running along both sides of the branch cut and around the singularity contributes to the integral for large $x$.

Figure 6

The integrand in (5) plotted in the complex $k$-plane, with $c_1=0$ and $c_2=1$ in (8). Color indicates the phase while brightness indicates the absolute value (brighter is bigger). The branch cut is placed along the positive imaginary axis. Parameter values are $\omega =1$, $\alpha =1/2$ and $k_c=5$. The top plot is for $x=4$, the bottom for $x=-4$. With $C^{\prime }$ as the contour in (5) this defines an exact solution of (2).

Figure 7

The same as Fig. 6, but with the branch cut along the negative imaginary axis and a different contour $C^{\prime \prime }$. This defines another independent solution of (2).

Figure 8

Heuristic ray picture for the wave solution (54), whose only incident asymptotic wave component is the normalized mode-2 wave (b22) on the left. The incident wave scatters into all outgoing modes, some of which have negative norm (modes 3 and 4) and some propagate to the left relative to the fluid (modes 1 and 4). As in Fig 1, the flow corresponds to a white-hole binary. Reversing $t$ (i.e. reading the plot top to bottom) corresponds to a black-hole binary, in which several modes in the past combine to give an outgoing low-$k$ mode 2 on the left.

Figure 9

Heuristic ray picture for the wave solution (70), whose only incident asymptotic wave component is the normalized mode-3 wave (b23) on the left. The incident wave has negative norm and scatters into all outgoing modes, some of which have positive norm (modes 1 and 2) and some propagate to the right relative to the fluid (modes 2 and 3 on the right).