Black hole analogues in two-dimensional flows with constant shear

Abstract

We review the Analogue Gravity description of a unidirectional water wave system, assuming no prior knowledge of General Relativity or differential geometry. In so doing, we generalize established results concerning an effective curved spacetime for surface waves on irrotational 2D flows, by including flows with constant shear. We show that such flows remain perfectly compatible with the existence of an effective curved spacetime and, in particular, of a metric description.

Figures (4)

• Figure 1: Space-time diagrams of two scattering processes occurring in a black-hole spacetime. The flow is from left to right, with the horizon occurring at the center. Red (blue) lines correspond to the co- (counter-) propagating mode. The negative-energy mode is shown in dashed line. (left) The Hawking effect: Governed by the bifurcation of the counter-propagating mode at the horizon, the relative amplitude between the two follows an exponential law that can be interpreted as a Boltzmann factor. (right) Scattering of an incident mode by inhomogeneities in the metric: The co-propagating mode simply crosses the horizon as it experiences nothing unusual there, and it is only on either side that it scatters into each of the counter-propagating modes.
• Figure 2: Scattering coefficients over a fixed obstacle with identical values of the flow rate $q = 0.2\,{\rm m}^{2}/{\rm s}$, across several values of the vorticity. The top row shows the background flow: the water depth (left), the Froude number (center), and the profiles of $\bar{u}$ and $c$ (right). On the bottom are shown the scattering coefficients for the reflected mode (left), the transmitted mode (center), and the negative-energy mode (right).
• Figure 3: Scattering coefficients over a fixed asymmetric obstacle (described in Eq. (\ref{['eq:ACRI2010_obstacle']})) with identical values of the flow rate $q = 0.2\,{\rm m}^{2}/{\rm s}$, across several values of the vorticity. The top row shows the background flow: the water depth (left), the Froude number (center), and the profiles of $\bar{u}$ and $c$ (right). On the bottom are shown the scattering coefficients for the reflected mode (left), the transmitted mode (center), and the negative-energy mode (right).
• Figure 4: Scattering coefficients over a fixed asymmetric obstacle (described in Eq. (\ref{['eq:ACRI2010_obstacle_smoothed']}), which is a smoothed version of that in Eq. (\ref{['eq:ACRI2010_obstacle']})) with identical values of the flow rate $q = 0.2\,{\rm m}^{2}/{\rm s}$, across several values of the vorticity. The top row shows the background flow: the water depth (left), the Froude number (center), and the profiles of $\bar{u}$ and $c$ (right). On the bottom are shown the scattering coefficients for the reflected mode (left), the transmitted mode (center), and the negative-energy mode (right).