frame. The green (light gray) curve is the local speed of sound $c(x)=\sqrt{g{n_{\rm bf}}(x)/m}$, where $g$ is a coupling constant defined in Appendix \[app:1dGPE\], and ${n_{\rm bf}}(x)$ is the local density of the background flow, which we identify here using a WFT of the density $n(x)$ (see Appendix \[app:FT\_vc\] for details).

### Locating the black and white hole horizons {#sec:Locating}

The black hole horizon is defined as the location where a right moving phonon is at rest in the step frame. This corresponds to the right intersection of $c(x)$ and $-v(x)$ in Fig. \[fig:exp\](c). The step frame is distinguished as the one in which the system is closest to being stationary near the step. In particular, $c(x)$ and $v(x)$ are nearly steady where the trapped BEC spills over the moving step.

The definition of the WH horizon is not as simple, because the step frame is not a global stationary frame of the system, due to the spatial variation in the trap potential and background condensate density. Instead, what is dynamically important is the location of the transition from supersonic to subsonic in the frame in which conditions are [*locally*]{} stationary. In particular, this is the locus of Čerenkov radiation, 