which arises from the accessibility of negative energy modes in a frame in which energy is conserved, i.e., in which conditions are stationary.

At early times in the sweep of the step, the WH horizon so defined is located where the BEC density, and therefore the sound speed, is significantly smaller. It therefore starts out moving much more slowly than the step. As the sweep progresses it accelerates smoothly, until it reaches a uniform velocity slightly less than that of the step. It is then approximately located at the left intersection of $c(x,t)$ and $-v(x,t)$, and the distance $L$ between the two horizons grows slowly and uniformly in time. This behavior can be seen in Figs. \[fig:stimulated\_HR\](b) and \[fig:M20\](b).

Cavity standing wave
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Figure \[fig:GP1\] shows comparisons of the simulated density profile with experiment. Figures (a)-(g) show the BEC density for $U_s = k \times 6$ nK after the launch of a sweep at 20 ms intervals, where $k$ is the Boltzmann constant. Figure (h) corresponds to the density profile at $t=$ 120 ms for $U_s = k \times 3$ nK. The coordinate origin in each panel has been displaced to coincide with $x_{\rm BH}$.

![\[fig:GP1\] (a-g) Density vs. time 