Is there a name for that moving blank space on an oscilloscope?
Unlike a scrolling display possible on digital scopes the analog CRT scope draws from left to right. The scope trigger circuit determines when the trace starts.



Figure 1. The electron beam is deflected by the charge on the horizontal and vertical deflection plates. A sawtooth wave applied to the horizontal plates to give a linear left to right trace. Image source: hit.bme.hu.

The phosphor glows when struck by the electron beam and then decays. The phosphor is chosen to give a reasonable balance between persistance at low trace speeds (where a long decay would be advantages) and quick decay to prevent blur on changing waveforms at high speed. This could be compensated for somewhat by adjusting the beam intensity.


  Is there a name for that moving blank space on an oscilloscope?


I like @EugeneSh's comment that the blank space is "the future". However, there is a ghostly line visible on the right half of the screen and this is the past (obviously from the previous trace).