I think for this to be a complete explanation, it needs to be shown that what was "the time" coordinate is not expanding in some sense.
Does volume even make sense in GR?
Thought experiment here would be: suppose me and you appeared stationery (from the point of view of the outside reference frame) just above the horizon of a non-rotating black hole in the manner of Boltzmann brains. We immediately start falling and cross the horizon. Now will you see me receding from you or going at you? Will I be red-shifted or blue-shifted?
Does volume even make sense in GR?
Thought experiment here would be: suppose me and you appeared stationery (from the point of view of the outside reference frame) just above the horizon of a non-rotating black hole in the manner of Boltzmann brains. We immediately start falling and cross the horizon. Now will you see me receding from you or going at you? Will I be red-shifted or blue-shifted?