On density variation

We can now also consider the dilation described above to be a valid choice of f, if we let g(x)=1/8. The total mapping doesn't preserve density as we required in our simple model of Newtonian mechanics, but now we have a mapping which can be used to describe a gas whose container has expanded in volume by a factor of 2 in all dimensions. (This transformation is still forbidden if we set g(x)=1, which corresponds to the matter-creating transformation which was forbidden in the structured-Newtonian regime.)

It's worth noting that the rotation and translation maps from the Newtonian regime cannot have a decrease in density everywhere in the matter-preservation model: that is, we can't have g(x)<1 for all x in S, because we need to preserve the mass. If we want to talk about ideal rotations and translations, we would probably want to leave the "composition" of the object constant: the most natural way to do this is to have g(x)=1. When we refer to rotations or translations in the matter distribution model, this is what we'll refer to.