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Hi
First if all - Lorentz contraction is geometric but not in Euclid space, but rather it is a rotation in Minkowski space time. Whilst this may seem a subtle difference it is huge in reality. Like many mind concepts like baloons in cosmology and strings for GUTs, the concept of contraction in Euclid geometry is flawed and will lead on up the wrong path conceptually. If you want to think about changes in a box then it is better to imagine a rotation in Euclid space and the corresponding contraction you would see, but of course this is ultimately misleading as well.
You do not mention the context in which the reference is made, but it will definitely be in some context associated with an observer at rest observing relativistic particles. Examples could be observing fast moving boson gas clouds or superconducting particles in a wire. In this instance there has to be adjustments for the perceived particle density for the observer at rest. Here we are talking about the perceived reduction in volume and therefore the perceived increase in density.
I have not read the book but would be interested as to how this relates to gravity.
I hope that helps.
Chris
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