I submitted this question on physics.stackexchange a couple of weeks ago; it received some upvotes but no answer, so I thought I would resubmit it here, by analogy with the standard practice for maths questions. I apologize for any breaches of etiquette.
The Bekenstein bound says that the maximum entropy which can be contained in a (spherical) volume of space with a given amount of energy is proportional to the amount of energy multiplied by the "length scale factor" (radius) of the volume - up to some dimensionless constant, it is just the aforementioned quantity expressed in natural units.
Where does this "dimensionality" come from? Is it of any particular significance? (I understand that the entropy itself is dimensionless, but the dimensionless terms making up the bound do nevertheless correspond to the energy multiplied by the length scale as expressed in natural units)
Wikipedia says "Note that while gravity plays a significant role in its enforcement, the expression for the bound does not contain the gravitational constant G." If the bound is independent of the value of G, in what sense is general relativity "necessary" for deriving it (as opposed to merely having been used in the standard derivation)? Is it possible that the bound could in fact be derived using only special relativity and statistical mechanics?
On a much more speculative note, I note that the "coupling constant" associated with an interaction of inverse-square type has the same "dimensionality" of force multiplied by distance squared. It's tempting to wonder whether this might somehow measure information shared between two entities via the interaction. Might this line of speculation have any merit?