### 8.4 Gravity cut-off

If we compare the two examples when the Bekenstein–Hawking entropy is correctly reproduced, the
model of induced gravity with minimally-coupled constituents and the brane world model, we find that the
success of the two models is strongly based on a precise relation between Newton’s constant, the number of
species and the UV cut-off. This relation can be reformulated in terms of the Planck mass
() and energy cutoff ,
where the precise numerical pre-factor depends on the concrete model. It is amazing to note that exactly
this relation (326) was proposed by Dvali [77, 79, 78] to hold in general in a theory of quantum gravity
coupled to a large number of matter species. The arguments, which were used to get this relation, although
they include some thought experiments with black holes, are, in fact, unrelated to (and are thus
independent of) the entropy. However, it is clear that relation (326), provided it is correct, helps to
reproduce precisely the Bekenstein–Hawking entropy as an entropy of entanglement and automatically
solves the species puzzle [80].