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 MPL (2 1∕GN ∼ M PL) and energy cutoff Λ ∼ 1∕ 𝜖,
√ --- Λ = MPL ∕ N , (326 )
where the precise numerical pre-factor depends on the concrete model. It is amazing to note that exactly this relation (326View Equation) was proposed by Dvali [77Jump To The Next Citation Point, 79Jump To The Next Citation Point, 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 (326View Equation), provided it is correct, helps to reproduce precisely the Bekenstein–Hawking entropy as an entropy of entanglement and automatically solves the species puzzle [80Jump To The Next Citation Point].
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