Severe constraints on the loop-quantum-gravity energy-momentum dispersion relation from the black-hole area-entropy law

We explore a possible connection between two aspects of loop quantum gravity which have been extensively studied in the recent literature: the black-hole area-entropy law and the energy-momentum dispersion relation. We observe that the original Bekenstein argument for the area-entropy law implicitly requires information on the energy-momentum dispersion relation and on the position-momentum uncertainty relation. Recent results show that in first approximation black-hole entropy in loop quantum gravity depends linearly on the area, with small correction terms which have logarithmic or inverse-power dependence on the area. And it has been argued that in loop quantum gravity the dispersion relation should include terms that depend linearly on the Planck length, while no evidence of modification of the position-momentum uncertainty relation has been found. We observe that this scenario with Planck-length-linear modification of the dispersion relation and unmodified position-momentum uncertainty relation is incompatible with the black-hole-entropy results, since it would give rise to a term in the entropy formula going like the square root of the area.