Holography, Entropy and Extra Dimensions

We show that higher dimensional models (brane worlds) in which the scale of quantum gravity $M_*$ is much smaller than the apparent scale $M_P \sim 10^{19}$ GeV are in conflict with bounds arising from holography and black hole entropy. The thermodynamic entropy of astrophysical black holes and sub-horizon volumes during big bang nucleosynthesis exceed the relevant bounds unless $M_* > 10^{(4-6)}$ TeV, so a hierarchy relative to the weak scale is unavoidable. We discuss the implications for extra dimensions as well as holography.