Embolic aspects of black hole entropy

We attempt to provide a mesoscopic treatment of the origin of black hole entropy in (3+1)-dimensional spacetimes. We treat the case of horizons having space-like sections $\mathrm{\Sigma }$ which are topological spheres, following Hawking’s and the Topological Censorship theorems. We use the injectivity radius of the induced metric on $\mathrm{\Sigma }$ to encode the linear dimensions of the elementary cells giving rise to such entropy. We use the topological entropy of $\mathrm{\Sigma }$ as the fundamental quantity expressing the complexity of $\mathrm{\Sigma }$ on which its entropy depends. We point out the significance, in this context, of the Berger and Croke isoembolic inequalities.