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In 1984, 't Hooft famously used a brickwall (aka stretched horizon) tocompute black hole entropy up to a numerical pre-factor. This calculation issometimes interpreted as due to the entanglement of the modes across thehorizon, but more operationally, it is simply an indirect count of thesemi-classical modes trapped between the stretched horizon and the angularmomentum barrier. Because the calculation was indirect, it needed both the massand the temperature of the black hole as inputs, to reproduce the area. A moreconventional statistical mechanics calculation should be able to get theentropy, once the ensemble is specified (say via the energy, in amicrocanonical setting). In this paper, we explicitly compute black hole normalmodes in various examples, numerically as well as (in various regimes)analytically. The explicit knowledge of normal modes allows us to reproduce$both$ the Hawking temperature as well as the entropy, once the charges arespecified, making this a conventional statistical mechanics calculation. Aquasi-degeneracy in the angular quantum numbers is directly responsible for thearea scaling of the entropy, and is the key distinction between the Planckianblack body calculation (volume scaling) and the 't Hooftian calculation (areascaling). We discuss the (rotating) BTZ case in detail and match thethermodynamic quantities $exactly$. Schwarzschild and Kerr normal modes arediscussed in less detail using near-horizon approximations. Our calculationsreveal a new hierarchy in the angular quantum numbers, which we speculate isrelated to string theory.

Normal Modes of the Stretched Horizon: A Bulk Mechanism for Black Hole Microstate Level Spacing