On horizon constraints and Hawking radiation

Questions about black holes in quantum gravity generally presuppose the presence of a horizon. Recently Carlip has shown that enforcing an initial data surface to be a horizon leads to the correct form for the Bekenstein-Hawking entropy of the black hole. Requiring a horizon also constitutes fixed background geometry, which generically leads to non-conservation of the matter stress tensor at the horizon. In this work, I show that the generated matter energy flux for a Schwarzschild black hole is in agreement with the first law of black hole thermodynamics, $8\pi G\mathrm{\Delta }Q=\kappa \mathrm{\Delta }A$.