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, $8 \pi G \Delta Q = \kappa \Delta A$.