Positivity Conditions for Generalised Schwarzschild Space-Times

We analyse the impact of positivity conditions on static spherically symmetric deformations of the Schwarzschild space-time. The metric is taken to satisfy, at least asymptotically, the Einstein equation in the presence of a non-trivial stress-energy tensor, on which we impose various physicality conditions. We systematically study and compare the impact of these conditions on the space-time deformations. The universal nature of our findings applies to both classical and quantum metric deformations with and without event horizons. We further discuss minimal realisations of the asymptotic stress energy tensor in terms of physical fields. Finally, we illustrate our results by discussing concrete models of quantum black holes.