Spherically symmetric loop quantum gravity: Schwarzschild spacetimes with a cosmological constant

We provide a quantization of the Schwarzschild spacetime in the presence of a cosmological constant, based on midisuperspace methods developed in the spherically symmetric sector of loop quantum gravity, using in particular the 'improved dynamics' scheme. We include both the deSitter and anti-deSitter cases. We find that the quantization puts a Planckian upper limit on the possible values of a positive cosmological constant similar to the bounds obtained earlier from studies of homogeneous spacetimes with a cosmological constant. Using semiclassical physical states, we obtain the effective metric and demonstrate the causal structure for various cases. Quantum gravity modifications ensure that the singularity is replaced by a transition surface in all the cases, where the curvature invariants approach mass-independent Planckian bounds. Analysis of the effective stress-energy tensor shows that the null energy condition is violated in the vicinity of the transition surface.