Mapping a shrinking black hole horizon with quantum corrections included

This sixth installment of a series extends a first-principles black hole perturbation scheme to second order, now focusing on apparent horizons — the observer-dependent boundaries that replace event horizons in evaporating black holes. The authors derive the horizon shape at second order and show how to reconstruct the full four-dimensional metric from reduced phase-space variables. Taking quantum expectation values of that metric produces an effective classical geometry, which can be drawn as a corrected Penrose diagram. The paper closes by outlining how quantisation in this reduced phase space should cause the apparent horizon area to decrease, consistent with Hawking radiation.