Negative-temperature Hawking radiation seen near the inner horizon, the outer horizon, and beyond

The near-horizon geometry of an astrophysical black hole can be drastically altered when quantum effects like Hawking radiation are included on top of the classical vacuum metric. Here we study particle production effects through two different means: first, an effective temperature can be defined that in the adiabatic limit exactly reproduces the Hawking effect but generalizes to any inertial observer at an arbitrary position inside or outside the black hole, looking in any direction. Second, in the near-horizon regimes, the exact Bogoliubov coefficients governing the vacuum expectation value of the number operator can be computed to find the full Hawking greybody spectrum. When applied to a charged black hole with a quantum scalar field, these two calculations can help answer several questions about the semiclassical nature of astrophysical black holes: does Hawking radiation contribute to mass inflation at the inner horizon, and is it confined to a single point in the radial direction? Can the Hawking temperature ever be negative, and how should that be properly interpreted?