Renormalized stress-energy tensor near the horizon of a slowly evolving, rotating black hole

The renormalized expectation value of the stress-energy tensor 〈T_μν{〉}ʳᵉⁿ$ of a quantum field in an arbitrary quantum state near the future horizon of a rotating (Kerr) black hole is derived in two very different ways: One derivation (restricted for simplicity to a massless scalar field) makes use of traditional techniques of quantum field theory in curved spacetime, augmented by a variant of the ``{η} formalism'' for handling superradiant modes. The other derivation (valid for any quantum field) uses the equivalence principle to infer, from 〈T_μν{〉}ʳᵉⁿ$ in flat spacetime, what must be 〈${T}_{{μ}{ν}}〉ʳᵉⁿ near the hole's horizon. The two derivations give the same result---a result in accord with a previous conjecture by Zurek and Thorne: 〈T_μν{〉}ʳᵉⁿ$, in any quantum state, is equal to that, 〈${T}_{{μ}{ν}}〉ZAMO, which zero-angular-momentum observers (ZAMO's) would compute from their own physical measurements near the horizon, plus a vacuum-polarization contribution T_μνᵛᵃᶜᵖᵒˡ, which is the negative of the stress-energy of a rigidly rotating thermal reservoir with angular velocity equal to that of the horizon ΩH, and (red-shifted) temperature equal to that of the Hawking temperature TH.A discussion of the conditions of validity for equivalence-principle arguments reveals that curvature-coupling effects (of which the equivalence principle is unaware) should produce fractional corrections of order α²{≡}(surface gravity of hole)²×{}(distance to horizon)² to T_μνᵛᵃᶜᵖᵒˡ; and since gravitational blue-shifts cause the largest components of T_μνᵛᵃᶜᵖᵒˡ in the proper reference frame of the ZAMO's to be of O(α^-2), curvature-coupling effects in T_μνᵛᵃᶜᵖᵒˡ and thence in 〈T_μν{〉}ʳᵉⁿ$ are of O(${{α}}⁰$) in the ZAMO frame.It is shown, by a quantum-field-theory derivation of the density matrix, that in the Hartle-Hawking vacuum the near-horizon ZAMO's see a thermal reservoir with angular velocity ${{Ω}}H$ and temperature ${T}H$ whose thermal stress-energy 〈${T}_{{μ}{ν}}〉ZAMO gets renormalized away by T_μνᵛᵃᶜᵖᵒˡ, annulling the O(α^-2) and O(α^-1) pieces of 〈T_μν{〉}ʳᵉⁿ$, and leaving only the O(${{α}}⁰$) vacuum-polarization, curvature-coupling contributions. This translates into 〈${T}ₗₗ〉^ren=〈T_lφ〉ʳᵉⁿ=0 on the future horizon in the Hartle-Hawking vacuum, where l and {φ} denote components on the horizon generator l^μ and on the generator of rotations {∂}/{∂}{φ}. In quantum states representing a black hole in the real Universe (with both evaporation and accretion occurring), the fluxes of red-shifted energy and angular momentum across the future horizon, per unit solid angle sin{θ} d{θ} d{φ}, are shown to equal the corresponding accretion fluxes into the hole's atmosphere from the external universe minus the fluxes evaporated by the hole. As a consequence, the hole's horizon evolves in accord with standard expectations. As an aside it is shown that the Hartle-Hawking vacuum state {}H〉 is singular at and outside the velocity-of-light surface scrSL, i.e., at sufficiently large radii that the rest frame of its thermal reservoir is moving at or faster than the speed of light. Its renormalized stress-energy tensor is divergent there, and its Hadamard function does not have the correct behavior. To make {}H〉 be well behaved (and have the properties described above), one must prevent its rotating thermal reservoir from reaching out to scrSL, e.g., by placing a perfectly reflecting mirror around the hole just inside scrSL.