Frolov, Valery P. and Thorne, Kip S. (1989) Renormalized stressenergy tensor near the horizon of a slowly evolving, rotating black hole. Physical Review D, 39 (8). pp. 21252154. ISSN 05562821. http://resolver.caltech.edu/CaltechAUTHORS:FROprd89

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Abstract
The renormalized expectation value of the stressenergy tensor〈Tμν〉^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〈Tμν〉^ren in flat spacetime, what must be〈Tμν〉^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: 〈Tμν〉^ren, in any quantum state, is equal to that, 〈Tμν〉^ZAMO, which zeroangularmomentum observers (ZAMO’s) would compute from their own physical measurements near the horizon, plus a vacuumpolarization contribution Tμνvac pol , which is the negative of the stressenergy of a rigidly rotating thermal reservoir with angular velocity equal to that of the horizon ΩH, and (redshifted) temperature equal to that of the Hawking temperature TH. A discussion of the conditions of validity for equivalenceprinciple arguments reveals that curvaturecoupling effects (of which the equivalence principle is unaware) should produce fractional corrections of order α^2≡(surface gravity of hole)^2×(distance to horizon)^2 to Tμνvac pol; and since gravitational blueshifts cause the largest components of Tμνvac pol in the proper reference frame of the ZAMO’s to be of O(α2), curvaturecoupling effects in Tμνvac pol and thence in 〈Tμν〉^ren are of O(α^0) in the ZAMO frame. It is shown, by a quantumfieldtheory derivation of the density matrix, that in the HartleHawking vacuum the nearhorizon ZAMO’s see a thermal reservoir with angular velocity ΩH and temperature TH whose thermal stressenergy 〈Tμν〉^ZAMO gets renormalized away by Tμνvac pol, annulling the O(α^2) and O(α^1) pieces of 〈Tμν〉^ren, and leaving only the O(α^0) vacuumpolarization, curvaturecoupling contributions. This translates into 〈Tll〉^ren=〈Tlφ〉^ren=0 on the future horizon in the HartleHawking vacuum, where l and φ denote components on the horizon generator lμ 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 redshifted energy and angular momentum across the future horizon, per unit solid angle sinθ dθ dφ, 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 HartleHawking vacuum state ‖H〉 is singular at and outside the velocityoflight 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 stressenergy 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.
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Additional Information:  ©1989 The American Physical Society Received 22 June 1988 For helpful discussions or correspondence we thank Sam Braunstein, Carlton M. Caves, Vitali L. Ginzburg, Bernard S. Kay, and Robert M. Wald. This research was supported in part by the Academy of Sciences, USSR, the Ministry of Higher Education, USSR, and The National Science Foundation, USA, Grant No. AST8514911. 
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Record Number:  CaltechAUTHORS:FROprd89 
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Alternative URL:  http://dx.doi.org/10.1103/PhysRevD.39.2125 
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