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

Abstract

The renormalized expectation value of the stress-energy 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 zero-angular-momentum observers (ZAMO's) would compute from their own physical measurements near the horizon, plus a vacuum-polarization contribution Tμνvac pol , 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 α^2≡(surface gravity of hole)^2×(distance to horizon)^2 to Tμνvac pol; and since gravitational blue-shifts cause the largest components of Tμνvac pol in the proper reference frame of the ZAMO's to be of O(α-2), curvature-coupling effects in Tμνvac pol and thence in 〈Tμν〉^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 ΩH and temperature TH whose thermal stress-energy 〈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) vacuum-polarization, curvature-coupling contributions. This translates into 〈Tll〉^ren=〈Tlφ〉^ren=0 on the future horizon in the Hartle-Hawking 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 red-shifted 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 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.

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