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

Abstract

The renormalized expectation value of the stress-energy tensor 〈${T}_{\ensuremath{\mu}\ensuremath{\nu}}$${〉}^{\mathrm{ren}}$ of a quantum field in an arbitrary quantum state near the future horizon of a rotating (Kerr) black hole is derived in two very different ways: One derivation (restricted for simplicity to a massless scalar field) makes use of traditional techniques of quantum field theory in curved spacetime, augmented by a variant of the ``\ensuremath{\eta} formalism'' for handling superradiant modes. The other derivation (valid for any quantum field) uses the equivalence principle to infer, from 〈${T}_{\ensuremath{\mu}\ensuremath{\nu}}$${〉}^{\mathrm{ren}}$ in flat spacetime, what must be 〈${T}_{\ensuremath{\mu}\ensuremath{\nu}}$${〉}^{\mathrm{ren}}$ near the hole's horizon. The two derivations give the same result---a result in accord with a previous conjecture by Zurek and Thorne: 〈${T}_{\ensuremath{\mu}\ensuremath{\nu}}$${〉}^{\mathrm{ren}}$, in any quantum state, is equal to that, 〈${T}_{\ensuremath{\mu}\ensuremath{\nu}}$${〉}^{\mathrm{ZAMO}}$, which zero-angular-momentum observers (ZAMO's) would compute from their own physical measurements near the horizon, plus a vacuum-polarization contribution ${T}_{\ensuremath{\mu}\ensuremath{\nu}}^{\mathrm{vac}\mathrm{pol}}$, which is the negative of the stress-energy of a rigidly rotating thermal reservoir with angular velocity equal to that of the horizon ${\ensuremath{\Omega}}_{H}$, and (red-shifted) temperature equal to that of the Hawking temperature ${T}_{H}$.A discussion of the conditions of validity for equivalence-principle arguments reveals that curvature-coupling effects (of which the equivalence principle is unaware) should produce fractional corrections of order ${\ensuremath{\alpha}}^{2}$\ensuremath{\equiv}(surface gravity of hole${)}^{2}$\ifmmode\times\else\texttimes\fi{}(distance to horizon${)}^{2}$ to ${T}_{\ensuremath{\mu}\ensuremath{\nu}}^{\mathrm{vac}\mathrm{pol}}$; and since gravitational blue-shifts cause the largest components of ${T}_{\ensuremath{\mu}\ensuremath{\nu}}^{\mathrm{vac}\mathrm{pol}}$ in the proper reference frame of the ZAMO's to be of O(${\ensuremath{\alpha}}^{\mathrm{\ensuremath{-}}2}$), curvature-coupling effects in ${T}_{\ensuremath{\mu}\ensuremath{\nu}}^{\mathrm{vac}\mathrm{pol}}$ and thence in 〈${T}_{\ensuremath{\mu}\ensuremath{\nu}}$${〉}^{\mathrm{ren}}$ are of O(${\ensuremath{\alpha}}^{0}$) 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 ${\ensuremath{\Omega}}_{H}$ and temperature ${T}_{H}$ whose thermal stress-energy 〈${T}_{\ensuremath{\mu}\ensuremath{\nu}}$${〉}^{\mathrm{ZAMO}}$ gets renormalized away by ${T}_{\ensuremath{\mu}\ensuremath{\nu}}^{\mathrm{vac}\mathrm{pol}}$, annulling the O(${\ensuremath{\alpha}}^{\mathrm{\ensuremath{-}}2}$) and O(${\ensuremath{\alpha}}^{\mathrm{\ensuremath{-}}1}$) pieces of 〈${T}_{\ensuremath{\mu}\ensuremath{\nu}}$${〉}^{\mathrm{ren}}$, and leaving only the O(${\ensuremath{\alpha}}^{0}$) vacuum-polarization, curvature-coupling contributions. This translates into 〈${T}_{\mathrm{ll}}$${〉}^{\mathrm{ren}=\mathrm{〈}{T}_{l\ensuremath{\varphi}}{〉}^{\mathrm{ren}}}$=0 on the future horizon in the Hartle-Hawking vacuum, where l and \ensuremath{\varphi} denote components on the horizon generator ${l}^{\ensuremath{\mu}}$ and on the generator of rotations \ensuremath{\partial}/\ensuremath{\partial}\ensuremath{\varphi}. 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\ensuremath{\theta} d\ensuremath{\theta} d\ensuremath{\varphi}, 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 \ensuremath{\Vert}H〉 is singular at and outside the velocity-of-light surface ${\mathrm{scrS}}_{L}$, 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 \ensuremath{\Vert}H〉 be well behaved (and have the properties described above), one must prevent its rotating thermal reservoir from reaching out to ${\mathrm{scrS}}_{L}$, e.g., by placing a perfectly reflecting mirror around the hole just inside ${\mathrm{scrS}}_{L}$.