A Planck-like problem for quantum charged black holes

Motivated by the parallelism existing between the puzzles of classical physics at the beginning of the XXth century and the current paradoxes in the search of a quantum theory of gravity, we give, in analogy with Planck’s black body radiation problem, a solution for the exact Hawking flux of evaporating ReissnerNordström black holes. Our results show that when back-reaction effects are fully taken into account the standard picture of black hole evaporation is significantly altered, thus implying a possible resolution of the information loss problem. [email protected] [email protected] [email protected] The remarkable discovery that black holes emit thermal radiation [1] has raised serious doubts on the unitarity of a quantum theory of gravity. Hawking argued [2] that the semiclassical approximation should be valid until the Planck mass is reached. This, in turn, implies that the black hole should shrink slowly during the evaporation. At the Planck mass there is not enough energy inside the black hole to radiate out the information of the collapsed matter, thus implying a loss of quantum coherence. However, it has been stressed [3] that gravitational back-reaction effects could change the standard picture of the evaporation process. It is clear that the back-reaction must be very important, at least at late times, in order to prevent the total emitted energy to diverge. In contrast, at early times one could expect these effects to be negligible and the radiation can be calculated using the classical space-time geometry. A natural scenario where one can exactly evaluate the emitted radiation at late times is in the scattering of extremal Reissner-Nordström (RN) black holes by massless neutral particles. If the incoming matter has a long-wavelength, and preserves spherical symmetry, the resulting near-extremal RN black hole can be described, by an infalling observer crossing the horizon, by means of the ingoing Vaidya-type metric [4] ds = − 2l r0 ( 2x l2q3 − lm(v) ) dv + 2dvdx+ (r 0 + 4lx)dΩ 2 , (1) where l = G is Newton’s constant and r0 = lq is the extremal radius. The function m(v) represents the deviation of the mass from extremality and it verifies the evolution law ∂vm(v) = − ~ 24πlq m(v) + (∂vf) 2 , (2) f being the null matter field related to the 2d stress tensor T f vv , and the corresponding 4d one T (4) vv , by (∂vf) ≡ T f vv = 4πr T (4) vv . Technically, this is the solution coming from the effective action Seff which is obtained by integrating out the field f . Due to spherical symmetry and considering the near-horizon region, Seff corresponds to the classical action plus the Polyakov-Liouville one [5]. In the absence of incoming matter (m(v) = 0), and in the very near-horizon limit r 0 >> lx, the metric (1) recovers the Robinson-Bertotti anti-de Sitter geometry [6]. In the dynamical situation the metric (1) implies the existence of a negative incoming quantum flux crossing the horizon that goes down exponentially for v → +∞. A “complementary” description can also be given from the point of view of an asymptotic observer, for whom there is no incoming radiation but there exists an outgoing (Hawking) evaporation flux. The geometry for the outside observer can be then described, at late times, by an outgoing Vaidya-type metric [7] ds ∼ − 2l r0 ( 2x l2q3 − lm(u) ) dv − 2dudx+ (r 0 + 4lx)dΩ 2 , (3)