Black hole complementarity and the physical origin of the stretched horizon.

We discuss the idea of black hole complementarity, recently suggested by Susskind et al., and the notion of stretched horizon, in the light of the generalized uncertainty principle of quantum gravity. We discuss implications for the no-hair theorem and we show that within this approach quantum hair arises naturally. PACS categories: 04.60, 12.25, 97.60. The problems related to the application of quantum mechanics to black holes rank between the most challenging in theoretical physics. Despite great effort, it has not yet been reached a consensus on the validity of Hawking’s claim [1] that the evolution of states in the presence of black holes violates unitarity. Recently, an extremely interesting proposal, close in spirit to previous work of ’t Hooft [2], has been put forward by Susskind and coworkers [3-6] and termed “black hole complementarity”. The basic observation is that physics looks very different to an observer in free fall in a black hole and to a “fiducial observer” at rest with respect to the black hole, outside the horizon. Crossing the horizon of a very massive black hole, the free falling observer should not experience anything out of the ordinary. If the mass of the hole M is much larger than Planck mass MPl a classical description of the black hole should be adequate, and in classical general relativity the horizon merely represents a coordinate singularity, while physical quantities like the curvature are non-singular. Furthermore, from the point of view of the free falling observer, the flux of Hawking radiation is switched off when he approaches the horizon. This can be shown observing that near the horizon, with an appropriate change of variables, the Schwarzschild metric approaches the Rindler metric, and a free falling observer in Schwarzschild spacetime becomes a free falling observer in flat Minkowski space – and certainly does not detect any radiation. The point of view of a fiducial observer is dramatically different. In Schwarzschild coordinates, a fiducial observer at a distance r from a Schwarzschild black hole measures an effective temperature T = (1− 2GM r ) TH , (1) where TH = h̄/(8πGM) is Hawking temperature. Climbing out of the gravitational potential well, the radiation is gravitationally red-shifted by a factor (1− 2GM r ) and is seen by an observer at infinity as having temperature TH . Instead, at the horizon r = 2GM the temperature measured by a fiducial observer diverges. For a fiducial observer, this temperature is certainly a very real effect. If too close to the horizon, he would be killed by the eccessive heat. From this point of view, a fiducial observer regards the black hole horizon as a physical membrane, endowed with real physical properties. More in general, within the membrane paradigm [7] all interactions of a black hole with the external environment, as seen by a fiducial observer, are described in terms of a two-dimensional membrane endowed with properties like electric