Hawking versus Unruh effects, or the difficulty of slowly crossing a black hole horizon

Received: August Hawking versus Unruh e ects, or the di culty of slowly crossing a black hole horizon Luis C. Barbado 0 1 2 3 5 6 7 8 Carlos Barcelo 0 1 2 3 5 6 7 8 Luis J. Garay 0 1 2 4 5 6 7 8 Gil Jannes 0 1 2 5 6 7 8 Open Access 0 1 2 5 6 7 8 c The Authors. 0 1 2 5 6 7 8 0 Universidad Complutense de Madrid , Ciudad Universitaria 1 Glorieta de la Astronom a s/n , 18008 Granada , Spain 2 Universitat Wien , Boltzmanngasse 5, 1090 Wien , Austria 3 Departamento de Astronom a Extragalactica, Instituto de Astrof sica de Andaluc a (CSIC) 4 Departamento de F sica Teorica II, Facultad de Ciencias Fsicas 5 eld. Under simplifying assump- 6 Calle Tajo s/n, 28670 Villaviciosa de Odon, Madrid , Spain 7 Serrano 121 , 28006 Madrid , Spain 8 Plaza Ciencias 1 , 28040 Madrid , Spain When analyzing the perception of Hawking radiation by di erent observers, the Hawking e ect becomes mixed with the Unruh e ect. The separation of both e ects is not always clear in the literature. Here we propose an inconsistency-free interpretation of what constitutes a Hawking e ect and what an Unruh e ect. An appropriate interpretation is important in order to elucidate what sort of e ects a detector might experience depending on its trajectory and the state of the quantum tions we introduce an analytic formula that separates these two e ects. Armed with the previous interpretation we argue that for a free-falling detector to cross the horizon without experiencing high-energy e ects, it is necessary that the horizon crossing is not attempted at low velocities. aQuantenoptik; Quantennanophysik und Quanteninformation; Fakultat fur Physik 1 Introduction E ective temperature function Hawking versus Unruh e ects Observers outside a black hole The di culty of crossing the horizon at low velocities Arguably, the two cornerstone results of Quantum Field Theory in curved spacetimes and non-inertial reference frames are the Hawking e ect in black holes [1] and the Unruh e ect [2]. The Unruh e ect is typically considered a subjective e ect, meaning that it is something that is not objectively there but shows up as detector perception e ects when subject to acceleration; it is typically presented, in its most basic form, as the perception of an accelerated detector in Minkowski spacetime. On the other hand, the Hawking e ect is interpreted as an objective e ect, something that happens to any black hole once it is formed. For an observer following an arbitrary trajectory outside a black hole, it is clear that these two e ects will in general be present together; one could then talk about perception e ects near radiating black holes (see e.g. [3, 4]). When looking at the literature on radiating black holes, however, there is often no consensus nor clarity on what constitutes a Hawking e ect and what constitutes an Unruh e ect in the net perception of the observer (see e.g. the discussions in [5{12]; even the very existence of the Unruh e ect is sometimes still questioned, see e.g. [13]). A controversy on whether the equivalence principle is preserved in the presence of these e ects is still unsettled [14]. Our work can help to clear up this controversy. More generally, the existence or not of a separation between the Hawking and Unruh e ects, and why and how this separation should be accomplished is the main theme of the present work. To see the problem clearly, let us consider an observer sustaining himself at rest at xed radius just outside the horizon of a radiating black hole. A standard calculation concludes that this observer perceives an outgoing ux of black-body radiation at the Hawking temperature associated with the black hole, corrected by an enormous gravitational blueshift due to the location of the observer. It is commonly stated that this tremendous temperature can be interpreted as an Unruh e ect (this association can be read of e.g. in Birrell-Davies book, page 282 [15]). As we will argue, this association is potentially misleading. The standard argument is that the Unruh vacuum state (the state of a radiating black hole) is perceived as vacuum by free-falling observers at the horizon; then, as the observer is strongly accelerating with respect to the local free-falling reference frame in order to sustain his position, he must experience an Unruh e ect and thereby perceive a thermal bath with a temperature proportional to his acceleration. This reasoning indeed yields the correct result for the temperature (strictly speaking, only in the horizon limit; see the discussion in [3]). The problems with this interpretation come about when one asks oneself whether this radiation could or could not lead to some buoyancy e ect, that is, when one tries to understand the feedback e ects of the radiation on the detector trajectory. Interpreted as an Unruh e ect, one will immediately conclude that the answer is negative: any back-reaction associated with the Unruh e ect should work against maintaining the level of acceleration and thus add (...truncated)