Uninformed Hawking radiation

We show in detail that the Parikh-Wilczek tunneling method (PWTM), which was designed for resolving the information loss problem in Hawking radiation (HR) fails whenever the radiation occurs from an isothermal process. The PWTM aims to produce a non-thermal HR which adumbrates the resolution of the problem of unitarity in quantum mechanics (QM), and consequently the entropy (or information) conservation problem. The effectiveness of the method has been satisfactorily tested on numerous black holes (BHs). However, it has been shown that the isothermal HR, which results from the emission of the uncharged particles of the linear dilaton BH (LDBH) described in the Einstein-Maxwell-Dilaton (EMD) theory, the PWTM has vulnerability in having non-thermal radiation. In particular, we consider Painlevé-Gullstrand coordinates (PGCs) and isotropic coordinates (ICs) in order to prove the aforementioned failure in the PWTM. While carrying out calculations in the ICs, we also highlight the effect of the refractive index on the null geodesics. Copyright c © EPLA, 2015 Introduction. – As is well known, Hawking [1] theoretically proved that BHs could emit radiation (often called HR), which implies that a BH would eventually evaporate away, leaving nothing over time. According to the principles of QM, complete information about a system is encoded in its wave function. The evolution of the wave function is determined by a unitary operator, and unitarity implies that information is conserved in the quantum sense. However, the combination of QM and general relativity suggests that physical information could permanently disappear in a BH, allowing many physical (pure quantum) states to devolve into the mixed state of HR. This phenomenon is called the BH information loss paradox (the reader may refer to [2] for the topical review), which underscores the apparent violation of unitarity in the process of HR. Among the many attempts at a resolution of this problem, the most promising one came at the turn of this century, belongs to Parikh and Wilczek (PW) [3]. The theorem states that when a virtual pair is created just inside the BH horizon, the positiveenergy particle (real particle) can tunnel out the BH horizon by a process similar to the QM tunneling, whereas the negative-energy particle (antiparticle) continues to stay in the BH. Conversely, as one would expect from particleantiparticle symmetry, if a virtual pair is created just outside the horizon, the antiparticle can tunnel inward, while the real particle will eventually escape to spatial infinity. In the PWTM, the conservation of energy is enforced. Therefore, the mass of the BH must continuously decrease while it radiates. Besides this, the information-carrying particle is modelled as a thin spherical shell with energy ω. Those shells could tunnel through the potential barrier, following the principles of QM. In short, the whole tunneling process is considered semiclassically, and the transmission coefficient is determined by the classical action of the particle with the aid of the Wentzel-Kramers-Brillouin (WKB) method [4]. As a result, the obtained spectrum is not precisely thermal, and this also leads to the unitarity of the underlying quantum theory and the conservation of information [5]. On the other hand, so far, the solution of the PWTM to the problem of the information paradox has not convinced everyone, and hence it has also remained debatable (one can see the extensive review on the PWTM analysis [6] and references therein). Furthermore, the PWTM has also extended to the HR analysis of the non-asymptotically flat (NAF) BHs (see for instance [7–9]). The PWTM through the quantum horizon of a LDBH geometry, which is the solution to the EMD theory [10–12], and its extended theories [13], was studied in [9,14,15]. This BH is a NAF, four dimensional, spherically symmetric and static dilatonic spacetime. It was shown by [14,15] that in the proposed PW setup, there is no correlation between different subsequently emitted particles, which