Hawking Fluxes, Back reaction and Covariant Anomalies

Hawking radiation is an important and prominent quantum effect that arises upon the quantization of matter fields in a background spacetime with an event horizon. This is due to the different vacuum states between the vicinity of the event horizon and the asymptotic infinity [1]. Apart from the original derivation [1], there are other approaches [2, 3] each having their own merits and demerits. Recently, Robinson and Wilczek [4] followed by Iso, Umetsu and Wilczek [5], gave a new derivation of Hawking effect based on anomaly cancellation approach. Soon after, the analysis of [4, 5] was reformulated completely by Banerjee and Kulkarni [7] in terms of covariant expressions only. Very recently, Banerjee and Kulkarni [6] gave a new derivation of the Hawking effect based solely on the structure of the effective action and boundary conditions at event horizon. None of these computations however, consider the effect of back reaction. Back reaction, it might be recalled is an effect of non-zero expectation value of the energy-momentum tensor on the spacetime geometry, which acts as a source of curvature. It is possible to include the effect of gravitational back reaction in the derivation of Hawking radiation. Indeed, using the conformal anomaly method the effect on the spacetime geometry by one loop back reaction was computed in [8, 9]. Based on this approach, corrections to the Hawking temperature were obtained in [9, 10]. Correction to the Hawking temperature using the back reaction equation for linearised quantum fluctuation was derived in [11]. Recently, more useful and intuitive way to understand the effect of back reaction through the quantum tunneling formalism [3] was developed in [12]. Naturally, it becomes interesting to incorporate the effect of back reaction in the analysis of [4, 6, 7]. A particularly useful way to understand the Hawking effect is through effective action formalism developed in [7]. Only the structure of effective action defined near the event horizon is sufficient to determine the Hawking flux. An important ingredient of [7] was to realize that effective theory become two dimensional and chiral near the event horizon [4]. Yet another important aspect in this approach was that, implementation of a specific boundary condition the vanishing of the covariant form of the current and energy momentum tensor [7]. Unlike the approaches [4, 5, 7], the important advantage of this method[6] was that, Hawking fluxes were obtained only by using the covariant anomaly near the event horizon. This was true not just for charged black holes, but for other spacetime geometries as well [14, 13, 15] and also for deriving the Hawking fluxes associated with higher spin[16]. In this paper we present a derivation for Hawking energy and charge flux for the Riessner-Nordstrom black hole in the presence of back reaction, by using the effective action approach developed in [6]. The expressions for charge and energy flux get modified as a consequence of the one loop effect of back reaction.