Hawking Radiation, Effective Actions and Covariant Boundary Conditions

Introduction: Hawking radiation arises upon the quantisation of matter in a background spacetime with an event horizon. It therefore plays an important role in black hole physics. Apart from Hawking’s [4] original derivation, there are other approaches [5, 6], although none is completely clinching or conclusive. This has led researchers to consider alternative derivations providing new insights into the problem. Here we discuss another approach that is based solely on the structure of the effective action and boundary conditions at the event horizon. We therefore guarantee the universality of Hawking radiation which ought to be determined by properties at the event horizon onlya feature that is usually lacking in approaches based on effective actions [7, 8]. To put our work in a proper perspective, however, it is desirable to elaborate on the recent approaches [1, 2, 3] to the Hawking effect which rely on the cancellation of gauge and gravitational anomalies. An anomaly, it might be recalled, is a breakdown of some classical symmetry due to the process of quantisation. For example, a gauge anomaly is an anomaly in gauge symmetry, taking the form of nonconservation of the gauge current. Similarly, a gravitational anomaly occurs from a breaking of general covariance, taking the form of nonconservation of the energy momentum tensor (For reviews see, [9, 10]). The simplest manifestation of these (gauge and gravitational) anomalies, which is also relevant for the present discussion, occurs for 1 + 1 dimensional chiral fields. Recently, Robinson and Wilczek [1], followed by Iso, Umetsu and Wilczek [2], gave a new derivation of the Hawking effect. They found that, by the process of dimensional reduction, effective field theories become two dimensional and chiral near the event horiazon of a black hole. This leads to the occurrence of gauge and gravitational anomalies. The Hawking flux is necessary to cancel these anomalies. An essential aspect of [1, 2] is that a two dimensional chiral theory admits two types of anomalous currents (and/or energy momentum tensors)the consistent and the covariant [9, 10, 11, 12, 13, 14]which are actually related by local counterterms. The covariant divergence of these currents and energy momentum tensors yields either the consistent or covariant form of the anomaly. Then the Hawking flux was derived in [1, 2] by a cancellation of the consistent anomaly but the boundary condition necessary to fix the parameters was obtained from a vanishing of the covariant current at the horizon [2]. It was also observed [16] that an incorrect result for the charge flux would be obtained if, instead, the vanishing of the consistent current at the horizon was taken as the boundary condition. The approach of [1, 2] was very recently generalised by us [3]. It was shown that the complete analysis was feasible in terms of covariant expressions only. The flux from a charged black hole was correctly determined by a cancellation of the covariant anomaly with the boundary condition being the vanishing of the covariant current (and energy momentum tensor) at the horizon. Apart from being conceptually clean and more natural (all expressions being covariant), it simplified the original analysis [1, 2] considerably. This was true not just for charged black holes, but for other black holes as well [17]. From the analysis of [2, 3] it appears therefore that covariant boundary conditions at the horizon play a fundamental role. We adopt the arguments of [1, 2] which imply that effective field theories are chiral and two dimensional near the horizon. Then, exploiting known structures of the two dimensional effective actions, the relevant expressions for the currents and the energy momentum tensors are derived by only imposing covariant boundary conditions at the horizon. The Hawking flux from charged black holes is correctly reproduced in this manner. Finally, we establish the connection of our approach with calculations based on the Unruh vacuum [15, 18].