Hawking Radiation and Hawking Flux from Spherical Reduction

In view of the recent progress in 2D quantum gravity [1] from spherical reduction of Einstein gravity (SRG) it should be possible to also derive the correct radiative flux to infinity related to the Hawking temperature at the horizon [2] from the SRG action in D = 2. For minimally coupled scalars the correct result had been obtained a long time ago [3]. As shown first in [4] a calculation based only upon the conformal anomaly in the presence of dilaton fields leads to an unacceptable (negative) flux at infinity, requiring the addition of a nonconformally invariant piece to the effective action. A more recent calculation [5] of that anomaly disagreed with [4] and further conflicting results appeared in the literature [6, 7, 8, 9]. We (together with H. Liebl) have derived that anomaly for completely general nonminimal interaction of the scalar fields in D = 2 and for general dilaton dependent measure [8] and used the integration of the standard energy momentum conservation with Unruh vacuum at the horizon [2] to arrive at the same result for the flux as [4]. Our computation of the anomaly agreed with [6, 9, 10] in the case of SRG. After paper [11] our computation of the anomaly now is generally accepted (cf. e.g. [12])

Using the proper "extended" energy momentum conservation in the presence of dilaton interaction for matter and calculating the corresponding "dilaton anomaly" we were able to show [13] that the correct positive flux at infinity follows, completing earlier attempts in this direction [14]. Our computation was based upon a novel application of the heat kernel technique [15].

As a by-product, but not to be used in our argument, we also gave for the first time the full effective action in the presence of a dilaton field which couples arbitrarily to the scalar field and which enters in a general way in the quantum measure. This action extends the one for minimal coupling [16] to the most general case of a dilaton interaction in D = 2. Possible criticisms of our result [17, 18, 19] are discussed.